Selected Papers of
P
M. Oya
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Selected Papers of
P
M. Ohya
K World Scientific
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Library of Congress Cataloging-in-Publication Data Ohya, Masanori. 1947[Papers. Selections] Selected papers of M. Ohya / edited by N.Watanabe. p. cm. Includes bibliographical references. ISBN-13: 978-981-279-419-2 (hardcover : alk. paper) ISBN-10: 981-279-419-0 (hardcover : alk. paper) 1. Quantum entropy. 2. Quantum teleportation. 3. Quantum theory. I. Watanabe, N. (Noboru), 11. Title. QC174.85.QS303582 2008 530.12-dc22 2007052697 The editor and the publisher would like to thank the following publishers for their assistance and their permission to reproduce the articles found in this volume: AkadCmiai Kiad6 The American Institute of Physics Elsevier IEEE Society Kluwer Plenum Publishing Co. Polish Scientific Publishers
The Royal Society of London Societi ltaliana di Fisica Springer Tokyo Institute of Technology University of California Wroclaw University and Wroclaw University of Technology
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Masanori Ohya
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Preface
1. Introduction
A few years ago 60 years were considered a threshold, just preceding retirement and a gradual disengagements from previous activities. Contemporary society has changed this situation drastically and the 60 years threshold now marks the entrance of a scientist into the age of synthesis, in which you look back in perspective to the topics that have been at the barycentre of your scientific interest and try to find out a pattern in the development of your interaction with these problems. In the case of Masanori Ohya, I would describe such a pattern as follows: use of advanced mathematical techniques for the development of quantum information theory and life sciences. The core inspiration for M. Ohya has undoubtedly been quantum information, of which he can be considered as one of the pioneering figures. As usual the effort to develop the quantum generalization of a discipline, leads to a new and deeper comprehension of its classical aspects. This naturally led M. Ohya to consider life science, where the mechanism of elaboration of information plays a crucial role. Chronologically, the latest development of this intellectual adventure was the merging of these two lines of research into the quantum bio-information program, a bold attempt to investigate the role of quantum mechanics, more specifically of quantum information theory, in the life sciences. Since quantum mechanics represents the deepest level of development of contemporary physical sciences, it is clear that such role should exist and the role of pioneers is precisely to move the first steps in this direction. One of the main merits of M. Ohya consists in having understood that such an ambitions scientific program could not be handled by a single individualism: the success of the program required coordination of the energies and enthusiasm of young generations with a solid network of experts in different fields. He succeeded remarkably well in both directions, first of all with an intensive activity as educator which produced hundreds (literally) of students some of whom now have achieved preeminent positions in different branches of science. Second with a careful selection of collaborators from all over the world, which has made Tokyo University of Science one of the main centers of quantum information theory since almost three decades. One highlight of the national and international network built by M. Ohya is the success of the international journal Open systems and information dynamics, which in a few years has achieved an impact factor higher than many much older scientific journals, and whose foundation was based on an early intuition of M. Ohya anticipating of several years the need for a quantum information journal (in the past
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few years many such journals have been founded). Another one is the monograph by M. Ohya and D. Petz: Quantum Entropy and his use, now at its second edition at Springer and well established as one of the classics of quantum information theory. In addition the series of annual workshops on Quantum Information Theory and Open Systems, organized by M.Ohya in collaboration with Izumi Ojima since 1992 at the Kyoto Research Institute of Mathematics (RIMS), were the first periodical meeting point for the Japanese researchers interested in the mathematical aspects of quantum information. These achievements parallel an impressive scientific production, marked by some highly original discoveries as well as the active participation in the editorial board of several international journals, an intense publishing activity including several books in different areas, and in particular the publication of the highly successful Encyclopedia of Information Sciences. Each of these activities would be more than enough for a normal carrier. There fore it is truly remarkable how M. Ohya could combine all of them and make them compatible with the additional duties of Dean of the Graduate School of Science and Technology, Director of the Education Research Center for Information Science and Technology, Member of the Committee of the International Institute for Advanced Studies, ... which he served during several years, sometimes in coincidence. Nowadays the pressure of specialization is very high. One of its psychological effects is that many scientists remain captured in a narrow horizon and become existentially unable to appreciate results beyond such an horizon. There is a sociological counterpart of this effect which makes some people unable to communicate, or to enjoy communication, with people outside a narrowly outlined community. Many scientists, even of very high technical level, do not succeed to avoid these traps. Therefore those, like M. Ohya, who succeed in maintaining a broad vision of science, while struggling to realize this vision without abdicating the luxury of having a wide human and scientific taste, are rare and precious examples for the future generations. 2. Comments on some of M. Ohya’s scientific results Classical information theory made its transition from a purely engineering and technological level to a well established mathematical theory, based on non trivial theorems, in the 1960’swith the work of C. Schannon who, developing earlier intuitions of N. Wiener, introduced the notion of entropy as a measure of information. The idea of a quantum extension of Shannon’s results naturally arose in the context of quantum signal processing and, already in the late 1960’s, was discussed by several authors (among whom: Gordon, Lax, Louisell, ...). Nowadays quantum information, enriched by the engineering appeal of the quantum computer programme, has become a widely practiced scientific discipline and the object of huge investments from all industrialized countries.
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But in the early 1980’s these investigations were cultivated by a handful of pioneer researchers among whom M. Ohya, who was one following the tracks of his former advisor, Professor Umegaki, the first one to extend the Kulback-Leibler relative information by defining the relative entropy in finite and sigma-finite von Neumann algebras. Later on M. Ohya extended von Neumann’s entropy to general C*-algebras 18. M. Ohya quantum extension of the notions of mutual entropy and compound channel [Ohya83a], whose classical counterparts play a basic role in Shannon’s a p proach to information theory, is one of the earliest, and still among the most important of his contributions to quantum information theory. A few years later the two notions of compound channel and of transition expectation were combined into the unifying notion of lifting which has now found several applications in different fields. Inspired by Shannon’s work, Kolmogorov introduced entropic type quantities as a measure of complexity and as a tool in approximation theory. These were generalized by M. Ohya to the quantum case l6 and also in the classical domain with the introduction of the notion of chaos degree. This notion found unexpected experimental support in econometrics, (joint papers of M.O. and T. Matsuoka) and in medicine (joint papers of M.O. and K. Sato). Kolmogorov also introduced a notion of dynamical entropy and proved that it provided the first example of invariants for dynamical systems finer than the spectral invariants introduced by Halmos and von Neumann. The fundamental step in Kolmogorov’s construction was to associate a family of finite Markov chains to the given dynamical system. He calculated the entropy of these chains, which can be done explicitly, and taking the supremum over all these chains, he obtained a number depending only on the dynamical system. Several quantum generalizations of Kolmogorov dynamical entropy have been proposed, but they were lacking this direct and canonical connection with the theory of Markov chains. This motivated the paper where, starting from a quantum dynamical system and extended to this case Kolmogorov’s original construction using quantum Markov chains rather than classical. The computation of the entropy of a quantum Markov chain is not so simple as in the classical case, however in several concrete examples it was possible to calculate this entropy or at least to produce a lower bound for it These results were extended to irreversible dynamical systems by Ohya, Kossakowski and Watanabe 12. Another anticipating intuition of M. Ohya was the recognition that the theoretical studies on quantum information and communication could find, in quantum computer and quantum cryptography a concrete realization and application. Thus he himself and his group begun to actively work in this field long before it became a fashionable scientific trend. I consider the channel theoretical formulation of teleportation, that M. Ohya had developed in collaboration with Kei Inoue, Hiroki Suyari and Noboru Watanabe 14,
’.
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17, as the first important contribution of Ohya’s group to quantum computer. This has now become the standard mathematical formulation of teleportation While the original formulation of this notion, due to Bennet and Brassard, looked like a very specific example and did not suggest any insight into its general structure, the channel theoretical formulation of Ohya’s group really captured the mathematical essence of teleportation thus opening the way to its realization in arbitrary dimensions. On this problem several groups were independently working in different parts of the world but, at that moment, it was still open. The first constructive solution of this problem was given in the paper which also included the first proof of the fact that the teleportation scheme must be based on a maximally entangled state in a given basis. Such a result was quite unexpected at that time and was later rediscovered and generalized in various ways by several authors. Another interesting result, obtained by Ohya in collaboration with Masuda 13, is the possibility to solve the SAT problem in polynomial time by quantum computer. Sharp estimates for the number of steps needed in the Ohya-Masuda algorithm can be found in 5 , 6 . However the complete realization of this algorithm still had a problem: one could not exclude the possibility that a positive solution (i.e. one guaranteeing satisfiability) would appear with such a small probability to be indistinguishable from a negative one. The proposal to amplify these probabilities with classical chaos methods, was advanced by Ohya and Volovich. A different, purely quantum method was later proposed in This method is based on the new idea of stateadaptive dynamics applied to the quantum state obtained as output of the Ohya-Masuda algorithm. This is used to construct a physically implementable interaction (hence the name stateadaptive dynamics) capable to drive a system to a stationary state. Thus, by discriminating the limit stationary states, one can discriminate between a positive or negative solution of the SAT problem. Finally the new notion of degree of entanglement, introduced by M. Ohya and T. Matsuoka l5 and providing a new criterion for entanglement, much easier to verify in concrete cases than previously introduced criteria, was applied in to prove the entangled nature of certain quantum Markov chains. The above mentioned results represent only a tiny fraction of M. Ohya’s scientific production. My selection criterion was very simple: I mentioned only those papers in which either I was directly involved or which inspired some of our joint papers. This kind of joint retrospective is a way to express my pleasure for a tradition of collaboration which is lasting since more that 15 years and which is now evolving in new directions.
Roma, November 2007
Luigi Accardi
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References 1. Accardi L., Ohya M.: Compound channels, transition expectations and liftings, Applied Mathematics Optimization 39 (1999) 33-59 Volterra preprint N. 75 (1991) 2. Accardi L., Ohya M., Watanabe N.: Note on quantum dynamical entropies, Reports on mathematical physics, 38 N. 3 (1996) 457-469 3. Accardi L., Ohya M., Watanabe N.: Dynamical entropy through quantum Markov chains, Open Systems and Information Dynamics 4 (1997) 71-87 4. Accardi L., Ohya M.: Teleportation of general quantum states, in: Quantum Information, T. Hida, K. Saito (eds.) World Scientific (1999) 59-70 Invited talk to the: International Conference on quantum information and computer, Meijo University 1998 Preprint Volterra N. 354 (1999) 5. Accardi L., Sabbadini R.: On the Ohya-Masuda quantum SAT Algorithm, in: Proceedings International Conference “Unconventional Models of Computations”, I. Antoniou, C.S. Calude, M. Dinneen (eds.) Springer (2001) Preprint Volterra, N. 432 (2000) 6. Accardi L., Sabbadini R.: A Generalization of Grover’s Algorithm, Proceedings International Conference: Quantum Information 111, Meijo University, Nagoya, 27-31 March (2001) World Scientific (2002) qu-phys 0012143 Preprint Volterra, N. 444 (2001) 7. Accardi L., Lu Y.G., Volovich I.: Quantum Theory and its Stochastic Limit, Springer Verlag (2002) Japanese translation Tokyo-Springer, to appear 8. Luigi Accardi, Masanori Ohya: A stochastic limit approach to the SAT problem, Open systems and Information Dynamics, 11 (3) (2004) 219-233 9. L. Accardi, T. Matsuoka, M. Ohya: Entangled Markov chains are indeed entangled, Infinite Dimensional Analysis, Quantum Probability and Related Topics 9 (2006) 379390 10. Ohya M., Petz D.: Quantum entropy and its use, Springer, Texts and Monographs in Physics (1993) 11. M. Ohya: Mathematical Foundation of Quantum Computer, Maruzen Publ. Company (1998) 12. Ohya M., Kossakowski A., Watanabe N.: Quantum dynamical entropy for completely positive map, Infinite dimensional analysis, quantum probability and related topics, 1 (2) (1999) 267-282 Preprint (1998) 13. Ohya M., Masuda N.: N P problem in quantum algorithm, Open Systems and Information Dynamics, Vo1.7, No.1, 33-39, 2000. arXiv:quant-ph/9809074 v2 13 dec (1998) 14. Ohya M., Watanabe N.: On the mathematical treatment of the Fredkin-ToffoliMilburn gate, Physica D 120 (1998) 206-213 15. M. Ohya, T. Matsuoka: Quantum Entangled State and Its Characterization, Foundation and Probability and Physics-3, AIP Proceedings 750 (2005) 298-306 16. Ohya M.: Complexities and their applications to characterization of chaos, Int. Journ. of Theort. Phy., 37 (1998) 495 17. Ohya M., Inoue K., Suyari H.: Characterization of quantum teleportation processes by nonlinear quantum channel and quantum mutual entropy, SUT Preprint (1997) 18. M.Ohya (1984) Entropy Transmission in C*-dynamical systems, J. Math. Anal.Appl., 100, No.1, 222-235“ 19. Ohya M.: On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29 (1983) 770-777
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Preface
I am indeed very glad to be able to write a few words as a preface to the collection of selected papers of Professor Ohya. I was in Noda many times on invitation of Prof. Ohya and this gave me many oppotunities of active scientific contacts with him. He is one of the creators and propagators of Information Dynamics as a new branch of modern science. He also came to Torun many times. Open Systems, increases of entropy and entropy as a measure of information, which were the main topics that united our scientific interests and collaboration. Tokyo University of Science was since XIX century, and is at present, an important center of research, especially in mathemetical physics. He is now the editor-in-chief of an international journal "Open Systems and Information Dynamics" which we organized together with Professors Accardi, Kossakowski, Jamiolkowski and others. Professor Ohya was always one of the most active in these fields. On the occasion of his 60th birth anniversary I wish him very cordially the further scientific activity and the best health and greatest satisfaction in science and personal life. His papers give a permanent contribution to Quantum Information, Information Dynamics and Mathematical Physics, but we hope that this will continue for many years in the future. I would like to invite the readers to the interesting papers and reviews of Professor Masanori Ohya as the important steps in modern development of information science.
Torun, June 2007
Roman. S. Ingarden
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Preface
On the occasion to celebrate the 60th anniversary of Prof. Masanori Ohya’s birth, I would like to recollect my personal history of friendship with him. After working on the formulation of quantum fields at finite temperatures (I.O., Ann. Phys. 1981), I became interested in the mutual relations between micro- and macroscopic aspects in quantum theory. So it was natural for me to be led to Prof. Ohya’s book, “Quantum Entropy”, written (in Japanese) with Prof. Umegaki, immediately after its publication in 1984, which impressed and attracted me very much. This book (which deserves still being translated into international languages) strongly convinced me that certain general mathematical method and framework should be established for treating the problems on the relations between micro- and macroscopic aspects in nature. Then I decided to meet him to make request for his cooperation in a research project along this line. He kindly accepted my request, and a series of workshops named “Quantum-Field Theoretical Approaches to Evolution Dynamics” started in 1986 at Research Institute for Fundamental Physics (= RIFP and, at present, Yukawa Institute for Theoretical Physics) in Kyoto University. This was really a new interdisciplinary project, covering quantum field theory, measurement theory, quantum theory of information and communications, quantum optics, cosmology and solid state physics, etc., which continued for five years until 1990 at RIFP. After 1991 its host institute was changed to Research Institute for Mathematical Sciences, Kyoto University, and Prof. Ohya served as its chief organizer for thirteen years from 1992 until 2004 when I succeeded the job from him until now. During this period, of more than twenty years, Prof. Ohya has made essential and important contributions to our science communities, as seen in this volume of his selected papers, including the proposals of such notions as information dynamics, lifting of channels, chaos degree, adaptive dynamics, degree of entanglement and so on, which have shed new lights on many fundamental intersecting areas of quantum and of information. If it were not for the fruitful communications and friendship with him, I could never have dared to step into my own new projects based upon“Micro-Macro duality”.
Kyoto, November 2007
Izumi Ojima
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Contents
Prefaces by L. Accardi by R. S. Ingarden by I. Ojima
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Introduction
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Curriculum Vitae
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(1) Adaptive Dynamics and its Applications to Chaos and NPC Problem, Quantum Bio-Informatics: From Quantum Information to Bio-Informatics (2008) 181-216
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1
(2) Teleportation Schemes in Infinite Dimensional Hilbert Spaces
(with K.-H. Fichtner and W. Freudenberg), Journal of Mathematical Physics 46 (2005) 102103-14
37
(3) Quantum Algorithm for SAT Problem and Quantum Mutual Entropy, Reports on Mathematical Physics 55 (2005) 109-125
51
(4) A Stochastic Limit Approach to the SAT Problem (with L. Accardi), Open Systems and Information Dynamics 11 (2004) 219-233
68
( 5 ) New Quantum Algorithm for Studying NP-complete Problems (with I. V. Volowich), Reports on Mathematical Physics 52 (2003) 25-32
83
(6) How Can We Observe and Describe Chaos (with A . Kossakowski and Y. Togawa), Open Systems and Information Dynamics 10 (2003) 221-233
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(7) On Quantum Capacity and Its Bound (with I. V . Volowich),
Infinite Dimensional Analysis, Quantum Probability and Related Topics 6 (2003) 301-310 (8) Entanglement, Quantum Entropy and Mutual Information (with V. P. Belaukin), Proceedings of the Royal Society of London. Series A , Mathematical and Physical Sciences 458 (2002) 209-231
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114
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(9) Semiclassical Properties and Chaos Degree for the Quantum Baker's Map (with K. Inoue and I. V. Volowich), Journal of Mathematical Physics 43 (2002) 734-755
137
(10) Quantum Teleportation and Beam Splitting (with K.-H. Fichtner), Communications in Mathematical Physics 225 (2002) 67-89
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(11) Quantum Teleportation with Entangled States Given by Beam Splittings (with K.-H. Fichtner) , Communications in Mathematical Physics 222 (2001) 229-247
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(12) NP Problem in Quantum Algorithm (with N . Masuda), Open Systems and Information Dynamics 7 (2000) 33-39
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(13) Application of Chaos Degree to Some Dynamical Systems (with K. Inoue and K. Sato), Chaos, Soliton and Fractals I 1 (2000) 1377-1385
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(14) Fundamentals of Quantum Mutual Entropy and Capacity, Open Systems and Information Dynamics 6 (1999) 69-78
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(15) Compound Channels, Transition Expectations, and Liftings (with L. Accardi), Applied Mathematics and Optimization 39 (1999) 33-59
227
(16) Quantum Dynamical Entropy for Completely Positive Map (with
A . Kossakowski and N . Watanabe), Infinite Dimensional Analysis, Quantum Probability and Related Topics 2 (1999) 267-282
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(17) Complexities and Their Applications to Characterization of Chaos, International Journal of Theoretical Physics 37 (1998) 495-505
270
(18) Analysis of HIV by Entropy Evolution Rate (with K. Sat0 and S. Miyazaki), Amino Acids 14 (1998) 343-352
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(19) On Capacities of Quantum Channels (with D. Petz and
N . Watanabe), Probability and Mathematical Statistics 17 (1997) 179-196
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(20) Complexity, Fractal Dimension for Quantum States, Open Systems and Information Dynamics 4 (1997) 141-157
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(21) Notes on Quantum Entropy (with D. Petz), Studia Scientiarum Mathematicarum Hungarica 31 (1996) 423-430
326
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(22) Note on Quantum Dynamical Entropies (with L. Accardi and N . Watanabe), Reports on Mathematical Physics 38 (1996) 457-469
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(23) Entropy Functionals of Kolmogorov-Sinai Type and Their Limit Theorems (with N. Muralci), Letters in Mathematical Physics 36 (1996) 327-335
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(24) Information Dynamics and Its Applications to Optical Communication Processes, Springer Lecture Notes in Physics, Vol. 378 (1991) 81-92
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(25) Information Theoretical Treatments of Genes, The Transactions of the IEICE E72 (1989) 556-560
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(26) Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes, Reports on Mathematical Physics 27 (1989) 19-47
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(27) On Compound State and Mutual Information in Quantum Information Theory, I E E E Transactions on Information Theory 29 (1983) 770-774
402
(28) Note on Quantum Probability, Lettere a1 Nuovo Cimento 38 (1983) 402-404
407
(29) Quantum Ergodic Channels in Operator Algebras, Journal of Mathematical Analysis and Applications 84 (1981) 318-327
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(30) Sufficiency, KMS Condition and Relative Entropy in von Neumann Algebras ( F . Hiai and M. Tsukada), Pacific Journal of Mathematics 96 (1981) 99-109
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(31) On Open System Dynamics
- An Operator Algebraic Study, Kodai Mathematical Journal 3 (1980) 287-294
431
(32) Dynamical Process in Linear Response Theory, Reports o n Mathematical Physics 16 (1979) 305-315
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(33) Stability of Weiss Ising Model, Journal of Mathematical Physics 19 (1978) 967-971
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List of Publications
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Introduction
Classical information theory cannot be applied to information communication for micro-scopic objects treated in quantum mechanics. For such study, quantum information theory has been developed. It is based on quantum probability and it is a theory for expressing information by quantum states. Prof. Masanori Ohya studied many topics, for more than thirty years, related to quantum entropy, quantum information, chaos dynamics and life science. His main accomplishments are as follows.
Elucidation of Mathematical Bases of Quantum Channels: M. Ohya rigorously studied quantum state change including irreversible processes (1970's) in quantum statistical physics and information communication (after 1980) by generalizing the state change to quantum channel. As a by-product of this study, he could succeed to derive the error probability for optical communication processes.
Formulation of Quantum Mutual Information (Entropy): M. Ohya formulated the quantum mutual entropy (information), which is a natural extension of Shannon's mutual information. Then he generalized it within C*-algebraic framework including several other definitions of both the entropy and the mutual entropy. Thereby, one could define and analyze the channel capacity rigorously. Various studies of quantum entropy including some of the above are discussed in the book "Quantum Entropy and its Use" with D. Petz.
Information Dynamics: M. Ohya proposed a new concept called Information Dynamics (ID) to integrate various dynamics of the systems with complexities. The mathematical basis and its applications of ID are summarized in the book "Information Dynamics and Open Systems" with R. Ingarden and A. Kossakowski.
Analysis of Quantum Teleportation: The quantum teleportation model transmitting a quantum state itself was proposed by Bennett et a1 in early go's, which is an epoch-making communication method for security. To overcome some difficulties of the previous models, K. Fichtner and M. Ohya made a new model of the quantum teleportation in Bose Fock space, where the incomplete teleportation has been introduced. In 2006, Kossakowski and M. Ohya proposed a new scheme of the teleportation in which the teleportation channel becomes linear and the complete teleportation is possible even for non-maximal entangled states.
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Quantum Algorithm: The quantum computer makes the computing speed extremely high. In 19992000, M. Ohya and Masuda wrote the quantum algorithm for the SAT problem, one of the NP-complete problems. Around 2000-2002, M. Ohya and I. Volovich found an algorithm solving the NP-complete problem in polynomial time by combining with the quantum algorithm and the state change in chaotic dynamics. Further, L. Accardi and M. Ohya showed another algorithm for the same problem based on the idea of Adaptive Dynamics in 2005. Recently these algorithms can be written in the language of generalized quantum Turing machine.
Proposal of Adaptive Dynamics: Existence of substance, more generally of any object, depends on how it is observed. How can one describe this fact mathematically? What is a philosophical basis for the mechanism of the existence? To answer these questions, M. Ohya proposed the Adaptive Dynamics. As the applications of the idea of the Adaptive Dynamics, he succeed to study chaos and quantum algorithms.
Life Science: M. Ohya started to study Life Science, in particular, Bic-Informatics about 20 years ago. He introduced new measures (entropy evolution rate, a measure of coding structure) to study the evolution (mutation) of species in the level of genome and protein (amino-acid sequence). He and coworkers applied these measures to make phylogenetic trees and to study the change of HIV virus. Moreover, he is now interested in making a model of brain to study its function with K. Fichtner and W. Fkeudenberg. M. Ohya founded a research center call "Quantum Bic-Informatics (QBIC)", where many researchers from different fields come together and look for new logic (theory) studying quantum information and life sciences. M. Ohya has been invited from several universities all over the world and he gave many invited talks at international conferences. He has been the editors of several international journals and the representatives of various research projects.
Tokyo, December 2007
Editors
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Curriculum Vitae
Date/Place of Birth: March 21, 1947, Chiba, Japan Address/Telephone: Tokyo University of Science, Department of Information Sciences, 2641 Yamazaki, Noda City, Chiba 278-8510, Japan TEL. +81/471/229357, FAX. +81/471/245948 E-Mail.
[email protected] Education: 1970, University of Tokyo, Dept. of Physics (68-70), Dept. of Mathematics (6768) 1976, Ph.D., University of Rochester (supervised by Prof. Gerard G. Emch) D.Sc., Tokyo Institute of Technology (supervised by Prof. Hisaharu Umegaki) Appointments: 1977-1978 Assistant Professor, Department of Information Sciences, Tokyo University of Science 1978-1982 Junior Associate Professor, Department of Information Sciences, Tokyo University of Science 1982-1987 Associate Professor, Department of Information Sciences, Tokyo University of Science 1987-present Professor, Department of Information Sciences, Tokyo University of Science 2000-2003 Director, Department of Information Sciences, Tokyo University of Science 2001-2005 Director, Frontier Research Center of Computational Science, Tokyo University of Science 2003-2006 Director, Education Reserach Organization for Information Science and Technology, Tokyo University of Science 2004-2006 Dean, Graduate School of Science and Technology, Tokyo University of Science 2002-2006 Committee, International Institute of Advanced Study 2006-present Dean, Science and Technology, Tokyo University of Science 2006-present Director, Quantum Bio-Informatics Center, Tokyo University of Science Academic Works: (1) He is (or was) the member of the following Societies: Mathematical Society of Japan; The Institute of Electronics, Information and Communication Engeeners (IEICE); The Physical Society of Japan; The Biophysical Society of Japan; American Physical Society (APS); The New York Academy of Sciences. (2) He is (or was); Vice-president of International Society of Information Dynamics; Committee of Information Theory Group of IEICE; Fellow of Mathematical Society of Japan; Chief of Analysis Group in
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Mathematical Society of Japan; Representative of Research Projects of RIMS (Research Institute for Mathematical Sciences, Kyoto University); Representative of Research Projects of International Institute of Advanced Study; Special Research Member of International Institute of Advanced Study; Some Committee of the Japanese Ministry of Education, Culture, Sports, Science and Technology. (3) He is (or was) the editors of the following International Journals: Reports on Mathematical Physics; Amino Acid; Infinite Dimensional Analysis, Quantum Probability and Related Topics; Open Systems and Information Dynamics (Chief Editor). Part Time Professor: Tokyo Institute of Technology, Ochanomizu University, Kyoto University, Hokkaido University, Chiba University, Tsukuba University and others Visiting Professor: Universitk di Roma 11,Copernicus University, Jena University and others. Research: Main scientific interests are connected with quantum entropy, quantum information theory, quantum computer, mathematical physics and information genetics.
Selected Papers O
h
M. Ohya
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A D A P T I V E D Y N A M I C S A N D ITS A P P L I C A T I O N S T O C H A O S A N D NPC P R O B L E M MASANORI OHYA
Department of Information Sciences, Tokyo University of Science, 8641, Yamazaki, Noda City, Chiba, Japan I will discuss the following four (1)-(4) below from both mathematical and philosophical views: (1) What is (or do we mean) the understanding of the existence ? (2) We propose ”Adaptive dynamics” to understand the existence. (3) The adaptive dynamics can be used to describe chaos. (4)The adaptive dynamics is applied to the SAT Quantum Algorithm to solve the NP complete problem.
1. Introduction
Natural science is not a copy of nature itself but is a mean to understand several natural phenomena for human beings. Thus it is a sort of a story which we made for recognition of nature, but it is a story beyond each person and personal experience, so that it should have a universality in that sense. Following Wilde, ”Nature imitates arts”. It is the only way for us to come face to face with nature, which is not our conceit but our limit. After discovery of quantum mechanics, we are forced to face with the facts like the above although there are not many people to feel this difficulty and to look for new description of nature overcoming this difficulty. In order to understand physical phenomena or other phenomena of human beings, it needs to examine, in various views not only physical but also observational, the ways how object exists and how we can recognize the object. (1) Existence itself, (2) its indicating phenomena and (3) their recognition have been extensively studied by philosophers and some physicists. Explaining (i.e., defining and describing) these three is essentially important for not only philosophy but also physics, information and all other sciences. It is a time for us to explain these three in more rigorous ways beyond usual philosophical and mathematical demonstrations, that is, by finding a method standing on a higher stage made from dialectic mix-
2
ing of philosophy, mathematics and some others, although its fulfillment is difficult. First of all, I will mention briefly ”phenomenology” of Husserl and ”existentialism” of Sartre. Before Husserl, in the theory of existence by like Kant or Hegel, philosopher could not neglect the existence transcendent (e.g., God), so that they had to distinguish the existence of essence and the existence of phenomenon. An appearance of the essence is a phenomenon and a description of phenomena only is not enough t o reach the essence. For instance, Hegel said, ”In order to reach the essence it is necessary for mind to develop itself dialectically”. In any case, the dualism of the existence of essence and phenomena has been a basis for several philosophies until materialism of Marx and phenomenology of Husserl appeared. Husserl was against to the idea that the essence of existence is transcendent objects, and he considered that the essence is a chain of phenomena and its integral. The essence is an accurate report of all data (of phenomena) obtained through the stream of consciousness. His consciousness has two characters, ”noesis” and ”noema”. The noesis is the operative part of consciousness to phenomena (objects), in other words, the acting consciousness on objects, and the noema is the object of consciousness experience, i.e., the results obtained by the noesis. His phenomenology is the new dualism of consciousness, but he avoids the existence transcendent, instead, he likes to go to the things themselves. Under strong influence of Husserl, there appeared several philosophies named ”exsitentialism” of Heidegger, of Sartre and of others. Sartre said, ”Existence precedes essence”. Sartre was affected by ” Cogito” of Descartes, and he found two aspects of existence (being) in his famous book ”L’Etre et le Neant (Being and Nothingness)” , one of which is the ”being in itself’ and another is the ”being for itself”. The first one is the being as it is, opaque (nontransparent) being like physical matter itself, being which does not have any connection with another being, being without reason for being, etc. Another one is the being as it is not, transparent being like consciousness, being with cause for being itself, being making any being-in-itself as being, etc. Sartre explained several forms of existence by his new dualism of existence; being-in-itself and being-for-itself. His main concern is being and becoming of human beings, various appearance of emotion, life and ethics, so his expression of philosophy is rather rhetoric and literal. However I will explain that his idea can be applied for the proper interpretation of quantum entropy and dynamics.
3
1.1. Entropy, Information i n Classical and Quantum
World Physics is considered as ”theory of matter” equivalently, ”theory of existence in itself’. Information theory (Entropy theory) is considered as ”theory of events” so that it will be considered as ”theory of changes”. Quantum Information can be regarded as a synthesis of these two. The key concept of quantum information bridging between matter and event so between two modes of being, is ”entropy”, which was introduced by Shannon in classical systems and by von Neumann in quantum systems. I will discuss how this concept of entropy has a deep connection with the mode of existence considered in the beginning of this section. According to Shannon, information is related to uncertainty] so it is described by entropy: Information=Uncertainty=Entropy, and dissolution of uncertainty can be regarded as acquisition of information. Historically the concept of entropy was introduced t o describe the flow of heat, then it is recognized that the entropy describes cham or uncertainty of a system. A system is described by a state such as probability measure or density operator] which is a rather abstract concept not belonging to an object (observable) to be measured but a mean to get measured values. Thus the entropy is defined through a state of a system, which implies that the entropy is not an object considered in usual classical physics and it is an existence coming along action of ”observation”. It is close t o (actually more) than a description of chaos which is a mode taken by consciousness to the being-in-itsel. (I will discuss on chaos in Sec.3.) Therefore the entropy can be considered as a representation (formulation) of consciousness involving an observation of a certain object. The concept of entropy is not a direct expression of phenomena associated with a being-in-itself but is a being having an appearance of consciousness to phenomena of a being-in-itself, so that the mode of existence for the entropy is different from two modes of being proposed by Sartre and this third mode is in between being in and for itself. The rigorous (mathematical) study with this third mode of being might be important to solve some problems which we face in several fields. ]
1.2. Schematic Expression o f Understanding
Metaphysics, idea, feeling, thought are applied to various existence (series of phenomena)] which causes understanding (recognition] theory). To understand a physical system, the usual method, often called ”Reductionism” is to divide the system into its elements and to study their relations and
4
combinations, which causes the understanding of the whole system. Our method is one adding “how to see objects (existence)” to the usual reductionism, so that our method is a mathematical realisation of modern philisophy. The fact “how to see objects” is strongly related to setting the mode for observation, such as selection of phenomena and operation for recognition. Our method is called “Adaptive dynamics” or “Adaptive scheme” for understanding the existence. In this paper, we discuss the conceptual frame of AD and some examples in chaos and quantum algorithm, which are first steps to go to our final aims making complete mahematics for “adaptivity” . 2. Adaptive Dynamics-Conceptual MeaningThe adaptive dynamics has two aspects, one of which is the ”observableadaptivity” and another is the ”state-adaptivity”. The idea of observable-adaptivity comes from the papers38)49>50 studying chaos. We claimed that any observation will be unrelated or even contradicted to mathematical universalities such as taking limits, sup, inf, etc. Observation of chaos is a result due to taking suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases. The idea of the state-adaptivity is implicitly started in constructing a compound state for quantum c o m m u r ~ i c a t i o n ~ ~ ~in~ Accardi’s ~~~~~~and Chameleon dynamics.’ This adaptivity can be used to solve a pending problem for more than 30 years whether there exists an algorithm solving a NP complete problem in polynomial time. We found such algorithms first by quantum chaos a l g ~ r i t h mand ~ ~ secondly ,~~ by the adaptive dynamics3 based on quantum algorithm of the SAT7>52. We will discuss a bit more on the meaning of the adaptivity for each topics mentioned above. 2.1. Description of chaos
There exist several reports saying that one can observe chaos in nature, which are nothing but to report how one could observe the phenomena in specified conditions. I t has been difficult to find a satisfactory theory (mathematics) to explain such various chaotic phenomena in a unified way. An idea describing chaos of a phenomenon is to find some divergence of orbits produced by the dynamics explaining the phenomenon. However to explain such divergence from the differential equation of motion describing
5
the dynamics is often difficult, so that one take (make) a difference equation from that differential equation, for which one has to take a certain time interval T between two steps of dynamics, that is, one needs a processing discretizing time for observing the chaos. In laboratory, any observation is done in finite size for both time and space, however one believes that natural phenomena do not depend on these sizes how small they are, so that most of mathematics (theory) has been made as free from the sizes taken in laboratory. Therefore mathematical terminologies such as " lim" , llsupll,"inf' are very often used to define some quantities measuring chaos, and many phenomena showing chaos have been remained unexplained. In the p a p e r ~ , we ~ ~took r ~the ~ ~opposite ~ ~ position, that is, any observation will be unrelated or even contradicted to such limits. Observation of chaos is a result due to taking suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases. In other words, as discussed in Section 1, it is very natural to consider that observation itself plays a similar role of "noesis" of Husserl and the mode of its existence is a "being-for-itself', that is, observation itself can not exist as it is but it exists only through the results (phenomena) of objects obtained by it. Phenomena can not be phenomena without observing them, so t o explain the phenomena like chaos it is necessary to find a dynamics with observation. We claimed that most of chaos are scale-dependent phenomena, so the definition of a degree measuring chaos should depend on certain scales taken and more generally it is important to find mathematics containing the rules (dynamics) of both object and observation, which is "Adaptive dynamics". Concerning the definition of a criterion measuring chaos, Information dynamic^,^*^^^ a scheme to describe many different types of complex systems, can be applied. I introduced a quantity measuring chaos by means of Using this dethe complexities of ID, and I called it a chaos degree.30i35>43 gree in adaptive dynamics, we can explain or produce many different types of chaos. 2 . 2 . Chameleon dynamics Accardi considered a problem whether it is possible to explain quantum effects (e.g., EPR(Einstein-Polodoski-Rosen) correlation) by a sort of classical dynamics.' He could find a dynamics positively solving the above problem, and he called it "Chameleon dynamics". He considered two systems having their own particles, initially correlated and later separated. After some time, each particle interacts with a measurement apparatus independently.
6
By the chameleon effect the dynamical evolution of each particle depends on the setting of the nearby apparatus, but not on the setting of the apparatus interacting with the other particle (locality). The he succeeded t o reproduce the E P R correlations by this "chameleon dynamics", which is one of surprising results which many physicists were searching for long time. The explicit construction of the dynamics was done in the paper.2 The interaction between a particle and an apparatus depends on the setting of the apparatus, so that the chameleon dynamics is an adaptive dynamics. 2.3. Quantum SAT algorithm
Although the ability of computer is highly progressed, there are several problems which may not be solved effectively, namely, in polynomial time. Among such problems, NP problem and NP complete problem are fundamental. It is known that all NP complete problems are equivalent and an essential question is "whether there exists an algorithm to solve an NP complete problem in polynomial time". They have been studied for decades and for which all known algorithms have an exponential running time in the length of the input so far. The P-problem and NP-problem are those to be considered as follows10>16>55 Let us remind what the P-problem and the NP-problem are: Let n be the size of input.
(1)A P-problem is a problem whose time needed for solving the problem is a t worst of polynomial time of n. Equivalently, it is a problem which can be recognized in a polynomial time of n by deterministic Turing machine. (2)An NP-problem is a problem that can be solved in polynomial time by a nondeterministic Turing machine. This can be understood as follows: Let consider a problem to find a solution of f (x) = 0. We can check in polynomial time of n whether zois a solution of f (x)= 0, but we do not know whether we can find the solution of f (x)= 0 in polynomial time of
n. (3) An NP-complete problem is a problem polynomially transformed NP-problem. There are a lot of NP complete problems, e.g., satisfiable (SAT) problem, travel salesman problem and so on. It is known that all NP complete (NPC for short) problems are equivalent and have been studied for decades, for which all known algorithms have an exponential running time in the length of the input so far. For an essential question t o be asked for more than
7
30 years, that is, the existence of an algorithm to solve an NP complete problem in polynomial time, we found two different algorithms.3~52~54~55 Ins2 we discussed the quantum algorithm of the SAT problem and pointed out that the SAT problem, hence all other NP problems, can be solved in polynomial time by quantum computer if the superposition of two orthogonal vectors 10) and 11)is physically detected. However this detection is considered not to be possible in the present technology. The problem to be overcome is how t o distinguish the pure vector 10) from the superposed one Q 10) ,B 11), obtained by our SAT-quantum algorithm, if ,B is not zero but very small. If such a distinction is possible, then we can solve the NPC problem in the polynomial time.
+
2.3.1. Chaos SAT algorithm It will be not possible t o amplify, by a unitary transformation (usual quantum algorithm), the above small positive q E I,BI2 into suitable large one to be detected, e.g., q > 1/2,with staying q = 0 as it is. In the we proposed to use the output of the quantum computer as an input for another device involving chaotic dynamics, that is, t o combine quantum computer with a chaotic dynamics amplifier. We showed that this combination (nonlinear chaos amplifier with the quantum algorithm) provides us with a mathematical algorithm solving NP=P. This algorithm of Ohya and Volovich is going beyond usual (unitary) quantum Turing algorithm, but there exists a generalized quantum Turing machine in which the OV chaos algorithm can be t ~ - e a t e d . ~ ~ > ~ ~ 2.3.2. Adaptive SAT algorithm We applied the ”adaptive dynamics” to the OM SAT a l g ~ r i t h m .That ~ is, the state-adaptive dynamics is applied t o the OM SAT algorithm and rescaled the time in the dynamics by the stochastic limit, then we could show that the same amplification (distinction between q > 0 and q =0) is possible by unitary adaptive dynamics with the stochastic limit. Its details will be discussed in Section 4. The A 0 adaptive algorithm can be treated in the frame of generalized quantum Turing machine as a linear TM. 2.4. Summary
We summarize our idea on the adaptive dynamics as follows: The mathematical definition of adaptive system proposed was in terms of observables (resp. states).
8
Two adaptivities are characterized (defined) as follows:
The observable-adaptive dynamics is a dynamics characterized b y one of the following two: (1) Measurement depends on how to see an observable to be measured. (2)The interaction between two systems depends on how a fixed observable exists. The state-adaptive dynamics is a dynamics characterized by one of the following two: (1)Measurement depends on how the state to be used exists. (2)The correlation between two systems interaction depends on the state of at least one of the systems at the instant in which the interaction i s switched on. Examples of the state-adaptivity are seen in compound states42>48 (or nonlinear liftings4) studying quantum communication and in an algorithm solving NP complete problem in polynomial time with stochastic limit.3 Examples of the observable-adaptivity are used to understand ~ h a o s ~ ~ ~ ' ~ and examine violation of Bell's inequality.2 Notice that the definitions of adaptivity make sense both for classical and for quantum systems. The difference between the property (2) of the state-adaptive system and nonlinear dynamical system should be remarked here: (i) In nonlinear dynamical systems (such a s those whose evolution is described by the Boltzmann equation, or nonlinear Schrodinger equation, etc) the interaction Hamiltonian depends on the state at each time t: H I = HI(Pt)
(W .
(ii) In state-adaptive dynamical systems, the interaction Hamiltonian depends on the state only at each time t = 0: H I = H I ( ~ o ) . The latter class of systems describes the following physical situation: at time t = -T (T > 0) a system S is prepared in a state $.-T and in the time interval [-T, 01 it evolves according to a fixed (free) dynamics UI-T,OI so that its state at time 0 is U\-T,OI$-T =: $0 At time t = 0 an interaction with another system R is switched on and this interaction depends on the If we interpret the system R as environment, we can state $0: H I = HI($JO). say that the above interaction describes the response of the environment to the state of the system S. Therefore the adaptive dynamics can be linear and it contains the non-linear dynamics in many occasions. 3. Adaptive Dynamics Describing Chaos
There exist several approaches in the study of chaotic behavior of dynamical systems using the concepts such as (1) entropy and dynamical entropy, (2) Chaitin's complexity, (3) Lyapunov exponent (4)fractal dimension (5)
9
bifurcation (6) ergodicity. However these concepts are rather independently used in each case. In 1991, the present author proposed Information Dyn a m i c ~ ~to~treat > ~ such ~ > ~ chaotic ~ behavior of systems from a common standing point, in which a chaos degree to measure the chaos appeared in dynamical systems is defined by means of two complexities in Information Dynamics.44>45)55 In particular, among several chaos degrees, the entropic chaos degree was introduced in43 and it has been applied to several dynamical systems.28i29>43 For instance, semiclassical properties and chaos degree for quantum baker's map have been considered in.27>28
3.1. Information Dynamics Information dynamics (ID for short) is a synthesis of the dynamics of state change and the complexity of states. It is a trial to provide a new view for the study of chaotic behavior of systems. We briefly review what ID is. Let (A,6,a ( G ) )be an input (or initial) system and (X,E,?i@))be an output (or final) system. Here A is a set of some objects to be observed and 6 is a set of some means to get the observed value, a(G)describes a certain evolution of system. Often we have A = 2,6 = ??, a = ? Therefore ?i. we claim [Giving a mathematical structure to input and output triples = Having a theory] --
Let (AT,6 ~ ~ T,( G T be ) ) the total system of (A,6 , a ) and (A,6 , E ) , and S be a subset of 6 in which we are measuring observables (e.g., S is the set of all KMS or stationary states in C*-system). The dynamics of state change is described by a channel sending a state to another state A: 6 + (sometimes 6 + 6).Moreover ID contains two complexities, which are denoted by C and T . C is the complexity of a state 'p measured from a reference system S, in which we actually observe the objects in A and T is the transmitted complexity associated with a state change 'p + A p , both of which should satisfy the following properties: (Axioms of complexities) (i) For any
'p
E Sc 6,
CS(p) 2 0, TS(v;A)2 0 (ii) For any disjoint (in a proper sense) bijection j : esS all extremal points of S ,
-+
ezS, the set of
10
TJ'(')(j(cp); A) (iii) For CP = cp 8 1c, E St c B t , $ E product)
= TS(cp; A)
c G (here @ is
c'qQ,P)= C'(cp)
a properly defined
+ C"(1c,)
(iv) 0 I TS(cp;A) I C'(cp) (v) TS(cp;id) = C'((p), where "id" is an identity map from 6 to 6.
r
Instead of (iii), when '' Q, E ST c GT,put cp = CP A, II, = CP 12 (i.e., C'(cp) Cg(lc,) the restriction of CP to A and 2,respectively), C't(Q,) I " is satisfied, C and T is called a pair of strong complexity. Then ID is defined as follows:
+
Definition 3.1. Information Dynamics is described by
(A,6,a(G);X,G,@);A; CS(cp),TS(cp; A)) and some relations R among them. Therefore, in the framework of ID, we have to
--
(i) mathematically determine (A,6,a ( G ) ;A, 6,E(??)) (ii) choose A and R, and (iii) define Cs((p),TS(cp;A). 3.1.1. State change and complexities ID contains the dynamics of state change as its part. A state change is mathematically described by a unitary evolution, a semi-group dynamics, generally, a channeling transformation (it is _ simply - called ''channel" ). Let input and output triples (A,6,a ( G ) )and ( A ,6 , F ( G ) )be C*-dynamical systems; that is, A is a C*-algebraZ2and 6 is its state space and a ( G ) is an inner evolution of A with a parameter group G (or semigroup) and so __ is the output system. Let a channel be a mapping from 6 ( A )to 6 ( A ) . Although there exist several complexities, one of the most fundamental pairs of C and T in quantum system is the von Neumann entropy and the mutual entropy. Other entropic complexities C and T are Eentropy, Kolmogorov-Sinai type dynamical entropy, dynamical mutual entropy.30?40>45 Here we remind that the quantum entropy and the quantum mutual entropy are examples of our complexities C and T , respectively.
11
Example 3.1. The entropy S and the mutual entropy I , in both classical and quantum, satisfy the conditions of the complexities C and T of ID. For a density operator p in a Hilbert space 'FI (the case d =B ('FI)) and a channel A, C ( p ) is the entropy S(p) and T ( p ; A) is the mutual entropy I ( p ; A):
c( p ) = 0) = -trplog
P,
where the supremum is taken over all Schatten decompositions { E k } of p; p = X X k E k . In Shannon's communication theory in classical Systems, k
p is a probability distribution p = ( p k ) = x k p k 6 k and h is a transition probability matrix ( t i , j ) , so that the Schatten decomposition of p is unique and the compound state of p and its output 7 (= p = (pi)= A p ) is the joint distribution T = ( r i , j )with ri,j = t i j p j . Then the above complexities C and T become the Shannon entropy and mutual entropy, respectively;
c(P)= s (P)= T ( p ;A) = I ( p ; A)
Pk
=
log Pk, ~ i ,log j S L. P j Pi
We can construct several other types of entropic complexities. For instance, one pair of the complexities is
s(.,
ck
.) is quantum relative entropy of Umegaki57 and p = P k p k is where a finite decomposition of p, over all of which the supremum is taken. Example 3.2. Generalizing the entropy S and the mutual entropy I , we can construct complexities of entropy type: Let (d,G ( d ) , a ( G ) ) , G(x)E , ( c ) )be C* systems as before. Let S be a weak *-compact convex subset of B(d) and M,(S) be the set of all maximal measures p on S with the fixed barycenter cp
(a,
Moreover let F , ( S ) be the set of all measures of finite support with the fixed barycenter cp. The following three pairs C and T satisfy all conditions
12
of the complexities:
T S (cp; A)
= SUP
{ s,S ( A w , Acp)dp;
p E M,(S)}
C$(cp)E TS(cp;i d ) IS(cp;A)
= sup
{
S
(s,
)
w @ A w d p , 'p 8 Acp ; p
E
M,(S)
Cf(cp) F IS(cp;i d ) S
J (cp; A)
E
sup
is,
S ( A w , Acp)dp; p
E
Cf(cp) = Js(cp;id)
F,(S)
1
sS
Here, the state w @ A w d p is the compound state exhibiting the correlation between the initial state and the final state Acp, and S ( . , . ) is quantum relative e n t ~ - o p y .This ~ > ~compound ~ state was introduced as a quantum generalization of the joint probability measure in CDS (classical dynamical ~ y s t e m ) .Note ~ ~ that ) ~ ~in the case of B =S, TS(resp.CS,IS, Js) is denoted by T (resp. C, I , J ) for simplicity. These complexities and the mixing Sentropy SS('p),40348 the CNT (Connes-Narnhofer-Thirring) entropy HJA) satisfy some relations.
'p
We review the definition of the mixing S-entropy here.44i53For a state 6(d),put
ESc
D,(S)
=
where 6(y) is the delta measure concentrated on {p}, and put k
for a measure p E D,(S). Then the S-entropy of a state cp E S is defined as SS(4 =
{
inf { H ( p ) ; p E D, (S)}when D , (S) # +03 otherwise
0
>
Theorem 3.1. (1) 0 5 IS(cp;A)5 TS(cp;A)5 JS(cp;A). (2) Cl('p) = C T ( ' ~=) C~(cp) = S(cp) = H,(d). (3) W h e n d = 2 = B('FI),for any density operator p
0 5 I S ( p ; A)
= T S ( p A) ;
5
J S ( p ;A ) .
13 3 . 2 . Entropic Chaos Degree
In quantum systems, if we take C ( p ) = S ( p ) =von Neumann entropy, T ( p ; R )= I ( p ; A )=quantum mutual entropy and linear channel A, then we have
since S (Rp) = -TrAp log Rp = -Tr (EnpnAEn log Ap) for any Schatten decomposition { E n } of p. Therefore the above quantity D ( p ; A) can be interpreted as the complexity produced through the channel A. We apply
this quantity D ( p ; A) to study chaos even when the channel describing the dynamics is not linear. D ( p ; A) is called the entropic chaos degree (ECD). In order to describe more general dynamics such as in continuous systems, we define the entropic chaos degree in C*-algebraic terminology. This setting will not be used in the sequel application, but for mathematical completeness we will discuss the C*-algebraic _ -setting. Let (A,6) be an input C* system and (A,6) be an output C* system; namely, A is a C* algebra with unit I and 6 is the set of all states on A. We assume 2 = A for simplicity. For a weak* compact convex subset S (called the reference space) of 6, take a state 'p from the set S and let
be an extremal orthogonal decomposition of 'p in S, which describes the degree of mixture of cp in the reference space S. In more detail this formula reads
AEA P ( A ) = /4A)dPL,(W)l s The measure pq is not uniquely determined unless S is the Schoque simplex, so that the set of all such measures is denoted by Mp ( S ).
14
Definition 3.2. The entropic chaos degree with respect to cp E S and a channel A is defined by
D S (cp; A)
= inf
{ S,ss
(AU) dp; p
1
E Mv (s)
where Ss (.) is the mixing S - e n t r ~ p y ~ ' in ? ~the ~ reference space S. When S =B, Ds (cp; A) is simply written as D (cp; A) . This Ds (cp; A) contains the classical chaos degree and the quantum above. The classical entropic chaos degree is the case that A is abelian and cp is the probability distribution of a orbit generated by a dynamics (channel) A; cp = pkdk,
ck
where
6k
is the delta measure such as
6k
(j)
(Ic
=
j ) . Then the
classical entropic chaos degree is
Dc (cp; A) = x
p k S ( A 6 k ) k
with the entropy S . Summarize that Information Dynamics can be applied to the study of chaos by using more general complexity C(cp): Definition 3.3. (l)$ is more chaotic than cp if C($) 2 C(cp). (2)When cp E S changes to Ap, the chaos degree associated to this state change(dynamics) A is given by
DS (cp; A) = inf
{ S,cs
( ~ c p )dp; p E
M+,( s ) }.
Definition 3.4. A dynamics A produces chaos iff Ds (cp; A) > 0. Remark 3.1. It is important to note here that the dynamics A in the definition is not necessarily same as original dynamics (channel) but is one reduced from the original such that it causes an evolution for a certain observed value like orbit. However for simplicity we use the same notation here. In some cases, the above chaos degree Ds (cp; A) can be expressed as
Ds (cp; A) = Cs (Acp) - TS(cp;A). 3.3. Algorithm computing Entropic Chaos Degree
In order to observe a chaos produced by a dynamics, one often looks at the behavior of orbits made by that dynamics, more generally, looks at the
15
behavior of a certain observed value. Therefore in our scheme we directly compute the chaos degree once a dynamics is explicitly given as a state change of system. However even when the direct calculation does not show a chaos, a chaos will appear if one focuses to some aspect of the state change, e.g., a certain observed value. In the later case, algorithm computing the chaos degree for classical or quantum dynamics consists of the following two cases: (1) Dynamics is given by = F t (x) with x E I E [ a ,bIN c RN : First find a difference equation xn+l = F (x,) with a map F on 1 E [ a ,bIN c RN into itself, secondly let A := {Ai} be a finite partition (i.e., I 3 U k A k , Ai n A j = 0 (i # j)).Then the state qdn) at time n of the orbit determined
%
0
by the difference equation is defined by the probability distribution p,!"'
with a given finite partition A = { A i } , that is, q(n)= C i p i n ) 6 i , where for an initial value x E I and the characteristic function 1~
k=m
Now when the initial value x is distributed due to a measure v on I after a proper time m, the above pj"' is given as
The joint distribution
(p.n'n+l)) between the time n and n + 1 is defined :j
by
.
mfn
or
Then the channel A, at n is determined by
A,
= PJ?(
~
: transition probability
==+p(n+l)= A , V ( ~ ) ,
16
and the entropic chaos degree is given by, for a finite partition A := { A i } ,
(1) We can judge whether the dynamics causes a chaos or not by the value of D A for the partition A = {Ai} as
DA > 0 DA = 0
chaotic stable.
This chaos degree was applied to several dynamical maps such logistic map, Baker's transformation and Tinkerbell map, and it could explain their chaotic characters. This chaos degree has several merits compared with usual measures such as Lyapunov exponent as explained below. The partition free chaos degree D is defined by the infimum of D A over all partitions A.Therefore it is said that the dynamics pruduces a chaos in the scale { A k } if D A is positive. (2) Dynamics is given by pt = f t p o on a Hilbert space: Similarly as making a difference equation for (quantum) state, the channel A, at n is first deduced from F t , which should satisfy p(,+') = R,p(,). By means of this constructed channel, (i) we compute the chaos degree D directly according t o the definition 3.2 or (ii) we take a proper observable X and put 2 , = p(")(X),then go back to the algorithm (1). The entropic chaos degree for quantum systems has been applied to the analysis of quantum spin system and quantum Baker's type transformation.27,28,31 Note that the chaos degree D A above does depend on a partition A taken, which is somehow different from usual degree of chaos. This is a key point of our understanding of chaos, from which the idea of adaptivity comes, which is discussed in Subsection 3.4.
Example 3.3. Logistic Map Let us apply the entropy chaos degree (ECD) to logistic map. Chaotic behavior in classical system is often considered as exponential sensitivity to initial condition. The logistic map is defined by
17
The solution of this equation bifurcates as Fig. 1.
x, 1
0.8
0.6
0.4
0.2
-
I
3.2
3.4
3.6
3.8
4
a
Fig. 1. Logistic map
In order to compare ECD with other measure describing chaos, we take Lyapunov exponent for this comparison: Fig. 2 and Fig. 3. We computed the entropic chaos degree for various maps in,29and it is shown that Lyapunov exponent and chaos degree have clear correspondence. Moreover the ECD resolves some inconvenient points of the Lyapunov exponent as: Lyapunov exponent takes negative and sometimes -m, but the ECD is always positive for all a 2 0. For some map f whose Lyapunov exponent is difficult t o compute (e.g., dynamics in Rn (n 2 2)), the ECD o f f is easily computed. Generally, the algorithm for the ECD is much easier than that for Lyapunov exponent.
I. ECD with memory Here we generalize the above explained ECD to take the memory effect into account. Although the original ECD is based on the choice of the base space C := { 1 , 2 , . . . ,N } , we here take another base; Em, instead of C.On
18
CD 0.7
3
3.2
3.4
Fig. 2.
3.6
3.8
Entropic Chaos Degree of Logistic Map
1 4
R
t'
0.5 F
'
"
"
'
'
*
"
"
"
'
'
"
"
"
'
=
'
0
-0.5
-1
-1.5
3
3.2
3.4 Fig. 3.
3.6
3.8
Lyapunov exponent
this base space, a probability distribution is naturally defined as
4
19 (n,n+1,...,Tl with its mathematical idealization, pioil...i, := limn+m pioil,,,im +,).The channel A, over Em is defined by a transition probability,
.
.
~ j o i l . . . i ~ + 1 & 1...6imj, jl = ~ ( i l j i 2 , ..,2rn,2rn+11jorjl,. . . . ~.im)Pjo,jl...jm~ Thus it derives the ECD with rn-steps memory effect,
It notes that this quantity coincides with the original ECD when 'm = 1. This m e m o r y effect shows a n interesting result, that is, the longer the m e m o r y is, the closer the ECD i s t o the Lyapunov exponent for its positive part5'
Theorem 3.2. For given f , x and A , there exists a probability space (0,F , v ) and a random variable g depending o n f , x , A such that 1immdm D T ( x ;f ) = g d u =the positive part of Lyapunov exponent. 3.4. Adaptive Chaos Degree In adaptive dynamics, it is essential to consider in which states and by which ways we see objects. That is, one has t o select phenomena and prepare mode for observation for understanding the whole of a system. Typical adaptive dynamics are the dynamics for state-adaptive and that for observable-adaptive as mentioned in the previous section. We will discuss how such adaptivities are appeared in dynamics which cause a chaos. First of all we examine carefully when we say that a certain dynamics produces a chaos. Let us take the logistic map as an example. The original differential equation of the logistic map is
dx = a x ( 1 - x ) ,0 dt
5a54
with initial value xo in [0,1].This equation can be easily solved analytically, whose solution (orbit) does not have any chaotic behavior. However once we make the above equation discrete such as
This difference equation produces a chaos.
20
Taking the discrete time is necessary not only to make a chaos but also t o observe the orbits drawn by the dynamics. Similarly as quantum mechanics, it is not possible for human being to understand any object without observing it, for which it will not be possible t o trace a orbit continuously in time. Now let us think a finite partition A = { A k ; lc = 1,.. . , N } of a proper set 1 = [a,bIN c RN and an equi-partition Be = { B i ;lc = 1,.. . , N } of 1.Here "equi" means that all elements BE are equivalent. We denote the set of all partitions by P and the set of all equi-partitions by Pe. Such a partition enables to observe the orbit of a given dynamics, and moreover it provides a criterion for observing chaos. There exist several reports saying that one can observe chaos in nature, which are very much related to how one observes the phenomena, for instance, scale, direction, aspect. It has been difficult to find a satisfactory theory (mathematics) to explain such chaotic phenomena. In the difference equation 2 we take some time interval T between n and n 1,if we take r + 0, then we have a complete different dynamics. If we take coarse graining to the orbit of xt for time during 7; 5, = $ x t d t , we again have a very different dynamics. Moreover it is important for mathematical consistency to take the limits n 00 or N (the number of equi-partitions)+ 00 , i.e., making the partition finer and finer, and consider the limits of some quantities as describing chaos, so that mathematical terminologies such as "lim", "sup", "inf" are very often used to define such quantities. Let u s take the opposite position, that is, any observation will be unrelated or even contradicted t o such limits. Observation of chaos is a result due t o taking suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases. It is claimed in38 that most of chaos are scale-dependent phenomena, so the definition of a degree measuring chaos should depend on certain scales taken. Such a scale dependent dynamics is nothing but adaptive dynamics. Taking into consideration of this view we modify the definitions of the chaos degree given in the previous section 3.2 as below. Going back to a triple (A,6,a (G)) considered in Section 2 and we use this triple both for an input and an output systems. Let a dynamics be described by a mapping rt with a parameter t E G from 6 to 6 and let an observation be described by a mapping 0 from (A,6,Q! (G)) to a triple (B,2,,B (G)). The triple (B,2,/3 (G)) might be same as the original one or its subsystem and the observation map 0 may contains several different types of observations, that is, it can be decomposed as 0 = 0,. ..OI.Let us list some examples of observations.
+
sc-l).r
--f
21
For a given dynamics several observations.
= F (cpt)
, equivalently, cpt
=
rtcp,one can take
%
Example 3.4. Time Scaling (Discretizing): 0, : t 4 n, (t) + pn+lr SO that = F ( P t ) + qn+l = F ( P n ) and q t = r t p vn = rnv. Here T is a unit time needed for the observation.
*
%
Example 3.5. Size Scaling (Conditional Expectation, Partition): Let (B,T,p (G)) be a subsystem of (A,6 ,Q (G)), both of which have a certain algebraic structure such as C*-algebra or von Neumann algebra. As an example, the subsystem (B,2,,B (G)) has abelian structure describing a macroscopic world which is a subsystem of a non-abelian (noncommutative) system (d, 6,Q (G)) describing a micro-world. A mapping OC preserving norm (when it is properly defined) from d to B is, in some cases, called a conditional expectation. A typical example of this conditional expectation is according to a projection valued measure
associated with quantum measurement (von Neumann measurement) such that
k
for any quantum state (density operator) p . When B is a von Neumann algebra generated by { p k } , it is an abelian algebra isometrically isomorphic to Lm (0)with a certain Hausdorff space R , so that in this case Oc sends a general state cp to a probability measure (or distribution) p . Similar example of OC is one coming from a certain representation (selection) of a state such as one Schatten decomposition of p ;
by one-dimensional orthogonal projections { E k } associated to the eigenvalues of p with x k E k = I . Another important example of the size scaling is due to a finite partition of an underlining space R, e.g., space of orbit, defined as
22
3.4.1. Chaos degree with adaptivity We go back to the discussion of the entropic chaos degree. Starting from a given dynamics cpt = rtp, it becomes = after handling the operation 0,. Then by taking proper combinations 0 of the size scaling operations like OC, OR and O p , the equation cpn = rncp changes to (3 (cp,) = 0 (rncp) I which will be written by O cp, = Ol?,0-1c3cp or cpf = rf'po. Then our entropic chaos degree is redefined as follows:
rncp
Definition 3.5. The entropic chaos degree of I' with an initial state cp and observation 0 is defined by
where po is the measure operated by 0 to a extremal decomposition measure of p selected by of the observation O (its part O R ) . The entropic chaos degree of l7 with a n initial state cp is defined by qcp;r)=inf {P(cp;r);OE~o},
where SO is a proper set of observations naturally determined by a given dynamics. In this definition , S O is determined by a given dynamics and some conditions attached to the dynamics] for instance, if we start from a difference equation with a special representation of an initial state, then SO excludes Or and OR. Then one judges whether a given dynamics causes a chaos or not by the following way.
Definition 3.6. (1) A dynamics I' is chaotic for a n initial state cp in an observation O iff Do (cp; r) > 0. (2) A dynamics is totally chaotic f o r a n initial state cp iff D (p;l?) > 0. The idea introducing in this section to understand chaos can be applied not only to the entropic chaos degree but also t o some other degrees such as dynamical entropy, whose applications and the comparison of several degrees will be discussed in.51 In the case of logistic map, z,+1 = az,(l -zn) = F (z,) we obtain this difference equation by taking the observation Or and take an observation O p by equi-partition of the orbit space R = (5,) so as to define a state
23
(probability distribution). Thus we can compute the entropic chaos degree in adaptive sense. As an example, we consider a circle map
en+, = f,(O,)
= 8,
+ w (mod 2 ~ ) ,
where w = 27rv (0 < v < 1). If v is a rational number N / M , then the orbit (0,) is periodic with the period M . If u is irrational, then the orbit (8,) densely fills the unit circle for any initial value 8 0 ; namely, it is a quasiperiodic motion.
Theorem 3.3. Let I = [0,27r] be partioned into L disjoint components with equal length; I = B1 n Bz n . . . n BL. (1) If v is rational number N / M , then the finite equi-partition P = { B k ; k = 1,.. . ,M } implies Do (Oo; f,) = 0. (2) If v is irrational, then Do (00;f v ) > 0 for any finite partition P = {Bk}.
Note that our entropic chaos degree shows a chaos t o quasiperiodic circle dynamics by the observation due to a partition of the orbit, which is different from usual understanding of chaos. However usual belief that quasiperiodic circle dynamics will not cause a chaos is not at all obvious, but is realized in a special limiting case as shown in the following theorem.
Theorem 3.4. For the above circle map, if v is irrational, then D (eo; fv) = 0. Such a limiting case will not take place in real observation of natural objects, so that we claim that chaos is a phenomenon depending on observations, environment or periphery, which results the adaptive definition of chaos as above. The detailed examination of a map of this type is done in the paper.13 Note here that the chaos degree and the adaptivity can be applied to understand quantum dynamics either.27>28i31
4. Adaptive Dynamics Solving SAT Problem. 4.1. SAT problem
We take the SAT (satisfiable) problem, one of the NP-complete problems, to study whether there exists an algorithm showing NPC=P.
24 Let x = {XI,...,x,} be a set. Then xk and its negation (k = 1 , 2 , . . . ,n) are called literals and the set of all such literals is denoted by X’ = { z 1 , f l , . . . , IC,, f,}. The set of all subsets of X’ is denoted ftk
is called a clause. We take a truth by F(X‘)and an element C E F(X’) assignment to all Boolean variables zk. If we can assign the truth value to at least one element of C , then C is called satisfiable. When C is satisfiable, the truth value t (C) of C is regarded as true, otherwise, that of C is false. Take the truth values as ”true ~ 1false , ++O”. Then Cis satisfiable iff t ( C )= 1. Let L = (0, l} be a Boolean lattice with usual join V and meet A, and t (z) be the truth value of a literal z in X . Then the truth value of a clause C is written as t (C) = V z E c t(z). Moreover the set C of all clauses Cj ( j = 1 , 2 , . . . , rn) is called satisfiable iff the meet of all truth values of Cj is 1; t (C) = A z l t (Cj) = 1. Thus the SAT problem is written as follows:
Definition 4.1. SAT Problem: Given a Boolean set X = ( 2 1 , . . . ,zn}and a set C = {Cl, . . . ,}C , of clauses, determine whether C is satisfiable or not. That is, this problem is to ask whether there exists a truth assignment t o make C satisfiable. It is known in usual algorithm that it is polynomial time to check the satisfiability only when a specific truth assignment is given, but we can not determine the satisfiability in polynomial time when an assignment is not specified. In52 we discussed the quantum algorithm of the SAT problem, which was rewritten in7 with showing that OM SAT-algorithm is combinatric. In54255it is shown that the chaotic quantum algorithm can solve the SAT problem in polynomial time. that the SAT problem, hence all other Ohya and Masuda pointed NP problems, can be solved in polynomial time by quantum computer if the superposition of two orthogonal vectors 10) and 11) is physically detected. However this detection is considered not to be possible in the present technology. The problem to be overcome is how to distinguish the pure vector 10) from the superposed one a 10) ,O [ I ) , obtained by the OM SAT-quantum algorithm, if ,b’ is not zero but very small. If such a distinction is possible, then we can solve the NPC problem in the polynomial time. In54,55it is shown that it can be possible by combining nonlinear chaos amplifier with the quantum algorithm, which implies the existence of a mathematical algorithm solving NP=P. The algorithm of Ohya and Volovich is not known to be in the framework of quantum Turing algorithm or not. So the next
+
25 question is (1) whether there exists a physical realization combining the SAT quantum algorithm with chaos dynamics, or (2) whether there exists another method to achieve the above distinction of two vectors by a suitable unitary evolution so that all process can be discussed by a certain quantum Turing machine ( c i r c ~ i t s ) . The ~ ~ - paper34 ~~ by Iriyama and Ohya of this volume briefly discusses the essence of the quantum SAT algorithm. In this paper, we will discuss the SAT problem with adaptive dynamics based on the work 0f13 which is another method of (2) above.
4.2. Quantum Algorithm
The quantum algorithms discussed so far are rather idealized because computation is represented by unitary operations. A unitary operation is rather difficult to realize in physical processes, more realistic operation is one allowing some dissipation like semigroup dynamics. However such dissipative dynamics very much reduces the ability of quantum computation because the ability is based on preserving the entanglement of states and the dissipativity destroys the entanglement. Keeping high ability of quantum computation and good entanglement, it will be necessary to some kinds of amplification in the course of real physical processes in physical devices, which will be similar as amplication processes in quantum communication processes. In this section, to search for more realistic operations in quantum computer, the channel expression will be useful, at least, in the sense of mathematical scheme of quantum computation because the channel is not always unitary and represents many different types of dynamics. Let 7-l be a Hilbert space describing input, computation and output (result). As usual, the Hilbert space is 7-l = @pC2,and let the basis of 7-l = @pC2be: eo (= 10)) = 10)8 . . . @ 10)@ 10), e l (= 11)) = 10)@ . . . 8 10) 8 1 ) , . . . ,e p - 1 (= )2N - 1)) = 11) @ - .. B 1 ) 8 1 ) . Any number t (0,. . . , Z N
N
-
1) can be expressed by t =
C k=l
aik) = 0 or 1, so that the associated vector is written by
And applying n-tuples of Hadamrd matrix A vacuum vector 10) , we get A 10)( = E (0))
= &”&
(’ )
=1 Jz
1-1 (10) 11)). P u t
+
to the
26
Then we have
which is called Discrete Fourier Transformation. Thus altogether of the above operations, it follows a unitary operator UF ( t ) =_ W ( t )A and the vector E ( t )= U F ( t )10) . 4.2.1. Channel expression of conventional unitary algorithm
All conventional unitary algorithms can be written as the following three steps: (1) Preparation of state: Take a state p (e.g., p = 10) (01) applying the unitary channel defined by the above UF (t): AF = AduF(t)
AF = Adu, ==+ h ~ =pU F ~ U G (2) Computation: Let U a unitary operator on X representing the computation followed by a suitable programming of a certain problem, then the computation is described by a channel A, = Adu (unitary channel). After the computation, the final state p j will be
Pf
= AUAFP.
(3) Register and Measurement: For registeration of the computed result and its measurement we might need an additional system K (e.g., register), so that the lifting Ern from S (X) to S (X@ K) in the sense of3 is useful to describe this stage. Thus the whole process is wrtten as ~f = Ern ( A u A F P ) .
Finally we measure the state in K: For instance, let jection valued measure (PVM) on K
{Pk;k
E J } be a pro-
I 8PkPfI 8p k1
AmPj = k€J
after which we can get a desired result by observations in finite times if the size of the set J is small.
27
4.2.2. Channel expression of general quantum algorithm
Since actual physical process is dispative, the above three steps have to be generalized so that the dissipative nature is involved. Such a generalization can be expressed by means of suitable channel, not necessarily unitary, which gives us a basois of the generalized quantum Turing m a ~ h i n e . ~ ~ ? ~ ~ (1) Preparation of state: We may be use the same channel AF = AduF in this first step, but if the number of qubits N is large so that it will not be built physically, then AF should be modified, and let denote it by A p . (2) Computation: This stage is certainly modified to a channel A c reflecting the physical device for computer. (3) Registering and Measurement: This stage will be remained as aobe. Thus the whole process is written as
4.3. Quantum Algorithm of SAT In this subsection, we review fundamentals of quantum computation (see, for instance,55) for the SAT problem. Let C be the set of all complex numbers, and 10) and 11) be the two unit vectors and , respectively. Then, for any two complex numbers a and ,O satisfying IaI2 IPI2 = 1, a 10) +/?11) is called a qubit. For any positive integer N , let 'H be the tensor product Hilbert space defined as (C2)'N and let {lei) ;0 5 i 5 2 N - ' } be the basis as above. For any two qubits) . 1 and Iy), Jz,y) and is written as Ix) @3 Iy) and Ix) 18. . . @ I%), respectively.
(A)
(y) +
1%")
J
N times
The computation of the truth value can be done by by a combination of the unitary operators on a Hilbert space 'H, so that the computation is described by the unitary quantum algorithm. The detail of this section is given in the papers,7~32~52~55 so we will discuss just the essence of the OM algorithm. Throughout this section, let n be the total number of Boolean variables used in the SAT problem. Let C = { C1, . . . , Cm}be a set of clauses whose cardinality is equal to m. Let 'H = (C2)@"+'+' be a Hilbert space and be the initial state 1110) = IOn,Ofi,O), where p is the number of dust qubits which is determined by the following proposition. Let Up' be a unitary operator for the computation of the SAT:
IWO)
28
where xfi denotes the p strings in the dust bits and tet(C) is the truth value of C with ei. In,7,34152U p ) was constructed. This final state vector (vf> is also written as
Theorem 4.1. C is S A T zf and only if,
Pn+p,lUc) . 1
#0
where P n + p ,denotes ~ the projector Pn+p,l
:= In+p-l@ 11
>< 11
onto the subspace of 3-1 spanned by the vectors J ~ n , ~ p -1l >, r
where
E~ E
(0, l } n and
~
~ E-
{ O1 , l } p - l .
According to the standard theory of quantum measurement, after a measurement of the event P n + p , ~ the l state p = I v ~f>< v ~ fbecomes l
Thus the solvability of the SAT problem is reduced t o check that p' The difficulty is that the probability of Pn+,,1 is
# 0.
where ITCo)] is the cardinality of the set T(Co),of all the truth functions t such that t(Co) = 1. We put q := with r := ( T ( C o ) (in the sequel. Then i f r is suitably
,&
large to detect it, then the SAT problem is solved in polynomial time. However, for small r, the probability is very small and this means we in fact don't get an information about the existence of the solution of the equation t(C0) = 1, so that in such a case we need further deliberation. Let us simplify our notations. After the quantum computation, the quantum computer will be in the state
29
m.
where 1 ~ 1 )and Ipo) are normalized n qubit states and q = Effectively our problem is reduced to the following 1 qubit problem. We have the state
I 4 = ale) + 4 11) and we want to distinguish between the cases q = 0 and q > O(smal1positive number). I t is argued in15 that quantum computer can speed up N P problems quadratically but not exponentially. The no-go theorem states that if the inner product of two quantum states is close to 1, then the probability that a measurement distinguishes which one of the two it is exponentia,lly small. And one could claim that amplification of this distinguishability is not possible. At this point we emphasized55 that we do not propose to make a measurement which will be overwhelmingly likely to fail. What we do it is a proposal to use the output I$) of the quantum computer as an input for another device which uses chaotic dynamics. The amplification would be not possible if we use the standard model of quantum computations with a unitary evolution. However the idea of the is different. In54255it is proposed to combine quantum computer with a chaotic dynamics amplifier. Such a quantum chaos computer is a new model of computations and we demonstrate that the amplification is possible in the polynomial time. One could object that we do not suggest a practical realization of the new model of computations. But at the moment nobody knows of how to make a practically useful implementation of the standard model of quantum computing ever, It seems to us that the quantum chaos computer considered in55 has a potential to be realizable. 4.4. Quantum chaos algorithm
Various aspects of classical and quantum chaos have been the subject of numerous studies, see43 and ref’s therein. Here we will argue that chaos can play a constructive role in computations (see54y55for the details). Chaotic behavior in a classical system usually is considered as an exponential sensitivity to initial conditions. It is this sensitivity we would like to use to distinguish between the cases q = 0 and q > 0 from the previous section. Using the logistic map z,+1 = az,(l-
2,)
f
f(z),
2,
E [O, 11.
30
The properties of the map depend on the parameter a. If we take, for example, a = 3.71, the trajectory is very sensitive to the initial value and one has the chaotic behavior. It is important to notice that if the initial value 50 = 0, then 5, = 0 for all n. It is known that any classical algorithm can be implemented on quantum computer. Our quantum chaos computer will be consisting from two blocks. One block is the ordinary quantum computer performing computations with the output I+) = 10) q 11).The second block is a computer performing computations of the classical logistic map. This two blocks should be connected in such a way that the state I+) first be transformed into the density matrix of the form
m +
d
p = q2P1
+ (1 - 42) Po
where PI and POare projectors to the state vectors 11) and 10).This connection is in fact nontrivial and actually it should be considered as the third block. One has to notice that PI and POgenerate an Abelian algebra which can be considered as a classical system. In the second block the density matrix p above is interpreted as the initial data p ~ and , we apply the logistic map as Pm =
(1+ f m ( p 0 ) f f 3 )
2 where I is the identity matrix and 0 3 is the z-component of Pauli matrix on C2. To find a proper value m we finally measure the value of o 3 in the state pm such that
Mm
E trpmo3.
We obtain
Theorem 4.2.
Thus the question is whether we can find such a m in polynomial steps of satisfying the inequality Mm 2 for very small but non-zero q2. Here we have to remark that if one has q = 0 then po = POand we obtain Mm = 0 for all m. If q # 0, the stochastic dynamics leads to the amplification of the small magnitude q in such a way that it can be detected as is explained below. The transition from po to pm is nonlinear and can be considered as a classical evolution because our algebra generated by POand PI is abelian.
TI
31
The amplification can be done within at most 2n steps due to the following propositions. Since f"(q2) is x , of the logistic map z,+1 = f(x,) with xo = q 2 , we use the notation x , in the logistic map for simplicity.
Theorem 4.3. For the logistic map x,+l = ax, (1 - z), with a E [0,4] and xo E [0,1], let xo be and a set J be { 0 , 1 , 2 , . . . , n , . . . 2 n } . If a is 3.71, then there exists an integer m in J satisfying x, >
&
i.
Theorem 4.4. Let a and n be the same in the above proposition. If there exists mo in J such that x,, > , then mo > ,og2n;;l-l.
i
According to these theorems, it is enough to check the value x , (M,) around the above mo when q is for a large n. More generally, when q=& with some integer k, it is similarly checked that the value x, (M,) becomes over within a t most 2n steps. The complexity of the quantum algorithm for the SAT problem was discussed in Section 3 to be in polynomial time. We have only to consider the number of steps in the classical algorithm for the logistic map performed on quantum computer. It is the probabilistic part of the construction and one has to repeat computations several times to be able to distinguish the cases q = 0 and q > 0. Thus it seems that the quantum chaos computer can solve the SAT problem in polynomial time.In conclusion ~ f , the~ quantum chaos computer combines the ordinary quantum computer with quantum chaotic dynamics amplifier. It may go beyond the usual quantum Turing algorithm, but such a device can be powerful enough t o solve the NP-complete problems in the polynomial time.
&
In the following subsections we will discuss the SAT problem in adaptive dynamics. Now from the general theory of stochastic limitg one knows that, under general ergodicity conditions, an interaction with an environment drives an adaptive dynamical (but not necessarily thermodynamical) equilibrium state which depends on the initial state of the environment and on the interaction Hamiltonian. If one is able to realize experimentally these state dependent Hamiltonians, one would be able to drive the system S t o a pre-assigned dynamical equilibrium state depending on the input state $JO.
4 . 5 . Stochastic limit and adaptive SAT Problem We illustrate the general scheme described in the previous section in the simplest case when the state space of the system is Xs = C 2 . We fix an
~
,
~
32
orthonormal basis of 7 - l ~as {eo, el}. The unknown state (vector) of the system at time t
+ := C
ace, = aoeo
+ a l e 1 ; 11+11
=0
= 1.
EE{O,1)
In the case of Sec. 3, alcorresponds to q and e j does to l j ) ( j = 0 , l ) . This vector after quantum computation of the SAT problem is taken as input and defines the interaction (adaptive) Hamiltonian in an external field
where X is a small coupling constant. Here and in the following summation over repeated indices is understood. The free system Hamiltonian is taken to be diagonal in the e,-basis
H S :=
&leE)(eEI= EoIeo)(eoI + - W l ) ( e l l &EtO,1)
and the energy levels are ordered so that Eo < &.The environment Hamiltonian is
H E :=
s
w ( k ) AzAkdk,
where ~ ( kis) a function satisfying the basic analytical assumption of the stochastic limit. Thus the total free Hamiltonian is Ho := H s + H E . The free field evolution is given by eitHoA*ge--itHo = where Stg(k) = eitw(k)g(k). We can distinguish two cases as below, whose cases correspond to two cases of Sec. 3, i.e., q > 0 and q = 0. Case (1).If ao,a1 # 0 , then, according to the general theory of stochastic limit (i.e., t 4 t / X 2 ) , ’ the interaction Hamiltonian H I is in the same universality class as
HI
= DNA;
+ D+ @A,
where D := leo)(ell. The interaction Hamiltonian at time t is then f i I ( t ) = e-ZtwoD@ A:t,
+ h.c. = D @ A+(eit(W(P)-Wo)g) + h.c.,
33
where wo = El - Eo. The white noise ( { b t } ) Hamiltonian equation associated, via the stochastic golden rule, to this interaction Hamiltonian is
&Ut = i(Db$
+ D+bt)Ut
Its causally normal ordered form is equivalent to the stochastic differential equation
dl/, = (iDdB,f
+ iD+dBt - rD+Ddt)Ut,
where dBt := btdt and y is a certain constant. Then we derive the master equation as follows: d
Pt = L*Pt,
where pt := etL*p and L,p := ( I m y ) i [ pD'D] ,
-
(Re y ) { p , D'D}
+ (Re y)DpD+
For p = po := leo)(eol one has L*po = 0 so po is an invariant measure. From the Fagnola-Rebolledo criteria,23 it is
the unique invariant measure and the semigroup exp(tL,) converges exponentially to it. Case (2). If a1 = 0, then the interaction Hamiltonian H I is
Hr = Aleo)(eol
@
(A:
+4)
and, according to the general theory of stochastic limit, the reduced evolution has no damping and corresponds to the pure Hamiltonian
Hs
+ leo)(eol
+
= (Eo l)leo)(eol
+ Eilei)(eil
therefore, if we choose the eigenvalues E l , EOto be integers (in appropriate units), then the evolution will be periodic. Since the eigenvalues E l , EOcan be chosen a priori, by fixing the system Hamiltonian H s , it follows that the period of the evolution can be known a priori. This gives a simple criterium for the solvability of the SAT problem because, by waiting a sufficiently long time one can experimentally detect the difference between a damping and an oscillating behavior. We used the resulting (flag) state after quantum computation of the truth function of SAT to couple the external field and took the stochastic limit, then our final evolution becomes "linear" for the state p described as above. The stochastic limit is historically important to realize macroscopic
34 (time) evolution a n d i t is now rigorously established as explained in,g a n d we gave a general protocol to study the distinction of two cases a1 # 0 and a1 = 0 by this rigorous mathematics. T h e macro-time enables us to measure the behavior of t h e outcomes practically. Thus we show t h a t it is possible to distinguish two different states, 417 10) q 11) ( q # 0) a n d 10) by means of t h e adaptive dynamics and the stochastic limit. This provides another algorithm solving NPC problem in realistic time.
+
Acknowledgment T h e author t h a n k Monka-Sho for financial support.
References 1. L.Accardi, Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria pantistica. I1 Saggiatore, Rome (1997) 2. L.Accardi, K.Imafuku, M.Refoli, On the EPR-Chameleon experiment, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 5, NO. 1 (2002) 1-2 3. L.Accardi, M.Ohya, A Stochastic limit approach to the SAT problem”, Proceedings of VLSI 2003, and Open systems and Information Dynamics (2004) 4. L.Accardi, M.Ohya, Compound channels, transition expectations, and liftings”, Appl. Math. Optim., vo1.39, 33-59 (1999) 5. L.Accardi, M.Ohya, N.Watanabe, Note on quantum dynamical entropies Reports on mathematical physics, vo1.38 n.3 457-469 (1996) 6. H.Araki, Relative entropy of states of von Neumann Algebras, Publ.RIMS, Kyoto Univ.Vol.11, 809-833 (1976) 7 . L.Accardi, R.Sabbadini, On the Ohya-Masuda quantum SAT Algorithm, in: Proceedings International Conference UMC’O1, Springer (2001) 8. K.T.Alligood, T.D.Sauer, J.A.Yorke, Chaos-An Introduction to Dynamical Systems-, Textbooks in Mathematical Sciences, Springer (1996) 9. L.Accardi, Y.G. Lu, I. Volovich: Quantum Theory and its Stochastic Limit. Springer Verlag 2002; Japanese translation, Tokyo-Springer 2003. 10. M. Garey and D. Johnson, Computers and Intractability - a guide to the theory of NP-completeness, Freeman, 1979. 11. R.Alicki, Quantum geometry of noncommutative Bernoulli shifts, Banach Center Publications, Mathematics Subject Classification 46L87 (1991) 12. R.Alicki, M.Fannes, Defining quantum dynamical entropy, Lett. Math. Physics, 32, 75-82 (1994) 13. M.Asano, M.Ohya and Y.Togawa, Entropic chaos degree of rotations and log-linear dynamics, QBIC proceedings (this volume), 2007. 14. F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Springer (1993) 15. C. H. Bennett, E. Bernstein, G. Brassard, U. Vazirani, Strengths and Weaknesses of Quantum Computing, quant-ph/9701001. 16. R. Cleve, A n Introduction to Quantum Complexity Theory, quantph/9906111.
35 17. D. Deutsch, Quantum theory, the Church- Turing principle and the universal quantum computer, Proc. of Royal Society of London series A, 400, pp.97117, 1985. 18. A. Ekert and R. Jozsa, Quantum computation and Shor’s factoring algorithm, Reviews of Modern Physics, 68 No.3,pp.733-753, 1996. 19. A.Connes, H.Narnhofer, W.Thirring, Dynamical entropy of C*-algebras and von Neumann algebras, Commun.Math.Phys., 112, pp.691-719 (1987) 20. R.L.Devaney, An Introduction t o Chaotic dynamical Systems, Benjamin (1986) 21. G.G.Emch, H.Narnhofer, W.Thirring and G.L.Sewel1, Anosov actions on noncommutative algebras, J.Math.Phys., 3 5 , No.11, 5582-5599 (1994) 22. G.G.Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley (1972) 23. F. Fagnola and R. Rebolledo, On the existence of Stationary States for Quantum Dynamical Semigroups, to appear in J. Math. Phys., 2001. 24. K-H.Fichtner and M.Ohya, Quantum teleportation with entangled states given by beam splittings, Communications in Mathematical Physics, 2 2 2 , 229 (2001). 25. K-H.Fichtner and M.Ohya, Quantum teleportation and beam splitting, Communications in Mathematical Physics, 2 2 5 , 67 (2002). 26. K-H.Fichtner, W. Freutenberg and M.Ohya,Teleportation Schemes in Infinite Dimensional Hilbert Spaces, J. Math. Phys. 46, No. 10 (2006). 27. K.Inoue, M.Ohya, I.V.Volovich, Semiclassical properties and chaos degree for the quantum baker’s map, J. Math. Phys., 43-2, 734-755 (2002) 28. K.Inoue, M.Ohya, I.V.Volovich, On quantum-classical correspondence for baker’s map, quant-ph/0108107 (2001) 29. K.Inoue, M.Ohya and K.Sato, Application of chaos degree to some dynamical systems, Chaos, Soliton & Fractals, 11, 1377-1385 (2000) 30. R.S.Ingarden, A.Kossakowski, M.Ohya, Information Dynamics and Open Systems, Kluwer Publ. Comp. (1997) 31. K.Inoue, M.Ohya, A.Kossakowski, A Description of Quantum Chaos, Tokyo Univ. of Science preprint (2002) 32. S.Iriyama and M.Ohya, Rigorous Estimate for OMV SAT Algorithm, to appear in OSID, 2007. 33. S.Iriyama and M.Ohya, Language Classes Defined by Generalised Quantum Turing Machine, TUS preprint, 2007 34. S.Iriyama and M.Ohya, Review on Quantum Chaos Algorithm and Generalized Quantum Turing Machine, QBIC proceedings (this volume), 2007 35. S.Iriyama, M.Ohya and I.V.Volovich, Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm, QP-PQ:Quantum Prob. White Noise Anal., Quantum Information and Computing, 19, World Sci. Publishing, 204-225, 2006. 36. A.Kossakowski, M.Ohya (2006) ; New scheme of quasntum teleportation process, to appear in Infinite Dimensional Analysis and Quantum Probability 37. A.Kossakowski, M.Ohya (2006) Can Non-Maximal Entangled State Achieve a Complete Quantum Teleportation?, Reconsideration of Foundation-3, Amer-
36 ican Institute of Physics, 810, 211-216. 38. A.Kossakowski, M.Ohya, Y.Togawa, How can we observe and describe chaos? Open System and Information Dynamics, 10(3):221-233 (2003) 39. A.Kossakowski, M.Ohya, N.Watanabe, Quantum dynamical entropy for completely positive maps, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2,No.2, 267-282 (1999) 40. N.Muraki, M.Ohya, Entropy functionals of Kolmogorov-Sinai type and their limit theorems, Letter in Mathematical Physics,36, 327-335 (1996) 41. M.Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, No.5, 770-774 (1983) 42. M.Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep.Math.Phys., Vo1.27, 19-47 (1989) 43. M.Ohya, Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics,Vol.37, No.1, 495-505 (1998) 44. M.Ohya, State change, complexity and fractal in quantum systems, Quantum Communications and Measurement, Plenum Press, New York, 309-320 (1995) 45. M.Ohya, Complexity and fractal dimensions for quantum states, Open Systems and Information Dynamics, 4, 141-157 (1997) 46. M.Ohya, Note on quantum proability, L.Nuovo Cimento, Vo1.38, NO1 0 , 203-206, (1983) 47. M.Ohya, Information dynamics and its applications to optical communication processes, Lecture Note in Physics, 378,81-92 (1991) 48. M.Ohya, Entropy Transmission in C*-dynamical systems, J.Math. Anal.Appl., 100, No.1, 222-235 (1984) 49. M.Ohya (2004) Foundation of Chaos Through Observation, Quantum Information and Complexity edited by T.Hida, K.Saito and Si Si,391-410. 50. M.Ohya (2005): Adaptive Dynamics in Quantum Information and Chaos, in “Stochastic Analysis: Classical and Quantum” ed. by Hida, 127-142. 51. M.Ohya et al, Adaptive dynamics its use in understanding of chaos, TUS preprint 52. M.Ohya, N.Masuda, N P problem in quantum algorithm, Open Systems and Information Dynamics, vo1.7, No.1, 33-39 (2000) 53. M.Ohya, D.Petz, Quantum Entropy and its Use, Springer-Verlag (1993) 54. M.Ohya, I.V.Volovich, New quantum algorithm for studying NP-complete problems, Rep.Math.Phys.,52, No.1,25-33 (2003) and Quantum computing and chaotic amplifier, J.0pt.B (2003) 55. M.Ohya, I.V.Volovich, Mathematical Foundations of Quantum Information and Quantum Computation, to be published in Springer-Verlag 56. P.W. Shor, Algorithm for quantum computation : Discrete logarithm and factoring algorithm, Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, pp.124-134, 1994. 57. H.Umegaki, Conditional expectation in operator algebra IV, Kodai Math. Sem. Rep., 14, 59-85 (1962)
37 JOURNAL OF MATHEMATICAL PHYSICS 46, 102103 (2005)
Teleportation schemes in infinite dimensional Hilbert spaces Karl-Heinz Fichtnera’ Instirut jiir Aitgewandte Marhemarik, Friedrich-Schiller-Unversitat Jena, 07740 Jena, Germany
Wolfgang Freudenbergb) Institut jiir Mathematik, PF 101344, Brandenburgische Technische Uitiversitat Cottbus, 03013 Cottbus, Germany
Masanori Ohyac) Department of Information Science, Tokyo University of Science, Noda City, Chiba 278-8510, Japan
(Received 21 June 2005; accepted 2 August 2005; published online 14 October 2005) The success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. We describe the teleportation process in an infinite dimensional Hilbert space by giving simple examples. 0 2005 American Institute of Physics. [DOI: 10.1063/1.2044647]
1. INTRODUCTION
In quantum communication theory, we code information by quantum states and send it through a quantum device that is properly designed. If one can send any quantum state from an input system to an output system as it is, that is, if one can find such a method sending an input state without changing it, then it will be an ultimate way for information transmission. It is in quantum teleportation that we can discuss such an ultimate communication system. The problem of quantum teleportation is whether there exists a physical device and a key (or a set of keys) by which a quantum state attached to a sender (Alice) is completely transmitted and a receiver (Bob) can reconstruct the state sent. Bennett et al.’ showed that such teleportation is possible through a device (channel) made from proper (EPR) entangled states of Bell basis. The basic idea behind their discussion is to divide the information encoded in the state into two parts, classical and quantum, and send them through different channels, a classical channel and an EPR channel. The classical channel is nothing but a simple correspondence between sender and receiver, and the EPR channel is constructed by using a certain entangled state. However the EPR channel is not so stable due to quick decoherence. Fichtner and OhyaZs3studied the quantum teleportation by means of general beam splitting processes in Bose Fock space so that it contains the EPR channel as a special case, and they constructed a stable teleportation process with coherent entangled states. However, all these discussions have been based on finite dimensionality of the Hilbert spaces, attached to Alice and Bob. As is well known, success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. This paper is a trial to describe the teleportation process in an infinite dimensional Hilbert space by
“Electronic mail:
[email protected] b’Electronicmail:
[email protected] ‘)Electronic mail:
[email protected] Reprinted with permission from K.-H. Fichtner, W. Freudenberg and M. Ohya, J. Math. Phys. 46,102103 (2005) 0 2005, American Institute of Physics.
38 102103-2
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J.
Math. Phys. 46, 102103 (2005)
giving simple examples. In Sec. II, we fix the notations based on Fock space discussion of the series of papers.24 In Sec. 111, the channel expression of the teleportation is reviewed and the entanglement between Alice and Bob is constructed by an isometry operator, on which an operator expression of the teleportation channel is given, and some extreme cases of the teleportation are considered. To be closer to usual teleportation schemes and to get simple and explicit results we consider in Sec. IV the case of product states. In Sec. V, the existence of unitary keys is discussed. II. BASIC NOTIONS AND NOTATIONS We consider three complex Hilbert spaces XI, HF,and X 3 . To Alice there are attached the Hilbert spaces ‘HI and 7-t2. Alice wants to teleportate a state p on 7-t1 to Bob to whom there is attached the Hilbert space X 3 . Usually it is assumed that all three Hilbert spaces are finitedimensional ones. This is also necessary for obtaining perfect teleportation. In the present article we will consider the case of Hilbert spaces being separable but not necessarily finite dimensional. We assume that all three spaces are either infinite-dimensional separable Hilbert spaces or finitedimensional ones with same finite dimension. The paper continues and generalizes results obtained in Refs. 2-4. Let us be given orthonormal bases,
in H I , XF, and 7-t3 where the at most countable index set G is an abelian (additive) group with operation $. An important case is that G is the set of integers Z where the group operation @ will be usual addition. Since we need the structure of a group it is more convenient for our purposes to choose only orthonormal bases consisting of two-sided infinite sequences. To include usual teleportation models (with finite index space G) we consider also the case G = { l , ... , N ) with N belonging to the set N of natural numbers. In this case the operation @ :G X G + G is defined by k @ Z : = ( k + l ) m o d N .The operation inverse to @ we denote by 8 . In the latter case keZ=k-Z if k > l and k G l = k - Z + N if k 6 1 . The algebra of all bounded linear operators on a Hilbert space 7-1 we will denote by B(’H). Throughout this article we will assume that all states on a Hilbert space 7-t are normal states (on B(7-t)).The set of all normal states we denote by S(7-t). Let V E B(7-t~)be an arbitrary unitary operator. Consequently, the sequence ( v,JnEG with v,,:=V(i,n E G is a second orthonormal basis in ‘H2. Thus there exists a sequence (bkl)k,lsGsuch that
Obviously, the sequence (bkl)k,ltG has to fulfill
where S,,,,, denotes the Kronecker symbol. Observe that for all m , k Consequently, we obtain
Since v* is again unitary and
E
G it holds ( vmr6;)
=G.
.$i=v*v,, we also have
Remark: To simplify notations only if there appear ambiguities we will separate multiple indices by “commas,” i.e., usually we write b,, instead of bk,p
39 102103-3
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J. Math. Phys. 46, 102103 (2005)
Further, we consider a sequence (Ul,,),,IEG of unitary operators on ‘H2 acting as shift operators on the elements of the (original) basis:
u,& = 6tdin,
(6)
( m k E G).
The Hilbert space ‘H2 is connected by simple isometries S, to ‘HI and S3 to ‘H3: S1:7&
--t
= 6: @
‘HI 8 ‘HT, S,(&
(k E G ) ,
(7)
Finally, we construct a new basis in 1-1, @ ‘H2 by setting
where 1 denotes the identical operator (from the context it always will be clear on which space 1 operates). Observe that for k,ni E G tkw
= (l @
uni)
x
bkl(‘$
= 2 bkl(‘$:
8
leG
@ ‘$am).
IEG
Proposition I : The sequence (&,r,)k,mEG is an orthonormal basis in ‘H18‘H2. Proof: For n , m , r , s E G we get using (10) and (3)
Then
Using ( 5 ) this implies
This ends the proof. We denote by F,,,, E B(Zl8 NH2) the projection onto Fnm :=
l6nm)(tnnil= ( 6 n m
’
)tiin,
0
&,,,, i.e., (n,m E G ) .
(12)
Remark: We will use as well the “scalar product” notation as also the “bra and ket” symbols,
40 102103-4
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Fichtner, Freudenberg, and Ohya
However, using the symbol I@)let us make the convention that the symbol has to denote the normalization of the vector @, i.e.,
I@) := -.@ 11@.[1 Observe that for d, E ‘Hl 8 ‘H2 given by (11) and for all n,m E G one obtains Fmi@ = ( t n i n > @ ) t n n z =
C &(ti 8
fk3gnz,@)~nm=
keG
=
(C
C
Gars([: 8
&n,,tj
8 &)trzm
k,r,s E G
(14)
6 f f k , k @ n i )&mz.
keG
Especially,
-
~ , z m ( t j@ ti)= 8ss,rem bnt-tnm.
(15)
In the subsequent sections we investigate concrete teleportation channels. For this we need an explicit expression for (F,,, 8 l)(l@ S3) which maps ‘ H I @ ‘Hz into ‘HI 8 ‘H2 8 ‘H3. Proposition 2: Let @ E ‘HI 8 H 2 be given by (11). For all n,m E G ir holds
This proves (17). Corollary 3 allows us to get explicit formulas for
where x is a normal state on L?(‘HFt,@‘H;yl). Let x be given in the form
41 102103-5
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The teleportation process
c
x=
~uul@uu~~@uul
u,u~G
with
(@uu)u,utG
being an orthonormal sequence in 3-1,@3-1?,
=
@uu
c %uTst:
r.s
and
G
@
s:
( X u u ) u , u E Gfulfilling
c
Xu,= 1,
Xu, 2 0 .
u,utG
Of course, we have for all u , C, u ,U E G
C
ZZaGcirs
= 6u.z 8uu,r.
r,ssG
As an immediate consequence of Corollary 3 we obtain the following result. Proposition 4: Let x be given by (19). For all n,m E G it holds
with
Remark: Observe that the vectors quu,II)I usually are not normalized. Further, in general the sequence ( ~ u u n n I ) u , u t Gis not an orthogonal one. 111. THE TELEPORTATION CHANNEL
A. The measurement
Now we apply the model which was used in several for the description of a teleportation scheme. The measurement will be done with the operator
~ ~ ~ ~ of real numbers and (F,zm)n,nlec is the family of orthogonal where ( z , , ~ ~ )is, ,a , sequence 1 ~ in Sec. I1 (cf. (12) and (10)). In the above-mentioned articles projections on 7-L1 ~ 3 - introduced one considers the case of a given state p on HI (that has to be teleportated to Bob) and an 3 Alice makes entangled state (T on H2@'H3.Thus the whole system is prepared in the state ~ € u. a measurement (restricted to HI @'If2) with the operator F given by (25), i.e., the operator F @ 1 is applied to the system being in the state p @u.As the result of the measurement Alice obtains a value z, for some n , m E G. Consequently, after the measurement the whole system will be in the state O,,, on 3-1, 8 3-1? €3 X 3 given by
Bob who is informed about the result where tr denotes the full trace with respect to H I @Z2@ 'H3. of the measurement controls the state in&) on H 3 being the partial trace of ON,,with respect to 3-11 €3 3-12:
42 102103-6
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J. Math. Phys. 46, 102103 (2005)
where trI3denotes the partial trace on 'HI 8 'H2. Of course we have to assume that the denominator in (26), respectively, in (27) is nonzero. Otherwise, the left-hand side has to be set equal to zero. The mapping in,,,: S('H1)+S('H3)is called a teleportation channel. The teleportation works perfectly if Bob is able to reconstruct the initial state p from x,lfll(p). We will return to this question in the subsequent sections. Teleportation channels A,,, of the above-mentioned type might be useful also for modeling other transformations of states. For instance in Ref. 4 we proposed an extremely simplified model of certain recognition processes based on a teleportation channel. In this model the spaces 'HI, 'H2, and 'H3 represent the processing part (brain), the memory before and after recognition. B. The entanglement
Instead of the state p @ u on 'HI €4 'H2 8 'H3 we will consider now states of the form
(18 S3) x(a 8 S;)
(28)
with x being a state on 'HI 8 'H2 and S3isometry (8) coupling 'HI to 'H3. The simple entanglement is achieved just by applying S3. Especially, if x has the form x=pl @ p2 with between 'H2 and 'H3 $ being a state on 'Hj,!= 1 , 2 then using the above cited notations from Refs. 2 and 3 we get p = p l and u=S3p2S;. This case of x being a product state we will discuss in Sec. IV. We will consider now the channel A,,n,:S('Hl@'HH?) -S('H3) given by
where as in (27) tr12 denotes the partial trace with respect to 'Hl@'H, and tr the full trace on 'HI €4'H2c37fH3. Again we have to assume that the denominator in (29) is greater than zero. 0therwise we set An,fl(x)=O.Observe that for product states x = p @ u one has the relation
A,,&) Reinark: Since t r l , 2 ( ( ~ l&n)(a l n l @sS,)x(n 8 s ; ) ( F , , 81)) ~ is a positive linear functional on z3
=A,,,,,(pc34.
the denominator in (29) can be equal to zero only if this functional is zero. In other words, in this case no information about the measured value z , , ~can be transmitted to Bob (in a brain model this would mean that no information about the input signal will be stored in the memory). The following result is an immediate consequence of Proposition 4. Theorem 5: Let x be a state on 'HI 8 'H2 given b y (19)-(21). Then for all n , i n E G
where P ', E 'H3 is given by (24), and An,,,(%)has to be set equal to zero if the denominator in (30) is equal to zero (tr, denotes the trace in 'H3). Sometimes it is more convenient (cf. the second remark in this article) to write (30) in the form
Of course, mixed states are not necessarily transformed into mixed ones-possibly there is only one pair u,u such that / * uunml l >O. However, immediately from (30) we may conclude the following proposition.
43 102103-7
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The teleportation process
Proposition 6: If x is a pure state on ' H I @ 'Hz then A,,,,(x) is a pure state on Zero.
'Ft3 or
equal to
C. Examples Example 1: Let x=l@)(@l be apure stare on 'Ft,@'Hl, 11@11=1, obtain
@=~Cr,s.~&@[~.
Then we
with -
*nm
=
Z bnrar,ren,&rn,
rtG
provided llqnmli > 0. Especially, for x=
I@)(@( with a=,$@[:we get
~Ltl(x)= l~;@,fl)(Ll
provided bfl,sen, # 0. Let us discuss this result. Measuring the value zllmmeans that there was made the projection Observe that ~ r s G ~ b n r ( 2 ~ ~ r , r a r n ~ 2 =if~ and ~ 9 ~only l n , / if~ 2there >0 onto (,,=XrtGbnr(d @ &,). exists at least one r E G such that b,,,# 0 and a,,,@,# 0. Obviously, there exists r E G such that b,, # 0. The number is the coefficient of the basis element @ in the expansion of x=I@)(@l. So only if x is receptive for the signal $ 8 ($,,, for at least one admissible r there will be an output transmitted to Bob. Exainple 2: We consider the case G=Z and.fir an N E N. Let the state x be afiiiite iiiixriire of basis elements with equal weights:
[:
Since in this case
cyuurs=
.&,
Su,r.Su,swe get from (24)
Especially,
ll*uul*m112
=
{
u,u E { - N ,...,N } , u - u = m elsewhere.
lb,,f,
The conditions u , v E { - N , ...,N) and u-u=m imply m 6 2 N . So tve $finally get for n E G and m C 2N ~ i m ( x= )
1 N-m
-E
c u=-N
IbnuI2It~+rn)(&+mI
provided N-rn
lb,,,I2
C=
>0.
u=-N
For nz > 2N the numerator in (30) is equal to zero. That means that measuring the value ,,z in >2N it is not possible to send any information to Bob.
with
44 102103-8
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In further investigations one should consider more refined measurements than simple projections onto one-dimensional subspaces which are obtained just by a change of the basis and a shifting procedure. The next examples elucidates that the whole procedure becomes trivial if we even do not change the basis. Example 3: Let us consider the (extreme) case that the second basis in ' H 2 is the old one, i.e., +/rl= for all iz E G. This implies bnk=Sn,k, and the projection operator F,,,, is the projection onto &,,,,=&8&,,,. Let x be an arbitrary normal state on 'HI @'H2 given in the f o r m (19)-(21). I f Cu,uXuu~~,u,,,,,a,,,~H?=O there will be no output, i.e., A ,,,,,( x ) = O . If~~,uAuv~~,,,,,,,~,,,~2>0 we obtain A , , ~ ( ~ ) = I t ~ Only ~ ~ )ifthe ~ ~ vector ~ ~ ~ &, @I . appears in at least one noniero component aUu of the state x some informarion will be transmitred and theJinal stale A,,,,,(%)will be the pure state
<:
[2ffi, ,
ltLl)(t:ffin,l. IV. TRANSFORMATION OF PRODUCT STATES
To be closer to usual teleportation schemes and to get simple and explicit results we want to consider in the sequel the case that the state x on 'HI @'H2 is a product state: (33)
x=pl@$
Assume where p' E S('Hl)and p' E S('HH?). respectively, 'H2 such that
are orthonormal sequences in 'HI, (@i)uEG,
with Xi20
A!=l,
(j=1,2,u E G )
(35)
UEG
and
having the representation
%= 2 dire
(i=1,2,u
E
G).
rtG
Using the notations from (19)-(21) we thus get in that case a,,, = a:, . a:,
A,, = xt . x:,
(u,u,r,s E G).
Obviously, we have
rsG
&&=
S,,, (i=1,2,u,v
E
G).
From (24) we conclude
and Theorem 5 can be written in the following form. Theorem 7:Let x=p'@p' be a state on 'HI @'H2 given by (34)-(36). Then for all n , m E G
A,,,,l(Pl@
2)=
xu,, EG
'3(E,,uEG
' 'Z('UUllnl9
xi
' A~(qUUllllZ,
' )*Ir,UIl?!!
(38) ' )qUVU?,l)
where '€'u,,,l, E 'H3is given by (37) and where we have to assume that the denominator in (30) is greater than zero (tr, again denotes the trace in ' H 3 ) .
45 102103-9
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The teleportation process
We want to express the channel Am,,, acting from S ( X ,@ 'H2) to S('H3)by a consecutive application of single channels having an intuitive meaning. First for each h E 'HZwe introduce an operator Ah:'HH3+'H3by
Especially, for all r E G we have
Obviously,
so, for each h E 3-1' it holds A h E B('H3). For each h E 'H2 the adjoint of Ah is given by A>=
'c mct;,nt; v E 'H3).
(42)
reG
On 'H3 we define for all m E G (in the same way as on H 2 - c f . (6)) the shift operator V,,,:'H3 'HH3characterized by
being a unitary operator on 'H3. Finally, let J l :X1+'H3 and J 2 : 'H2 +'H3 be the isomorphisms exchanging the corresponding basis elements, i.e.,
J1t:=t:, J2t;=t:( r E GI. Lemma 8: For all u , u , n , m
E
G it holds
'In =
'c IEG
(c$ dejiiiition ( 2 ) of 17"). Proof: From
we conclude
(44)
46 102103-10
J. Math. Phys. 46, 102103 (2005)
Fichtner, Freudenberg, and Ohya
what proves because of (37) Lemma 8. Observe that for all h E ‘H2,f E ‘H3 and r E G
(5;,h)(<:,n = t P . J 3 ( t % h ) . From definition (39) of the operator Ah we thus immediately conclude Leinina 9: For all h E H2,f E ‘H3
A d = AjrXJ2h).
(47)
The following property of the operators A, will be basic for our description of the channel A,,,,,. Leinina 10: Let r be a positive trace-class operator on ‘H2 (possibly also the operator being identically zero) and let us be given two different representations of r
with $ 3 0 and (h$kEG,j = 1 , 2 being orthonorinal bases in H2. Then for each positive trace-class operator (Y on ‘H3 we have
Proof: It is sufficient to consider the case
( Y = B( ~f ‘H3,llfl\=1). ~ Using Lemma 9 we get for all g
x
&%&?(flA;g
ktG
E
7 1 3 and j = 1 , 2
= 2 Y&+f(Ar+f4) = 2 +dJ~2h’,(A@’,,L?) = keG
keG
ksG
ydJy21%)(h’,/Jh&g
=AJfi, dlg)()l‘,\J>;;fgy =A~>J~,dh;;fg. ksG
Consequently, left- and right-hand sides of (49) coincide-the on the special representation of T in some basis in 712.
expression in (49) does not depend
0 Let us remark that r may be a finite rank operator since we did not exclude $=O. Denote by 5 the set of all positive trace-class operators on Hj, j = 1,2,3. For each r E 12; we define a mapping K,: ?;-+ ?; by the ansatz
where
with yk>o and (hk)kEG being an orthonormal basis in Z2.Lemma 10 guarantees that the mapping K , is well defined, i.e., the definition (50) does not depend on the special representation of T in some basis of H2.
47 102103-11
J. Math. Phys. 46, 102103 (2005)
The teleportation process
In the sequel let us make the convention to denote the normalization of a positive trace-class operator x by 2, i.e., x=
-
(52)
trx
provided tr x >0. Now, we are able to express the channel An,,,(%) as a consecutive application of single channels. This explicit construction will be done only for product states x=pl @ p 2 . Besides the channel K , defined by (50) we still need two other channels-one lifting the state from X1 to 'H3 by the isomorphism J, given by (44) and one shifting the states on 'H3 with the shift operator V,,,defined by (43). We define K : ?; +% by
and for nz E G the mapping K, :T3-+
13
by
Theorem 11: For all p1E S(H1),p 2 E S(7-1,) and n , m E G it holds A,zn,(P1 @ P 2 ) = i
p 2
K,,,KI.l,,(J K b ' )
<
O
(55)
provided tr K P ~ ~ K r n ~ K l ; i ; ; ) >~Ol .~ K Hereby, (pl) is given by (46). Proof: Let p1@ p' be given by (34)-(36). we obtain for all u,u,n,n? Using (38) in Theorem 7 and the representation (45) of 9',u,,m EG
=~
p0 K 2
m 0 K I ~ J K(P'). ~ I O
Since Anm(pl@ p') is just the normalization of the previous expression using the notation (52) we finally obtain (55). 0 The aim of teleportation models is to find methods of transmission of states on 7-11 to states on 'H3 in such a way that Bob is able to reconstruct perfectly or in a nearly perfect way the emitted state-having only knowledge about the result of a certain measurement. In Sec. III we already gave some examples illustrating the model. For the states considered in these examples it will be impossible to reconstruct all original states. However, one is not interested only in perfect teleportations but also in the deformation of the state, the degree of destruction of the input signal, etc. In Ref. 4 we considered the special case of all spaces being equal and finite dimensional. For examples in this case we refer to this article. Further, one can ask for subsets of S(7-11) for which perfect teleportation can be achieved. We continue to illustrate the channels and operations previously introduced by simple examples connected with product states. The aim of the examples below is just to illustrate the mechanism of the procedure.
48 102103-12
Fichtner, Freudenberg, and Ohya
J. Math. Phys. 46, 102103 (2005)
In what follows, summands for which II*uu,,nIII=O just have to be cancelled. If it is equal to zero no information about the input signal (the measurement of the value z,,,,) will pass to Bob. Such cases of non-perfect teleportation were considered in Ref. 3. Example 4: Let us consider the extreme case that pl=l$)(& for some k E G. Then K ( p ' ) =I&(@, which implies Kl,,)(%.,i M P ' ) = lb,A21~;,(~;1. Consequently,
0 We try to give again a possible interpretation of the result. Measuring zllm we made a projection onto ~ ~ , r ~ = ~ . r t G b n r (At ~ ~least c . 3 one . ~ ~ofn the l ) . coefficients b,,,r E G has to be different : since from zero. The state Alice wants to teleportate consists only of the elementary signal 6 p'=l,$)(t;l. This implies that only the signal tie,,! is able to pass. So it is necessary that bltki0. The information about the index n is not contained in the elementary signal c.3 tie,,, and will not be contained in the output sent to Bob. However, b,, # 0 is necessary to get an output at all. Further, for the state p' given by (34)-(36) one easily checks
x
~ 2 ( P 2 l ~ : e l n ) ( t ; e=m l ) clff:,kerl1l2. utG
Xi>O
Consequently, to obtain the output l $ e, l ) ( & l , l at least for one with it must hold a;,ke, # 0. If Bob would know that only states of the form p'=l&(til for some k E G are sent by Alice he obviously has unitary keys to reconstruct the original state. This of course requires that all bnk# 0 and that the state p' is such that for all k E G there exists at least one u E G such that Xz > 0 and a;, # 0. Example 5: Now we consider the other extreme case that p2 is just an elementary signal, i.e., there exists an k E G such that p'=1,$)(621. Analogous calculation as in Example 4 shows that Ln(P1 @
$1 = Ie;,(dL
where we have to assume that t r l ( p ' l ~ i ~ , , , ) ( ~>O. ~ ~ nWe , l )already observed in the example above that p2 should contain as much infonnation as possible. In the case p2=l&)([il Bob obtains no infonnation at all abut the state p'. Independently of the result z,,, of the nzeasureinent pe$oniaed So we have the most extreme case of loss of by Alice either Anm(p1@p')=0 or equal to I&)(,$\. information.
V. EXISTENCE OF UNITARY KEYS We will consider again the case of product states x = p ' @ p2 on H Ic.3 ?-L2 where p' E S(?tl) and p2 E S(?-L2). That means that the entangled state u considered in Refs. 2 and 3 has the form u =&PIS;. For fixed state p2 on H 2 we consider the channel ~n,,t:S(?-Ll)+S('H3) given by
49 102103-13
J. Math. Phys. 46, 102103 (2005)
The teleportation process
Let GCS('H1) be a fixed set of states on 'HI. Dejinirion 12: We say tliar the teleportation scheme is perfect iffor each n,m E G there exists a unitary operator Vn,rl: ?-tl+Ti3 such that
and
C
u((Fnrn @ 1 ) @~~
3 18 ~ P' 'U
@ $)(~nm
8 1)) = 1
(P'
E
8).
(58)
n,meG
The unitary operators V,,, are called unitary keys. Usually, perfect teleportation means that the above-mentioned conditions hold for all p l ~ S ( 7 - i (cf. ~ ) Refs. 2, 3, and 5). The restriction of the set of states that have to be teleportated enlarges the possibilities for perfect or "nearly" perfect models. Condition (57) means that Bob can reconstruct the original state p' from the knowledge of the result z, of the measurement and of the received state ~ , , , , , ( p ' )where it is assumed that Bob possesses for each n , m E G the appropriate key V,,,. This follows obviously form p' = l(,,l,&,,,,(pl)Vlln,. Finally, (58) means that with probability one Bob will find the proper key, i.e., with certainty there will be a result z , of~ ~ measurement for which he has unitary keys. Let us have a closer look to formula (55). The channel K,,, is built with the help of the unitary Vnt.The same is true for K since J I J ; = J ; J l = l . The channel Klx)(xldestroys any hope for unitary keys. An easy calculation shows that for each f E ?-t3 it holds
We see that only if G is finite and all lbrlklare equal A will be unitary (up to a constant that vanishes after normalization). This is in accordance with (and another proof of) the fact that perfect teleportation requires finite dimensional spaces and maximal entanglement. VI. CONCLUDING REMARKS
The aim of this article is to touch the problem of teleportation schemes in infinite-dimensional spaces. The previous results still have to be supplemented by calculations of fidelity and other characteristics. Just to achieve simple explicit expressions we illustrated the model on the most simple sequence of elementary signals To obtain more interesting models one has to refine the above-mentioned models:
(d).
(1) consider more complex measurements than simple one-dimensional projections F,,, (2) if the filbert space is a symmetric Fock space take truncated coherent vectors (exponential vectors with "removed" vacuum part) as basis and beam splittings for the entanglement, (3) replace the isomorphisms J1,J2 and the trivial isometries S1, S3by more complex ' C . H. Bennett, G . Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). 'K.-H. Fichtner and M. Ohya, Commun. Math. Phys. 222, 229 (2001). 'K.-H. Fichtner and M. Ohya, Commun. Math. Phys. 225, 67 (2002). 4K.-H. Fichtner, W. Freudenberg, and M. Ohya, Prnceediri,qv nf the Cnnferenre nil Qiurntuni Prnbabiliry and Infillire Dimensional Analysis, Burg, March 2001, edited by W. Freudenberg (World Scientific, Singapore 2001); Quantum
50 102103-14
Fichtner, Freudenberg, and Ohya
J. Math. Phys. 46, 102103 (2005)
Probabilit)?and White Noise Analysis 15, 85-105 (2003). 'L. Accardi and M. Ohya, Quantum Information (World Scientific, 1999). pp. 59-70. 6L.Accardi and M. Ohya, Appl. Math. Optim. 39, 33 (1999). 'K.-H. Fichtner and W. Freudenberg, Commun. Math. Phys. 137, 315 (1991). 'K.-H. Fichtner, W. Freudenberg, and V. Liebscher, Infinite Dimen. Anal., Quantum Probab., Relat. Top. 1, 511 (1998). 9M. Ohya, Mathematical Foundation of Qzrantum Computer (Maruzen, 1999). 'OM. Ohya, Quantum lnfommfion 11, edited by T. Hida and K. Saito (World Scientific, Singapore, ZOOO), pp.149-160. "M. Ohya and I. V. Volovich, Mathematical Foundation of Quantum Information and Computation (Springer, New York, in press).
51 Vol. 55 (2005)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY* MASANORIOHYA Department of Information Sciences, Tokyo University of Science, 278 Noda City, Chiba, Japan (e-mail:
[email protected]) (Received November 4, 2004)
It is von Neumann who opened the window for today’s information epoch. He defined quantum entropy including Shannon’s information more than 20 years ahead of Shannon, and he explained what computation means mathematically. In this paper I discuss two problems studied recently by me and my coworkers. One of them concerns a quantum algorithm in a generalized sense solving the SAT problem (one of NP complete problems) and another concerns quantum mutual entropy properly describing quantum communication processes.
Keywords: quantum mutual entropy, quantum algorithm.
1. Introduction This paper consists of two parts, one (Sections 2 and 3) of them is about quantum algorithm solving the SAT problem, it is based on a series of papers [30, 7, 32, 33, 41. The other one (Section 4) is about quantum mutual entropy applying quantum communication processes, it is based on [27, 28, 351. Although the ability of computers is quickly progressing, there are several problems which may not be solved effectively, namely in polynomial time. Among such problems, nonpolynomial (NP) problems and NF’ complete problems are fundamental. It is known that all NP complete (NPC for short) problems are equivalent and have been studied for decades, for which all known algorithms have an exponential running time in the length of the input so far. An essential question asked for more than 30 years has been whether there exists an algorithm solving an NP complete problem in polynomial time. We found two different algorithms solving the NPC problems in polynomial time [30, 32, 33, 31. In first two sections of the present paper we discuss the essence of these algorithms. After von Neumann introduced the concept of quantum entropy [23] for density operators, many concepts on various quantum entropies have appeared [23, 311, among which quantum mutual entropy plays an important role. That is, the mutual *Lecture given at the von Neumann Centennial Conference, Budapest, October 15-20, 2003 [1091
52
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M. OHYA
entropy expresses the amount of information sent from input to output, so that it is a basic quantity measuring the ability of a communication channel. I defined the quantum mutual entropy for density operators in 1983 [27] by using Umegaki’s relative entropy [40] and I extended it to general C*-dynamical systems by means of Araki’s or Uhlmann’s relative entropy [6, 39, 311. Recently several types of quantum mutual entropies have been introduced [38, 8, 101, and they are used to discuss communication processes. In Section 4 of this paper, we compare these mutual entropies from the point of view of infomation communication based on [35]. 2. Quantum chaos algorithm of SAT Let us remind what the P-problem and the NP-problem are [16, 131: Let n be the size of input.
(1) A P-problem is a problem in which time needed for solving the problem is at worst of polynomial time of n. Equivalently, it is a problem which can be recognized in polynomial time of n by deterministic Turing machine. (2) An NP-problem is a problem that can be solved in polynomial time by a nondeterministic Turing machine. This can be understood as follows: Let us consider a problem to find a solution of f ( x ) = 0. We can check in polynomial time of n whether xo is a solution of‘ f (K) = 0, but we do not know whether we can find the solution of f (x) = 0 in polynomial time of n.
(3) An NP-complete problem is a problem of polynomially transformed NPproblem. We take the SAT (satisfiable) problem, one of the NP-complete problems, to study whether there exists an algorithm showing that NPC becomes P. It is known that the SAT problem is equivalent to any other NPC problem. Let x = {XI,. . . , x,} be a set. Then xk and its negation j k ( k = 1,2, . . . , n ) are called literals and the set of all such literals is de_noted by X = {XI, 21, . . . ,x,X,}. The set of all subsets of X’ is denoted by 3 ( X ) and an element C E F ( X ) is called a clause. We take a truth assignment to all Boolean variables xk. If we can assign the truth value to at least one element of C , then C is called satisjable. When C is satisfiable, the truth value t (C) of C is regarded as true, otherwise, that of C is false. Take the truth values as “true ++ 1, false t, 0”. Then C is satisfiable iff t (C) = 1. Let L = (0, 1) be a Boolean lattice with usual join V and meet A, and t ( x ) be the truth value of a literal x in X . Then the truth value of a clause C is written as t (C) 3 vXECt( x ) . Moreover the set C of all clauses C j ( j = 1 , 2 , . . . , m ) is called satisfiable iff the meet of all truth values of C, is 1; t (C) = Ay=lf (C,) = 1. Thus the SAT problem is written as follows:
DEFINITION1. SAT Problem: Given a Boolean set X = {XI, . . . , x,} and a set C = {Cl, . . . ,em}of clauses, determine whether C is satisfiable or not.
53
QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
111
That is, this problem is to ask whether there exists a truth assignment to make
C satisfiable. It is known for usual algorithms that it is polynomial time to check the satisfiability only when a specific truth assignment is given, but we cannot determine the satisfiability in polynomial time when an assignment is not specified. In [30] we discussed the quantum algorithm of the SAT problem, which was rewritten in [7] with showing that the OM SAT-algorithm is combinatoric. Ohya and Masuda pointed out [30] that the SAT problem, and hence all other NP problems, can be solved in polynomial time by quantum computer if the superposition of two orthogonal vectors 10) and 11) is physically detected. However, this detection is considered not to be possible in the present technology. The problem to be overcome is how to distinguish the pure vector 10) from the superposed one a! 10) fi \I), obtained by the OM SAT-quantum algorithm, if fi is not zero but very small. If such a distinction is possible, then we can solve the NPC problem in polynomial time. In [32, 331 it is shown that it can be possible by combining nonlinear chaos amplifier with the quantum algorithm, which implies the existence of a mathematical algorithm solving NP = P. The algorithm of Ohya and Volovich is going beyond usual (unitary) quantum Turing algorithm. So the next question is (1) whether there exists more general Turing machine scheme combining the unitary quantum algorithm with chaos dynamics, or (2) whether there exists another method to achieve the above distinction of two vectors by a suitable unitary evolution. In [4], we showed that the stochastic h u t , recently extensively studied by Accardi and coworkers [2], can be used to find another method of (2). In this paper, we review mathematical frame of quantum algorithm in Section 2 and the OV-chaos algorithm. In Section 3, based on the idea of quantum adaptive dynamics [l, 25, 41, we discuss how it can be used to solve the problem NP = P.
+
2.1. Quantum algorithm
The quantum algorithms discussed so far are rather idealized because computation is represented by unitary operations. A unitary operation is rather difficult to realize in physical processes, more realistic operation is the one allowing some dissipation like semigroup dynamics. For such a realization, we have to generalize the concept of quantum Turing machine so that the generalized one contains nonunitary operations. This work has been done in [4, 21, 51, about which we will not discuss here. In this paper we will explain the algorithms solving the SAT problem in polynomial time. First we remind the procedure of usual quantum algorithm which is needed to compute the truth value t (C) of the SAT. Let 7-1 be a Hilbert space describing input, computation and output (result). As usual, the Hilbert space is 3-t = @yCz,and let the basis of 7-l = @yC2be: eo(= 10)) = ~ ~ ) ~ . . . ~ ~ o ) @ ~ o ) , = e l 1( 0=) I@l .). > . @ l o ) @ l, .l .). , e z h i _ l ( = 1 2 N - 1 ) ) = ll)@~~~@ll)c3ll).
54
112
M. OHYA
Any number t (0,. . . , 2N - 1) can be expressed by t =
or 1, so that the associated vector is written by
It)
(=
et>
N
= @k=llat
And applying n times the Hadamard matrix H vector
lo), we
get H 10)( = 6 (0)) =
BY&
(10)
(k)
k= 1
).
(
=-
+
N
C at(k)2k-1,a,(k)-0
Jz
1 -1 1 1 ) ) .Put
)
to the vacuum
Then we have
which is called discrete Fourier transformation. Thus from all of the above operations it follows a unitary operator U F ( t ) ZE W ( t ) H and the vector 6 ( t ) = UF ( t ) 10). All conventional unitary algorithms can be written as the following three steps by means of certain channels on the state space in 'Ft (i.e. a channel is a map sending state to another state):
(1) Preparation of state: Take a state p (e.g. p = 10) (01) applying the unitary channel defined by the above UF ( t ) : A; = A; = AdU,
==+A ; p
=UF~U:.
( 2 ) Computation: Let U be a unitary operator on 'Ft representing the computation followed by a suitable programming of a certain problem, then the computation is described by a channel A; = AdU (unitary channel). After the computation, the final state pf will be pf = A*,A*,p.
(3) Registering and measurement: For registration of the computed result and its measurement we might need an additional system K (e.g. register), so that the lifting 8; from S('Ft) to S('H €3 K ) in the sense of [3] is useful to describe this stage. Thus the whole process is written as Pf = & : (A*,A*,P).
Finally we measure the state in measure (PVM) on K,
K:For instance, let {Pk;k E J } be a projection-valued
55 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
113
after which we can get a desired result by observations in finite times whether the size of the set J is small. REMARK1. When dissipation is involved, the above three steps have to be generalized so that dissipative nature is involved. Such a generalization can be expressed by means of a suitable channel, not necessarily unitary. (1) Preparation of state: We may use the same channel A> = AdU, in this first step, but if the number of qubits N is large so that it will not be built physically, then A> should be modified, and let us denote it by A > . (2) Computation: This stage is certainly modified to a channel A: reflecting the physical device for computer. (3) Registering and measurement: This stage will remains as above. Thus the whole process is written as
2.2. Quantum algorithm of SAT
We explain the algorithm of the SAT problem which has been introduced by Ohya and Masuda [30] and developed by Accardi and Sabbadini [7]. This quantum algorithm is described by a combination of the unitary operators discussed in the previous section on a Hilbert space 3.1. The detail of this section is given in the papers [30, 7, 331, so we will discuss just the essence of the OM algorithm. Throughout this subsection, let n be the total number of Boolean variables used in the SAT problem. Let 0 and 1 of the Boolean lattice L be denoted by the vectors 10) =
11) =
(:)
(3
and
in the filbert space C2, respectively. That is, the vector 10) corresponds
to falseness and 11) to truth. As we have explained in the previous section, an element x E X can be denoted by 0 or 1, so by 10) or 11). In order to describe a clause C with at most n length by a quantum state, we need the n-fold tensor product Hilbert space 3.1 = 6$C2. For instance, in the case of n = 2, given C = {XI,x2} with an assignment x1 = 0 and x2 = 1, then the corresponding quantum state vector is 10) 8 11), so that the quantum state vector describing C is generally written by IC) = 1x1) 8 1x2) E 3.1 with xk = 0 or 1 ( k = 1,2). Once X ZE (XI,. . . ,x,} and C = {Cl, C2, . . . , C m } are given, the SAT is to find the vector It 0 ) = V " € C j t(x>, where t(x) is 10) or 11) when x = 0 or 1, respectively, and t(x) A t ( y ) = t(x I(X) v t ( y ) = t(x v y ) .
A
y),
56
114
M. OHYA
For any two qubits Ix) and Iy), Ix, y ) and Ix") are defined as Ix) 8 Iy) and Ix) 8 . . 8 Ix), respectively. The usual (unitary) quantum computation can be
-
N times
formulated mathematically as the multiplication by unitary operators. Let UNOT,UCN and UCCNbe the three unitary operators defined as UNOT
UCN UCCN
11) (01 f 10) (11
7
+ 11) (118 UNOT, 10) (01 @ I @ I + 11) (11 8 10) (01 8 I + 11) (11 @ 11) (11 8 UNOT. 10) (01 8 I
UNOT,UCN and UCCNare often called NOT-gate, Controlled-NOT gate and ControlledControlled-NOT gate, respectively. For any k E N, U i N )(k) denotes the k-fold Hadamard transformation on (C2)@" defined as
57 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
UCOPY
c
= El
=
{IEl, E l ) (El,
01
+
IE1,
1- El)
(El,
115
Ill
€{Ox 11
n o ) (0,OI + 10, 1) (0, 11 + 11, 1) ( 1 , O I + I1,O) (1, 11.
Here ~1 and ~2 take the value 0 or 1. We call UANI),UOR and Ucopy, the AND gate, OR gate and COPY gate, respectively, whose extensions to (C2)@Nare denoted which are expressed as by U z , Uk!) and U$&,
+ p u - 1 ( E l ) (61
I z@u--LI--I (1- E l ) (11z@N-u--Lf.
where u , v and w are positive integers satisfying 1 5 u < v < w 5 N . These operators can be written, in terms of elementary gates, as ug)( u , 21, w)= ug)(24, w ) . UCN ( N ) (v, w) . ug; ( u , v , w), (N)
(u, u,
(N) w >= UCCN ( u , U, w > 1
Let C be a set of clauses whose cardinality is equal to m . Let 'H. = (C2 )@n+w+l be a Hilbert space and Ivo) be the initial state Ivo) = lo", O w , O), where p is the number of dust qubits (the details can be seen in [19]). Let U F ) be a unitary operator for the computation of the SAT,
where X K denotes a p strings of binary symbols and tCi(C) is a truth value of C with ei.
58
116
M. OHYA
Let {sk;k = 1, . . . ,m } be the sequence defined as s1
=n
+ 1,
+ a i , c N d ( c l ) - 1, Si = Si-1 + cXd(Ci-1) + a l , c N d ( c i - l ) ,
S2
= S1 f card (Cl)
3 ii I m,
where card(Ci) means the cardinality of a clause Ci. Take a value s as = sm - 1
+ card ( c m ) + al,card(C,Tt).
Note that the number m of the clause is at most 2n. Then we have [19]: The total number of dust qubits p is p=s-1-n m
for m 3 2. In order to construct U p ) concretely, we use the following unitary gates for this concrete expression [30,71:
I
x,,x,
E
ck
where ZI, Z2,13, l4 are positive integers such that xzE Ck or 2, E Ck (z = 11, . . . , Z4). THEOREM1. The unitary operator U p ) is represented as
uc( n ) - u(n+p+l) (m - 1) . u w l ) (m - 2 ) . . . u w 1 ) (1) AND . u$p+')(m) . u,!$?+') (m - 1).. . u,,(n+P+1)(1) . u;+~+l)( n ) . Applying the above unitary operator to the initial state, we obtain the final state (C)) in the last section of the final vector, which will be taken out by a projection Pn+@,1 = Z@n+P 8 11) (11 onto the subspace of H ' spanned by the vectors Ien,d', 1). The following theorem is easily seen. p . The result of the computation is registered as It
59 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL. ENTROPY
117
THEOREM2. C is SAT i f and only i f Pn+p,lu;)
+ 0.
1~01
According to the standard theory of quantum measurement, after a measurement of the event P,+@,J,the state p = Ivf)(ufl becomes
Thus the solvability of the SAT problem is reduced to checking that p’ difficulty is that the probability
# 0. The
is very small in some cases, where IT(Co)l is the cardinality of the set T(Co), of all the truth functions t such that t(C0) = 1. We put q = with r = IT(Co)l . Then if r is suitably large to detect it, then the SAT problem is solved in polynomial time. However, for small I, the probability is very small so that we in fact do not get any information about the existence of the solution of the equation t(C0) = 1, hence in such a case we need further discussion. Let go back to the SAT algorithm. After computation, the quantum computer will be in the state
6
IUf) =
G-? 190) c3 10) + q 1401) c3 I I ) ,
m.
where Ip1) and 190) are normalized IZ (= n + p ) qubit states and q = Effectively our problem is reduced to the following 1 qubit problem: The above state Ivf) is reduced to the state
I*)
= J1-q210) f q I1)7
and we want to distinguish between the cases q = 0 and q > 0 (small positive number). It will not be possible to amplify, by a unitary transformation, the above small positive q into suitable large one to be detected, e.g. q > 1/2, having q = 0 as it is. The amplification would be not possible if we use the standard model of quantum computations with a unitary evolution. What we did in [32, 331 was to propose to use the output I@) of the quantum computer as an input for another device involving chaotic dynamics. That is, it was proposed to combine quantum computer with a chaotic dynamics amplifier in [32, 331. Such a quantum chaos computer is a new model of computations and we could demonstrate that the amplification was possible in the polynomial time. 2.3. Chaos algorithm of SAT Here we will argue that chaos can play a constructive role in computations (see 132, 331 for the details). Chaotic behaviour in a classical system usually is
60
118
M. OHYA
considered as an exponential sensitivity to initial conditions. We would like to use this sensitivity to distinguish between the cases q = 0 and q > 0 mentioned in the previous section. Consider the so-called logistic map
x,+1 = ax,(l - x,) = g(x), x, E [O, 11. The properties of the map depend on the parameter a. If we take, for example, a = 3.71, then the Lyapunov exponent is positive, the trajectory is very sensitive to the initial value and one has the chaotic behaviour [26]. It is important to notice that if the initial value xo = 0, then x, = 0 for all n . It is known [14] that any classical algorithm can be implemented on quantum computer. Our quantum chaos computer will consist of two blocks. One block is the ordinary quantum computer performing computations with the output I+) = ,/10) q 11). The second block is a computer performing computations of the classical logistic map. These two blocks should be connected in such a way that the state I+) first be transformed into the density matrix of the form
+
+
P = q2P1 (1 - q2) Po, where PI and PO are projectors to the state vectors 11) and 10) . This connection is in fact nontrivial and actually it should be considered as the third block. One has to notice that P1 and Po generate an abelian algebra which can be considered as a classical system. In the second block the above density matrix p is interpreted as the initial data PO, and we apply the logistic map as Pm
=
(1 -k gm( P O b 3 )
2
where I is the identity matrix and c ~ 3is the z-component of Pauli matrix on C2. To find a proper value m we finally measure the value of a3 in the state pm such that M , = trp,,a3. THEOREM3.
Thus the question is whether we can find such an m in polynomial steps of n satisfying the inequality M , 2 for very small but nonzero q 2 . Here we have to remark that if one has q = 0 then po = PO and we obtain M , = 0 for all m. If q f 0, the stochastic dynamics leads to the amplification of the small magnitude q in such a way that it can be detected as is explained below. The transition from po to pm is nonlinear and can be considered as a classical evolution because our algebra generated by PO and P1 is abelian. The amplification can be done within at most 2n steps due to the following propositions. Since g"(q2) is x, of the logistic map xm+l = g(xm) with xo = q 2 , we use the notation x, in the logistic map for simplicity.
61 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
119
THEOREM4. For the logistic map x,+1 = ax, (1 - x,) with a E [0,4] and xo E [O, 11, let xo be and a set J be (0, 1 , 2 , . . ., n , . . . , 2 n } . I f a is 3.71, then there exists an integer m in J satisfying x , >
zfr
i.
THEOREM5. Let a and n be the same as in the above theorem. I f there exists mo in J such that xm0 > 21 , then mo > log2 3.71-1 ' According to these theorems, it is enough to check the value x , (M,) around the for a large n. More generally, when 4 = & with some integer k, above mo when q is it is similarly checked that the value x,(M,) becomes over within at most 2n steps. The complexity of the quantum algorithm for the SAT problem was discussed in Section 3, it was of polynomial time. We have only to consider the number of steps in the classical algorithm for the logistic map performed on quantum computer. It is the probabilistic part of the construction and one has to repeat computations several times to be able to distinguish the cases q = 0 and q > 0. Thus it seems that the quantum chaos computer can solve the SAT problem in polynomial time. In conclusion of [33], the quantum chaos computer combines the ordinary quantum computer with quantum chaotic dynamics amplifier. It may go beyond the usual quantum Turing algorithm, but such a device can be powerful enough to solve the NP-complete problems in polynomial time. The detail estimation of the complexity of the SAT algorithm is discussed in [19]. In the next two sections we will discuss the SAT problem in a different view, that is, we will show that the same amplification is possible by unitary dynamics defined in the stochastic limit.
3.
Quantum adaptive algorithm of SAT
The idea to develop a mathematical approach to adaptive systems, i.e. to systems whose properties are in part determined as responses to an environment [ l , 251, was born in connection with some problems of quantum measurement theory and chaos dynamics. The mathematical definition of adaptive system is in terms of observables, namely: an adaptive system is a composite system whose interaction depends on a fixed observable (typically in a measurement process, this observable is the observable one wants to measure). Such systems may be called observable-adaptive. In [4] we extended this point of view by introducing another natural class of adaptive systems which, in a certain sense, is the dual to the above defined one, namely the class of state-adaptive systems. These are defined as follows: a state-adaptive system is a composite system whose interaction depends on the state of at least one of the subsystems at the instant in which the interaction is switched on. We applied the state-adaptivity to quantum computation. The difference between state-adaptive systems and nonlinear dynamical systems should be emphasized:
120
M. OHYA
(i) In nonlinear dynamical systems (such as those whose evolution is described by the Boltzmann equation, or nonlinear Schrodinger equation, . . . , ) the interaction Hamiltonian depends on the state at each time t : H I = H1(pt); Vt .
(ii) In state-adaptive dynamical systems (such as those considered in the present paper) the interaction Hamiltonian depends on the state only at time t = 0: HI = H I ( P 0 ) . Now, from the general theory of stochastic limit [2] one knows that, under general ergodicity conditions, interaction with an environment drives the system to a dynamical (but not necessarily thermodynamical) equilibrium state which depends on the initial state of the environment and on the interaction Hamiltonian. Therefore, if one is able to realize experimentally these state dependent Hamiltonians, one would be able to drive the system S to a pre-assigned dynamical equilibrium state depending on the input state $0. In the following subsection we will substantiate the general scheme described above with an application to the SAT problem described in the previous sections. 3.1. Stochastic limit and SAT problem We illustrate the general scheme described in the previous section in the simplest case when the state space of the system is ?-ts = C2. We fix an orthonormal basis of ZS as {eo, el}. The unknown state (vector) of the system at time t = 0 is
+ := 1a!&?, = aOeO + q e l ;
\I*\\ = 1.
&NII
In Section 3, a1 corresponds to q and ej to l j ) ( j = 0, 1 ) . This vector is taken as input and defines the interaction Hamiltonian in an external field HI = 41c. ($1) 63 (A;
+ Ag) 8
= C ~ ~ . Z E I ~ E ) ( ~ ~ / I
+ Ag),
where h is a small coupling constant. Here and in the following summation over repeated indices is understood. The free system Hamiltonian is taken to be diagonal in the eE-basis,
HS :=
C EEleE)(e&l= Eoleo)(eol + EII~I)(~II EE(O,1)
and the energy levels are ordered so that Eo < El. Thus there is a single Bohr frequency wo := El - Eo > 0. The 1-particle field Hamiltonian is S , g ( k ) = ei'o(k)g(k), where w ( k ) is a function satisfying the basic analytical assumption of the stochastic limit. Its second quantization is the free field evolution e i t H ~e - ~i t H o g
- AS,g.
63 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
121
We can distinguish two cases as below, which correspond to two cases of Section 3, i.e. q > 0 and q = 0. Case 1. If ao, a1 # 0, then, according to the general theory of stochastic limit (i.e. t -+ t / h 2 ) [ 2 ] , the interaction Hamiltonian H I is in the same universality class as
E?;=D@A;+D+@A~,
where D := leo)(ell. The interaction Hamiltonian at time t is then E l ( ? )= e-"m'JD @
A g g + h.c. = D
+
@ Af(ei'(W(P)-mO)g)
h.c.,
and the white noise ({b,})Hamiltonian equation associated, via the stochastic golden rule, to this interaction Hamiltonian is
a,U, = i(DbT
+ D+b,)U,.
Its causally normal ordered form is equivalent to the stochastic differential equation dU, = (iDdB,?
+ iD+dB, - y-D+Ddt)U,,
where d B , := b,dt. The causally ordered inner Langevin equation is
+
dj,(x) =dU:xU, U:xdU, +dU:xdU, = U:(-iD+xdB, - iDxdB,? - T-D'Dxdt +ixD+dB, - y-xD+Ddt
+
+ ixDdB,?
+ y-D+xDdt)U,
= i j , ( [ x , D+l)dB, i j t ( [ x ,Dl)dB: -(Re y-)j,((D+D, x})dt + i(Irny-)j,([D+D,x ] ) d t +j,(D+xD)(Re y - W , where j , ( x ) := U;"xU,. Therefore the master equation is d dt
- P ' ( x ) = ( I m y ) i [ D + D ,P ' ( x ) ] - (Rey-)(D+D, P ' ( x ) }
+(Re y-) D + P ' ( x )D , where D+D = lel)(ell and D+xD = (eo,xeo)lel)(ell. The dual Markovian evolution P,' acts on density matrices, and its generator is
L,p = ( I m y - ) i [ p , D+D] - (Re y - ) { p , D+D}
Thus, if po = leo)(eol one has
+ (Re y-)DpD+.
L*Po = 0,
so po is an invariant measure. From the Frigerio-Fagnola-Rebolledo criteria, it is the unique invariant measure and the semigroup exp(tl,) converges exponentially to it. Case 2. If ctl = 0, then the interaction Hamiltonian HI is
HI = hleo) (eol 8 (A;
+ Ag)
64
122
M. OHYA
and, according to the general theory of stochastic limit, the reduced evolution has no damping and corresponds to the pure Hamiltonian
ffs
+ leo)(eol = (Eo + l>leo)(eol+ Ellel)(ell.
Therefore, if we choose the eigenvalues E l , Eo to be integers (in appropriate units), then the evolution will be periodic. Since the eigenvalues E l , Eo can be chosen a priori, by fixing the system Hamiltonian Hs, it follows that the period of the evolution can be known a priori. This gives a simple criterion for the solvability of the SAT problem because, by waiting a sufficiently long time one can experimentally detect the difference between damping and an oscillating behaviour. A precise estimate of this time can be achieved either by theoretical methods or by computer simulation. Both methods will be analyzed in the complete paper [5]. CONCLUSION1. We pointed out that it was possible to distinguish two different states, ,/10) q 11) (q $0) and (0) by means of the adaptive dynamics with the stochastic limit.
+
CONCLUSION 2. Finally we remark that our algorithm can be described by a deterministic generalized quantum Turing machine [21, 51. 4.
Comparison of various quantum mutual type entropies
There exist several different types of quantum mutual entropy. The classical mutual entropy was introduced by Shannon to discuss the transmission of information from an input system to an output system [20]. Then Kolmogorov [24], Gelfand and Yaglom [17] gave a measure theoretic expression for the mutual entropy by means of the relative entropy defined by Kullback and Leibler. Shannon’s expression for mutual entropy was generalized for the finite-dimensional quantum (matrix) case by Holevo [l8, 221. Ohya took the measure theoretic expression of KGY and defined quantum mutual entropy by means of quantum relative entropy [27, 281. Recently Shor [38] and Bennett et al. [lo] took the coherent information and defined new types of mutual entropy in order to discuss Shannon’s coding theorem. In this section, we compare these types of mutual entropies. The most general form of quantum mutual entropy defined by Ohya, generalizing the KGY measure theoretic mutual entropy, is given as
11(rp; A) = sup
{
SAU (Am, AYJ) dp; p
E
I
MP( S ) .
Here S is the set of all states in a certain C*-algebra (or von Neumann algebra) describing a quantum system, SAU (., .) is the relative entropy of Araki [6] or Uhlmann [39] and p is a measure decomposing the state rp into extremal orthogonal Wdp, in S, whose set is denoted by MP ( S ). states, i.e. lp =
65 QUANTUM ALGORITHM FOR SAT PROBLEM AND QUANTUM MUTUAL ENTROPY
123
In the case that the C*-algebra is B (X)and S is the set of all density operators, the above mutual entropy goes to
where p is a density operator (state), Su (., .) is Umegalu’s relative entropy and p = EnA, En is the Schatten-von Neumann (one-dimensional spectral) decomposition. The SN decomposition is not always unique unless S is Choque simplex, so we take the supremum over all possible decompositions. It is easy to show that we can take orthogonal decomposition instead of the SN decomposition [29]. These quantum mutual entropies are completely quantum, namely, they describe the transmission of information from a quantum input to a quantum output. When the input system is classical, the state p is a probability distribution and the Schatten-von Neumann decomposition is unique with delta measures 6, such that p = C, A,&. In this case we need to code the classical state p by a quantum state, whose process is a quantum coding described by a channel r such that ran= a n (quantum state) and D = r p = EnL D ~ Then . the quantum mutual entropy ZI( p ; A) becomes Holevo’s one, that is,
C1.S ( A D , )
11 ( p ; A r ) = S (ACT)-
n
when EnAnS ( A D , ) is finite. Let us discuss the entropy exchange [8]. For a state p , a channel A is defined by an operator-valued measure ( A j } such as A (.) = C , AT . A,. Then define a trATpAj , by wl-uch the entropy exchange is defined matrix W = (Wi,) with Wzj = tr A P by Se ( p , A) = -tr Wlog W. Using the above entropy exchange, two types of mutual entropies are defined as below and they are applied to the study of the quantum version of Shannon’s coding theorem [8, 38, 101. The first one is called the coherent information Z2 ( p ; A) and the second one is 13 ( p , A), which are defined by 12
(
~
A) E S (AP)- Se 7
13 ( P , A) = s ( P I
(
~
A) 3
7
+ S (Ap) - Se ( P >A ) .
By comparing these mutual entropies for information communication processes, we have the following theorem [35]. THEOREM6. When { A j } is a projection-valued measure and dim(ran Aj) = 1, for arbitrary state p we have (1) I1 ( p , A) I min { S ( p ) , S ( A p ) } , ( 2 ) Z2 ( p , A) = 0, (3) 13 ( P , A) = S ( P ) . From this theorem, the entropy I1 ( p , A) only satisfies the inequality held for classical systems, so that only this entropy can be a candidate as the quantum
66
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M. OHYA
extension of the classical mutual entropy. The other two entropies can describe a sort of entanglement between input and output, such a correlation can be also described by quasi-mutual entropy, some generalization of ZI( p , A ) , discussed in [29, 91.
Acknowledgements The author thanks IIAS and SCAT for financial support of this work. REFERENCES [l] L. Accardi and K. Imafi~ku:Control of quantum states by decoherence, to appear in Open Systems and Information Dynamics, 2003. [2] L. Accardi, Y. G. Lu and I. Volovich: Quantum Theory and its Stochastic Limit, Springer, Berlin 2002;
Japanese translation, Tokyo-Springer 2003. [3] L. Accardi and M. Ohya: Compound channels, transition expectations, and liftings, Appl. Math. Optim. 39 (1999), 33-59. L. Accardi and M. Ohya: A stochastic limit approach to the SAT problem, to appear. L. Accardi and M. Ohya: Generalized quantum Turing machine and stochastic limit for the SAT problem, in preparation. H. Araki: Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ. 11 (1976), 809-833; Relative entropy for states of von Neumann algebras It, Publ. RIMS, Kyoto Univ. 13 (1977), 173-192. L. Accardi and R. Sabbadini: On the Ohya-Masuda quantum SAT Algorithm, Proceedings Intern. Con$ “Unconventional Models of Computations”, I. Antoniou, C.S. Calude, M. Dinneen (eds.) Springer 2001; Reprint Volterra, N. 432, 2000. H. Barnum, M. A. Nielsen and B. W. Schumacher: Information transmission through a noisy quantum channel, Phys. Reu. A 57 (1998), 4153-4175. V. P. Belavkin and M. Ohya: Quantum entropy and information in discrete entangled states, infinite dimensional analysis, quantum probability and related topics, Vol. 4, No. 2 (2001), 137-160; Quantum entanglements and entangled mutual entropy, Proc. R. SOC.Lond. A. 458 (2002). 209-231. C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyalz: Entanglement-assisted capacity of a quantum channel and the reverse Shannon Theorem, quant-ph/0106052. E. Bernstein and U. Vazuani: Quantum complexity theory, Proc. of the 25th Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 11-22 (1993), SIAM Jounral on Computing 26 (1997), 1411. C. H. Bennett, E. Bernstein, G. Brassard and U. Vazirani: Strengths and Weaknesses of Quantum Computing, quant-ph/9701001. R. Cleve: An Introduction to Quantum Complexity Theory, quant-pW9906111. D. Deutsch: Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Royal Societv. of ” London series A. 400 (1985). . ,. 97-117. A. Ekert and R. Jozsa: Quantum computation and Shor’s factoring algorithm, Rev. Mod. Phys. 68 (1996), 733-753. M. Garey and D. Johnson: Computers and Intractability-a Guide to the Theory of NP-completeness, Freeman, 1979. I. M. Gelfand and A. M. Yaglom: Calculation of the amount of information about a random function contained in another such function, Ame,: Math. SOC. Transl. 12 (1959), 199-246. A. S. Holevo, Some estimates for the amount of information transmittable by a quantum communication channel (in Russian), Problemy Peredachi Informacii 9 (1973), 3-1 1. S. Iriyama and S. Akashi: Complexity of Ohya-Masuda-Volovich algorithm, to appear. R. S. Ingarden, A. Kossakowski and M. Ohya: Information Dynamics and Open Systems, Kluwer 1997.
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[211 S. Iriyama, M. Ohya and I. Volovich Generalized quantum Turing machine and its application to the SAT chaos algorithm, TUS (Tokyo University of Science) preprint, 2003. [22] R. S. Ingarden: Quantum information theory, Rep. Math. Phys. 10 (1976), 43-73. [23] J. von Neumann: Die mathematischen Gmndlagen der Quantemechanik, Springer, Berlin, 1932. [24] A. N. Kolmogorov, Theory of transmission of information, Amer. Math. SOC. Translation, Ser. 2, 33 (1963), 291-321. [25] A. Kossakowski, M. Ohya and Y. Togawa: How can we observe and describe chaos?, Open System and Information Dynamics lO(3) (2003), 221-233. [261 M. Ohya, Complexities and their applications to characterization of chaos, Int. J. Theor. Phys. 37 (1998), 495. [27] M. Ohya: On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory 29 (1983). 770-777. [283 M. Ohya: Some aspects of quanmm infomation theory and their applications to irreversible processes, Rep. Math. Phys. 27 (1989). 19-47. [29] M. Ohya: Fundamentals of quantum mutual entropy and capacity, Open Systems and Information Dynamics 6 (1999), 69-78. [30] M. Ohya and N. Masuda: NP problem in quantum algorithm, Open Systems and Information Dynamics 7 (2000), 33-39. [31] M. Ohya and D. Petz: Quantum Entropy and its Use, Springer 1993. [32] M. Ohya and I. Volovich: Quantum computing, NP-complete problems and chaotic dynamics, Quantum Information II, eds. T. Hida and K. Saito, World Scientific 2000; quant-pW9912100 and J. Opt. B, 5 (2003). 639-642. [33] M. Ohya and I. Volovich: New quantum algorithm for studying NP-complete problems, Rep. Math. Phys. 52 (2003), 25-33. [34] M. Ohya and I. Volovich: Quantum Information, Computation, Cryptography and Teleportation, Springer (to appear). [35] M. Ohya and N. Watanabe: Remarks on quantum mutual entropy, TUS preprint. [36] D. Petz and M. Mosonyi: Stationary quantum source coding, J. Math. Phys. 42 (2001), 4857-4864. [37] P. Shor, Algorithm for quantum computation, Discrete logarithm and factoring algorithm, Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, pp. 124-134, 1994. [38] P. Shor: The quantum channel capacity and coherent information, Lecture Notes, MSRI Workshop on Quantum Computation, 2002. [39] A. Uhlmann: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory, Commun. Math. Phys. 54 (1977), 21-32. [40] H. Umegaki: Conditional expectations in an operator algebra N (entropy and information), Kodai Math. Sem. Rep. 14 (1962), 59-85.
68 Open Sys. &Information Dyn. 11: 219-233, 2004 @ 2004 Kluwer Academic Publishers
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A Stochastic Limit Approa4chto the SAT Problem Luigi Accardi Centro V. VoJterra Universiti di Roma Torvergata Via Orazio Raimondo, 001 73 Roma, Itdia email: accardiQvoJterra.mat.uniroma2.it
Masanori Ohya Department of Information Sciences Tokyo University of Science Noda City, Chiba 278-8510, Japan email: ohya&.noda.tus.ac.jp
(Received: January 26, 2004) Abstract. There exists an important problem whether there exists an algorithm to solve an NP-complete problem in polynomial time. In this paper, a new concept of quantum adaptive stochastic systems is proposed, and it is shown that it can be used to solve the problem above.
1. Introduction Although the performance of computers is highly progressed, there are several problems which may not be solved effectively, namely, in polynomial time. Among such problems, so-called NP-problems and NP-complete problems are fundamental. It is known that all NP-complete problems are equivalent and an essential question is whether there exists a n algorithm t o solve an N P complete problem in polynomial tame. Problems of this kind have been studied for decades and so far all known algorithms have an exponential running time in the length of the input. The standard definition of P- and NP-problems is the following [14,17,20]: DEFINITION 1 Let n be the size of input. (1) A P-problem is a problem such that the number of elementary steps needed to solve it is polynomial in n. Equivalently, it is a problem which can be recognized in time which is polynomial in n by a deterministic Turing machine. (2) An NP-problem is a problem which can be solved in polynomial time by a nondeterministic Turing machine. This can be understood as follows: Let us consider a problem to find a solution of f (z) = 0. We can check in time polynomial in n whether zo is a solution of f (x) = 0, but we do not know whether we can find the solution of f (x) = 0 in such time.
69
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DEFINITION 2 An NP-complete problem is a problem to which any other NPproblem can be polynomially transformed. We take the SAT (satisfiability) problem, one of the NP-complete problems, to study whether there exists an algorithm showing NPC = P. Our aim of this paper and the previous ones [lo, 12,131 is to find a quantum algorithm solving the SAT problem in polynomial time in the size of data. G {XI,.. . ,z,} be a set. Then xk and its negation Tk (k = 1 , 2 , .. . , n) are Let called literals and the set of all such literals is denoted by X’= {z1,51,. . . ,z, 5,). The set of all subsets of X’ is denoted by .F(X’) and an element C E F ( X ’ ) is called a clause. We take a truth assignment to all Boolean variables Xk. If we can assign the truth value to at least one element of C , then C is called satisfiable. When C is satisfiable, the truth value t (C) of C is regarded as true, otherwise, it is is false. Taking the truth values as “true -1, false -0”. Then C is satisfiable iff t ( C )= 1. Let L = (0, l} be a Boolean lattice with usual join V and meet A operations, and t ( 2 ) be the truth value of a literal z in X . Then the truth value of a clause C is written as t (C) = VzECt (z). Moreover the set C of all clauses Cj ( j = 1 , 2 , . . . ,rn) is called satisfiable iff the meet of all truth values of Cj is 1;t (C) = A F l t (Cj) = 1. Thus the SAT problem is written as follows:
x
DEFINITION 3 SAT Problem: Given a Boolean set X E ( 5 1 , . . . ,z,} C = {el,.. .C}, of clauses, determine whether C is satisfiable or not.
and a set
That is, this problem is to ask whether there exists a truth assignment which makes C satisfiable. It is known that one needs polynomial time to check the satisfiability when a specific truth assignment is given, but we cannot determine the satisfiability in polynomial time when an assignment is not specified. In [lo] we have discussed the quantum algorithm of the SAT problem, which was rewritten in [18] and we have showed that OM SAT-algorithm is combinatoric. In [12,13] it is shown that the chaotic quantum algorithm can solve the SAT problem in polynomial time. Ohya and Masuda pointed out [lo] that the SAT problem, and hence all other N P problems, can be solved in polynomial time by a quantum computer if the superposition of two orthogonal vectors 10) and 11) can be physically detected. However this detection is considered impossible with the present day technology. The problem to be overcome is how to distinguish the pure vector 10) from the superposed one a! 10) /?11), obtained by the OM SAT-quantum algorithm, if ,B is not zero but very small. If such a distinction is possible, then we can solve the NPC problem in the polynomial time. In [12,13]it is shown that it can be possible by combining nonlinear chaos amplifier with the quantum algorithm, which would imply the existence of a mathematical algorithm solving N P = P. It is not known if the algorithm of Ohya and Volovich lies in the framework of quantum Turing algorithms or not. So the next question is (1) whether there exists a physical realization combining the SAT quantum algorithm with chaos dynamics, or (2)
+
70 A Stochastic Limit Approach to the SAT Problem
221
whether there exists another method to achieve the above distinction of two vectors by a suitable unitary evolution so that all process can be modeled by a certain quantum Turing machine (circuits). In this paper, we argue that the stochastic limit, recently extensively studied by Accardi and coworkers [l],can be used to find another method of (2) above. In Sect. 2, we review mathematical frame of quantum algorithm and the OM SATalgorithm following the representation of Accardi and Sabaddini [18]with a quick review of OV-chaos algorithm in Sect. 3. In Sect. 4, a new concept - quantum adaptive stochastic system - is proposed, and in Sect. 5, we show that it can be used to solve the problem N P = P. 2.
Quantum Algorithm
The quantum algorithms discussed so far are rather idealized because computation is represented by unitary operations. A unitary operation is rather difficult to realize in physical processes, a more realistic operation is the one allowing some dissipation like semigroup dynamics. However such dissipative dynamics destroys the entanglement and hence they essentially reduce the ability of quantum computation to preserve the entanglement of states. In order t o keep the power of quantum computation and good entanglement, it will be necessary to introduce some kind of amplification in the course of real physical processes in physical devices, which will be similar to the amplication processes in quantum communication. In this section, to look for more realistic operations in a quantum computer, the channel expression will be used, at least, in the sense of mathematical scheme of quantum computation because a channel is not always unitary and represents many different types of dynamics. Let 7-l be a Hilbert space describing the input, computation and the output (result). As usual, the Hilbert space is 'If = @ y C 2 ,and let the basis of IFt = &'"iN2 be:
eo el
... ep-1
Any number t E (0,.
= =
10) = 10) 8 . . . @ 10) 8 10) , 11) = l O ) 8 . . . @ l O ) @ . I 1 ) ,
...
=
12N-1)
. . , 2N - 1)
= I1>~...~11>811>.
can be expressed by
a,(") = 0 or a,(") = 1, so that the associated vector is written by N
71
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L. Accardi and M. Ohya
Applying n-tuples of the Hadamard matrix A vector
EZ 2-
lo), we get
(
1'
)
to the vacuum
N 1
Put
Then we have
which is called Discrete Fourier Transformation. The combination of the above operations gives a unitary operator UF ( t ) W ( t )A and the vector ( t )= UF ( t )10) .
t
2.1.
CHANNEL EXPRESSION OF CONVENTIONAL UNITARY ALGORITHM
All conventional unitary algorithms can be written as a combination of the following three steps: (1) Preparation of state: Take a state p (e.g., p = 10) (01) and apply the unitary channel defined by the above UF ( t ) : A> = Ad,(,)
A> = AduF
==+
Agp = U ~ p u ; .
(2) Computation: Let U be a unitary operator on 3-1 representing the computation followed by a suitable programming of a certain problem, then the computation is described by a channel A; = AdU (unitary channel). After the computation, the final state p j will be pf = h;A>p.
(3) Registration and Measurement: For the registration of the computed result and its measurement we may need an additional system K (e.g., register), so that the lifting && from S ('H) to S ('FI 8 K) in the sense of [2] is useful to describe this stage. Thus the whole process is wrtten as Pf = && ( G J G P ) '
Finally, we measure the state in K: For instance, let {pk;k E J } be a projection valued measure (PVM) on K
ALP,
=
c
I 8 PkPfI 8 93
k€ J
after which we can get a desired result by observations in finite times if the size of the set J is small.
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A Stochastic Limit Approach to the SAT Problem
2.2.
C H A N N E L EXPRESSION OF T H E GENERAL QUANTUM ALGORITHM
When dissipation is involved the above three steps have to be generalized. Such a generalization can be expressed by means of suitable channel, not necessarily unitary. (1) Preparation of state: We may use the same channel A*, = Adu, in this first step, but if the number of qubits N is large so that it will not be built physically, then A*, should be modified; let us denote it by A;. (2) Computation: This stage is certainly modified to a channel A; reflecting a physical device realizing it. ( 3 ) Registration and Measurement: This stage is the the same as above. Thus the whole process is written as
3. Q u a n t u m Algorithm of S A T
(h)
Let 0 and 1 of the Boolean lattice L be denoted by the vectors 10) = and 1 ) = in the Hilbert space C?, respectively. That is, the vector 10) represents false and 11) truth. This section is based on [lo, 18,3]. As we explained in the previous section, an element x E X can be denoted by 0 or 1, i.e. by 10) or 11) in the present context. In order to describe a clause C of length at most n by a quantum state, we need the n-tuple tensor product Hilbert space ‘Ft = @C2. For instance, in the case of n = 2, given C = {x~,xz} with an assignment X I = 0 and x2 = 1, the corresponding quantum state vector is 10) @ \I),so that the quantum state vector describing C is generally written as IC) = 1x1) 8 1x2) E IH with x k = 0 or 1 ( k = 1,2). The quantum computation is performed by a unitary gate constructed from several fundamental gates such as “Not” gate, “Controlled-Not” gate, “ControlledCz, . . . ,Cm} Controlled” Not gate [22,11]. Once X = {XI,.. . ,rc,} and C = {CI, are given, the SAT is to find the vector
(y)
m
It(C))
3
A
v
t(x) 1
j=1 X E C ,
where t(x) is 10) or 11) when x = 0 or 1, respectively, and t(x) A t ( y ) f t(x A y),
t(x)v t ( y ) = t(x v y). 3.1.
LOGICAL NEGATION
DEFINITION 4 Let X be a set. A negation on X is an involution without fixed points, i.e. a map X 3 x H x‘ E X such that (x’)’= x ; x # x‘ Vx E X. x’ is called the negation of x.
73 224
L. Accardi and M. Ohya
PROPOSITION 1 Given a nonempty set X with a negation (x H x’) and denoting
I’ := (x’EX : xEI},
for I 2 X,there exists a set I 2 X such that X
= I U I’.
Thus a finite set with a negation must be even. Let X be a finite set with 2n elements and with a negation (x H x’). A partition X = I U 1’, 111 = n can be constructed with an n-step algorithm. Not all n-step algorithms are equivalent. DEFINITION 5 Given a set X with a negation x H x’,a “clause” is a subset of X. A minimal clause is a subset I 5 X such that I n I’ = 0 (i.e. if I contains x, it does not contain the negation of x). h
In any set X of cardinality 2n there are 2n minimal clauses. Given a set Co of clauses, if there are non-minimal clauses in it, then we can eliminate them from CO because any truth function must be identically zero on_a non-minimal clause. However, to eliminate the non-minimal clauses from Co, one has to “read” all its elements. Their number can be of order 2n.
3.2.
TRUTH FUNCTIONS
The set (0,1} is a Boolean algebra with the operations EVE’
:= m a x ( E , E ’ } ,
:= min(e,E’},
EAE’
E,E’
E {o,I}.
A clause truth function on the clauses on the set X = ( 2 1 , . . . ,xn,x;,. . . , &} is a boolean algebra homomorphism
t
:
x
+
(0,1}
with the property (principle of the excluded third):
t ( x j )v t ( z > ) = 1, V j
= 1,.. . , n .
(1)
Because of (l),such a function is uniquely determined by values ( t ( s l ).,. . ,t ( z n ) } , hence the number of such functions is 2n. For this reason, in the following we will simply say tmth function on ( X I , .. . ,xn} meaning by this a truth function on the clauses of the set (xi,.. . ,x , , ~ ; , ... ,xi}. Conversely given any n-tuple E = ( ~ 1 , .. . , E ~ )E (0, l}n,there exists only one truth function on (21,. . . ,zn}, with the property that
t(Xj) =
E j ,
v j = 1,.. . , n .
In what follows, given a truth function t , we denote the string in { t ( q ) ,. . . , t ( x n ) } uniquely associated to that function by E t . Let 7 be the set of truth functions on {XI,.. . , xn}. The function
t E7
++ It(x1), . . . ,t(z,)) E
@C2
defines a one-to-one correspondence between 7 and the set (0, l}, that is, a oneto-one correspondence between truth functions and vectors of the computational basis of @C2
74
A Stochastic Limit Approach to the SAT Problem
225
PROPOSITION 2 Let C C_ X be a clause and I , I’ the sets associated to it through the procedure explained in Sect. 1. L e t t be a truth function on {XI,.. . ,x n } . Then
t(C) =
I,,
I
Vt(Zi) v
V ( 1-t(x.j)) [jEI/
]
.
Therefore, as stated in Introduction, a set of clauses CO is said to be SAT if there exists a truth function t , on (51,.. . ,x,} such that t ( C 0 ) :=
A c) =
t(
3.3.
QUANTUM ALGORITHM
FOR THE
n
t ( C )= 1 .
CECo
CECo
SAT PROBLEM
We review here a technique developed in [lo],which shows that the SAT problem can be solved in polynomial time by a quantum computer. Given a set of clauses CO = {Cl,. . . , Cm}on X , Ohya and Masuda constructed a Hilbert space 7-1 = @+T2, where p is a number that can be chosen linear in mn, and a unitary operator Uc, : 7-1 -+ 7-1 with the property that, for any truth function t , U C o I E t , 0,) = 1% 57-1,t ( C 0 ) ) , where ~t is the vector of the computational basis of BnC2 corresponding to t , and 0, (resp. is a string of p zeros (resp. a string of ( p - 1) binary symbols depending on E ) . Furthermore Uco is a product of gates, namely of unitary operators that act at most on two qubits at a time. Let CO and Uco be as above and, for every E E ( 0 , l}n,let t, be the corresponding truth function. Applying the unitary operator Uc, to the vector
XZ-~)
one obtains the final state vector
THEOREM 1 CO is satisfiable if and only i f
where Pn+H,~ denotes the projector
on the subspace of 7-1 spanned by the vectors
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L. Accardi and M. Ohya
According t o the standard theory of quantum measurement, after a measurement of the event Pn+p,l,the state p = Ivf)(vufl becomes
Thus the solvability of the SAT problem is reduced to check that p’ difficulty is that the probability of Pn+p,~ is
#
0. The
where IT(C0)l is the cardinality of the set T ( C o ) , of all the truth functions t such that t(&)= 1. with T := IT(C0)l in the sequel. Then if T is suitably We put q := large to detect it, then the SAT problem is solved in polynomial time. However, for small T , the probability is very small and this means we in fact don’t get an information about the existence of the solution of the equation t(C0)= 1, so that in such a case we need further deliberation. Let us simplify our notations. After the quantum computation, the quantum computer will be in the state
m.
where Iql) and Ipo) are normalized n qubit states and q = Effectively our problem is reduced to the following 1 qubit problem. We have the state
and we want to distinguish between the cases q = 0 and q > 0 (small positive number). It is argued in [16] that quantum computer can speed up NP problems quadratically but not exponentially. The no-go theorem states that if the inner product of two quantum states is close to 1, then the probability that a measurement distinguishes then is exponentially small. And one could claim that amplification of this distinguishability is not possible. At this point we emphasized [13] that we do not propose to make a measurement which will be overwhelmingly likely to fail. What we do it is a proposal to use the output I I$) of the quantum computer as an input for another device which uses chaotic dynamics. The amplification would be not possible if we use the standard model of quantum computations with a unitary evolution. However the idea of the paper [12,13] is different. In [12,13] it is proposed to combine quantum computer with a chaotic dynamics amplifier. Such a quantum chaos computer is a new model of computations and we demonstrate that the amplification is possible in the polynomial time. One could object that we do not suggest a practical realization of the new model of computations. But at the moment nobody knows of how to make a
76 A Stochastic Limit Approach to the SAT Problem
227
practically useful implementation of the standard model of quantum computing ever. It seems to us that the quantum chaos computer considered in [13] deserves an investigation and has a potential to be realizable. Here we mention two works on non-linear quantum evolution to study NPproblems done by Abrams-Lloyd [8] and Czachor [9]. The former was based on the Weinberg model of nonlinear quantum mechanics and the latter was done by means of the Polchinski type description. Czachor’s work looks similar to our approach (stochastic limit). Their works are very artificial and conceptually different from ours.
3.4.
CHAOTIC DYNAMICS
Various aspects of classical and quantum chaos have been the subject of numerous studies, see [19] and references therein. Here we will argue that chaos can play a constructive role in computations (see [ l a, 131 for the details). Chaotic behaviour in a classical system is usually considered as an exponential sensitivity to initial conditions. It is this sensitivity we would like to use to distinguish between the cases q = 0 and q > 0 from the previous section. Consider the so called logistic map which is given by the equation
The properties of the map depend on the parameter a. If we take, for example, a = 3.71, then the Lyapunov exponent is positive, the trajectory is very sensitive to the initial value and one has the chaotic behaviour [19]. It is important to notice that if the initial value z o = 0, then z, = 0 for all n. It is known [21] that any classical algorithm can be implemented on a quantum computer. Our quantum chaos computer will consist of two blocks. One block is an ordinary quantum computer performing computations with the output I$) = 10) q 11). The second block is a computer performing computations of the classical logistic map. This two blocks should be connected in such a way that the state I$) first be transformed into the density matrix of the form
d m +
where Pi and Po are projectors to the state vectors 11) and 10) . This connection is in fact nontrivial and actually it should be considered as the third block. One has to notice that PI and POgenerate an Abelian algebra which can be considered as a classical system. In the second block the density matrix p above is interpreted as the initial data p a , and we apply the logistic map as
Pm =
(I+ f r n b 0 ) 6 3 ) 2
,
where I is the identity matrix and 0 3 is the z-component Pauli matrix on C2. To find the proper value m we finally measure the value of 0 3 in the state p m such that Mm = Trpmcr3.
77
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L. Accardi and M. Ohya
We obtain THEOREM 2
Thus the question is whether we can find such an m in polynomial number steps in n satisfying the inequality Mm >_ for very small but non-zero g 2 . Here we have to remark that if one has q = 0 then po = Po and we obtain Ad, = 0 for all m. If q # 0, the stochastic dynamics leads to the amplification of the small magnitude q in such a way that it can be detected as is explained below. The transition from po to pm is nonlinear and can be considered as a classical evolution because our algebra generated by POand PIis abelian. The amplification can be done within at most 2n steps due to the following propositions. Since f m ( q 2 ) is x, of the logistic map x,+l = f (2,) with xo = q 2 , we use the notation x, in the logistic map for simplicity. THEOREM 3 For the logistic map xn+l = ax, ( 1 - 2,) with a E [0,4] and xo E [O, 11 , let xo be l / Z n and the set J be { 0 , 1 , 2 , . . . ,n, . . .2n}. I j a is 3.71, then there exists an integer m in J satisfying x, > l / 2 . THEOREM 4 Let a and n be the same as in the above proposition. If there exists rno in J such that xm0 > l / 2 , then mo > ( n - 1)/ log23.71. According to these theorems, it is enough to check the value x, (M,) around the above mo when q is 1/2, for a large n. More generally, when q = k/2" with some integer k , it is similarly checked that the value z, (M,) becomes over l / 2 within at most 2 n steps. The complexity of the quantum algorithm for the SAT problem was discussed in Sect. 3 to be polynomial in time. We have only to consider the number of steps in the classical algorithm for the logistic map performed on a quantum computer. It is the probabilistic part of the construction and one has to repeat computations several times to be able to distinguish the cases q = 0 and q > 0. Thus it seems that the quantum chaos computer can solve the SAT problem in polynomial time. In conclusion of [12,13], the quantum chaos computer combines the ordinary quantum computer with quantum chaotic dynamics amplifier. It may go beyond the usual quantum Turing algorithm, but such a device can be powerful enough to solve the NP-complete problems in the polynomial time. The detailed estimation of the complexity of the SAT algorithm is discussed in [23]. In the next two sections we will discuss the SAT problem from a different point of view, that is, we will show that the same amplification is possible by unitary dynamics defined in the stochastic limit. 4.
Quantum Adaptive Systems
The idea to develop a mathematical approach to adaptive systems, i.e. those systems whose properties are in part determined as responses to an environment
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A Stochastic Limit Approach to the SAT Problem
229
17,251, was born in connection with some problems of quantum measurement theory and chaos dynamics. The mathematical definition of an adaptive system is in terms of observables, namely: a n adaptive system is a composite system whose interaction depends o n a fixed observable (typically in a measurement process, this observable is the observable one wants to measure). Such systems may be called observable-adaptive. In the present paper, we want to extend this point of view by introducing another natural class of adaptive systems which, in a certain sense, is the dual to the one defined above, namely the class of state-adaptive systems. These are defined as follows: a state-adaptive system is a composite system whose interaction depends o n the state of at least one of the sub-systems at the instant in which the
interaction is switched on. Notice that both definitions make sense both for classical and for quantum systems. Since in this paper we will be interested in an application of adaptive systems to quantum computation, we will discuss only quantum adaptive systems, but one should keep in mind that all the considerations below apply to classical systems as well. The difference between state-adaptive systems and nonlinear dynamical systems should be emphasized: (i) in nonlinear dynamical systems (such as those whose evolution is described by the Boltzmann equation, or nonlinear Schrodinger equation, etc.) the interaction Hamiltonian depends on the state at each time t , i.e. H I = Hr(pt)
v t.
(ii) in state-adaptive dynamical systems (such as those considered in the present paper) the interaction Hamiltonian depends on the state only at each time t = 0, i.e. HI = H z ( p 0 ) . The latter class of systems describes the following physical situation: at time
t = -T (T > 0) the system S is prepared in the state $-T and in the time interval \-T,O] it evolves according to a fixed (free) dynamics U [ - T , ~so I that its state at time 0 is u[-T,O]$-T =: $0. At time t = 0 an interaction with another system R is switched on and this interaction depends on the state $ 0 , i.e. H I = H I ( & ) . If we interpret the system R as environment, we can say that the above inter-
action describes the response of the environment to the state of the system s. Now from the general theory of stochastic limit [l]one knows that, under general ergodicity conditions, an interaction with an environment drives the system to a dynamical (but not necessarily thermodynamical) equilibrium state which depends on the initial state of the environment and on the interaction Hamiltonian. Therefore, if one is able to realize experimentally these state dependent Hamiltonians, one would be able to drive the system S to a pre-assigned dynamical equilibrium state depending on the input state $0. In the following section we will substantiate the general scheme described above with an application to the quantum computer approach to the SAT problem described in previous sections.
79
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L. Accardi and M. Ohya
5.
Stochastic Limit and SAT Problem
We illustrate the general scheme described in the previous section in the simplest case when the state space of the system is l-ls = C2. We fix an orthonormal basis of xs as {eo, el). The unknown state (vector) of the system at time t = 0
In the case of Sect. 3, a1 corresponds to q and e j does to lj) ( j = 0 , l ) . This vector is taken as input and defines the interaction Hamiltonian in an external field HI
XI+)(?ll@ (A;
=
+ A,)
where X is a small coupling constant. Here and in the following summation over repeated indices is understood. The free system Hamiltonian is taken to be diagonal in the e,-basis
~s
:=
C
E,Ie,)(e,I
= EoIeo)(eoI
+ EiIei)(eiI
4OJI and the energy levels are ordered so that Eo < El. Thus there is a single Bohr frequency wg := El - EO> 0. The one-particle field Hamiltonian is
Stg(k)
= ei t 4 k ) g ( k ) ,
where w ( k ) is a function satisfying the basic analytical assumption of the stochastic limit. Its second quantization is the free field evolution
We can distinguish two cases as below, which correspond to two cases of Sect. 3, i.e., q > 0 and q = 0. Case 1
If a ~ , a # i 0, then, according to the general theory of stochastic limit (i.e., t
+
t / X 2 ) [l],the interaction Hamiltonian H I is in the same universality class as
-
H~= D 8 A;
+ D+ 8 A , ,
where D := \eo)(eI\ (this means that the two interactions have the same stochastic limit). The interaction Hamiltonian at time t is then
-HI(^) = e-itwoD @ A st+ + h.c. 9
=
D @A+(eit(w(p)-Wo)g)+h.c.
80
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A Stochastic Limit Approach to the SAT Problem
and the white noise ( { b t } ) Hamiltoniaii equation associated, via the stochastic golden rule, to this interaction Hamiltonian is
i3tUt = i(Dbr + D+bt)Ut. Its causally normal ordered form is equivalent to the stochastic differential equation
dUt
=
(iDdB:
+ iD+dBt - y-D+Ddt)Ut,
where d B t := bldt. The causally ordered inner Langevin equation is
+
+
djt(x) = dU;xUt U,"xdUt dU,*xdUt = U,"(-iD+xdBt - i D x d B r - T-D'Dxdt ixDdBz ixD'dBt - y-xD+Ddt -t y-D+xDdt)Ut = i j t ( [ x D'])dBt , i j t ( [ xD , ])dBj -(Re y - ) j t ( { D + D , z } ) d t + i ( I m y - ) j t ( [ D + Dx, ] ) d t +jt(D+xD)(Rey-)dt ,
+
+
+
where j t ( x ) := U{xUt. Therefore the master equation is
d dt
-P t ( x ) =
( I m y ) i [ D + DP, t ( x ) ]- ( R e y - ) { D + D ,P t ( x ) }
+ (Re ? - ) D + P t ( x ) D ,
where D+D = leI)(ell and D+xD = (eo,xeo)lel)(ell. The dual Markovian evolution P$ acts on density matrices and its generator is L*P = (Imy-)i[p, D+D] - (Re y - ) { p , D'D}
+ (Re y - ) D p D + .
Thus, if po = Ieo)(eol one has L*po = 0 so po is an invariant measure. From the Fagnola-Rebolledo criteria [26],it is the unique invariant measure and the semigroup exp(tL,) converges exponentially to it. Case 2
If
a1 = 0 ,
then the interaction Hamiltonian H I is
HI
=
4eo)(eol@(A:
+ A,)
and, according to the general theory of stochastic limit, the reduced evolution has no damping and corresponds to the pure Hamiltonian
H s + Ieo)(eol = (Eo + l>leo)(eol+ E i J e i ) ( e i J therefore, if we choose the eigenvalues E l , EOto be integers (in appropriate units), then the evolution will be periodic.
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Since the eigenvalues E l , Eo can be chosen a priori, by fixing the system Hamiltonian H s , it follows that the period of the evolution can be known a priori. This gives a simple criterion for the solvability of the SAT problem because, by waiting a sufficiently long time one can experimentally detect the difference between damping and an oscillatory behaviour. The precise estimate of this time can be achieved either by theoretical methods or by computer simulation. Both methods will be analyzed in the expanded paper 131. Czachor [9] gave an example of a nonlinear Schrodinger equation to distinguish two cases, similar to a1 # 0 and a1 = 0 given above, in a certain oracle computation. We used the resulting (flag) state after quantum computation of the truth function of SAT to couple the external field and took the stochastic limit, then our final evolution becomes “linear” for the state p described as above. The stochastic limit is historically important to realize macroscopic (time) evolution and it is now rigorously established as explained in [l],and we gave a general protocol to study the distinction of two cases a1 # 0 and a1 = 0 by this rigorous mathematics. The macro-time enables us to measure the behavior of the outcomes practically. Thus our approach is conceptually different from Czachor’s. Moreover Czachor discussed that some expectation value is constant for the case a1 = 0 and oscilating for a1 # 0, and ours gives the detail behavior of the state w.r.t the macro-time; damping (a1 # 0 case) and oscilating (a1 = 0 case) 6.
Conclusion
We showed in [lo, 12,131 that we can find an algoritlmi solving the SAT problems in polynomial number of steps by combining a quantum algorithm with chaotic dynamics. We used the logistic map there, however it is possible to use other chaotic maps if they can amplify one of two coefficients. In this short paper we pointed out that it is possible to distinguish two different states, 10) + q 11) ( q # 0) and 10) by means of an adaptive dynamics and the stochastic limit. Finally we remark that our algorithms can be described by deterministic general quantum Turing machine [24,4], whose result is based on the general quantum algorithm mentioned in Sect. 2.
Jq
Acknowledgment The authors thank SCAT for financial support of this joint work. We thank the referee for informing us about the paper of Czachor.
Bibliography [l] L. Accardi, Y .G. Lu, I. Volovich, Quantum Theory and its Stochastic Limit, Springer Verlag 2002, Japanese translation, Tokyo-Springer, 2003. [2] L. Accardi and M. Ohya, Compound channels, transition expectations, and laftings, Appl. Math. Optim. 39,33 (1999).
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[3] L. Accardi and M. Ohya, A stochastic limit approach t o the S A T problem, in preparation. [4] L. Accardi and M. Ohya, Generalized Quantum Turing machine and stochastic limit f o r the SAT problem, TUS preprint. [5] L. Accardi, R. Sabbadini, O n the Ohya-Masuda quantum S A T Algorithm, in: Proceedings 1ntern.Conf. ”Unconventional Models of Computations”, I. Antoniou, C. S. Calude, M. Dinneen, eds., Springer 2001; Preprint Volterra, N. 432, 2000 [6] L. Accardi, R.Sabbadini, A Generalization of Grover’s Algorithm, Proceedings Intern. Conf.: Quantum Information 111, Meijo University, Nagoya, 27-31 March, 2001; World Scientific 2002; qu-phys 0012143; Preprint Volterra, N. 444, 2001. [7] L. Accardi and K. Imafuku, Control of Quantum States by Decoherence, Volterra Center Preprint No. 542. [8] D. S. Abrams and S. Lloyd, Nonlinear quantum mechanics implies polynomial time solution f o r NP-complete and # P problem, Phys. Rev. Lett. 81,3992 (1998). [9] M. Czachor, Notes o n nonlinear quantum algorithm, Acta Phys. Slov. 48, 157 (1998). [lo] M. Ohya and N. Masuda, N P problem i n Quantum Algorithm, Open Sys. Information Dyn. 7,33 (2000). [ll] M. Ohya, Mathematical Foundation of Quantum Computer, Maruzen Publ. Company, 1998. [12] M. Ohya and I.V. Volovich, Quantum computing, NP-complete problems and chaotic dynamics, in: Quantum Information 11, eds. T.Hida and KSaito, World Sci. 2000; quantph/9912100 and J. Opt. B 5, 639 (2003). [13] M. Ohya and I.V. Volovich, New quantum algorithm f o r studying NP-complete problems, Rep. Math. Phys.52, 25 (2003). 1141 M. Garey and D. Johnson, Computers and Intractability - a guide to the theory of N P completeness, Freeman, 1979. [15] P. W. Shor, Algorithm for quantum computation: Discrete logarithm and factoring algorithm, Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, pp. 124-134, 1994. [16] C. H. Bennett, E. Bernstein, G. Brassard, U. Vazirani, Strengths and Weaknesses of Quantum Computing, quant-ph/9701001. [17] R. Cleve, An Introduction to Quantum Complexity Theory, quant-ph/9906lll. [18] L. Accardi, R. Sabbadini, O n the Ohya-Masuda quantum S A T Algorithm, in: Proceedings International Conference ”Unconventional Models of Computations”, I. Antoniou, C. s. Calude, M. Dinneen, eds., Springer, 2001. [19] M. Ohya, Complexities and Their Applications to Characterization of Chaos, Int. Journ. of Theort. Phys. 37,495 (1998). IZO] M. Ohya and I.V. Volovich, Quantum information, computation, cryptography and teleportation, Springer, to appear. I211 D. Deutsch, Quantum theory, the Church-Thing principle and the universal quantum computer, Proc. of Royal Society of London series A, 400, pp. 97-117, 1985. [22] A. Ekert and R. Jozsa, Quantum computation and Shor’s factoring algorithm, Reviews of Modern Physics 68, 733 (1996). [23] S. Iriyama and S. Akashi, Complexity of Ohya-Masuda- Volovich algorithm, to appear. [24] S. Iriyama and M. Ohya, O n generalized Turing machine, TUS (Tokyo University of Science) preprint, 2003. [25] A. Kossakowski, M. Ohya and Y . Togawa, How can we observe and describe chaos?, Open Sys. Information Dyn. 10, 221 (2003). [26] F . Fagnola and R. Rebolledo, O n the existence of Stationary States f o r Quantum Dynamical Semigroup, to appear in J. Math. Phys., 2001.
83 REPORTS ON MlXEMAllCAL PHYSICS
Vol. 52 (2003)
No. I
NEW QUANTUM ALGORITHM FOR STUDYING Np-COMPLETE PROBLEMS MASANON OHYA Tokyo University of Science, Department of Information Sciences, Noda City, Chiba 278-8510. Japan (e-mail: [email protected])
and IGOR
v. VOLOVICH
Steklov Mathematical Institute. Gubkin St. 8, 117% Moscow, Russia (e-mail: [email protected]) (Received December 6, 2002)
ordinary approach to quantum algorithm is based on quantum niring machine or quantum circuits. It is known that this approach is not powaful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using ow new quantum algorithm. Keywo~xhQuantum algorithm, NP-complete problem, chaotic dynamics.
1. Introduction
ordinary approach to quantum algorithm is based on quantum 'Ruing machine or quantum circuits [l-31. It is known that this approach is not powerful enough to solve NP-complete problems [4, 51. In [6] we have proposed a new approach to quantum algorithm which goes beyond the standard quantum computation paradigm. This new approach is a sort of combination of the ordinary quantum algorithm and a chaotic dynamics. This approach was based on the results obtained in the paper [71. There are important problems such as the knapsack problem, the travelling salesman problem, the integer programming problem, the subgraph isomorphism problem, the satisfiability problem that have been studied for decades and for which all known algorithms have a running time that is exponential in the length of the input. These five and many other problems belong to the set of NP-complete problems [4]. Many NP-complete problems have been identilied, and it s e m s that such problems are very difficult and probably exponential. If so, solutions are still needed,
84
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M. OHYA and I. V. VOLOVICH
and in this paper we consider an approach to these problems based on quantum computers and chaotic dynamics as mentioned above. As in the previous papers [7, 61, we again consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm. It is widely believed that quantum computers are more efficient than classical computers. In particular, Shor 18, 91 gave a remarkable quantum polynomial-time algorithm for the factoring problem. However, it is known that this problem is not NP-complete but is “-intermediate. Since the quantum algorithm of the satisfiability problem (SAT for short) has been considered in [7],Accardi and Sabbadini showed that this algorithm is combinatoric one and they discussed its combinatoric representation [lo]. It was shown in [7]that the SAT problem can be solved in polynomial time by using a quantum computer under the assumption that a special superposition of two orthogonal vectors can be physically detected. The problem one has to overcome here is that the output of computations could be a very small number and one needs to amplify it to a reasonable large quantity. In this paper we construct a new model (representation) of computations which combine ordinary quantum algorithm with a chaotic dynamical system and prove that one can solve the SAT problem in polynomial time. For a recent discussion of computational complexity in quantum computing see [ll-141. Mathematical features of quantum computing and quantum information theory are summarized in [151.
SAT Problem Let X 3 { X I , . . .,x,) be a set. Then xk and its negation Xk (k = 1,2,. . . ,n) are called literals and the set of all such literals is denoted by X‘ = { X I , XI,. . . ,x n , En). The set of all subsets of X’ is denoted by F(X’) and an element C E F(X’)is called a clause. We take a truth assignment to all variables xk. If we can assign the truth value to at least one element of C, then C is called sutisfmble. When C is satisfiable, the truth value f(C)of C is regarded as true, otherwise, that of C is false. Take the truth values as true “l”, false “0”. Then
2.
C is satisfiable iff r (C) = 1. Let L = {0,1} be a Boolean lattice with usual join v and meet A, and let r ( x ) be the truth value of a literal x in X. Then the truth value of a clause C is written as
t ( C ) = vxECr(x).
Further, the set C of all clauses C, ( j= 1,2, . . . rn) is called satisfiable iff the meet of all truth values of Cj is 1,
f(c) 3 Ay=lf(Cj) = 1.
85
m WQ
U
W ALGORlTHM FOR STUDYING NP-COMPLETE PROBLEMS
27
Thus the SAT problem is defined as follows.
DERNITION 1. SAT Problem: Given a set X = (xl, . .. ,x n ] and a set C = {Cl,Cz, . . . ,C,) of clauses, determine whether C is satisfiable or not. That is, this problem is to ask whether there exists a truth assignment to make C satisfiable. It is known [4] for usual algorithm that the time to check the satisfiability is polynomial only when a specific truth assignment is given, but we cannot determine the satisfiability in polynomial time when an assignment is not specified. Note that a formula made by the product (AND A) of the disjunction (OR V) of literals is said to be in the product of s u m (POS) form. For example, the formula (Xi V z2) A (Ti) A (Xz V 7 3 ) is in the POS form. Thus a formula in the POS form is said to be satisfiable if there is an assignment of values to variables so that the formula has value 1. Therefore, the SAT problem can be regarded as determining whether or not a formula in the POS form is satisfiable. The following analytical formulation of the SAT problem is useful. We define a family of Boolean polynomkds fd, indexed by the following data. Let A be a set A = {&, . . . , SN,T I , .. . , TN},
where Si, T E ( 1 , . , . , n}, and
fd
be defined as
We assume here the addition modulo 2. The SAT problem now is to determine whether or not there exists a value of x = (XI,. . . ,x,) such that fd(x) = 1.
3. Quantum algorithm Although the quantum algorithm of the SAT problem is needed to add the dust bits to the input n bits, the number of dust bits is of the order of n [7, lo]. Therefore for simplicity we will work in this paper in the (n 1)-tuple tensor product Hilbert space 'H = @y+1@2 with the computational basis
+
1x1,.* .
rxnr
Y ) = @=l Ixi) Q-IY)
9
where X I , . . . , x n , y = 0 or 1. We denote 1x1,. ..,xnry ) = Ix, y) . The quantum version of the function f(x) := fa(x) is given by the unitary operator UfIx, y) = Ix, y +- f ( x ) ) . We assume that the unitary matrix Uf can be build in the polynomial time, see [7]. Now let us use the usual quantum algorithm:
86
28
M. OHYA and I. V. VOLOVICH
(i) using the Fourier transform produce from l0,O) the superposition
(ii) use the unitary matrix Uf to calculate f ( x ) ,
Now if we measure the last qubit, i.e., apply the projector P = Z 8 11) (11 to the state I u f ) , then we obtain that the probability to find the result f(x) = 1 is llP Iuf)1I2 = r/2", where r is the number of roots of the equation f ( x ) = 1. If r is suitably large to detect it, then the SAT problem is solved in polynomial time. However, for small r, the probability is very small and this means that in fact we do not get any information about the existence of the solution of the equation f(x) = 1, so that in such a case we need further discussion. Let us simplify our notation. After the step (ii), the quantum computer will be in the state = ((Po) Q 10) qI dc 3 t1)
)f.1
m
+
I
m.
Effectively our where Iql) and [fi)are normalized n qubit states and q = problem is reduced to the following 1-qubit problem. We have the state
and we want to distinguish between the cases 4 = 0 and q > 0 (a small positive number). It is argued in [5] that quantum computer can speed up NP problems quadratically but not exponentially. The no-go theorem states that if the inner product of two quantum states is close to 1, then the probability that a measurement distinguishes which one of the two occurs is exponentially small. And one could claim that amplification of this distinguishability is not possible. At this point we emphasize that we do not propose to make a measurement (not read) which will be overwhelmingly likely to fail. What we do is a proposal to use the output I+) of the quantum computer as an input for another device which uses chaotic dynamics in the sequel. The amplification would not be possible if we used the standard model of quantum computations with a unitary evolution. However, the idea of our paper is different. We propose to combine quantum computer with a chaotic dynamics amplifier. Such a quantum chaos computer is a new model of computations going beyond usual scheme of quantum computation and we demonstrate that the amplification is possible in the polynomial time. One could object that we do not suggest a practical realization of the new model of computations. But at the moment nobody knows how to make a practically useful
87 NEW QUANTUM ALGORITHM FOR STUDYING NF'COMF'UTE PROBLEMS
29
implementation of the standard model of quantum computing. Quantum circuit or quantum ' k i n g machine is a mathematical model, though a convincing one. It seems to us that the quantum chaos computer considered in this paper deserves investigation and has a potential to be realizable. In this paper we propose a mathematical model of computations for solving SAT problem by refining our previous paper [6]. A possible spedic physical implementation of quantum chaos computations with some error correction will be discussed in a separate paper 1161, which is somehow related to the recently proposed atomic quantum computer [17].
Chaotic dynamics Various aspects of classical and quantum chaos have been the subject of numerous studies, see El81 and references therein. The investigation of quantum chaos by using quantum computers has been proposed in [19-211. Here we will argue that chaos can play a constructive role in computations. Chaotic behaviour in a classical system usually is considered as an exponential sensitivity to initial conditions. It is this sensitivity we would like to use to distinguish between the cases q = 0 and q r 0 from the previous section. Consider the so-called logistic map which is given by the equation
4.
xn+1
= a ~ n ( l -x")
g(x)l
Xn E
[O,I]
*
The properties of the map depend on the parameter a. If we take, for example, a = 3.71, then the Lyapunov exponent is positive, the trajectory is very sensitive to the initial value and one has the chaotic behaviour [18]. It is important to notice that if the initial value xo = 0, then xn = 0 for all n. It is known [2] that any classical algorithm can be implemented on quantum computer. Our quantum chaos computer will consist of two blocks. One block will be the ordinary quantum computer performing computations with the output I$) = ,/10) q 11). The second block will be a computer performing computations of the classical logistic map. These two blocks should be connected in such a way that the state I@)should first be transformed into the density matrix of the form
+
P = 4%
+ (1 - q 2 )Po,
where PI and PO are projectors to the state vectors 11) and lo). This connection would in fact be nontrivial and actually should be considered as the third block. One has to notice that PI and PO generate an abelian algebra which can be considered as a classical system. In the second block the density matrix p above is interpreted as the initial data po, and we apply the logistic map as Pm
=
(1
+f % o ) ~ 3 ) 2
7
88
30
M.OHYA and I. V. VOLOVICH
where I is the identity matrix and u3 is the z-component of Pauli matrix on C2. This expression is different from that of our first paper [6].To find a proper value m we finally measure the value of q in the state pm such that
M,
= trpmu3.
After simple computation we obtain
Thus the question is whether we can find such m in polynomial steps of n satisfying the inequality M,,, 2 for very small but nonzero q2. Here we have to remark that if one has q = 0 then po = Po and we obtain Mm= 0 for all m. If q # 0, the stochastic dynamics leads to the amplification of the small magnitude q in such a way that it can be detected as is explained below. The transition from po to pm is nonlinear and can be considered as a classical evolution because our algebra generated by PO and PI is abelian. The amplification can be done within at most 2n steps due to the following propositions. Since gm(q2) is x, of the logistic map = g(xm) with xo = q2, we use the notation xm in the logistic map for simplicity. PROPOSITION 2. Fur the logistic map n,+l = ax, (1 - x,) with a E [O, 41 and and the set J be {O, 1,2, ...,n ,..., Zn}. If a is 3.71, then there exists an integer m in J satisfying X m =-
xo E [0, 11, let xo be
&
i.
Proof: Suppose that there does not exist such m in J. Then xm 5 m E J. The inequality x m 5 f implies
Thus we have 1 3.71 - >xm 2 22
1.*. 2
(3;")"
-
from which we get Zflfm-' 1 (3.71)'". According to the above inequality, we obtain
Since logz3.71 = 1.8912, we have
xo= (3):'"-
1 2" '
4 for any
89 NEW QUANTLTM ALGORITHM FOR STUDYING NPCOMPLETE PROBLEMS
31
which is definitely less than 2n-1 and it is contradictory to the statement “Xm 5 4 r! for any m E J”. Thus there exists m in J satisfying X m >
i.
P R O F J O S ~ O N3. Let a and n be the same as in the above proposition. If there exists mo in J such that xmo > , then mo > &.
Proof: Since 0 5 x, 5 1, we have X,
= 3.71(1 - X,-l)Xm-l
5 3.71xm-1,
which reduces to X,
For mo in J satisfying xmo > xo 2
5 (3.71),~0.
, it holds
1
1
(3.71)m0xm0> 2 x (3.71)”O.
It foHows from xo = $ that
10g22 x (3.71)”O > n, which implies n-1 logz 3.71 ‘
’
..
~
0
mo According to these propositions, it is enough to check the value x, (M,) around the above rno when q is & for a large n . More generally, when q=$ with some integer k, it is easily checked that the above two propositions hold and the value
4
(M,) becomes over around the mo above. One can think about various possible implementations of the idea of using chaotic dynamics for computations, which is an open and very interesting problem. For this problem, realization of nonlinear quantum gates wiU be essential; it will be discussed in [161. Finally, we show in Fig. 1 how we can easily amplify the small q in several steps. Xm
Conclusion The complexity of the quantum algorithm for the SAT problem has been considered in [7] where it was shown that one can build the unitary ma& Us in the polynomial time. We have also to consider the number of steps m in the classical algorithm for the logistic map performed on quantum computer. It is the probabilistic part of the construction and one has to compute several times to be able to distinguish the cases q = 0 and q > 0. Thus we conclude that the quantum chaos algorithm can solve the SAT problem in polynomial time according to the above propositions. 5.
32
M. OHYA and I. V. VOLOVICH
xn
I 0.9 0.8 0.7
0.6 0.5 0.4
0.3
0:; 0
t
1
I
0
5
I
U
‘
1 10
15
20
25
1 30
35
40
45
n
50
Ng. 1. Change of xn w.r.t. time n
In conclusion, in this paper the quantum chaos algorithm is proposed. It combines the ordinary quantum algorithm with quantum chaotic dynamics amplifier. We argue that such an algorithm can be powerful enough to solve the NP-complete problems in the polynomial time. Our proposal is to show the existence of algorithm to solve NP-complete problem. The physical implementation of this algorithm is another question and it is strongly desirable to study it further. RFFERENCES
[I] D. Bouwmecster, A. Ebrt and A. Zeilinger: l%e Physics of Q u a n m Information, Springer, Berlin 2001. [2] D. Deutsch Quantum theory, the Church-”brhg principle and the universal quantum computer. Proc. Royal SOC. London series A, OOO (1985), 97-117. [3] E. Bemstein and U. Vazirani: Quantum Complexity Theory, in Proc. the 25th Annul ACM Symposium on Theory of Computing, ACM Press, New York 1993. 11-20. [4] M. Garey and D. Johnson: Computers and Intmctability-a Guide to the Theory of NP-completeness, Freeman, 1979. [5] C. H. Bennett, E. Bernstein, G. Brassard and U. Vazirani Strengths and Weaknesses of Quantum Computing, quant-ph19701001 M. Ohya and I. V. Volovich: Quantum computing, NP-complete problems and chaotic dynamics, in T.Hida and K. Saito (eds.). Q w t u m Information II, World Scientific, Singapo~.2000; quant-ph/9912100. M. Ohya and N. Masuda: NP problem in Quantum Algorithm, Open Sys&ms and Information Dynamics 7 No.] (zooO), 33-39. P. W. Shor: Algorithm for quantum computation: Discrete logarithm and factoring algorithm, Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, 1994, 124-134. A. JZkert and R Jozsa: Quantum computation and Shor’s factoring algorithm, Rev. Mod. Phys. 68, N0.3 (1996), 733-753. L. A d . R. Sabbadini: On the Ohya-Masuda quantum SAT Algorithm, in Proceedings International Confervnce “Unconventional M o ~ f e bof Computations”, I. Antoniou, C . S. Calude, M. Dinneen (eds.). Springer 2001.
91 Open Sys. &Information Dyn. 10: 221-233, 2003 @ 2003 Kluwer Academic Publishers
221
How Can We Observe and Describe Chaos?+ Andrzej Kossakowski Institute of Physics N.Copernicus University, Grudzigdzka 5, 87-100 Toruli, Poland
Masanori Ohya and Yosliio Togawa Department of Information Sciences Tokyo University of Science, Noda City, Chiba 27g8510, Japan
(Received: January 31, 2003) Abstract. We propose a new approach to define chaos in dynamical systems from the point of view of Information Dynamics. Observation of chaos in reality depends upon how to observe it, for instance, how to take the scale in space and time. Therefore it is natural to abandon taking several mathematical limiting procedures. We take account of them, and cham degree previously introduced is redefined in this paper.
1. Introduction There exist several attempts to describe chaos appearing in classical or quantum dynamical systems [l- 161. One of the present authors introduced Information Dynamics (ID for short) [18] as a frame to discuss complexity and chaos appearing in various fields, in which he tried to find a common basis by synthesizing the state change (dynamics) and the complexity associated with dynamical systems. Since then ID has been applied t o several different topics [9, 191, among which chaos degree, a quantity measuring the degree of chaos associated with a dynamics, was introduced by means of the complexities in ID and its entropic version (called Entropic Chaos Degree (ECD for short)) has been computed numerically for rather famous chaotic dynamics such as logistic map, baker’s transformation, Tinkerbel map. It is surprising that the result of the ECD exactly matches that of Lyapunov exponent in the case when the later can be computed. Moreover, the algorithm computing the ECD is much easier than that of Lyapunov exponent, so that the ECD is almost always computable even when the Lyapunov exponent is not. However, there are some unclear points, both conceptual and mathematical, why the ECD should be so successful for computational experiments. In this paper we study these points and propose a new description of chaos. In Sect. 2, we briefly review information dynamics and chaos degree, and in Sect. 3 the entropic chaos degree and its algorithm are recalled with a computational result. In Sect. 4, a new way of detecting chaos from a given dynamics is ~~~~
t
This is an invited paper for the anniversary 10th volume of OSID.
92 222
A. Kossakowski, M. Ohya, and Y . Togawa
discussed based on the ECD, that is, we propose a new approach to define chaos in dynamical systems. 2.
Information Dynamics and Chaos Degree
We briefly review what ID is. Let (A,6,a ( G ) )he an input (or initial) system be an output (or final) system. Here A is the set of objects to and (x,g,?i(c)) be observed and 6 is the set of means to get the observed value, a ( G ) describes evolution of system with a parameter g in a certain set G. Often we have A = 2, 6 = ?$?, LY = ?i, G = G. Therefore it can be said that: giving a mathematical structure to input and output triples
= having a theory
The dynamics of state change is described by a channel, that, is, a map A*: (sometimes 6 6 ) . The fundamental point of ID is that ID contains (At,6.t,at(Gt))be the total system of (A,6,a ) and two complexities in itself. Let __ (A,6 , E ) , and S be a subset of 6 in which we are measuring observables (e.g., S is the set of all KMS or stationary states in C*-system). The two complexities are denoted by C and T . C is the complexity of a state p measured from a reference system S, in which we actually observe the objects in A and T is the transmitted complexity associated with the state change cp ---$ A*cp, both of which should satisfy the following properties:
6
+
-+
Axioms of complexities (i) For any cp E S
c 6,
c ~ (2 ~ 0, ) (ii) For any orthogonal bijection j : e x 6 of 6,
T ~ ( ~ ; A2*0). -+
e x 6 , the set of all extremal points
Cj(S)(j(cp)) = CS(p)
Tj(')(j(cp);A*) = TS(cp;A*) . (iii) For @
= p @ 4 E Si C Bt, CSt(@)= C S ( q )+ C"($)
(iv) 0 5 TS(cp;A*)I CS(p) (v) TS(cp;id) = C'(p), where "id" is the identity map from 6 to 6 . Instead of (iii), when (iii') @ E St c 6t, put p = @ CSt(@)L CS(p) C"($)
+
A (i.e., the restriction of @ to A), $ z @
3,
is satisfied, C and T is called a pair of strong complexity. Therefore ID is defined as follows:
93 223
How Can We Observe and Describe Chaos?
DEFINITION 2.1 Information dynamics is described by
and some relations R among them. In the framework of ID, we have to _ _
(i) mathematically determine A, G, a ( G ) ;A , G , Z(??), (ii) choose A* and R, and (iii) define Cs(cp),TS(cp;A*). In ID, several different topics can be treated on a common footing so that we can find a new clue bridging several fields. We assume 2 = A for simplicity in the sequel. For a certain subset S (called the reference space) of G and a state cp E S , there exists a decomposition of the state cp into a mixture of extreme (pure) states such that
This extremal decomposition of cp describes the degree of mixture of cp in the reference space S. The measure p is not always unique, so that the set of all such measures is denoted by M+,(S). For instance, when ( A , 6 )and is a C*-system containing both classical and quantum systems, that is, A is a C* algebra and 6 is the set of all states on A, the reference space S is a weak* compact convex subset of 6 and the measure p is not uniquely determined unless S is the Choquet simplex. In this paper we will not go to the details of such general mathematical discussion. A measure of chaos produced by the dynamics A* is defined in [21,22]: DEFINITION 2.2 (1) 1c, is more chaotic than cp if C ( $ ) 2 C(p). (2) When cp E S changes to hay,the chaos degree associated to this state change (dynamics) A* is given by
Ds (cp; A*)
= inf
I
(J,Cs ( A * w )d p ; p E M+,( S )
.
DEFINITION 2.3 The dynamics A* produces chaos iff Ds (cp; A*) > 0. It is important to note here that the dynamics A* in the definition is not necessarily the same as the original dynamics (channel) but is the one reduced from the original one such that it causes an evolution for a certain observed value like a n orbit. However for simplicity we often use the same notation in this paper. In some cases, the above chaos degree Ds (cp; A*) can be expressed as
Ds (cp; A*)
=
Cs (A'cp) - TS(cp;A * ) .
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A. Kossakowski, M. Ohya, and Y. Togawa
3. Entropic Chaos Degree and its Algorithm Although there exist several complexities [20], one of the most useful examples of C and T are Shannon's entropy and mutual entropy in classical systems (von Neumann entropy and quantum mutual entropy in quantum systems [as]),respectively. The concept of entropy was introduced and developed to study the topics such as irreversible behaviour, symmetry breaking, amount of information transmission, so that it originally describes a certain chaotic property of state. Let us recall the simplest case of C and T , that is, Shannon's entropy and mutual entropy. In classical communication systems, an input state cp is a probability distribution p = ( p k ) = x k p k 6 k and a channel h* is a transition probability ( t i , j ) so that the compound state of cp and its output p (= j? = (Fi)= h * p ) is the joint distribution T = ( r i , j ) with ri,j G t i , j p j . Then the complexities C and T are given as
Thus the entropic chaos degree of the channel A* becomes DEFINITION 3.1
D ( p ;A*)
=
S ( A * p )- I ( p ;A * ) .
Quantum version of the above entropic chaos degree was discussed in [lo,221, which we will briefly review here in the case of usual Hilbert space formulation. Let p be a quantum state, namely, a density operator on a Hilbert space 'H, and A* be a channel sending the set 6 of all states on 'H into itself. Then the entropic chaos degree is defined by
where & is the set of all Schatten decompositions (i.e., onedimensional spectral decompositions) of the state p := x k X k E k , and s is the von Neumann entropy.
3.1.
AN ALGORITHM COMPUTING CHAOSDEGREE
In order to observe chaos produced by a dynamics, one often looks at the behaviour of orbits produced by that dynamics, more generally, looks at the behaviour of a certain observed value. Therefore, in our scheme we directly compute the chaos degree once the dynamics is explicitly given as a state change of the system. However, even when the direct calculation does not show chaos, it will appear
95 225
How Can We Observe and Describe Chaos?
if one focuses at sonie aspect of the state change, e.g., a certain observed value which may be called an orbit, as usual. The algorithm computing the chaos degree for a dynamics falls into the following two cases [ 2 1 , 2 2 , 1 2 ,l o ] :
1. The dynamics is given by d x / d t = f t (x)with x E I 5 [a,b]" c R": First find a difference equation xn+l = f ( x n ) with the map F on I 5 [a,b]" C R" into Ak be a finite partition with Ai n Aj = 0 (i # j ) . Then itself, secondly let I = the state p(n)of the orbit determined by the difference equation is defined by the probability distribution (p?'), that is, q(n)= p?)&, where for a given initial value x E I and the characteristic function 1~
uk
xi
Now, when the initial value x is distributed according to a measure v on I , the above p?' is given as Pi
=
~
n f l
/
m+n
1~~( F k x )d v .
Ikzm
The joint distribution p$'n+l) between the times n and n
+ 1 is defined by
or
Then the channel A; at n is determined by
and the entropic chaos degree is given by the Definition 3.1;
We can judge whether the dynamics is chaoting or not by the value of D as in the Definition 2.2:
D > 0 a chaotic D = 0 stable.
96
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A. Kossakowski, M. Ohya, and Y. Togawa
a 3.2
3.6
3.4
3.8
Fig. 1: The bifurcation diagram for logistic map This chaos degree was applied to several dynamical systems such as logistic map, Baker's transformation and Tinkerbel map, and it could explain their chaotic characters. This chaos degree has several advantages when compared with the usual measures, such as Lyapunov exponents, as explained below. 2. The dynamics is given by yt = FZyo on a Hilbert space: Similarly as converting it into the difference equation of state, the channel A: at n is first deduced from F : , which should satisfy p(ntl) = h*,~p(~). By means of this constructed channel ( a ) we compute the chaos degree D directly according to the Definition 3.2 or (p) we take a proper observable X and put 5 , = p(")(X),then go back to the algorithm (1). Note that the chaos degree D does depend on a partition A taken, which is somehow different from usual degree of chaos (cf., dynamical entropy [l,4,3,14]). This is a key point of our understanding of chaos, which will be discussed in the next section.
3.2.
LOGISTICMAP
Let us explain how well the entropy chaos degree (ECD) describes the chaotic behaviour of the logistic map. The logistic map is defined by 5,+1
=
ax, (1- 5 , )
,
5,
E [O, 11 ,
0 5 a 54.
The bifurcation diagram for this equation is shown in Fig. 1. In order to compare ECD with other measures describing chaos, we choose Lyapunov exponents. Definition of Lyapunov exponent (1) Let f be a map on
W and
let
TO
E
W. Then
the Lyapunov exponent X o (f)
97
227
How Can We Observe and Describe Chaos?
for the orbit 0 = {f" ( s o ) ; n = 0,1,2,. . .} is defined by
(2) Let f = (fi,f2,. matrix Jn = D f
. . , fm) be (TO)
at
a map on R" and let s o E R". defined by
The Jacobi
T O is
Then, the Lyapunov exponent A 0 ( f ) of f for the orbit
0
= {f" (so) ; n = 0,1,2,. . .}
is defined by A0
(f) =
logfi1,
fik =
n 1/n lim (&)
n+m
,
k = I,. . . ,m .
Here, p i is the k-th largest square root of the m eigenvalues of the matrix J,' Jn ,
> 0 A,(f) A 0 (f) 5 0
+ +
orbit 0 is chaotic. orbit 0 is stable.
The properties of the logistic map depend on the parameter a. If we take a particular constant a , for example, a = 3.71, then the Lyapunov exponent and the entropic chaos degree are positive, the trajectory is very sensitive to the initial value and one has the chaotic behaviour. From the above example and some other maps (see [ll]),Lyapunov exponent and the entropic chaos degree have clear correspondence, but the ECD can resolve some inconvenient properties of the Lyapunov exponent as follows: (1) Lyapunov exponent can take negative values and sometimes even -m, but the ECD is always positive for any a 2 0. (2) It is difficult to compute the Lyapunov exponent for some maps like the Tinkerbell map f because it is difficult to compute f" for large n. On the other hand, the ECD of f is easily computed. (3) Generally, the algorithm for calculating the ECD is much easier than that for the Lyapunov exponent. 4.
New Description of Chaos
First of all, we have to examine carefully when a certain dynamics produces chaos. Let us take the logistic map as an example. The original differential equation of the logistic map is -dx_ - ax(1-s), OIa54 (2) dt
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A. Kossakowski, M. Ohya, and Y . Togawa
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.7
0.6 [
3
3.2
3.4
3.6
3.8
4
a
Fig. 2: Chaos degree for logistic map
109
3
3.2
3.4
3.6
3.8
4
Fig. 3: Lyapunov exponent for logistic map with initial value 20 in [0,1].This equation can be easily solved analytically, whose solution (orbit) does not have any chaotic behaviour. However, once we convert it into a discrete equation such as %+I
= u2,(1-2,),
o
(3)
this difference equation produces chaos. Taking the time discrete is necessary not only for observing chaos but also for visualising orbits drawn by the dynamics. Similarly as in quantum mechanics, it is not possible for a human being to understand any object without observing it, for which it will not be possible to trace an orbit continuously in time. Now let us think about finite partition A = {Ak ; Ic = 1,. . . ,N } of a proper set c B and a equi-partition Be = {Q; k = 1,.. . , N } of I . Here “equi” I E [u,b]“ means that all elements Bg are identical. We denote the set of all partitions by P
99
229
How Can We Observe and Describe Chaos?
and the set of all equi-partitions by Pe.In Sect. 3, we specify a special partition, in particular, an equi-partition for computer experiment calculating the ECD. Such a partition enables us to observe the orbit of a given dynamics, and moreover it provides a criterion for observing chaos. There exist several reports saying that one can observe chaos in nature, which are very much related to how one observes the phenomena, for instance, scale, direction, aspect. It has been difficult to find a satisfactory theory (mathematics) to explain such chaotic phenomena. In the difference equation we can take some time interval r between n and n 1. If we take r + 0, then we have a completely different dynamics. If we take coarse graining of the orbit zt for time interval 7,
+
we again have a very different dynamics. Moreover it is important for mathematical consistency to take the limits n + 03 or N (the number of equi-partitions)+ 00, i.e., making the partition finer and finer, and to consider the limits of some quantities describing chaos, so that mathematical operations such as “lim”, “sup”, “inf” are used very often in definitions of such quantities. In this paper we take the opposite position, that is, any observation will be unrelated or even in contradiction with such limits. Detection of chaos is a result of assuming suitable scales of, for example, time, distance or domain, and it will not be possible in the limiting cases.
W e claim in this paper that most of chaotic phenomena are scale-dependent, so the definition of a degree measuring chaos should depend o n certain scales taken. Taking this view into consideration we modify the definitions of the chaos degree given in the previous sections as follows. We go back to the triple (A,6,a (G)) considered in Sect. 2 and we use this triple both for an input and for an output systems. Let the dynamics be described by the mapping I’t from 6 to 6 with a parameter t E G and let an observation be described by the mapping 0 from (A,6,a: (G)) to the triple ( B , 2 ,p (G)). The 2,p (G)) might be the same as the original one or the one of its subsystem triple (B, and the observation map 0 may contains several different types of observations, that is, it can be decomposed as 0 = 0,. . . 01. Let us list some examples of observations. For a given dynamics dcp/dt = F (pt) , equivalently, cpt = l?;cp, one can take several observations. EXAMPLE 1 Time Scaling (Discretizing): 0, : t + n, d p / d t ( t )-+ pn+l,so that dcpjdt = F (cpt) +- yn+1 = F (cpt) and cpt = I’fp +- cpn = rzcp. Here r is the unit of time needed for the observation. EXAMPLE 2 Size Scaling (Conditional Expectation, Partition): Let (B,2, fl (G)) be a subsystem of (A,6,a ( G ) ) ,both of which have a certain algebraic structure such as C*-algebra or von Neumann algebra. As an example, the subsystem (13,2,p (G)) has abelian structure describing macroscopic world which is
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A . Kossakowski, M. Ohya, and Y . Togawa
a subsystem of a non-abelian (non-comniutative) system (A,6 ,a (G)) describing micro-world. A mapping O c preserving norm (when it is properly defined) from A to I3 is, in some cases, called a conditional expectation. A typical example of this conditional expectation is according to a projection valued measure {Pk; PkPj = Pk6kj = P,*6kj 2 0 , c k Pk = I } associated with a quantum measurement (von Neumann measurement) such that O c ( p ) = x k PkpPk for any quantum state (density operator) p . When 13 is a von Neumann algebra generated by {Pk} , it is an abelian algebra isometrically isomorphic to L" (a) with a certain Hausdorff space 0, so that in this case O c sends a general state cp to a probrtr bility measure (or distribution) p . Similar example of O c is one coming from a certain representation (selection) of state such as e.g. Schatten decomposition of p ; p = 8 ~ = px k XkEk by one-dimensional orthgonal projections {Ek} associated to the eigenvdues of p with x k Ek = I . Another important example of size scaling is due to a finite partition of an underlining space Cl, e.g., space of orbit, defined as N
op(n)=
{Pk; P k n P j = P k 6 k j ,
k , j = 1 , ... N ,
uPk=.>. k=l
We go back to the discussion of the entropic chaos degree. Starting from a given dynamics cpt = I',"cp, it becomes qn = I'iq after handling the operation 0,. Then by taking proper combinations O of the size scaling operations like Oc, OR and O p , the equation cpn = rgcp changes to 0 (9,)= O (I'Ecp), which will be written by Ocp, = Ol?~O-lOcpor cp: = I'Eocpo. Then our entropic chaos degree is redifined as follows: DEFINITION 4.1 The entropic chaos degree of I" with an initial state cp and observation 0 is defined by
Do (9; r*) where is the measure induced by the operation (3 when applied to an extremd decomposition measure of cp selected by the observation (3 (its part OR). DEFINITION 4.2 The entropic chaos degree of I" with an initial state qi is defined by D (cp; I?') = inf { D o (9;I?*) ; O E SO} where S O is a proper set of observations natually determined by a given dynamics. Then one judges whether a given dynamics is chaotic or not in the following way. DEFINITION 4.3 (1) The dynamics I'* is chaotic for an initial state cp in an observation 0 if and only if D" (cp;??') > 0. (2) r' is totally chaotic for an initial state cp if and only if D (cp; r")> 0.
101
231
How Can We Observe and Describe Chaos?
In Definition 4.2 S O is determined by a given dynamics and some conditions attached to the dynamics, for instance, if we start froin a difference equation with a special representation of an initial state, then S O excludes 0, and OR. The idea introduced in this paper to understand chaos can be applied not only to the entropic chaos degree but also to some other degrees such as dynamical entropy, whose applications and comparison with several other chaos indicators will be discussed in a forthcoming paper. In the case of logistic map, z,+1 = az,(l - z), = F (2,) , we obtain this difference equation by taking the observation 0, and an observation OP by equipartition of the orbit space R = {z,} so as to define a sta,te (probability distribution). Thus we can compute the entropic chaos degree as discussed in Sect. 3. It is important to notice here that the chaos degree does depend on the choice of observations. As an example, we consider the circle map 0,+1
=
f,(e,)
= &+w
(modb),
(4)
where w = 27rv(O < v < 1). If v is a rational number N / M , then the orbit {On} is periodic with the period M . If v is irrational, then the orbit (0,) densely fills the unit circle for any initial value 00; namely, it is a quasiperiodic motion. We proved in [lo] the following theorem. THEOREM 1 Let I = [ 0 , 2 ~be ] partitioned into L disjoint components with equal length; I = Bln Ba n... n BL.
(I) If v i s a rational number equal t o N / M , then the finite equi-partition P = {Bk;k = 1,.. . ,M } implies Do (00; fv) = 0. (2) If v is irrational, then Do (00; fv) > 0 f o r any finite partition P = { B k } . Note that our entropic chaos degree indicates chaos for the quasiperiodic circle dynamics by the observation according t o a partition of the orbit, which is different from the usual understanding of cham. However, the usual belief that quasiperiodic circle dynamics will not cause chaos is not at all obvious, but is realized in a special limiting case as shown in the following proposition. PROPOSITION 1 For the above circle map, i f v i s irrational, then D (00; fv) = 0.
Proof. Let us take an equipartition P = { B k } as
k = 1,2,. . .
~
where 1 is a certain integer and Bk+l = Bk. When u is irrational, put vo = [b] with Gaussian [.I. Then fu(Bk)intersects only two intervds B k + v o and Bk+,,o+l, so that we can denote by (1-s) : s the ratio of the Lebesgue measure of f,(Bk)nBk+,, and that of fv(Bk)n Bk+vo+l. This s is equal to lv - [Iv]and the entropic chaos degree becomes D p = -slogs - (1 - s ) log (1- S ) .
232
A. Kossakowski, M. Ohya, and Y. Togawa
Take the continued fraction expansion of u and denote its j-th approximate by b j l c j . Then we have bj 1 u-5 -. cj c? I
I
I
For the above equi-partition B = { B k } with 1 = c j , we find
and bj
when
u-
bj-1
when
u-
[lu] =
5 >0 Ci Ci
< 0.
It implies
which goes to 0 as j
-+ 03.
Hence D = inf { D p ;P } = 0.
0
Such a limiting case will not appear in real observation of natural objects, so we claim that chaos is a phenomenon depending on observations, which results in the definition of chaos as above. In the forthcoming paper [24], we will discuss how obtain chaotic dynamics starting from general differential dynamics in both classical and quantum systems. That is, it is demonstrated how we can get chaotic dynamics by considering observations introduced in this paper, and we calculate the entropic chaos degrees in each case.
Acknowledgment The authors thank JSPS and SCAT for financial support.
Bibliography [l] L. Accardi, Ad. Ohya, and N. Watanabe, Dynamical entropy through quantum Markov chain, Open Sys. Information Dyn. 4, 71 (1997).
[Z] K . T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos-An Introduction to Dynamical Systems,
Textbooks in Mathematical Sciences, Springer, 1996.
[3] R. Alicki and M. Fannes, Defining quantum dynamical entropy, Lett. Math. Physics 32, 75 (1994). [4] F . Benatti, Deterministic Chaos in Infinite Quantum Systems, Springer, 1993. [5] R. L. Devaney, An Introduction to Chaotic dynamical Systems, Benjamin, 1986. [6] G. Casati and B. Chirikov, Quantum Chaos: Between Order and Disorder, Cambridge University Press, 1995.
103 How Can We Observe and Describe Chaos?
233
[7] G. G. Emch, H. Narnhofer, W. Thirring, and G. L. Sewell, Anosou actions on noncommutatiue algebras, J. Math. Phys. 35,No. 11, 5582 (1994). IS] H. Hasegawa, Dynamical formulation of quantum level statistics, Open Sys. Information Dyn. 4,359 (1997). [9] R. S. Ingarden, A. Kossakowski, and M. Ohya, Information Dynamics and Open Systems, Kluwer Academic Publishers, 1997. [lo] K. Inoue, M. Ohya, and A. Kossakowski, A Description of Quantum Chaos, preprint, 2002. [ll] K. Inoue, M. Ohya, and K. Sato, Application of chaos degree to some dynamical systems, Chaos, Solitons & Fractals 11, 1377-1385 (2000). [12] I(. Inoue, M. Ohya, and V. Volovich, Semiclassical properties and chaos degree f o r the quantum baker's map, Journal of Mathematical Physics 43,734 (2002). [13] K. Inoue, M. Ohya, and I. V. Volovich, O n quantum-classical correspondence for baker's map, quant-ph/0108107. [14] A. Kossakowski, M. Ohya, and N. Watanabe, Quantum dynamical entropy f o r completely positive maps, Infinite Dimensional Analysis, Quantum Probability and Related Topics 2, 267 (1999). [15] W . A. Majewski, Does quantum chaos exitst? A quantum Lyapunou exponents approach, quant-ph/9805068. I161 N. Muraki and M. Ohya, Entropy functionals of Kolmogorou Sinai type and their limit theorems, Lett. Math. Phys. 36,327 (1996). [17] M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys. 27,19 (1989). [18] M. Ohya, Information dynamics and its applications to optical communication processes, Lecture Note in Physics 378,81 (1991). [19] M. Ohya, State change, complexity and fractal in quantum systems, Quantum Communications and Measurement, Plenum Press, New York, 1995, pp. 30+320. [20] M. Ohya, Complexity and fractal dimensions f o r quantum states, Open Sys. Information Dyn. 4,141 (1997). [21] M. Ohya, Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics 37,495 (1998). [22] M. Ohya, Complexity an Quantum System and its Application to Brain Function in: Quantum I n f o m a t i o n II, T. Hida and K. Saito, ed., World Scientific, 2000. [23] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer-Verlag, TMP, (1993). [24] A. Kossakowski, M. Ohya and Y. Togawa, in preparation.
104 Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 6, NO. 2 (2003) 301-310 @ World Scientific Publishing Company
World Scientific www.worldscientific.com
ON QUANTUM CAPACITY A N D ITS BOUND
MASANORI OHYA
Department of Information Sciences, Science University of Tokyo, 278-8510, Noda City, Chiba, Japan o h y a o k n o d a . tw.ac.jp IGOR V. VOLOVICH Steklov Mathematical Institute, Gubkan St. 8, GSP-1 11 7966, Moscow, Russia volovichQmi. m s . ru Received 3 December 2002 The quantum capacity of a pure quantum channel and that of classical-quantum-classical channel are discussed in detail based on the fully quantum mechanical mutual entropy. It is proved that the quantum capacity generalizes the so-called Holevo bound.
Keywords: Quantum mutual entropy; quantum capacity; Holevo bound. AMS Subject Classification: 94A15, 81P99
1. Introduction Measure theoretic formulation of the mutual entropy (information) in classical systems was done by Kolmogorov' and Gelfand, Yaglom,2 which enabled one to define the capacity of information channel. In quantum systems, there have been several definitions of the mutual entropy for classical input and quantum In 1983, Ohya defined6 the fully quantum mechanical mutual entropy, i.e. for quantum input and quantum output, by means of the relative entropy of Umegaki,7 and he extended it8 to general quantum systems by using the relative entropy of Arakig and Uhlmann.l0 In this short note, we prove that the quantum capacity'' of a quantum channel derived from the fully quantum mechanical mutual entropy generalizes the so-called Holevo bound. 2. Mutual Entropy The quantum mutual entropy was introduced in Ref. 6 for a quantum input and quantum output, namely, for a purely quantum channel, and it was generalized for a general quantum system described by C*-algebraic terminology.8 We briefly 301
105 302
M. Ohya
€9 I. V.
Volovich
review the mutual entropy in the usual quantum system described by a Hilbert space. Let 3-1 be a Hilbert space for an input space, B(3t) be the set of all bounded linear operators on 3-1 and S(3-1)be the set of all density operators on 3-1. An output space is described by another Hilbert space ‘I? but , often 3t = %. A channel from the input system to the output system is a mapping A* from S(3t) to S(%).12 A channel A* is said to be completely positive if the dual map A satisfies the following condition: E l j = , AiA(B;Bj)Aj 2 0 for any n E N and any Aj E I?(%), Bk
E B(&).
An input state p E S(3t) is sent to the output system through a channel A*, so that the output state is written as jj = A*p. Then it is important to ask how much information of p is sent t o the output state A*p. This amount of information transmitted from input to output is expressed by the quantum mutual entropy. The quantum mutual entropy was introduced on the basis of the von Neumann entropy ( S ( p ) e -tr plogp) for purely quantum communication processes. The mutual entropy depends on an input state p and a channel A*, so it is denoted by A*), which should satisfy the following conditions: The quantum mutual entropy is well-matched to the von Neumann entropy. Furthermore, if a channel is trivial, i.e. A* = identity map, then the mutual entropy equals to the von Neumann entropy: I ( p ;id) = S ( p ) . When the system is classical, the quantum mutual entropy reduces to classical one. Shannon’s fundamental inequality13 0 5 I ( p ;A*) 5 S ( p ) is held. To define such a quantum mutual entropy extending Shannon7sand GelfandYaglom’s classical mutual entropy, we need the quantum relative entropy and the joint state (it is called “compound state” in the sequel) describing the correlation between an input state p and the output state A * p through a channel A*. A finite partition of measurable space in classical case corresponds to an orthogonal decomposition { E k } of the identity operator I of 3t in quantum case because the set of all orthogonal projections is considered to make an event system in a quantum system. It is known14 that the following equality holds:
and the supremum is attained when { E k } is a Schatten decomp~sition’~ of p. Therefore the Schatten decomposition is used to define the compound state and the quantum mutual entropy following the formulation of the classical mutual entropy by Kolmogorov, Gelfand and Yaglom.2
106 On Quantum Capacity and its Bound
303
The compound state CTE(corresponding t o joint state in CS) of p and A*p was introduced in Refs. 6 and 16, which is given by OE
=
AkEk
(2.1)
@A*Ek,
k
where E stands for a Schatten decomposition { E k } of p, so that the compound state depends on how we decompose the state p into basic states (elementary events), in other words, how to observe the input state. The relative entropy for two states p and CT is defined by Umegaki7 and Lindblad,17 which is written as when EiT@
cEDB,
otherwise.
(2.2)
Then we can define the mutual entropy by means of the compound state and the relative entropy,6 i.e.
where the supremum is taken over all Schatten decompositions because this decomposition is not unique unless every eigenvalue is not degenerated. Some computations reduce it to the following form6:
This mutual entropy satisfies all conditions (i)-(iii) mentioned above. It is important to note here that the Schatten decomposition of p is unique when the input system is classical. That is, when an input state p is given by a probability distribution or a probability measure. For the case of probability distribution; p = { A k } , the Schatten decomposition is uniquely given by
k
where
6k
is the delta measure;
Therefore for any channel A*, the mutual entropy becomes
I ( p ;A*)
=
AkS(A*dk,
A*p)7
(2.7)
k
which equals to the following usual expression of Shannon when the minus is welldefined:
304 M. Ohya €4 I . V. Volovzch
The above equality has been taken as the definition of the mutual entropy for a classical-quantum ~ h a n n e l . ~ - ~ Note that the definition (2.3) of the mutual entropy is written as
where F,(p) is the set of all orthogonal finite decompositions of p. The proof of the above equality is given in Ref. 18 by means of the fundamental properties of the quantum relative entropy.
3. Communication Processes
We discuss communication processes in this ~ e c t i o n .Let ~?~ A ~= {a l, a 2 ,. . . ,a,} be a set of certain alphabets and R be the infinite direct product of A: R = AZ = n T m A calling a message space. In order to send an information written by an element of this message space to a receiver, we often need to transfer the message into a proper form for a communication channel. This change of a message is called a coding. In other words, a coding is a measurable one-to-one map E from R to a proper space X . Let (0,FQ, P ( Q ) )be an input probability space and X be the coded input space. This space X may be a classical object or a quantum object. For instance, X is a Hilbert space 7-l of a quantum system, then the coded input system is described by (B(7-lLS(7-l)). An output system is similarly described as the input system: The coded output space is denoted by X and the decoded output space is fi made by another alphabets. A transmission (map) from X to X is described by a channel reflecting all properties of a physical device, which is denoted by y here. With a decoding the whole information transmission process is written as
i,
RE X r , X L . f i .
(3.9)
That is, a message w E R is coded to [ ( w ) and it is sent to the output system through a channel y, then the output coded message becomes y o [(w)and it is decoded to o y o [ ( w ) at a receiver. This transmission process is mathematically set as follows: M messages are sent to a receiver and the lcth message d k )occurs with the probability A k . Then the occurrence probability of each message in the sequence (&I, ~ ( ~ 1. ., ,.u ( ~ of ) )M messages is denoted by p = {Ak}, which is a state in a classical system. If [ is a classical coding, then [ ( w ) is a classical object such as an electric pulse. If 6 is a quantum coding, then E(w) is a quantum object (state) such as a coherent state. Here we consider such a quantum coding, so that [( w ( ' ) ) is a quantum state, and we denote [(w('))) by o k . Thus the coded state for the sequence (d), ~ ( ' 1 , . . . ,dM))
108 On Quantum Capacity and its Bound
305
is written as C7=xXk(Tk.
(3.10)
k
This state is transmitted through a channel y. This channel is expressed by a completely positive mapping r*,in the sense of Sec. 1,from the state space of X to that of X ,hence the output coded quantum state 5 is r*n. Since the information transmission process can be understood as a process of state (probability) change, when R and fl are classical and X and X are quantum, the process (3.9) is written as
P(R) - 5 S (X)A S ( ii)5 P (fl) ,
(3.11)
where Z* (resp. g*)is the channel corresponding to the coding E (resp. () and S ( X ) (resp. S ( f i ) )is the set of all density operators (states) 011 X (resp. %). We have to be careful in studying the objects in the above transmission process (3.9) or (3.11). Namely, we have to make clear which object is going to be studied. For instance, if we want to know the information capacity of a quantum channel y(= I'*), then we have to take X so as to describe a quantum system like a Hilbert space and we need to start the study from a quantum state in quantum space X not from a classical state associated to a message. If we like to know the capacity of the whole process including a coding and a decoding, which means the capacity of a channel [ o y o E ( = g*or*oE*), then we have to start from a classical state. In any case, when we are concerned with the capacity of channel, we only have to take the supremum of the mutual entropy I ( p ;A*) over a quantum or classical state p in a proper set determined by what we like to study with a channel A*. We explain this more precisely in the next section. 4. Channel Capacity
We discuss two types of channel capacity in communication processes, namely, the capacity of a quantum channel I?* and that of a classical (classical-quantumclassical) channel E*o r*o E*. (1) Capacity of quantum channel: The capacity of a quantum channel is the ability of information transmission of a quantum channel itself, so that it does not depend on how to code a message being treated as a classical object and we have to start from an arbitrary quantum state and find the supremum of the mutual entropy. One often makes a mistake at this point. For example, one starts from the coding of a message and compute the supremum of the mutual entropy and he says that the supremum is the capacity of a quantum channel, which is not correct. Even when his coding is a quantum coding and he sends the coded message to a receiver through a quantum channel, if he starts from a classical state, then his capacity is not the capacity of the quantum channel itself. In his case, usual Shannon's theory is applied because he can easily compute the conditional distribution by the usual (classical) way. His supremum is the capacity of a classical-quantum-classical channel, and it is in the second category discussed below.
109 306
M. Ohya & I. V . Volovich
Let SO(cS(3t)) be the set of all states prepared for expression of information. Then the capacity of the channel I?* with respect to SOis defined as:
Definition 1. The capacity of a quantum channel r*is
cso(r*)= SUP{I(~; r*);p E s o } .
(4.12)
Here I ( p ; r * )is the mutual entropy given in (2.3) or (2.4) with A* = r*. When SO= S(%), Cs(Rfl)(r*) is denoted by C(r*) for simplicity. In Refs. 8, 19 and 18, we also considered the pseudo-quantum capacity C,(r*) defined by (4.12) with the pseudo-mutual entropy I P ( p ; r * )where the supremum is taken over all finite decompositions instead of all orthogonal pure decompositions:
(4.13) However the pseudo-mutual entropy is not well-matched to the conditions explained in Sec. 2, and it is difficult to compute numerically.20From the monotonicity of the mutual entropy,I* we have
(2) Capacity of classical-quantum-classical channel: The capacity of C-Q-C channel E*o r * o Z* is the capacity of the information transmission process starting from the coding of messages, therefore it can be considered as the capacity including a coding (and a decoding). As is discussed in Sec. 3, an input state p is the probability distribution { X k } of messages, and its Schatten decomposition is unique as (2.5), so the mutual entropy is written by (2.7):
I ( p ;g*0 r*0 z*)=
XkS(s*
0
r*
0
2 * 6 k , 2. A*
0
r* z*p). 0
(4.14)
k
If the coding Z* is a quantum coding, then 2 * 6 k is expressed by a quantum state. Let us denote the coded quantum state by f J k and put = E*p = E k x k f J k . We denote the set of such quantum codings by C. Then the above mutual entropy becomes I ( p ; g*0 I'*0 =*)
=
XkS(s*
0
r*gk,
g* r*g). 0
(4.15)
k
This is the expression of the mutual entropy of the whole information transmission process starting from a coding of classical messages. Hence the capacity of C-Q-C channel is as follows:
Definition 2. The capacity of C-Q-C channel is
cpy? r*o z*)= SUP{I(P; E* or*o e*);p E p0},
(4.16)
110
O n Quantum Capacity and its Bound
307
where PO(CP ( R ) ) is the set of all probability distributions prepared for input (u priori) states (distributions or probability measures). Moreover the capacity for coding free in C is found by taking the supremum of the mutual entropy (4.15) over all probability distributions in PO and all codings in C:
c,'(S* r*)=
{ q p ; S* r* z*); E p0,z* E C } . 0
(4.17)
There are several ways to decode quantum states such as quantum measurements, so that such decodings and denoted by V.The capacity for decoding free in D is c pd o (
r* z*)= sup { q p ; S* r* z*);p E p0,e*€2)). 0
(4.18)
The last capacity is for both coding and decoding free and it is given by
-* (4.19) cz(r*) = sup { q p ; S* r* s*);p E p0, E c, e*E V } . These capacities Cp , C p , C z do not measure the ability of the quantum channel r*itself, but measure the ability of I?*through the coding and decoding. The above 0
0
three capacities Cpo, C?, C z satisfy the following inequalities:
o 5 cpo(G* o r *z*)5 c?(Z*or*) ,
c,'o(r*z*)5 cz(r*)5 ~ ~ p { s ( p )E; pp 0 } , where S ( p ) is not the von Neumann entropy but the Shannon entropy: Remark that if X k S ( r * O k ) is finite, then (4.15) becomes
ck
qp;S* r* z*)= s(S* r*a)- C A&*
o r*Ok).
XI, log X k .
(4.20)
k
Further, if p is a probability measure having a density function f(X) and each X corresponds to a quantum coded state .(A), then o = f(X)o(X)dX and
qp;e* r* z*)= S(S* 0 r*a)- f(x) s( ~0 *r*+))dX,
(4.21)
which is less than (4.22) This bound is computed in several following inequality
case^.^^^^^
This bound is a special one of the
qp;G*or* =*) 5 qp;r* E*), which comes from the monotonicity of the relative entropy. When the decoding is not taken into account, we only have to consider the mutual entropy I ( p ; r*o E*) above.
111 308
M . Ohva €9 I. V . Volovich
Let us define an extension of the functional of the relative entropy. If A and B are two positive Hermitian operators (not necassarily the states, i.e. not necessarily with unit traces) then we set
S (A ,B ) = tr A(1og A - log B ) There is the following Bogoliubov inequality. S(A, B ) 2 tr A(1og tr A - log t r B ) . The following theorem gives us the bound of the mutual entropy I ( p ; I?* o 3.).
Theorem 1. For a probability distribution p = {A,) and a quantum coded state a = E*p X k a k , Xk 2 0, c k Xk = 1, one has the following inequality for a n y quantum channel decomposed as F* = r; 0 rz such that I';a = EiaEi by a projection valued measure { Ei}:
xi
k
k r
1
(4.23)
ck
Proof. The equality I ( p ;r*o E*)= XkS(r*Ok,r*a)is the case of the equality (15), and the first inequality comes from the monotonicity of the relative entropy. Furthermore, by applying again the monotonicity of the relative entropy, we have
k
Here the second inequality is due to the Bogoliubov inequality.
0
In the case that the channel r; is trivial; rfa = a, the above inequality reduces to the bound obtained by Holevo3:
112 On Quantum Capacity and its Bound X k S ( a k , 0)=
-tr
C 7
+
309
XI, tr Uk log
log fJ
k
k
> I ( ~r* ;
-
2
z*)= I ( ~r;; *;I
[-tr(OEi) log tr(gEi)
+
1
XI, tr(&?i) log tr(ffk&) Ic
Remark that the right-hand side in the inequality is sometimes called the accessible information.
Using the above upper and lower bounds of the mutual entropy, we can compute these bounds of the capacity in many different cases.
Acknowledgments The authors thank SCAT for finacial support of this work.
References 1. A. N. Kolmogorov, Theory of transmission of information, Amer. Math. SOC. Trans. Ser. 2 33 (1963) 291-321. 2. I. M. Gelfand and A. M. Yaglom, Calculation of the amount of information about a random function contained in another such function, Amer. Math. SOC. Trans. 12 (1959) 199-246. 3. A. S. Holevo, Some estimates f o r the amount of information transmittable by a quantum communication channel (in Russian), Prob. Pered. Infor. 9 (1973) 3-11. 4. R.S. Ingarden, Quantum information theory, Rep. Math. Phys. 10 (1976) 43-73. 5. L. B. Levitin, Physical information theory for 90 years: basic concepts and results, Springer Lect. Note in Phys., Vol. 978 (Springer, 1991) pp. 101-110. 6. M. Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Inf. Theory 29 (1983) pp. 770-777. 7. H. Umegaki, Conditional expectations in an operator algebra I V (entropy and information), Kodai Math. Sem. Rep. 14 (1962) 59-85. 8. M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys. 27 (1989) 19-47. 9. H. Araki, Relative entropy f o r states of von Neumann algebras, Publ. RIMS Kyoto Univ. 11 (1976) 809-833. 10. A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory, Commun. Math. Phys. 54 (1977) 21-32. 11. M. Ohya, Fundamentals of quantum mutual entropy and capacity, Open Systems Infor. Dyn. 6 (1999) 69-78. 12. M. Ohya, Quantum ergodic channels in operator algebras, J . Math. Anal. Appl. 84 (1981) 318-327. 13. C. E. Shannon, Mathematical theory of communication, Bell System Tech. J. 27 (1948) 379-423. 14. M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, 1993). 15. R. Schatten, Norm Ideals of Completely Continuous Operators (Springer, 1970). 16. M. Ohya, Note on quantum probability, L. Nuovo Cimento 38 (1983) 402-406. 17. G. Lindblad, Entropy, Information and quantum measurements, Commun. Math. Phys. 33 (1973) 111-119.
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18. N. Muraki, M. Ohya and D. Petz, Note on entropy of general quantum systems, Open Systems Infor. Dyn., 1, (1992) 43-56. 19. M. Ohya, D. Petz and N. Watanabe, On capacities of quantum channels, Probab. Math. Statist. 17 (1997) 179-196. 20. M. Ohya, D. Petz and N. Watanabe, Numerical computation of quantum capacity, h t . J . Theor. Phys. 38 (1998) 507-510. 21. H. P. Yuen and M. Ozawa, Ultimate information carrying limit of quantum systems, Phys. Rev. Lett. 70 (1993) 363-366.
THE ROYAL SOCIETY
10.1098/rspa.2001.0867
Entanglement, quantum entropy and mutual information BY VIACHESLAV P. B E L A V K I NA~N D M A S A N O ROI H Y A ~ Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, UK Department of Information Sciences, Science University of Tokyo, 278 Noda City, Chiba, Japan Received 18 M a y 2001; accepted 24 M a y 2001; published o n l i n e 29 November 2001
The operational structure of quantum couplings and entanglements is studied and classified for semi-finite von Neumann algebras. We show that the classicalquntum correspondences, such as quantum encodings, can be treated as diagonal semi-classical (d-) couplings, and the entanglements, characterized by truly quantum (4-) couplings, can be regarded as truly quantum encodings. The relative entropy of the d-compound and entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the maximum of mutual information over all d-entanglements, and the dimensional entropy, which is achieved at the standard entanglement (true quantum entanglement) coinciding with a d-entanglement only in the case of pure marginal states. The d- and q-information of a quantum noisy channel axe, respectively, defined via the input d- and q-encodings, and the q-capacity of a quantum noiseless channel is found to be the logarithm of the dimensionality of the input algebra. The quantum capacity may double the classical capacity, achieved as the supremum over all d-couplings (or encodings) bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra. Keywords: entanglements; compound states; quantum entropy and information
1. Introduction The entanglements, specifically quantum correlations first considered by Schradinger (1935), aze now used t o study quantum information processes, in particular, quantum computations, quantum teleportation and quantum cryptography (Bennett et al. 1993; Ekert 1993; Jozsa & Schumacher 1994). There have been mathematical studies of the entanglements in Werner (1989, 1998), Bennett et aE. (1996) and Schumacher (1993a,b), in which the entangled state of two quantum systems is d e h e d a s a compound state which is a convex combination C, p, 8 cn,p(n) with some states en and q,, on the corresponding algebras A and B. However, it is obvious that there exist several types of correlated states, written as ‘separable’ forms above. Such correlated, or classically entangled, states have also been discussed in several contexts in quantum probability, such as quantum measurement and filtering (Belavkin 1980, 1994), quantum compound states (Ohya 1983a,b) and lifting (Accardi & Ohya 2002). Proc. R. Soc. Lond. A (2002) 458, 209-231
@ 2002 The Royal Society
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In this paper, we study the mathematical structure of classical-quantum and quantum-quantum couplings to provide a finer classification of quantum separable and entangled states. We also discuss the informational degree of entanglement and entangled quantum mutual entropy and quantum capacity. The latter are treated here solely as quantities arising in certain maximization problems for quantum mutual information which is generalized here for arbitrary semi-finite algebras. The term entanglement was introduced by Schrodinger in 1935 out of the need to describe correlations of quantum states not captured by mere classical, statistical correlations which are always the convex combinations of non-correlated states. In this spirit, the by now standard definition (Werner 1989) of the entanglement in physics is the state of a compound quantum system ‘which cannot be prepared by two separated devices with only correlated classical data as their inputs’. We show that the entangled states can be achieved by quantum (9-) encodings, and that the non-separable couplings of states, in the same way as the separable states, can be achieved by classical (c-) encodings. The compound states, called o-coupled, are defined by orthogonal decompositions of their marginal states. This is a particular case of a so-called diagonal (d-compound) state of a compound system which is achieved by the classical-quantum correspondences, called encodings. The d-compound states, as convex combinations of the special product states, are most informative among c-compound states, in the sense that the maximum of the mutual entropy over all c-couplings of probe systems A to the quantum system B, with a given normal state c, is achieved on the extreme dcoupled (even o-coupled) states. This maximum is the von eumann entropy, which is bound by the rank capacity LnrankB, the supremum of (q) over all q. The rank rankB of the algebra B is a topological characteristic of B defined as the dimensionality of the maximal Abelian subalgebra A C 2 3 (in the case of the simple B it coincides with the dimensionality d i m z of the Hilbert space ;Ft of representation for B). The von Neum capacity defined as the maximal von Neumann entropy, i.e. as the maximum InrankB of mutual entropy over all c-couplings of the classical probe systems A to the quantum system B, is finite only if rank23 < 00. Due to dimB < (rankB)2 (the equality is only for the simple algebras B), it is achieved on the normal tracial density operator ff = (rankB)-II only in the case of hite-dimensional B. We prove that the truly entangled compound states are most informative, in the sense that the maximum of the mutual entropy over all couplings including entanglements of the quantum probe systems A to the quantum system B is achieved on a non-separable q-compound skate. It is given by the standard entanglement, an extreme entanglement of A = B with the marginal state p = f, where (g, is r). The maximal information gained the transposed (time-inversed) system to (23, or such extreme q-compound states defines another ype of entropy, the q-entropy ‘(51, which is bigger than t h von Neumann entropy ‘(q) in the case of mixed q. The maximum of the q-entropy (c) over all states c defines the dimensional capacity In dim B. The dimensionality dim B of the algebra B is the major topological characteristic of 23,and it gives true quantum capacity of I3 achieved at the standard entangleme t with the maximal chaotic c. Thus, the true quantum capacity is the maximum q‘ = 1ndimB of the mutual entropy over all, not only classical-quantum, couplings of the probe systems A to the quantum system B, and it is finite only
Y
r=
c)
a
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H for the finite-dimensional algebra t3. The q-entropy (c), also called the dimensional entropy, can b considered as the true quantum entropy, in contrast to the von Neumann entropy 3 (q), called also rank-entropy, or c-entropy (semi-classical entropy) as the supremum of p u t u a l entropy ov couplings with only classical probe systems A. The capacity coincides with only in the class'cal case of the Abelian B, and it is strictly larger than the semi-classical capacity - lnrank B for any noiselndim B is achieved as less quantum channel. We shall show that the capacity the supremum of the quantum Shannon information for th noiseless channel over the entanglements as q-encodings similar to the capacity ' c , which is achieved as the supremum over c-encodings described by the classical-quantum correspondences
[
'
cql
A -+ B. In this paper we consider the case of semi-hite quantum systems which are described by the von Neumann algebras A and B with normal, faithful semi-finite trace. Such quantum systems include all simple quantum systems described by full operator algebras as well as all classical systems as the commutative case. The particular cases of simple and discrete decomposable algebras are considered in Belavkin & Ohya (1998, 2000). 2. Pairings, couplings and entanglements In this section, we give mathematical characterizations of entanglement in terms of quantum coupling, which is described in terms of transpose-completely positive operations extending individual states to a compound state of a composed quantum system. We show how any normal compound state can be achieved in this way, m d introduce the standard entanglement as an operation giving rise to the standard entangled compound state. Let 7-l denote the Hilbert space of a quantum system, and B = C(R)be the algebra of all linear bounded operators on X.Note that B consists of all operators A : 7-l 4 'FI having the adjoints At on X.A linear functional c on I3 with complexvalues r ( B ) € C is called a state on B if it is positive (i.e. c ( B ) > 0 for any positive operator B = A t A in 23) and normalized (i.e. r(1) = 1 for the identity operator I in A). A normal state can be expressed as
<(B) = tr X + B H = (B,(T), B E B ,
(2.1)
where H is a linear Hilbert-Schmidt operator from 3-1 to (another) Hilbert space 4 , and xt is the adjoint operator from 4 to 'H. Here, tr stands for the usual trace in 4 (in cases of ambiguity it will also be denoted as trg). This K is called the amplitude operator, which can always be considered on 9 = 'FI as the square root of the operator xxt (it is called simply the amplitude if G is the one-dimensional s p x e C ,x = q E 3-t with Z ~ K = [1q1I2= 1, in which case xt is the functional qt from 7-l to C). We can always equip 3-1 (and will equip all auxiliary Hilbert spaces, e.g. 4 ) with an isometric involution J = J t , J2 = I having the properties of complex conjugation
VAj E c, and denote by ( B , a ) the tilde-pairing trB3 of B with the trace class operators u E I(7-l) such that 5 = J o t J . We shall call (T = J x x t J = kt2 the probability density of the state (2.1) with respect to this pairing and assume that the support E, of u Proc. R. SOC.Lond. A (2002)
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is the minimal projector E = Et E B for which s ( E ) = 1, i s . that-z := JE, J = E,. The latter can also be expressed as the symmetricity property E, = E, with respect to the tilde operation (transposition) B = JBtJ on L(7-1).One can always assume that J is the standard complex conjugation in an eigenrepresentation of a such that 5 = zzt = Z coincides with a as the real element of the invariant m&mal Abelian subalgebra A c C(7-1) of all diagonal (and thus symmetric) operators in this basis. The auxiliary Hilbert space G and the amplitude operator in (2.1) are not unique; however, w is defined uniquely up t o a unitary transform wt ++ Uwt in 4. 4 can always be taken t o be minimal by identifying it with the support X u = E,X for a, defined as the closure of a X (E,is the minimal orthoprojector in B such that aE = a).In general, 4 is not one dimensional, the dimensionality d i m g must not b e less than rankwt = r a n k a , and the dimensionality of the r a n p = ranwt of p coincides with the support Gp for p = d w N 5. Given the amplitude operator w : 4 ---i 7-1, one can define not only the state q but also the normal state
Q(A) = t r GtAG = (A, p),
A
E A,
(2.2)
on A = L(G),as the marginal of the p u r e compound state,
w ( A 8 B ) = t r & + B x = trGtAGg, where w is defined on the algebra A 8 B of all bounded operators on the Hilbert tensor product space G 8 7-1. Indeed, the defined bilinear form, with = J A t J ,is uniquely extended to such a state given on C(G 8 7-1) by the amplitude $J = A?,where A? is uniquely defined by (C 8 q ) t d = q t w ~ cfor , all E G,q E 7-1. This pure compound state w is the so-called entangled state (Schrodinger 1935) unless its marginal state c (and e) is pure corresponding t o a rank-one operator wt = Cqt, in which case w = e 8 c is given by the amplitude v = C @ q. The amplitude operator x corresponding to mixed states on A and B will be called the entangling operator of p = wtw to a = G ~ G . As follows from the next theorem, any pure entangled state
A
c
w ( A 8 B )= q t ( A @ B ) $ ,
A @ B EC(G@X)
given by an amplitude $J E G 87-1can , be described by a unique entanglement w to the algebra A = L(G) of the marginal state c on 17 = C(7-1). Before formulating this theorem in the generality required for further considerations, let us introduce the following notation. Let A be a +-algebra on G with a normal, faithful and semi-finite trace p ; let A' denote the commutant {A' E L(G) : [A', 41 = 0, VA E A } of A; and let (A,,G) denote the transposed algebra of the operators A with D(A) = p(d),which may not coincide with (A,p ) (nor with d').We can always assume that d = J A t J, with respect to an involution J on G representing on the same Hilbert space G and, in most cases, A = A and G, = p , but not in the standard representation, unless A is Abelian algebra. We denote by A, C_ A the space of all operators A E A in the form xtz, where x, z E a,, with a, = {z E A : p ( z t z ) < 00). (Gp, L, J,) denotes the standard representation L : A -t L(G,) given by the left multiplication L(A)E= Ax on a, with the standard isometric involution J, : x ++xt defining the representation
A
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;(a)= j P L ( A t ) jofp 2 on the completion gp of the module
Up with respect to the inner product .( I z ) , = p ( z t z ) . We recall that the von N e u m a q algebra A defined by A” = A i s anti-isomorphic to (A)‘ = J,L(A)J,-and thus 4 N L(A)’ and that the space of all continuous functionals A : 4 +-+ (4, A)p with respect to A A*, the *-norm (1411, = sup{(p(A+)( : ( ( A (<( 1) on Ap and the Pairing N
( x t z ,A), = p,(zAxt)= ( A ,ztz),,
z f z E A,,
A E A.
The completion of A, with respect t o the norm /I (1, is the pre-dual Banach space, denoted as A, (if p = r I A is the usual trace r = trg on A, then A, coincides with A, as the class A, = A n T ( 4 )of trace operators 7(g) = { x t z : 2,z E S ( g ) } ,where S(G)= (2 E C(G) : trg z t x < 00)). If A is not the algebra of all operators C(B), the density operator p for a normal state (2.2) is not unique with respect t o r = trg. However, it is uniquely defined as the bounded probability density p = J x ~ z J= ??t%with respect t o the restriction p = r I A (i.e. as the density operator with respect t o j ) , describing this state as ( A ,p ) , = p ( z A d ) by the additional condition M = 5 EA,. Note that each probability density p€AA,,describing the normal state e(A)=(A,p),on A 3 A, is positive and2ormalized as (I,p), = 1. However, the pre-dual space A,, as the *-completion of A,, ma2 consist not only of the bounded densities with resEct to p (although each p E A, can always be approximated by the bounded pn. E A,). In the following formulation, B can also be the more general von Neumann algebra, rather than C(?-t), with a normal, faithful and szmi-finite trace v : By w C,defining the pairing ( B , U ~ Z L= ) , v(GtBG), where u E b, (23, = bLb, coincides with 23, in the case of the standard trace v(BCr)-= trB5 = ( B , ( T )when ~ , b, is the space of Hilbert-Schmidt operators y E B and 23 = a).
Theorem 2.1. Let w : A @B -+ C be a normal compound state
-
w(A 8 B ) = T ( u ( A8 E ) v t ) := ( A8 B , u ~ u ) , described by an amplitude operator v : G 8 X -+ Hilbert spaces E and 3,satisfying the condition
ut, E A @
a,
(2.3) E 8 3 on the tensor product of
r ( v d )= 1,
where T fi fi 8 i is the trace r(uut)= ( I @ I , utw),defined in (2.3) by the pairing for A 8 B With respect to p @ u. Then this state is achieved by ~ L Ientangling operator x:G@F+&@Xas
( A , ~ ( x ~ ( I @ B= ) xw)()A~@ B )= ( B , p ( G t ( A 8 1 ) 2 ) ) , ,
(2.4)
for all A E A and E E B , such that
v ( x t ( I €4 B ) x ) C A, p ( G t ( A 8 I ) G ) C_ g, The operator
M
together with k = J z t J is uniquely d e h e d by u = UNl, where
( 6 @ d ) t d ( C 8 J v=) ( C @ v ) t ~ ( I € 4 J d )5, E &, up to a unitary transformation PWC.R. SOC.Lond. A (2002)
17’ E F,
C E ’2,
U of the minimal subspace space ranu
77 E 7-1, (2.5)
c & €4 F .
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Proof. Without loss of generality, we can assume that & = Gpl F = 7&, and W t = v ( E p@ E,) askhe sgpport (G @ 31)v+v= ranwt for utw is contained in G p @ 3-1,. Due to w t w E (A' @ B')', the range of w is invariant under the action
(A g)'
BE@
V A E s?;,
( A@ B)Y= w(AE, @ BE,),
g'.
of the commutant 8 = 8 Let us equip G and X with the iiivolutions J leaving invariant G, = EpG and jFt, = E,'H, denoting J p = E,J, J , = E, J , and & @ F = G p @ 31, with the induced involution J(C @ q ) = JpC 8 J,q. It is easy to check for such Y and x = v', defined by w = x' in (2.5), that for any A E A' and
BE@ (& @ q)tx(<@ BJq') = (& =
where
A = JAJ
E
@B
v ' ) ~ ~@(J[q ) = (E @ q')tw(At
JBq)
(E @ B9)tx(& @ Jq'),
s?;, B = JBJ
E B'. Hence, for any
B
E
B,
(A@B')xt(I@B)x= x t ( A @ B ' B ) x =x t ( l @ B ) x ( A @ B ' ) , := Z E p , B' E
where A E
Bk
:= B'E,, and, for any
A E A,
(A' @ B)%t(A@ I ) % = x(A'A @ B)xt = % t ( A@ I)%(A'@ B ) , where A' E A' and B E
@. Thus, for all A E A and B
x t ( I @B)x E
(2, @ Bk)',
Moreover, due to A: = E , U ,
E A,
E B,
Gt(A@ I ) % E (A: @
and B: = E,BE,
R)'.
B,,
(&@ B-, ) P B ~ ,
xt(I@ B ) x C JpApJp @ E,B,E,
:=
Gt ( AEJ I ) % C E p A p E fEJ J,&J,
:= (Af @ B
u)p~~,
as bounded by IlBllxtx and by llAll%t%,respectively. The partial traces v and p on these reduced algebras are defined as
v(xt(I@ B)x)= (B, wtw),, according to ( A ,(B,W
p(%t(A@ I ) % )= ( A ,W
~ W ) ~ ,
(2.6)
~ W )= ~ () A~@ B , Y ~ Y = )( B ,( A ,W ~ W ) ~where ) ~ , _v
( B ,wtwjv = v ( ( I @B)K),
-
_v
( A ,U
~ Y= ) ~ p((A
I)wtu).
In particular,
v ( x L )= G(WtY)= p1 Any other choice of
Y
p ( d % ) = G(YtW)= c7.
with the minimal & €3 F
N
4, €9 X, is unitary
d. Note that the entangled state (2.3) is written in (2.4) as
( B l 4 A ) ) V= 4 4 @ B)= ( A ,aT (B))u, Proc. R. SOC.Lond. A (2002)
equivalent to
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E n t a n g l e m e n t , q u a n t u m entropy and m u t u a l i n f o r m a t i o n in terms of the mutually adjoint maps w : A -+ 13, and wT are given in (2.6) as
where the linear map traces
T
:B
4
215
A,. These maps
:t 3 t A, and the adjoint r* : A + t3, are defined a s partial
rr(B) = v((1El B ) Z ) ,
.rr*(A)= p ( ( AEl I ) v f . ) .
The linear normal map w in (2.6) is written in the Kraus-Stinespring form (Stinespring 1955) and thus is completely positive (CP). It is not unital but normalized to the density operators u = w ( I ) with respect t o the weight v. A linear map T : B -+ A, is called tilde-positive if the map T - defined as r-(B):= J7r(B)tJ is positive for any positive (and thus Hermitian) operator B 0 in the sense of non-negative definiteness of B . It is called tilde-completely positive (TCP) if the operator-matrix n - ( B )= J r ( B ) t J is positive for every positive operator-matrix B = [&k] = B*,where At = [Afk],B* = [Bli] (and, thus, At = [Aki] for A = [Aik] 2 0, and B" = B for B 2 0). Obviously, every tildepositive and tilde-completely positive 7r is positive, since positive is = JAt J for every positive A, but it is not necessarily completely positive unless A = A , for all A E A, in which case A is Abelian (or the Abelian is B). The map rr defined in equation (2.8) a s a TCP t-map, T ( B ~=) r ( B ) t ,is obviously > any B > 0 , transpose-CP in the sense of positivity of r r ( ~ )=t [X(Bki)]= . r r ( ~ tfor but it is in general not CP. Because every transpose-CP map can b e represented as tilde-CP, there might be a positive-definite matrix B for which .rr(B)is not positive. Note that the adjoint map T* = ;irT is also TCP, as well as the maps ii = ?i and 7 r ~= ?i*, where ?i(B)= Jr(B)J,obtained from (2.6) as partial tracings,
a
?i(B)= v ( J ( I @B)x),
7rT
( A ) = p ( % t ( A @I)%).
(2.8)
In these terms, the compound state (2.4) is written as (Al7@))p
= ++
@ B)=
(T*(A)I~),,
where (sly)= (y,Z) defines an inner product which coincides in the case of traces with the GNS product (z 1 y). In the following definition, the pre-dual space B, = & (as well as A, = A,) is identified by the pairing (B, a ) , = r(B) with the space of generalized density operators (T,which are thus uniguely d e b e d as selfadjoint (unbounded) operators in 'FI. Note that B, = B, if B = B and u = t r x = 3.
Definition 2.2. A TCP map 7r : B + A, (or B -+ .Ap A*) normalized as p(x(1)) = 1 and having an adjoint with 7r*(A)C_ 23, (7r*(d) B,) is called normal coupling (bounded coupling) of the state = c- p 0 7r on B to the state Q = u o .rr* on A. The CP map w : A -+ B, (or A + B, BT),normalized t o the probability
c
c
density
(T
= a ( I ) of q with mT( I ) E
a*
( a 1
( I ) E &),
will be called normal entanglement (bounded entanglement) of the system (A,e) with the probability density p = a T ( I ) to (a,c). The coupling 7r (entanglement Q) Proc. R. SOC.Lond. A (2002)
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is called truly quantum if it is not CP (not TCP). The self-adjoint entanglement wq = w: on (A,Q) = (g,?)(or symmetric coupling rq= 7r: into A, = BT) is called standard for the system ( B ,c) if it is given by
Note that the standard entanglement is true as soon as the reduced algebra B, = on the support X, = E,X of the state q is not Abelian, i.e. is not one corresponding t o a pure normal q on B = C(’H). dimensional in the case B = C(X), Indeed, 7 r q , restricted to B,, is the composition of the non-degenerated multiplication B, 3 B w 31/2Bg1l2 (which is CP) and the transposition B = JBtJ on B, (which is TCP but not CP if dimX, > 1). The standard entanglement in the purely quantum case B = B(X)= g,v = t r = fi corresponds to the pure standard compound state
E,BE,
t r Aa1/2Ba1/2 = ws(A8 B)= tr B?i1/2r?61/2
(2.10)
on the algebra B @ 23. It is given by the amplitude w’ N lo1/’) = +, with = x’ = (a1/21defined in (2.5) as d([8 J q ) = qtx< for x = a’/’. Any entanglement on A = ,C(G),p = t r corresponding to a pure compound state is true if rankp = r a n k a is not one. If the space 4 is also minimal, 4 = 4,, 7 r T is unitary equivalent t o the standard one 7rq. Indeed, m(A)= 2 t A 2 can be decomposed as a ( A )= d/2UtAUa1/2= m,(UtAU), where U : 01/2q H 2q is a unitary operator from X u onto the support 4, of p = UaUt with non-Abelian A, = L(4,) and B, = UtA,U = L(7tFt,). Note that the compound state (2.4) with 2 = all2, corresponding to the standard w = wq, can always be extended to a _vector state on B V B in the standard representation (Xv, L , Jv) of B L ( B ) when , B = J,BJ, = 13‘. However, it cannot be extended to a normal state on B 8 B in the case of con-atomic B. If 23 is a factor, this state is pure, given in the standard representation B V B = L(7-t”)by the unit vector E ;Ft,. However, it is not normal on B @ B unless B is type I: B N ,C(X). y=
3. c-, d- and o-couplings and encodings In this section, we discuss the operational meaning of couplings corresponding t o different types of encodings, which are treated here solely in terms of coupling maps on input of a quantum physical system. We hope that this mathematical treatment will provide a new physical insight for the corresponding asymptotic problems of quantum information. The compound states play the role of joint input-utput probability measures in classical information channels and can be pure in the quantum case, even if the marginal states are mixed. The pure compound states achieved by an entanglement of mixed input and output states exhibit new, non-classical type correlations, which are responsible for the EPR-type paradoxes in the interpretation of quantum theory (Werner 1989). However, mixed, so-called separable, states on A @ B, defined as convex product combinations w, (A 8 B ) =
C en(A)sn( ~ ) p ( n ) , 7l
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which we refer as the c-compound states, do not exhibit such paradoxical behaviour. Here, p ( n ) > 0, C p , = 1 is a probability distribution, and en : A -+ c,cn : I3 --t (C are usually normal states defined by the product densities pn @ un E A, 8 BT of W, =-en @ un. Such compound states are achieved by c-couplings r, : B -+ A,, given by r C= w: , where w,(A) =
C en(A)unp(n),
.IJ ( B:)=
C cn(B)PnP(n). 7L
n
Here, pn E A, and u, E B, are the probability densities for en and qn, with respect to given traces p and v on A and B. Note that the c-entanglement w,,being the convex combinations of the primitive CP-TCP maps wn(A)= en(A)unE BT, is not truly quantum. The separable states of the particular form
A,
where Q,(A) = (nlAln),are pure states on A = C(G) = given by an orthonormal system {In)} c 4 , and q(n, B ) = ( B ,u(n)), with a ( n ) = unp(n), are usually considered as the proper candidates for the input-utput states in the communication channels involving the classical-quantum (c-q) encodings. Such a separable state was introduced by Ohya (1983a, 1989) using a Schatten decomposition p = C I n ) ( n l p ( n ) of the input density operator p E T(G) into the orthogonal onedimensional projectors pn = In)(nl.Here, we note that such a state is the mixture of the classicalquantum correspondences n H \n)(nl@gn, which can be described as the composition of quantum channelling In)(AOl)and the errorless encodings n ++ In)(nl,in the sense that they can be inverted by the measurements In)(nlH n as input decodings. MTe shall call such separable states d-compound as they are achieved by the diagonal couplings r d = wi (d-couplings) to the subalgebra Ad C A of the diagonal operators A = C a(n)ln)(nl,where wd(A) = C ( n l A l n ) u ( n ) ,
wA(B)
= xq(%B)ln)(nl,
(3.2)
n
n
with respect t o the standard transposition (nlAlrn)= (mlAln) in the eigenbasis of p. Actually, Ohya obtained the compound states wd as the result of the composition. Wd(A €4 B ) = w0(A €4 d(B)) of quantum channels as normal unital CP maps A : B + A and the special, compound states wo(A @ B ) = C(nlAln)P(n)(nlBIn),
0-
(3.3)
n
corresponding to the orthogonal decompositions
%(A) = Cb-4Aln)P(n)In)(nl = a&% n
such that qn,(B)= (nlA(B>In>, Proc. R. SOC.Lond. A (2002)
un = AT(In)(nl)t
(3.4)
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218 where
( B ,AT( P ) ) = ~ trg A(B)ij. Assuming that ( A ,p ) = trg Aij, we can extend this construction to any discretely decomposable algebra A = on the Hilbert sum G = e4i with invariant components giunder the standard complex conjugatio2 J in the eigenbasis of the density operator i j = J p J = p. In particular, the von Neumann algebra- A might be Abelian, as it is in the case A = A , for all A E A,-e.g. when A = A is the diagonal algebra of pointwise multiplications Ag = ag = Ag by the bounded functions n t+ a(n) E C on the functional Hilbert space 4 = E2 3 g with the standard complex conjugation J g = 9. In this case, the densities p E A, are given by the summable functions p E Cl with respect t o the standard trace p(p) = C p ( n ) , and any compound state has the separable form with @,(A)= a(n),corresponding to the Kronecker &densities p, 'Y 6,. The normal states on the A N C" are described by the probability densities p(n) 2 0, Cp(n)= 1, with respect t o the standard pairing
A
( A ,P), =
c
a E 1-
a(n)p(n), P E
of A, = A, with the commutative algebra A. Every normal compound state w on A 8 23 is d e h e d by
wc(A8 B)=
1a(n)(B,
a(n))u,
n
where g ( n ) = a,p(n) is the function with positive values ~ ( n E) BT normalized to the probability density p(n) = ( I ,~ ( n ) Thus, ) ~ . all normal compound states on Cw @ B are achieved by c-couplings rc = w: : B -t C1 with T: = wc given by convex combinations of the primitive C P (and TCP) maps w,(a) = a(n)a, E B,,
n
n
where r(n,B ) = ( B ,a(n)),. Note that any d-coupling can be regarded as quantum-classical c-coupling, achieved by the identification a(n) = (nlA[n)of ern 3 a and the reduced diagonal algebra Ao = {C ln)a(n)(nl: A E A}. This follows simply from the commutativity of the density operators p = C In)(n\p(n) for the induced states @(A) = w ~ ( A @ Iidentified ) with p E E l . Ln the case A = L ( Q ) and pure elementary staies w, described by probability 8 $, where gn = [X,) E G, $, = I$,) E 'H, we have density amplitudes u, = operators p, = and a, = of rank one. The total compound amplitude is obviously u = ln)w(n),where w ( n ) = ~ ~ @ $ , p ( n ) lare / ~ the amplitude operators @ 7-t ---t C2 satisfying the orthogonality relations
xn
XLX,
$A$,
u(n)+u(m)= Pn B Unp(n)bF
corresponding to the decomposition utu = C pn @a,p(n). The 'entangling' operator for the separable state x can be chosen as either x : C [n)x(n)-or as M C x(n)(n[, or even as x = C In)x(n)(nlwith x ( n ) = 8 $ ( n ) ,where $,(n) = $ ~ , p ( n ) l / ~ In. particular, a d-entangling operator x corresponding to d-encodings (3.2) is diagonal, M = C [n)@(n)(nl on 4 = E 2 , corresponding to the orthogonal 2, = In). Thus, we have proved theorem 3.1, below, in the case of pure states q, and en. But, before
xn
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formulating this theorem in a natural generality, let us introduce the following notations. The general c-compound states on A @ B aTe defined as integral convex combinations w(A @ B) = e , ( A ) ~ ( B b ( d z ) ,
/
given by a probability distribution p on the product states ex @ qX. Such compound states are achieved by convex combinations of the primitive CP (and TCP) maps T ; = w; with w,(A) = p,(A)a,: %(A) =
/
@,(A)azP(dz),a, (B 1 -
/
&(B)PzP(dz).
(3.5)
This is always the case when the von Neumann algebra A is Abeliam, and, thus, can be identified with the diagonal algebra of multiplications ( A g ) ( z )= a ( z ) g ( z )by the functions a E L y on the functional Hilbert space D = L; with respect to a (not necessarily bite) measure 1-1 on X.It defines trace p on A, 21 Lb fl L y as the integral p(p) = Jp(z)p(dz) for the bounded multiplication densities ( p g ) ( z ) = p(z)g(z).The normal states on A are given by the probability densities p E Li with respect to the standard pairing
(A, P ) =~
1
a(z)p(z)p(dz), P E L i ,
a EL r
I
of A, = A, N L i and A = A N Lr corresponding to the trivial transposition 6 = a. Any normal compound state w on A @ B N L y ( X -+ B ) is the c-compound state, defined on the diagonal algebra A by
Wd(A@ B ) =
1
a(z)c(z:, B)p(dz),
(3.6)
where ~ ( zB) , = (B,u(z)), is an absolutely integrable function with density operator values u(z)= u,p(z) normalized to the probability density p ( z ) = ( I , u ( z ) ) ,= ~ ( xI). , It corresponds to d-couplings 7rd = w: = 7rd with T: = a d decomposing into m ( z ,A ) = a ( z ) c ~ ( z ) :
ad(A) = /a(z)'(z)p(dz),
pL(B)
=
1
c(z, B)6zp(dz),
(3.7)
where ~5~is the (generalized) density operator of the Dirac state @,(A)= {A,6 z ) p = a(.) on the diagonal algebra A.
Theorem 3.1. Let w, : A 8 I3 + &: be a normal c-compound state given as
wc(A8B ) = S p , ( X t A ~ . ) ~ , ( ~ ~ ~ ~ , ) p ( d z ) , (3.8)
JZJ,,G--+ Ex,
where
2, =
:
qjz : 7-1 ---t .Fz are linear operators having bounded transpose = ~ q j J. ! on Hilbert spaces E. =
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8
1
8
E,p(dz),
X
=
FXp(dz)
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V. P. Belavkin and M. Ohya
with respect to pointwise involution J. = J!. We also assume that
xLxx fz2, ?b:$x
E
B,
px(X!X.) = 1 = I.,(@&)
with respect to the traces
I.z(4!4.) = (17?b5/5x)v.
PX(xt2.)= (LX:Xx)pr
Then this state is achieved by a decomposable entangling operator $,p(dz) defining c-entanglement (3.5) with
eZ(A) = pX(XfAX.),
cx(B)= vX(q!A&),
$xZ,
N
(3.9) = J@xx8
(3.10)
$LQx.
corresponding to the probability densities px = ux= Lnparticular, every d-compound state (3.6) corresponding top(dx) = p(x)p(ds) with the Abelian algebra A can be achieved by the orthogonal sum of entangling operators zx= 6, €3 $x defining d-entanglement (3.7) with = $51ClxP(s)1
+,B) = V X ( ! P A & ) P ( 4 .
s"
Proof. The amplitude operator u = w,p(dx), corresponding to c-com ound €3 $x on 4 8 Tt into J &x 8 state (3.8), is defined as the orthogonal sum of u, = Fzp(dx). Without loss of generality, we can aSsume that Ex = G f 1 Fx = Flu and = wx(E, 8 E,) because the support ( 4 8 ; Y ) J ~ ,= ran.: for
xx
ui
t u:ux = xxxz c9 1ClL$,
is in
= p x €3 c x
4, @'Hu.Due to
xLxx E X f l E for almost all 2 , the operators xx and qX commute with A E tively, and GZ commutes with B E 8' for almost all 2 . Thus, %:AXx E A,
and B E @, respec-
13xB$x E ,131
which d e h e s the traces (3.9) on Lp" €3 A and Lp" €3 B for almost all x. The rest of the proof is a repetition of the prooi of theorem 2.1 for each x, with the addi8 qZfor each x. The total estangling operator tion that zxis the product wk = x : 4 €3 F,---f E. 8 ?-t acts componentwise as N,([ €3 71.) = xxC €3 lClxqz. In the case of d-compound state (3.6), one should take 4 = L;, Ex = @, and xxg = g(x). Thus, the entangling operator in this case is given as
xz
Note that c-entanglements W , in (3.5) are both CP and T C P and are thus not true quantum entanglements. The map a, : A + BT with Abelim algebra A in (3.7) is described by a &-valued measure g(dx) = u(x)p(dx) normalized to the input probability measure as p(dx) = ( I ,u(dx))y. This gives the concise form for the description of random classical-quantum state correspondences zM axwith the given probability measure p , called encodings of ~7= J a(dx). Proc. R. Sac. Land. A (2002)
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221
Definition 3.2. Let both algebras A and B be non-Abelian. The map w : A -t B, is called a c-encoding of ( B ,<) if it is a convex combination of the primitive maps unen given by the probability densities an E B, and normal states en : A -+ c. It is called d-encoding if it has the diagonalizing form (3.2) on A, and it is called o-encoding if all density operators an are mutually orthogonal: amu, = 0, for all # n as in (3.4). The entanglement which is described by non-separable cp map w : A -+ B, will be called q-encoding. Note that, due t o the commutativity of the operators A 8 I with I 8 3 on 4 @ can treat the encodings as non-demolition measurements (Belavkin 1994) in A with respect to B. The corresponding compound state is the state prepared for such measurements on the input 8. It coincides with the mixture of the states corresponding to those after the measurement, without reading the message sent. The set of all d-encodings for a Schatten decomposition of the input state p on A is obviously convex with the extreme points given by the pure output states qn on B, corresponding t o the not necessarily orthogonal (not Schatten) decompositions u = C a ( n )into the one-dimensional density operators a ( n ) = p ( n ) g n . The Schatten decompositions (T = C,q(n)a, correspond to eencodings, the extreme d-encodings (T, = q,vi, p ( n ) = q ( n ) characterized by the orthogonality umun= 0, m # n. For each Schatten decomposition of (T they form a convex subset of d-encodings with mixed commuting ( T ~ .
K,one
4. Quantum versus von Neumann entropy.
As we have seen in the previous section, the encodings w : A + a,, which are described in (3.7), usually with a discrete Abelian A, correspond to the cme (3.2) when the generalized entanglement (2.7) is d-encoding, with the diagonal coupling T
= WT
in the eigenrepresentation of a discrete probability density p on non-Abelian
A. The true quantum entanglements with non-Abelian A cannot be achieved by d-, or more generally, c-encodings even in the case of discrete A. The non-separable, true entangled states w called q-compound states in Ohya (1989), can be achieved by q-encodings, the quantum+pantum non-separable correspondences (2.6) which are not diagonal in the eigenrepresentation of p. As we shall prove in this section,-the self-dual standard true entanglement wq = w; to the probe system (Ao,eo) = ( B ,C), which is defined in (2.9), is the most informative for a quantum system (B,c), in the sense that it achieves the maximal mutual information in the coupled system (d@ 23,w)when w = wq,is given in (2.10). Let us consider entangled mutual information and quantum entropies of states by means of the above three types of compound states. To define the quantum mutual entropy we need to apply a quantum version of the relative entropy t o compound states on the algebra M = A @ B ,also called the information divergency of the state w with respect to a reference state p on M . The relative entropy was defined in Lindblad (1973), Araki (1976) and Umegaki (1962), even for the most general von Neumann algebra M , but, for our purposes, we need the following explicit formulation. Let M be a s e m i - h i e algebra with normal states w and p having the density operator wtw and q5 E M with respect to the pairing
( M ,wtw) = -r(wA?wt), Proc. R. SOC.Lond. A (2002)
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given by a normal, faithful trace r on the transposed algebra G=J M J (not necessary decompos ble as r = fi 8 V in (2.3) in the case of M = A 8 23). Then the relative entropy (w; p) of the state w , with respect t o p, is given by the formula
R
R(w : p) = T(v(1nvtv - 1n6)vt) = r(w(1nw - 1.4)).
(4.1) (For notational simplicity here and be1 w we identify the state w with its density 'k operator wtv.) It has a positive value (w : p) E [O,OO] if the states are equally normalized, say (as usual) T ( W ) = 1 = ~ ( q 5 ) , and it can be finite only if the state w is absolutely continuous with respect to the reference state p, i.e. if w ( E ) = 0 for the maximal null-orthoprojector E E M , Eq5 = 0. This definition does not depend on the choice of the semi-finite trace r , and it can also be extended to the arbitrary normal w and p with unbounded self-adjoint density operators#tv and 4. The most important property of the information divergence is its monoto 'city Y(wo : property (Lindblad 1973; Uhlmann 1977), i.e. non-increase in the divergency b po)after the application of the pre-dual of a normal, completely positive unitd map K : M t M o to the states uo and p o on a von Neumann algebra M o : (w = woK,p = poK) =+ R ( :~p) < R (wo : y o ) . (4.2) I I The mutual information (n)= (n*)in a compound state w , achieved by a coupling n : B t A,, or by T* : A + B, with the marginals
@ ( A )= w ( A 8 I ) = ( A ,P ) ~ , ~ = (B ~ )( 1B8) = (B, u )
~ ,
is defined by the relative entropy I(7r) = + ( h w - h ( p @ I) - h(I8a ) ) )= R (w : p 8 q)
(4.3)
of the state w on M = A823 with respect to the product state p = e@qfor r = ,5817. This quantity, generalizing the classical mutual information corresponding to the case of Abelian A, B, describes an information gain in a quantum system (A,e) via the entanglement a T = n-,or in ( B ,5) via an entanglement a : A -+ BT. It is naturally treated as a measure of the strength of the generalized entanglement having zero value only for completely disentangled states w = e 8 q. Proposition 4.1. Let ( d o , p o ) be a quantum system with a normal, faithful semi-hite trace, and no : AD ---t B, be a normal coupling of the state eo = v o r0 on Ao to q = p o T , d e h i n g an entanglement P = T*- of (A,e) to ( B ,q) by the composition * - roK with a normal completely positive m'tal map K : A -+ do. Then '(n)< (no),where no = xi.In particular, for each normal c-coupling given by
T -
(3.5), such as T - = a; , there exists a not less informative d-coupling no = a,!with , Abelian A0 corresponding to the encoding a. = ni of (B,q), and the standard qcoupling no = rq,n,(B) = d/zl?d/z to po = 5: on Ao = is the maximal coupling in this sense.
a
proof. The fist follows from the monotonicity property (4.2) applied to the extension K(A @ B ) = K(A) @ B of the CP map K from A t Ao to d 8 B t Ao 8 a. The compound state wo(K @ I) (I denotes the identity map B 4 B ) is achieved by the entanglement a = WOKand po(K @ I) = @ c, e = poK, corresponding to yo = eo 8 q. It corresponds to the coupling n = K*nO,which is defined by K* : d: t A, as K*io = J(KT po)t J , where
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This monotonicity property proves, in particular, that, for any separable compound state (3.8) on A 8 B,which is prepared by the c-entanglement T , = WE, there exists a d-entanglement w; = 7ro with (A0,po) having the same, or even larger information gain (4.3). One can even take a classical system (AD,po), say the diagonal subalgebra Ao N Lp" on Go = L i with the state po, induced by the measure p. = p , and consider the classical-quantum correspondence (encoding)
assigning the states c ( B ) = ( B , g , ) v to the letters E with the probabilities p(dx). In this case, the state e is described by the density p = I , the multiplication by identity function in L;, w is multiplication by u.(x)= 6x in Lg 8 f i and the mutual information (4.3) is given as
where S(a)= -V(nlnu). The achieved information gain I (no)is larger than I ( x ) , corresponding to w = S p , 8 u,p(dx), because the c-entanglement w,in (3.5) is represented as the composition WOK of the encoding wo : Ao --t 17, with the CP map
K(A) = given by
U(E)
e,(A)p(dz), A E A
= @,(A) for each A 6 A. Hence,
- -
n*(A) = w(A) = wOKA = no(KA), VA E A, where n o = a,, and thus, I (no)>, I(K*7r0)= ' ( T ) , where = T: = w:, The inequality (4.2) can also be applied t_O the standard entanglement corresponding to the compound state (2.10) on B @ B . Indeed, any normalentanglement w(A) = p ( k t ( A@ I ) % )on A into B, , described by a C P map A --t B,,can be decomposed as
p ( d ( A@ I ) % )= U ' / ~ ~ ( X @ ~ (I )AX ) O ' / ~= wo(KA), where KA = p ( X t ( A@ I)X) is a normal unital CP map A 4 8. It is uniquely given by an operator X : E @ ;Ft t B @ 3 with 8 = G p , X = 3gsatisfying the condition X ( I @ u)lI2= 2, and thus, X E A @a',due to the commutativity of ii. with A' @ B and u with B. Moreover, the partial trace p of XtX is well defined by p(%t%) = a as p(XtX) = I . Thus, a_ = w,K and T = K*7rTTq, where K is a normal unital CP map A -+ and K* : B, = B, -+ A,. Hence, the stand rd ent glement ( oupling) (2.9) corresponds to the maximal mutual information, i (rq) " "> p(K*n,) = F( T ) .
g,
Note that the mutual information (4.3) is written as
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denotes the entropy of t h e density o p e r a t o r y t v E of the state w , 9 h respect to the trace 'p on M . Note that the entropy (w/p),coinciding with - (w : p) (see with (4.1) in the case T = @),is not generally positive, and may not even be bounded from below as a function of w. However, in the case of irreducible M it can always b e made positive by the choice of the standard trace T = t r on M , in which c is called the von Neumann entropy of the state w (= v b ) , denoted simply as ( w ) . S ' ( w / T ) = - trwlnw = ( w ) . (4.6)
.Ye 't
In the following, we shall - aSsume that B is a discrete decomposition of the irreducible = L($li) = f3i with the trace v = t r x = 17 induced on B, = B,. The entropy (c)= ( ~ / vof) the density operator elf r the nor al state q on B, can b e found in this case as the maximal information '(c) = sup (nc)achiev_edvia all c-encodings P : A H 23, of the system (23, F), such that m(1) = c,a T = nc. Ind ed, as follows from the proposition above, it is sufficient to find the maximum of (n) over all d-couplings n = wT , mapping B into Abelian A with fixed a(1)= c,i.e. to find maximum of (4.4) under the condition r , p ( d z ) = LT. Due to positivity of the d-conditional entropy
9
Y
'i
the information ' ( n o )= '(Td) has t h e p a x i m u m '(c), which is achieved on an extreme d-coupling n: when almost all (0,) are zero, i.e. when almost all ex are one-dimensional projectors g2 = P, corresponding to pure states qz. One can take, for 3 for example, the maximal Abelian subalgebra Ao & B generated by P, = In)(nl E f a Schatten decomposition LT = C , I n ) ( n l p ( n ) of c E B,. The maximal value lnrankB of the von Neumann entropy is d e h e d by the dimensionality rank23 = dimAo of the maximal Abelian subalgebra of the decompo able a1 ebra B i.e. by dim7-t. However, if 7r is not c-coupling, the difference (n)= ( r )- (n) can achieve the negative value and may not serve as a measure of conditional entropy in such a case.
5
S
r
Definition 4.2. The supremum of the mutual information
I
H(<) = sup{ (n): p which is achieved on A = coupling nq(B) =
0 7r
I
= F} = (7Tq),
(4.8)
a for a fixed state F(B) = t r x BLTby the standard qis called q-entropy of the state F. The maximum
over all c-couplings nc corresponding t o c-encodings (3.5), which is achieved on an extreme d-coupling 7r& is called the c-entropy of the state q. The differences,
H
T n )= (s) -
I
(T),
s(4= S (F) - I (TI1
are called, respectively, the q-conditional entropy on B with respect to A and the (degree of) disentanglement f r the couplin5n : B + A. A compound state is said . (n)< 0, and (n)2 0 for a c-coupling 7~ = 7rc (this is to be essentially entangled if ? called the c-conditional entropy on B, with respect to A). Proc. R. Sac. Land. A (2002)
225 Entanglement, quantum entropy and mutual information S Obviously, '(5) and (q) are both positive and (unlike (u) = (q/v)) H do not depend on the choice of the faithful trace v on B , w d obey t h inequality (q) 2 (q). The same is true for the conditional entropies (n)and ( T ) , where ' ( K ) always hM a positive value 374 3 S(nO)3 0,
s
s
f
s
in the case of a c-coupling 7r = K = , due to KE = K ~ for K a normal unital CP map K : t .Ao, where K O = 7rd is a d-coupling with Abelian AO. But the disentanglement K ) can also achieve the negative value inf{
s( K > : p
0
n = q} = s (q) - H (q) = -
C x(i)s(ui),
(4-9)
i
as the following theorem states in the case of discrete B. Here, the ui E L(Xi) are the density operators of the normalized factor-states q = x(i&ls.I L.'Hi) with x(i)= <(Ii),where I' are the orthoprojectors onto 'Hi.Note that (q) = (q) if the algebra B is ompletely decomposable, i.e. Abelian. In this case, the aximal value lnrankB of (q) can be p t t e n p lndimB. The disentanglement is always
TK)
3
positive in this case, and
(K)
=
(T)
as in the case of Abelian A.
Theorem 4.3. Let B be a discrete decomposable algebra on E = @i'&, with a normal state given by the density operator CT = $o(i), with respect to the trace p = t r x on B , and let C 5 B be its centre with the state x = q I C induced by the probability distribution x(i)= tru(i). Then the c-entropy (q) is given as the von Neumann entropy (4.6) of the density operator u and the q-entropy (4.8) is given by the formula '( l n u ( i ) ) . (4.10)
s
i
This can be written as H (s) = H q ~ ( s+) H~ ( s )where , HC(F) Hqc(r) = -2
c
= - C i x ( i ) h x ( iand ),
sB I C ( S ) ,
x(i)tr7-t; (Ti hcri = 2
i
with ui = u ( i ) / x ( i ) .'(s) is h i t e if 5 (s) < 00, and i f B is finite dimensional, it is bounded, with maximal value H (q') = LodimB, achieved for u' = @uPx'(i): C T ~=
(dirn'Hi)-'Ii,
where dimB(i) =
~ ' (= i )dimB(i)/ dimB,
dimB = C i d i m B ( i ) .
Proof. We have already proved that
s(q)
= S(u),where
' ( u ) = - E t r x r u ( i ) ~ n f f ( i= ) Sc(s->+ 5Blc(q),
"~(4,
i
with 5c(q) = s~lc(c)= cx(i) S (ui)= 5i HBIc(5). The q-entropy (5) is the supremum (4.8) of the mutual information (4.3) which is achieved on the standard entanglement, corresponding to the density operator w = @w(i,k) with w(i, k) = %(i)((T:'2)(a;'2\m; Proc. 2%. SOC. Lond. A (2002)
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-
of the standard compound state (2.10) with 23 = 23, p = u.Thus, H(s) = I (7rq), where ’(nq)= trw(1nw - 1n(u @ 11 - I n ( I @u))= s ( w ) - 2s ( a ) =
C x ( i )~nx ( i )
-
2 t r u In u = -
C x(i)(In x(i>+ 2 trxi uiIn ui>.
i
i
Here, we used that trwlnw = Ci x(i)lnx(i)due to wlnw = c ~ i , k w ( i , ~ ~ ) ~ n w=( i@,i~xct. ) (i)1u.i1’2)(ut’211nxzt(i), and that tr u l n a =
xix(i)(lnx(i)
-
S
a,(si)) due to
+
u l n a = @iu(i)lnu(i) = @ix(i)ui(lnx(i) Inui) for the o p g o n s decomp ition u = @iw(i)u. where $i) = tr u(i). S 3c(c) < 2 (c), and it is bounded by c(s) = 2 nlc(s) Thus, (s) = qc(s)
+ ?7
+
=-inf~x(i)(lnx(i)-21ndim?ti) =lndimB. X
i
s
Here, we used the fact that the supremum of von Neumann entropies (ui)for the simple algebras B ( i ) = L(?ti), with dimB(i) = < 00, is achieved on the tracial density operators ui = (dim’H;)-’Ii = ur,and the infimum of the relative entropy R(x : x’)= X(Z)(ln x(i)- Inx0(i)),
C i
where xo(i) = dim 23(i)/ dim 23, is zero, achieved at x = x’.
rn
Note that, as sh wn in Ohya & Petz (1993) for the case of the simple algebra B,the quantum ntropy 7-l(c) can also be achieved as the supremum of the von Neumann entropy .f( e ) over all pure couplings given by the isometries X : ?t -+ @ X, X t X = I , preserving the state s. The latter means that the density operator w of the corresponding compound states with the marginals p = t r n w and u = trg w is given as w = X U X ~ . 5. Quantum channel and entropic capacities In this section we describe a noisy quantum chaanel in terms of normal unital CP maps and their duals, and introduce an analogue of Shannon information for general semi-finite algebras. We consider the maximization problems for this quantity with various operational constrains on encodings, and define the entropic capacities which serve as upper bounds for the operational capacities corresponding to these constrains. The question of’asymptotic equivalence of the entropic and operational capacities is not touched on here. Let Elbe a Hilbert space describing a quantum input system and let 7-l describe its output Hilbert space. A quantum channel is an affine operation sending each input state defined on 7-tl to an output state defined on ‘H, such that the mixtures of states are preserved. A deterministic quantum channel is given by a linear isometry Proc. R. Soe. Lond. A (2002)
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227
U : 7j1 --+ 'F1 with U t U = I' (1' is the identify operator in
Z1), such that each input state vector q1 E H I , llq111 = 1, is transmitted into an output state vector q = Uql E 7 j , llqll = 1. The orthogonal sums c1 = @cl(n)of pure input states cl(B,n) = ql(n)tBql(n)are sent into the orthogonal sums c = ~ ( nof )pure states on B = L ( X ) , corresponding to the orthogonal state vectors q(n)= U q l ( n ) . A noisy quantum channel sends pure input states 51 on an algebra B1 L ( X l ) into mixed ones c = 5111 given by the composition with a normal completely positive unital map A : B + B1.We shall assume that B1 (as well as B ) is equipped with a normal faitgd semi-finite trace y d e b i n g the pairing (B,utu)1 = v ~ ( G ~ B G of ) Bl and B; = B i . Then, the input-output state transformations are described by the transposed map AT : B$ --+ BT:
(RAT(a1)) = ( A ( B ) , U l ) l ,
E
a,
01
E
q,
defining the output density operators a = AT ( a l )for any input normal state ql(B) = ( B ,al)l. Without loss of generality, the input algebra B1 can be assumed to be the smallest decomposable algebra generated by the range A(B) of the channel map A is Abelian if A(B) consists only of commuting operators on XI). The input generalized entanglements w1 : A -+ B;, including encodings of the will be defined by the couplings IC* : Bl -+ A, state with the density a1 = ..'(I), as = K - . Here, IC : A -+ B; is a normal TCP map defining the state e = v1 o K of a probe system (A,p ) which is entangled t o (B', el) by K - ( A = ) Jn(At)J,and the adjoint map K* is defined as usual by
(a'
(Alfi*(B)), = wl(At 8 B)= (n(A)lB)i, 'dA E A,
B E B1,
where w1 is the corresponding compound state on A 8 B1. These (generalized) entanglements describe the quantum-quantum correspondences (q-, c-, or o-encodings) of the probe systems (A,p ) with the density operators p = KT (I1),t o the input (B1,cl) of the channel A. In particular, the most informative standard input entanglemenLmi : Bl --+ B: is the entanglement of the transposed input system (Ao,eo) = (B1, el), corresponding t o &he TCP map K ~ ( A=) J c ~ / ~ AJ .~Incthe ~ case / ~ of discrete decomposable Ao = Bl = Bl with the density operator a1 = $ial(i), this extreme input q-encoding defines the following density operat or wq =
(I @ AT)(Wqi),
wqi
= $~~~i(~)1~2)(~~(i)1~z~
(5.1)
Ao @ B = Bl @ B. qt The other extreme case of the generahzed input entanglements, the pure c-encodings corresponding to (3.2), are less informative then the pure d-encodings wi = K: given by the decompositions IC: = Cln)(nlc1(n)with pure states cl(B,n) = q(n)tBq(n)on Bl. They define the density operators of the input-output compound state w A on
wd=(I@AT)(wdl),
wdl=
~ln)(nl@ql(n)"Il(n)',
(5.2)
n
of the f3l C3 B-compound state w d l A = w d l o (I 8 A ) . These are the Ohya compound states w, = wOlA (Ohya 1983a) in the case
Proc. R. SOC.Lond. A (2002)
133
V. P. Belavkin
228
and
M. Ohya
of orthogonality of the density operators u l ( n ) normalized to the eigenvalues pl(n) of u1. The o-compound states are achieved by pure Dencodings w: = IC, described by the couplings I C ~= C In)(nlcy(n)with qi' corresponding t o 7:. The input-output density operator wo = (1@ ATb o l ,
wo1 =
1In)(nl@77:(+7y(n)f
(5.3)
n
of the Ohya compound state wo is achieved by the coupling X = n*A of the output (a,c) to the extreme probe system (A",po) = ql) as the composition of n* and the channel A. If K : A + A" is a normal completely positive unital map
(a1,
Z A ~ +A, E A,
K(A) = trFwhere IC
=
X is a bounded operator F-@ 6 0
noK,
7r
4
G with trF- XtX = I",the compositions
= A*K describe the entanglements of the probe system ( A , Q )to the
channel input (a1, 51) and the output (a,q) via this channel, respectively. The state p = poK is given by KT( P O ) = X ( I - @ P O ) , Xt E A* for each density operator p o E A:, where I - is the identity operator in 3-.The resulting entanglement T = X*K defines the compound state w = wol o (K @ A ) on A @ B with wol(A" @ B 1 ) = t r Ao~E(B1) = trC&(A" @ B1)dol on A" @ B1.Here, wol : GO @ + Fol is the amplitude operator uniquely defined by the input compound density operator wol E A: @ B: up to a unitary operator Uo on Fol. The effect of the input entanglement IC and the output channel A can be written in terms of the amplitude operator of the state w as 'u
= (X@ Y ) ( I - @ W O l c3 I+)U
U in 3 = 3- @ 3 0 1 @ 3+.Thus, the density operator of the input-output compound state w is given by wol(K @ A ) with the density
up to a unitary operator
(K@A)*(Woi) = (X@Y)woi(X@Y)t,
(5.4)
t where w01 = wolwol. Let Kt b e the set of all normal T C P maps K : A -+ Bi with any probe algebra A normalized as t r & ( I )= 1, and let Icq(sl)be the subset of all IE E K: with ~ ( 1=)TI. Each IC E Ki can be decomposed as sqK, where I C :~ Ao -+ 2 3 ' definesthe standard input entanglement wi = K ; , and K is a normal unital C P map A -+ B1. Further, let K: be the set of all CP-TCP maps IE described as the combinations
K(A) =
C en(A)n(n)
(5.5)
n
o f t h e primitive maps A H Qn(A)al(n), and let entanglements IC, i.e. the decompositions
Proc. R. SOC.Lond. A (2002)
KA be the subset of the diagonalizing
Entanglement, quantum entropy and mutual information
As in the first cme,
&(I) = cl, Each
229
and &(GI)
denote the subsets corresponding to a fixed K = n&, where Kd describes a pure d-encoding wi = K: of (B1, ~ 1 )for a proper choice IC,(s1)
x,(ql)can be represented as the composition
normalized to Q of the C P map K : A -+B'. Furthermore, let ICA (and ICo(c,-l)) be the subset of all decompositions (5.6) with orthogonal a l ( n ) (and fixed C , u1(n) = ai): ul(m)crl(n) = 0, m # 72.
Each K E I C , ( C ~ ) can also be represented as K = K ~ Kwith , K,, describing the pure ~ 1 ) = (Ao,eo). o-encoding wi = K, of (B1, Now, let us maximize the entangled mutual entropyfor a given quantum channel A (and a fixed input state ~1 on the decomposable B1 = Bl) by means of the above four types of entanglement K . The mutual information (4.3) was defined in the previous section by the density operators of the corresponding compound state w on A 8 B, and the product-state 'p = e 8 c of the marginals e, c for w. In each case, w = woi(K @ A ) ,
'P = ' ~ o i ( K@ A ) , where K is a C P map A -+ do = B1, wo1 is one of the corresponding extreme compound states wql, w,1 = W d l , wol on 2 3 ' 8 B1 and 'pol = po @ ql. The density operator w = (K 8 A)T ( ~ 0 1 )is written in (5.4), and 4 = p @ cr can be written as
4 = KT ( I )€3AT (I), where
AT
= AT
IT!.This proves the following proposition.
P r o p o s i t i o n 5.1. The entangled mutual information achieves the following m a imal values.
where K. axe the corresponding extremd input couplings dot They are ordered as Iq(Cl,A) 2 IC(CllA> = ' d k l 4 2 ' O ( C 1 , A ) .
In the fo owing definition, the maximal information denoted as l(q1,A).
11
lc(s-l, A)
with p
o6 : = Q.
(5.9) = 'd(q1,
A ) is simply
Definition 5.2. The suprema C I I ,(A> = SUP (.*A) = S U P q(Cl,A>, ffiEKi
51
(5.10)
are called the q-, c- or d-, and 0-capacities, respectively, for the quantum channel defined by a normal unital CP map A : B --f B l . Proc. R. SOC.Lond. A (2002)
135
V. P. Belavkin and M. Ohya
230
Obviously, the capacities (5.10) satisfy the inequalities
'.(A) G cl(A) G ' , ( A ) . Theorem 5.3. Let A ( B ) = UtBU be a unital CP map B quantum deterministic channel. Then
Ilk1,A) =
I
o(Flr4 =
S
',(Cl,A)
( d i
--f
B1 describing
a
= sq(
where sq(ql) = H(ql), and, thus, in this case 'I(A)
= C.(A) = InrankB',
',CA) = IndimB'
Proof. It was proved in the previous section for the case of the identity channel
A = I and is thus also valid for any isomorphism A : B ++ UtBU describing the state transformations AT : u YuYt by a unitary operator U = Y.In the case of non-unitary Y , we can use the identity t r Y ( u l @ I + ) ~ t I n Y ( u@l I + ) Y ~ =trS(ul@I+)lnS(u181+), where S = YtY. Due to this, '(qlA) = - tr S(u1@ I+)lnS(o1@I+), and S ( w o l ( ~@ A ) ) = - tr(R 8 s)(I-@ wol @ I + > I ~ (8Rs ) ( I - 8 wol 8 I+), where R = X t X . Thus, '(q1A) = s(a),s( q l ( K @ A ) ) = S(w01(K@I))if Y t Y = I , and b 1 4 ) =
G
s(QoK) + s(cl) - S(wol(K S
S(eo) + (4-
S
I
@ 1))
(wo1) = (WOl),
for IC = I C ~with K any normal unital CP map I< : A + doand a compound state wO1 on Ao 8 B1.The supremum (5.7), which is achieved at t standard entanglement, corresponding to w01 = w q l , coincides with q-entropy a), and the supremum (5.8), coinciding with S( Q ) , is achieved for a pure 0-entanglement, correspondi to wO1 = wO1 given by any Schatten decomposition for u1. Moreover, the entropy (a) is also achieved by any pure d-entanglement, corresponding to w01 = w d l give by any extreme decomposition for u1 and thus is the m y a l mutual information l(
F
3
in the case of deterministick. Thus, the capacity , ( A ) of the determinist'c channel is given by he maximumC, = IndimX1 of the von Neumann entropy ', and the q-capacity ,(A) is equal BI = IndimB'. w
8
In the general case, d-entanglements can be more informative than 0-entanglements, as can be shown by an example of a noisy quantum channel for which 'l(F1,A)
>IdF14,
cl(4 > co(A,.
The last equalities of the above theorem are related to the work on entropy by Voiculescu (1995). The authors acknowledge the support under the JSPS Senior Fellowship Program and The Royal Society scheme for UK-Japan research collaboration. Proc. R. SOC.Lond. A (2002)
136
Entanglement, quantum entropy and mutual information
231
References Accardi, L. & Ohya, M.2002 Compound channels, transition expectations and liftings. J . A p p l . Math. Uptimiz.(In the press.) Araki, H. 1976 Publications Research Institute of Mathematical Sciences, Kyoto University, V O ~ . 11, pp. 809-833. Belavkin, V, P. 1980 Radio Engng Electron. Phys. 25, 1445-1453. Belavkh, V. P. 1994 Found. Phys. 24, 685-714. Belavkin, V. P. & Ohya, M. 1998 Quantum entanglements and entangled mutual entropy. Los Alamos Archive, quant-Ph/9812082, pp. 1-16. Belavkin, V. P. & Ohya, M. 2000 Entanglements and compound states in quantum information theory. Los Alamos Archive, quant-Ph/0004069, pp. 1-20. Bennett, C. H., Brassard, G., CrBpeau, C., Jozsa, R., Peres, A. & Wootters, W. K. 1993 Phys. Rev. Lett. 70, 1895-1899. Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A. & Wootters, W. K. 1996 Phys. Rev. Lett. 76,722-725. Ekert, A. 1993 Phys. Rev. Lett. 67,661-663. Jozsa, R. & Schumacher, B. 1994 J. M o d . Opt. 41, 2343-2350. Lindblad, G. 1973 Carnmzn. Math. Phys. 33, 305-322. Ohya, M. 1983a IEEE Trans. Inform. Theory 29, 770-774. Ohya, M. 1983b Nuovo Cim. 38, 402-406. Ohya, M. 1989 Rep. Math. Phys. 27 19-47. Ohya, M. & Petz, D. 1993 Quantum entropy and its use. Springer. Schrodinger, E. 1935 Naturwissenschaften 23, 807-812, 823-828, 844-849. Schumacher, B. 1993a Ph.ys. Rev. A51, 2614-2628. Schumacher, B. 19936 Phys. Rev. A51, 2738-2747. Stinespring, W. F. 1955 Proc. Am. Math. SOC.6, 211. Uhlmann, A. 1977 Commun. Math. Phys. 54, 21-32. Umegaki, H. 1962 Kodai Mathematical Seminars Report, vol. 14, pp. 59-85. Voiculescu, D. 1995 Cammun. Math. Phys. 170, 249-281. Werner, R. F. 1989 Lett. Math. Phys. 17, 359-363. Werner, R. F. 1998 Phpls. Rev. A 58, 1827-1832.
Proc. R. SOC. Lond. A (2002)
137 JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 2
FEBRUARY 2002
Semiclassical properties and chaos degree for the quantum Baker’s map Kei lnoue and Masanori Ohya Deparinient of Information Sciences, Science University of Tokyo, Noda City? Chiba 278-8510, Japan
lgor V. Volovicha) Steklov Mathematical Insiiiuie, Russian Academy of Science, Gubkin Si. 8, Moscow, GSPl, 11 7966, Russia
(Received 27 February 2001; accepted for publication 3 October 2001) We study the chaotic behavior and the quantum-classical correspondence for the Baker’s map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered. 0 2002 American Instiiufe of Physics. [DOI: 10.1063/1.1420743]
1. INTRODUCTION
The study of chaotic behavior in classical dynamical systems dates back to Lobachevsky and Hadamard, who have studied the exponential instability property of geodesics on manifolds of negative curvature, and to Poincare, who initiated the inquiry into the stability of the solar system. One believes now that the main features of chaotic behavior in the classical dynamical systems are rather well understood (see, for example, Refs. 1 and 2). However, the status of “quantum chaos” is much less clear although s i e c a n t progress has been made on this fiont. Sometimes one says that an approach to quantum chaos, which attempts to generalize the classical notion of sensitivity to initial conditions, fails for two reasons: kst, there is no quantum analog of the classical phase space trajectories and, second, the unitarity of linear Schrhdinger equations precludes sensitivity to initial conditions in the quantum dynamics of state vector. Let us remind, however, that in fact there exists a quantum analog of the classical phase space trajectories. It is quantum evolution of expectation values of appropriate observables in suitable states. Also, let us remind that the dynamics of a classical system can be described either by the Hamilton equations or by the linear Liouville equations. In quantum theory the linear Schrhdinger equation is the counterpart of the Liouville equation while the quantum counterpart of the classical Hamilton equation is the Heisenberg equation. Therefore, the study of quantum expectation values should reveal the chaotic behavior of quantum systems. In this article we demonstrate this fact for the quantum Baker’s map. If one has the classical Hamilton equations d q l d f zp ,
dpldt=
- V‘(q),
then the corresponding quantum Heisenberg equations have the same form dqh/dt=ph,
dph/dt=-y’(qh),
where qh m d p h are quantum canonical operators of position and momentum. For the expectation values one gets the Ehrenfest equations a ) ~ ~ e ~ t rmoainl: i [email protected] ~ Reprinted with permission from K. h u e , M Ohya and V. Volovich, J. Math. Phys. 43 (2), 734 (February 2002). 0 2002, American Institute of Physics.
138 J.
Math. Phys., Vol. 43, No. 2, February 2002
Semiclassical properties and chaos degree
735
Note that the Ehrenfest equations are classical equations but for nonlinear V‘(qh) they are neither Hamilton equations nor even differential equations because one can not write (V‘(qh))as a function of ( q h ) and ( P h ) . However, these equations are very convenient for the consideration of the semiclassical properties of quantum system. The expectation values ( q h ) and (P,,) are functions of time and initial data. They also depend on the quantum states. One of important problems is to study the dependence of expectation values from the initial data. In this article we will study this problem for the quantum Baker’s map. The main objective of “quantum chaos” is to study the correspondence between classical chaotic systems and their quantum counterparts in the semiclassicallimit?34The quantum-classical correspondence for dynamical systems has been studied for many years (see for example Refs. 5-10 and reference therein). A significant progress in understanding this correspondence has been achieved in the Wentzel-Kromers-Brillouin (WKB) approach when one considers the Planck constant h as a small variable parameter. Then it is well known that in the limit h+O quantum d theory is reduced to the classical one.” However, in physics the Planck constant is a ~ e constant although it is very small. Therefore, it is important to study the relation between classical and that a characquantum evolutions when the Planck constant is k e d . There is a ~onjecture’~-’~~* teristic timescale T appears in the quanta1 evolution of chaotic dynamical systems. For time less than T there is a correspondence between quantum and classical expectation values, while for times greater that r the predictions of the classical and quantum dynamics no longer coincide. The important problem is to estimate the dependence 7 on the Planck constant h. Probably a universal formula expressing in terms of 11 does not exist and every model should be studied case by case. It is expected that certain quantum and classical expectation values diverge on a timescale inversely proportional to some power of h.” Other authors suggest that a breakdown may be The characteristictime r associated with anticipated on a much smaller logarithmic time~cale.’~-’~ the hyperbolic jixed points of the classical motion is expected to be of the logarithmic form T = (lA)ln(C/h), where X is the Lyapunov exponent and C is a constant which can be taken to be the classical action. Such a logarithmic timescale has been found in the numerical simulations of some dynamical models? It was shown also that the discrepancy between quantum and classical evolutions is decreased by even a small coupling with the environment, which in the quantum case leads to decohe~ence.~ The chaotic behavior of the classical dynamical systems is often investigated by computing the Lyapunov exponents. An alternative quantity measuring chaos in dynamical systems, which is called the chaos degree, has been suggested in Ref. 24 in the general fi-amework of information dynamic^?^ The chaos degree was applied to various models in Ref. 26. An advantage of the chaos degree is that it can be applied not only to classical systems but also to quantum systems as well. In this work we study the chaotic behavior and the quantum-classical correspondence for the Baker’s map.15327 The quantum Baker’s map is a simple model invented for the theoretical study of quantum chaos. Its mathematical properties have been studied in numerical works. In particular its semiclassicalproperties have been ~onsidered,’~-’~ quantum computing and optical realizations have been p r o p o ~ ed ; ~ -various ~~ quantization procedures have been d i s c ~ s s e d , ’ ~ and , ~ ~a- ~ ~ symbolic dynamics representation has been g i ~ e n . 3 ~ It is well known that for the consideration of the semiclassical limit in quantum mechanics it is very useful to use coherent states. We d e k e an analog of the coherent states for the quantum Baker’s map. We study the quantum Baker’s map by using the correlation functions of the special form which corresponds to the expectation values, translated in time by the unitary evolution operator and taken in the coherent states. To explain our formalism we fist discuss the classical limit for correlation functions in ordinary quantum mechanics. Correspondencebetween quantum and classical expectation values for the Bakei’s map is investigated and it is numerically shown that it is lost at the logarithmic timescale. The chaos degree for the quantum Baker’s map is computed and it is demonstratedthat
139 736
Inoue, Ohya, and Volovich
J. Math. Phys.. Vol. 43, No. 2, February 2002
it describes the chaotic features of the model. The dependence of the chaos degree on the Planck constant is studied and the correspondence between classical and quantum chaos degrees is established.
II. QUANTUM VERSUS CLASSICAL DYNAMICS In this section we discuss an approach to the semiclassical limit in quantum mechanics by using the coherent states (see Ref. 6). Then in the next section an extension of this approach to the quantum Baker’s map will be given. Consider the canonical system with the Hamilton function H= in the plane @ , x )
E
P2
+V(X) 2
R2.We assume that the canonical equations i ( t ) = p ( t ) , j ( t ) = - V‘(x(t))
(2)
have a unique solution ( x ( t ) , p ( t ) ) for times It( < T with the initial data
x( 0) = x o ,
p ( 0 )=uo.
(3)
Tbis is equivalent to the solution of the Newton equation (4)
f ( t ) = - V’(x(t)),
with the initial data x( 0) = x o ,
(5)
i ( 0 )= u o .
We denote 1 a=-(xo+iuo).
dz
The quantum Hamilton operator has the form
where ph and q h satisfy the commutation relations
[Pfi,qhl=-ih. The Heisenberg evolution of the canonical variables is dehed as
p h ( t ) = u(t)Ph u(t )* I
h (t ) =
u(t )
h
u(1) *
9
where
u(t)=exp(-itHh/h). For the consideration of the classical h i t we take the following representation,
ph=-ihlna/dX,
qh=hlnX,
acting to functions of the variable x E R. We also set
140 Semiclassical properties and chaos degree
J. Math. Phys.,Vol. 43, No. 2, February 2002
a=
A(
1 (qh+iph)=fih'"
737
1
, a*=-
x+-
fih1/2
Then, [a,a*]=1.
The coherent state la) is d e b e d as la)= W(a)IO),
where a i s acomplexnumber, W(a)=exp(ma*-aa*) vacuum vector is the solution of the equation
(4h+ iPh) lo)
(7)
and 10) is thevacuumvector, a10)=0. The
=o-
(8)
In the x-representation one has
10)
= exp( - x2/2)/
6.
(9)
The operator W( a ) one can write also in the form
w ( ~= c,iqpo ) lhine-iphxo/h'n
(10)
where C=exp(-v~d2h). The mean value of the position operator with respect to the coherent vectors is the real valued function
q( t , cY,h)= (h-'"al
q h ( t )177-l"(y).
(11)
Now one can present the following basic formula describing the semiclassical limit
limq( t,a,h) = X ( t , a ) . h+O
Here x ( t , a ) is the solution of (4) with the initial data (5) and a is given by (6). Let us notice that for time f = 0 the quantum expectation value q ( t ,a , h ) is equal to the classical one:
q(O,a,h) =x(O,a) =xg
(13)
for any h. We are going to compare the time dependence of two real functions q(t,a,l?)and x ( t , a ) ; these functions are approximately equal. The important problem is to estimate for which
t the large difference between them wiU appear. It is expected that certain quantum and classical expectation values diverge on a timescale inversely proportional to some power of 1 1 . ' ~ Other authors suggest that a breakdown may be anticipated on a much smaller logarithmic time~cale.'~-'~ One of very interesting examples5 of classical systems with chaotic behavior is described by the Hamilton function
Pi P; H= -+ -+hxfx;. 2
2
The consideration of this classical and quantum model within the described framework will be presented in another publication.
141 738
Inoue, Ohya. and Volovich
J. Math. Phys., Vol. 43, No. 2, February 2002
111. COHERENT STATES FOR THE QUANTUM BAKERS
MAP
The classical Baker's transformation maps the unit square OCq, p G 1 onto itself according to
(q,p)--t
1
( 2 q , p / 2 ) , if o s q s f , ( 2 q - l , ( p + l ) / 2 ) , if
f
This corresponds to compressing the unit square in the p direction and stretching it in the q direction, while preserving the area, then cutting it vertically and stacking the right part on top of the left part. The classical Baker's map has a simple description in terms of its symbolic dynami~s.3~ Each point ( q , p ) is represented by a symbolic string
5=. ..5-25-150 .5152... where &E{0,1},
3
(14)
and m
m
The action of the Baker's map on a symbolic string 5 is given by the shift map (Bernoulli shift) U dejined by U(=.$', where (;=&+I. This means that, at each time step, the dot is shifted one place to the right while the entire string remains ~ e dAfter . n steps the q coordinate becomes m
This relation d e h e s the classical trajectory with the initial data m
Quantum Baker's maps are d e h e d on the D-dimensional Hilbert space of the quantized unit square. To quantize the unit square one defkes the Weyl unitary displacement operators 6 and P in D-dimensional Hilbert space, which produces displacements in the momentum and position directions, respectively, and the following commutation relation is obeyed,
6P= EGG, where E= exp(Zm7D). We choose D = ZN, so that our Hilbert space will be the N qubit space CBN. The constant h = 1/D= 2-N can be regarded as the Plank constant. The space Cz has a basis
The basis in CBN is
We write
Semiclassical properties and chaos degree
J. Math. Phys., Vol. 43, No. 2, February 2002
739
where
is the basis in C*N,
and j+ f qj=F,j=O,l, ...,2N-1.
The momentum operator is defhed as I ~ = F N ~ F,; * ,
where FN is the quantum Fourier transform acting to the basis vectors as D-1
where D =2N. The symbolic representation of quantum Baker’s map T was introduced by Schack and Caves33 and studied in Refs. 35 and 36. Let us explain the symbolic representation of the quantum Baker’s map as a special case:33BY applying a partial quantum Fourier transform m
G,= I@. . .@I@ FN-”,
to the position eigenstates, one obtains the following quantum Baker’s map T: TI-61. * ‘ e N ) 161
-
52.
.’e.v),
where
T= G ~ 1 and
~
~
~
1
143 740
J. Math.
Phys., Vol. 43, No. 2,
Inoue, Ohya, and Volovich
February 2002
161"'~N-m.6N-m+l"'6N)"GmlSN-m+l"'SNSN-m"'61) -
- 1 6N-m + 1) '8. ' ' @ I 6N)@FN-ml 6N-m)@ ' * ''8I 61).
The quantum Baker's map T is the unitary operator in CaN with the following matrix elements,
l~)=l~l~2.*.[N),
where l ~ ) = l ~ 1 ~ 2 . *and * ~ 8N( x)) is the Kronecker symbol, 8(0)=1; 8 ( x ) =o, x f O . We d e h e the coherent states by
ICY) = c e z d u e - z 4 x l
+,,).
(18)
+
Here CY = x iu, x and u are integers, C is the normalization constant and I Go) is the vacuum vector. This dejlnition should be compared with (10). The vacuum vector can be d e h e d as the solution of the equation (qh+iPh)I+d=O
[compare with (S)]. We will use the simpler deiinition which in the position representation is (4jI + ~ ) = C E X P ( - ~ ; / ~ )
[compare with (9)]. Here C is a normalization constant.
IV. CHAOS DEGREE
Let us review the entropic chaos degree defined in Ref. 24. This entropic chaos degree is given by a probability distribution 'p and a dynamics (channel) A* sending a state to a state; 'p = Z p J k , where 8, is the delta measue such as
Then the entropic chaos degree is defined as
with the von Neumann entropy S, equivalent to the Shannon entropy because the probability distribution 'p is a classical object. A dynamics T o f the orbit produces the above channel A*, so let {x,} be the orbit and 3be a map fiom x , to x,,+1. Take a h i t e partition {Bk} ofI=[a,b]' ( a , b ER)CR' such as
I=
U B ~( ~ ~ f h ~ = r a , i # j ) k
for a map 7on I with x , + ~= F((x,,)(a difference equation). The state $) of the orbit determined by the difference equation is d e h e d by the probability distribution that is, 'p(")=p(") =X,pj")Gi, where for an initial value x € 1 and the characteristic function l A
@I")),
Semiclassical properties and chaos degree
J. Math, Phys., Vol. 43, No. 2. February 2002
.
741
m+n
when the initial value x is distributed due to a measure v on I , the above p p ) is given as
In the case that 3is a classical Baker's transformation, if the orbit is not stable and periodic, then it is shown that the m-rm limit ofpj") exists and equals a natural invariant measure for a fixed n E N. 37 The joint distribution @$'sJ1+l)) between the time I z and n + 1 is defined by
or
,
,m+n
Then the channel A: at n is determined by
and the chaos degree is given by
We can judge whether the dynamics causes a chaos or not by the value of D as
D > Oechaotic, D = 0-stable. Therefore, it is enough to k d a partition {Bk} such that D is positive when the dynamics produces chaos. This classical chaos degree was applied to several dynamical maps, such as a logistic map, a Baker's transformation and a Tinkerbel map, and it could explain their chaotic ~ h a r a c t e r s ?Our ~~~~ chaos degree has several merits compared with usual measures such as the Lyapunov exponent.
V. EXPECTATION VALUES AND CHAOS DEGREE In this section, we show a general representation of the mean value of the position operator 4^ for the time evolution, which is constructed by the quantum Baka's map. Then we give the algorithm to compute the chaos degree for the quantum Baker's map, To study the time evolution and the classical limit h-tO which corresponds to N - m ofthe quantum Baker's map T , we introduce the following the mean value of the position operator 4 for time r~ E N with respect to a single basis
1 e}:
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J. Math. Phys., Vol. 43, No. 2,February 2002
where It)= 15162.. . t N ) . From (17), the following formula of the matrix elements of T" for any tained:
11
E N is easily ob-
where A is the 2 x 2 matrix with the element AxlX2=exp((7i-/2)ilxl-x2l) for x l , x 2 = 0 , 1, p =1,...,N-1 andmEN. Using these formulas, the following theorems are obtained and their proofs are given in the Appendix. Theorem 5.1:
where A is the 2 x 2 niatrix with the element AXlXZ=exp((7i-/2) ilxl-x2() for x l , x 2 = 0 , 1, p = 1,...,N- 1 and m EN. By diagonalking the matrix A , we obtain the following formula of the absolute square of the matrix elements of A" for any n E N. Lemma 5.2: For any n EN, we have
Z"cos2( I(A")kj["
2"sin2(
7) 7)
if k = j
if k # j
Combining Theorem 5.1 and Lemma 5.2, we obtain the following two theorems with respect to the mean value rLw of the position operator. Theorem 5.3: For the case n = m N + p , p = 1,2,...,N - 1 and m E N , we have
Semiclassical properties and chaos degree
J. Math. Phys., Vol. 43, No. 2, February 2002
743
whei-e vk= &+ 1(rnoa), k= I ,...,N . Theorem 5.4: For the case n = mN, n z EN, we have
T$)=
-
if m=1,3(mod 4),
2
?'k2-kf
1 z" f
k= 1
if nz=2(mod 4 ) .
Using formulas ( 2 3 ) - ( 2 5 ) , the probability distribution @?)) of the orbit of mean value rbv of the position operator 4 for the time evolution, which is constructed by the quantum Baker's map, is given by ,
m+n
for an initial value rim ~ [ 0 , 1 ]and the characteristic function 1,. The joint distribution ( p P n + l ) ) between the time n and nf 1 is given by
.
m+n
I 0.9
0.8 0.7
0.6 0.5 0.4
0.3 0.3
0.I
n
pz
n
200
40(1
6W
800
FIG. 1. The distribution of r$) for the case N=500.
IWO
147 744
J. Math. Phys.. Vol. 43,No. 2, February 2002
Inoue,
Ohya, and Volovich
n 0
?in)
400
FIG. 2. The distriiution of the classical value q(") for fie case N= 500.
Thus the chaos degree for the quantum Baker's map is calculated by
whose numerical value is shown in the next section. VI. NUMERICAL SIMULATION OF THE CHAOS DEGREE AND CLASSICAL-QUANTUM CORRESPONDENCE
i-iw
We compare the dynamics of the mean value of position operator with that of the classical value q n in the q direction. We take an initial value of the mean value as
D
(i) N=100
(iii) N=500
D
(ii) N=300
(iv) N=700
FIG. 3. The change of the chaos degree for several N's up to time n = 1000.
Semiclassical properties and chaos degree
J. Math. Phys., Vol. 43,No. 2, February 2002
745
11.2 11.1s
lLU5 101)
:IN)
EX)
<XI
5W
hM1
JIXI
SIX1
91xJ
N
FIG. 4. The Werence ofthe chaos degree between quantum and classical for the case n = 1000.
N
where t i is a pseudo-random number valued with 0 or 1. At the time zero we assume that the classical value q o in the q direction takes the same value as the mean value rbv of position operator 4. The distribution of for the case N=500 is shown in Fig. 1 up to the time n =1000. The distribution of the classical value qn for the case N=500 in the q direction is shown in Fig. 2 up to the time n = 1000. Figure 3 represents the change of the chaos degree for the case N = 100,300,500,700up to the time n= 1000. The correspondencebetween the chaos degree D , for the quantum Baker's map and the chaos degree D,for the classical Baker's map for some k e d N's (100,300,500,700 here) is shown for the time less than T=logz (llh) = 10g,2~=N,and it is lost at the logarithtic time scale T. Here we took a iinite partition {Bk} of I=[O,l] such as Bk=[k/lOO,(k+l)/lOO))(k=O,l, ...,98) and B,, =[99/lOO,l) to compute the chaos degree numerically. The difference of the chaos degrees between the chaos degree D, for the quantum Baker's map and the chaos degree D , for the classical Baker's map for a iixed time n (1000, here) is displayed wxt. N in Fig. 4. Thus we conclude that the dynamics of the mean value rn(N)reduces the classical dynamics q,, in the q direction in the classical limit N+m(h-+O). The appearance of the logarithmic timescale have been proved rigorously in our recent paper.j*
ACKNOWLEDGMENTS
The main part of this work was done during the visit of I.V. to the Science University of Tokyo. I.V. is grateful to JSPS for the Fellowship award. Our work was also partially supported by RFFI 99-0100866, INTAS 99-00545 and SCAT.
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J. Math. Phys.. Vol. 43, No. 2, February 2002
APPENDIX A: Proof of Theorem 5.1: By a direct calculation, we obtain
Using (22),the mean value
r-k, in the case n
can be expressed as
150 J. Math. Phys., Vol. 43, No. 2, February 2002
Semiclassical properties and chaos degree
For the case n =N , we similarly obtain
=
For n = niN+p, p = 1,2,.
and for n=mN, m E N ,
..,N-
2
-
1 1 1 1 -( 2N- 1)2N+ =2 2 2‘ 1, ni E N ,
747
151 I n o w Ohya, and Volovlch
J. Math. Phys., Vol. 43, NO. 2, February 2002
748
E
Proof of Lemma 5.2: By a direct calculation, the matrix A is diagonalized as foUows:
where
From (Al), we have A"= FD"F*
Using (M),it follows that for any k = j , k= 1,2,
I(A")kjIz= +{( 1+ i)" + ( 1- i)"} g(1+ i)" + ( 1- i)"} = :{(
+
1 i)"+(l -i)"}{(I - i ) " + ( l + i)"} = ${(1+ i ) " + ( l - '}")i
='( ( 4
=;[
4 7 ' 3'
fi!3)"+(
(fi)"( ?)"+(d)"(
[=: { : y - yi)] [ : [ ,,,(7) y ) +[) y ) y)]IZ (exp(:i))"+(exp(
=
=
and for any k # j , k= 1,2,
exp(
i ) +exp(
+isin(
-:i))"]' 2
WS(
-isin(
152 J. Math. Phys., Vol. 43, No. 2, February 2002
Semiclassical properties and chaos degree
Proof of Theorem 5.3: For the case 1'1= niN+p, p = 1,. ..,N- 1 and 771 E N,
By a direct calculation, we obtain
N-P
(i) rn=O(mod4)
P
749
153 750
J. Math. Phys., Vol. 43, No. 2, February 2002
Inoue, Ohya, and Volovich
From the above lemma, we have
for any I= l;.. , N - p and k = N - p + l;.. , N . Using this formula the product of absolute squares can be expressed as N- D
N
2mN if j l = t p + r for all I = I;.. , ~ - p , 0
otherwise.
Equation (A3) can be rewritten as
- 2 ( m + l ) N + p I.N - p + l
N
N-p
2mN 9'''
.
JN
(ii) m = 1(mod 4) From the above lemma, we have
[ ( Zl
*p+k2N-k)
+ (k=N-p+l
ikZN-')
+
i]
(A4)
154 J. Math. Phys., Vol. 43, No. 2, February 2002
for any I= 1; .. ,N-p and k= N - p can be expressed as N-
N-
N
D
D
(iii)m=2(rnod4)
N
Semiclassical properties and chaos degree
751
+ 1,; .. ,N. Using this formula the product of absolute squares
155 752
J. Math. Phys., Vol. 43,No. 2, February 2002
Inoue. Ohya, and Volovich
From the above lemma, we have
-
+ 1;. . ,N. Using this formula the product of absolute squares
for any I = 1;. , N - p and k= N - p can be expressed as N-P
N
rI I(Arn)
l= 1
=I
*p
,=gI
+JIl
P+l
(2m)N-p(2m)p 0
if
( A m+
1*k--(N-p)ikI
ji+ tP+{for all I= 1,. . . , ~ - p ,
otherwise,
if j r # t p + / for all I= 1,...,N - p , = ( 2"N 0 otherwise. Let ~ p + l = ~ p + ~ + l ( m o dI=1, 2 ) ,...,N - p . It follows that
N-
N
D
Substituting v p + k for
tp+k
(A4), we get
(iv) m = 3 (mod 4) From the above lemma, we have
for any 1 = 1 , ...,N - p and R=N-p+ 1,...,N . Using this formula the product of absolute squares can be expressed as N-P
N
Equation (A3) can be rewritten as
156 J. Math. Phys.,Vol. 43, No. 2, February 2002
N-v
Semiclassical properties and chaos degree
N
Proof of Theorem 5.4: For any n = mN, m EN,
By a direct calculation, we obtain
(i) rn=O(mod4) From the above lemma, we have
Using this formula the product of absolute squares can be expressed as for any k= 1,...,N.
Using this formula the mean value rkw of the position operator can be expressed as
(E) 7n= 1,3(mod4)
753
157
754
Inoue, Ohya, and Volovich
J. Math. Phys., Vol. 43. No. 2, February 2002
From the above lemma, we have
I (Am)fdk12=2 m - 1 for any k = 1 , ...,N . Note that
Using this formula the mean value r:? of the position operator can be expressed as
1
=p q
zo F =
2N- 1
k+
1
1
2
2N-'(2N-
1)+
1 2
=
1 -. 2
(iii) m = 2 (mod 4) From the above lemma, we have
for any k= 1 ,..., N . Using this formula the product of absolute squares can be expressed as
Let v k = ' $ k f l ( m O d 2 ) ,
k=1,...,N. It follows that
'Qvnomical System, edited by D. V. Anosov and V. I. Amold (VJNITI, Moscow, 1996). 'Ya. G. Sinai, Inhoduciion to Bgodic Theory (Fasis. Moscow, 1996). 'M. C. Gutzwiller, Chaos in CIassical and Quanfum Mechanics (Springer, Berlin, 1990). Quantum Chaos: Between Order and Disorder, edited by G.Casati and B. V. Chirikov (Cambridge Udversity press, Cambridge, 1995). 51. Y.Arefev5 P. B. Medvedev, 0. A. Rytchkov, and I. V. Volovich, Chaos, Solitons Fractals lO(2-3), 213 (1999). 'K Hepp, Commun. Math. Phys. 35,265 (1974). 'W. H. Zurek, Phys. Rev. D 24, 1516 (1981). 'W. It Zurek, quant-ph/ol5127.
J. Math. Phys., Vol. 43, No. 2, February 2002
Semiclassical properties and chaos degree
755
'G. G. Emch, H. Namhofer, W. Thining, and G. L. Sewell, J. Math. Phys. 35, 5582 (1994). Hasegawa, Open Syst. I d Dyn. 4, 359 (1997). "V. P. Maslov, Perturbuiiou Theoiy und Asyiilpiotic Methodr (MGU, Moscow, 1965). "G. P.Berman and G. M. Zaslavsky, PhysicaA 91A, 450 (1976). l3 G. M. Zaslavskii, Stochasiiciv of @i?mnicaZ $sterns (Nauka, Moscow, 1984). V. Berry, Some Quuntum io Clussicul Asynipioiics, Les Houches Summer School "Chaos and quantum physics," edited by M. J. Giannoni, k Voros, and 2. Justi (North-Holland,Amsterdam, 1991). "N. L. Balazs and A. Voros, AM. Phys. 190, 1 (1989). I6A. M Omrio de Almeida and M. Saraceno, Ann. Phys. 210, 1 (1991). "F. M. Dittes, E. Doron, andU. Smilansky, Phys. Rev. E 49, R963 (1994). "M. Saraceno and A. Voros, Physica D 79, 206 (1994). "L. Kaplan and E. J. Heller, Phys. Rev. Lett 76, 1453 (1996). 'OM. G. E. da Luz and k M Ozorio de Almeida, Nonlinearity 8,43 (1995). "R Schack and C. M. Caves, Phys. Rev. Lett.71, 525 (1993). =R Schack and C. M. Caves, Phys. Rev. E 53, 3257 (1996). nP. W. O'Comor and S. TomsoVic, Ann. Phys. 207, 218 (1991). 24M. Ohya, Int. J. Theor. Phys. 37(1), 495 (1998). 25RS. Ingarden,k Kossakowski, and M. Ohya, Jlfoimution Dynaniics uizd Opeir Systems (Kluwer Academic, Dordrecht, 1997). 26KInoue, M. Ohya, and K Sato, Chaos, Solitons Fractals 11, 1377 (2000). 27M. Saraceno, Ann. Phys. 199, 37 (1990). 28J. H. Hamay, J. P. Keating, and A. M. Ozorio de Almeida, Nonlinearity 7, 1327 (1994). "R Schack, Phys. Rev.A 57, 1634 (1998). "T. B m and R Schack, Phys. Rev. A 59,2649 (1999). "A. Lakshminarayan and N. L. Balazs, Ann. Phys. 226, 350 (1993). "A. Lakshminarayan, Ann. Phys. 239, 272 (1995). 33RSchack and C. M. Caves, Applic. Alg. Eng. Commue Comput. AAECC 10, 305 (2000). 34V.M. Alekseev and M N. Yakobson, Phys. Rep. 75,287 (1981) 3sA. N. Soklakov and R Schack, quant-pW9908040. 36k N. Soklakov and R Schack, quant-pb/0107071. 37B.0% Chaos in Dynumical Systems (Cambridge University Press, Cambridge, 1993). 38KJnoue, M Ohya, and L V. VoloVich, quant-pWO108107.
10H.
159 Commun. Math. Phys. 225,67 - 89 (2002) With kind permission of Springer Science and Business Media
Communications in
Mathematical Physics 0 Springer-Verlag 2002
Quantum Teleportation and Beam Splitting Karl-Heinz Fichtner’, Masanori Ohya2
*
Friedrich-Schiller-UniversitatJena, Fakultat f i r Mathematik und Informatik, Institut fir Angewandte Mathematik, 07740 Jena, Germany. E-mail: [email protected] Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278-85 10, Japan. E-mail: [email protected]
Received: 1 February 2001 /Accepted 19 July 2001
Abstract: Following the previous paper in which quantum teleportation is rigorously discussed with coherent entangled states given by beam splittings, we further discuss two types of models, the perfect teleportation model and non-perfect teleportation model, in a general scheme. Then the difference among several models, i.e., the perfect models and the non-perfect models, is studied. Our teleportation models are constructed by means of coherent states in some Fock space with counting measures, so that our model can be treated in the frame of usual optical communication. 1. Introduction
Following the previous paper [ 121, we further discuss non-perfect teleportation. The notion of non-perfect teleportation is introduced in [ 121 to construct a handy (i.e., physically more realizable) teleportation, although its mathematics becomes a little more complicated. For the completeness of the present paper, we quickly review the meaning of teleportation and some basic facts of Fock space in tlus section. Then we dicuss perfect teleportation in a very general (more general than that given in [ 121) scheme with our previous results, and we state the main theorems obtained in [ 121 for non-perfect teleportation, both in Sect. 2. The main results of this paper are presented in Sect. 3, where we discuss the difference among three models, i.e., the perfect model, the non-perfect one given in [12] and that discussed in the present paper. The proofs of the main results are given in Sect. 4.
1.1. Quantum teleportation. The study of quantum teleportation was started in paper [3], whose scheme can be mathematically expressed in the following steps [ I 1,121: Step 0: A girl named Alice has an unknown quantum state p on (a N-dimensional) Hilbert space and she was asked to teleport it to a boy named Bob.
160 68
K.-H. Fichtner, M.Ohya
Step 1: For this purpose, we need two other Hilbert spaces 7-12 and 'U3,7-12 is attached to Alice and 'U3 is attached to Bob. Prearrange a so-called entangled state (T on 7-12 '8 'U3 having certain correlations and prepare an ensemble of the combined system in the state p '8 (T on 7-11 '8 7-12 '8 7-13. Step 2: One then fixes a family of mutually orthogonal projections ( F , z l n ) ~ m on = l the ,~ Hilbert space 7-11 '8 7-12 corresponding to an observable F := C z , ~ Frlm, n,m
and for a fixed pair of indices n , m, Alice performs a first kind incomplete measurement, involving only the ' U l @ 7-12 part of the system in the state p '8 (T, which filters the value zIlm,that is, after measurement on the given ensemble p '8 (T of identically prepared systems, only those where F shows the value znm are allowed to pass. According to the von Neumann rule, after Alice's measurement, the state becomes
where tr123 is the full trace on the Hilbert space 'UI '8 7-12 @ 'U3. Step 3: Bob is informed which measurement was done by Alice. This is equivalent to transmitting the information that the eigenvalue Z,rm was detected. This information is transmitted from Alice to Bob without disturbance and by means of classical tools. Step 4: Making only partial measurements on the third part of the system in the state p,1f3' means that Bob will control a state An,,z( p ) on 'H3 given by the partial trace on 7-11 '8 7-12 of the state p,1Lf3' (after Alice's measurement) (123)
Anm(p> = tr12 P,lm
= tr12
( F n m '8 1 ) @ ~ ( T ( F n m '8 1)
tr123(Fnm '8 1)p '8 ( T ( F n m '8 1 ) '
Thus the whole teleportation scheme given by the family (Fnm)and the entangled state (T can be characterized by the family ( A n m ) of channels from the set of states on 7-11 into the set of states on 7-13 and the family (prim) given by Pnm(p> := tr123(Fnm '8 1)p '8 a(Fnm @ 1)
ofthe probabilities that Alice's measurement according to the observable F will show the value znm. The teleportation scheme works perfectly with respect to a certain class 6 of states p on 7-11 if the following conditions are fulfilled:
(El) For each n, m there exists a unitary operator v,,,,, : 7-11 + 7-13 such that A,wn(p) = unm P
nm
UlTm
(P E
6>,
161 Quantum Teleportation and Beam Splitting
69
(El) means that Bob can reconstruct the original state p by unitary keys (u,,,) provided to him. (E2) means that Bob will succeed to find a proper key with certainty. Such a teleportation process can be classified into two cases [ 11, i.e., weak teleportation and general teleportation, in which the solutions of the teleportation in each case and the conditions of the uniqueness of the unitary key were discussed. The solution of the weak teleportation is a triple F(I2),U } such that
holds for any state ,o(’) E S(Z1) . Once a weak solution of a teleportation problem is given, we can construct the general solution for all n , m above [ 11. In [l2], we considered a teleportation model where the entangled state (T is given by the splitting of a superposition of certain coherent states, although this model doesn’t work perfectly, that is, neither (E2) nor (El) hold. In the same paper, we estimated the difference between the perfect teleportation and this non-perfect teleportation by adding a further step in the teleportation scheme: Step 5: Bob will perform a measurement on his part of the system according to the projection
where lexp(0)) (exp(O)(denotes the vacuum state (the coherent state with density 0). Then our new teleportation channels (we denote it again by A,,,,) have the form
and the corresponding probabilities are p,trn(P) := tr123(Fnrn 8 F+> P 8 “(Fnrn 8 F+>.
For this teleportation scheme, (El) is fulfilled but (E2) is not, which we review in the next section.
1.2. Basic notions and notations. We collect some basic facts concerning the (symmetric) Fock space in a way adapted to the language of counting measures. For details we refer to [6-8,2,9]. Let G be an arbitrary complete separable metric space. Further, let p be a locally finite diffuse measure on G, i.e. p ( B ) i+co for bounded measurable subsets of G and p ( ( x ) ) = 0 for all singletons x E G. We denote the set of all finite counting measures on G by M = M ( G ) . Since cp E M
can be written in the form cp =
C 6,j
for some n = 0, 1 , 2 , . . . and x j
E G
with the
j=I
Dirac measure 6, corresponding to x
E
G, the elements of M can be interpreted as finite
162 K.-H. Fichtner, M. Ohya
70
(symmetric) point configurations in G. We equip M with its canonical a-algebra D(cf. [6,7]) and we consider the a-finite measure F by setting
where X y denotes the indicator function of a set Y and 0 represents the empty configuration, i. e., O ( G ) = 0. Since p was assumed to be diffuse one easily checks that F is concentrated on the set of simple configurations (i.e., without multiple points) M := {cp E
M ;cp({x)) I 1 for all x
E
GI.
M
= M ( G ) := L 2 ( M ,D,F ) is called the (symmetric) Fock space over G. In [6] it was proved that M and the Boson Fock space r ( L 2(G)) in the usual definition are isomorphic. For each Q, E M with Q, # 0 we denote by I@) the corresponding normalized vector
Further, I Q,) (Q, I denotes the corresponding one-dimensional projection describing a pure state given by the normalized vector I@).Now, for each n 1 1 let M@' be the n-fold tensor product of the Hilbert space M , which can be identified with L 2 ( M " , F"). For a given function g : G + C the function exp (8) : M + C defined by
is called an exponential vector generated by g. Observe that exp ( g ) E M if and only if g E L 2 ( G ) and one has in this case 2 llexp (g)1I2 = ellgll and lexp ( g ) ) = e-411gt12exp(g).Theprojection lexp (g))(exp (g)l is called the coherent state corresponding to g E L 2 ( G ) .In the special case g := 0 we get the vacuum state lexp(0)) = 'q0) . The linear span ofthe exponential vectors of M is dense in M , so that bounded operators and certain unbounded operators can be characterized by their actions on exponential vectors. The operator D : dom(D) -+ M B 2 given on a dense domain dom(D) c M containing the exponential vectors from M by
+
D@(YJI, ( ~ 2 ):= @ ( a ~
2
)(@ E
dom(D), P I , ID^
E
M)
is called the compound Hida-Malliavin derivative. On exponential vectors exp (g) with g E L 2 ( G ) ,one gets immediately
163 Quantum Teleportation and Beam Splitting
71
The operator S : dom(S) -+ M given on a dense domain dom (S) containing tensor products of exponential vectors by S@(cp) :=
@($,cp - Cp)
c M@’
(@ E dom(S), cp E M )
050
is called the compound Skorohod integral. One gets
For more details we refer to [lo]. Let T be a linear operator on L 2 ( C )with 11 T 11 5 1. Then the operator r ( T ) called the second quantization of T is the (uniquely determined) bounded operator on M fulfilling
Clearly, it holds
It follows that r ( T ) is a unitary operator on M if T is a unitary operator on L 2 ( G ) . In [ 121 we proved. Lemma 1.1. Let K1, K2 be linear operators on L 2 ( C )with aproper-ty
Then there exists exactly one isometry u K I , K 2from M to MB2 = M 8 M with
(at least on dom( D ) but one has the unique extension). The adjoint u i l ,K 2 of characterized by u i , , ~ ~ ( e (xhp) 8 exp (g)) = exp(KTh
+ K;g)
(8, h E L 2 ( G ) )
vK1, K is ~
(8)
and it holds u ; , , ~=~ S(r(KT) @ r(K;)).
(9)
164 K.-H. Fichtner, M. Ohya
72
From K 1 , K2 we get a transition expectation ~ K , K: M~ €3 M + M , using V K ~ , K and the lifting C i l K 2 may be interpreted as a certain splitting (cf. [2]). The property (5) implies IIKig1I2
+ IIK2g1I2 = I1g1l2
(g E L 2 ( G ) > .
(10)
Let U ,V be unitary operators on L2(G).If operators K1, K2 satisfy ( 5 ) , then the pair K1 = U K 1 , K 2 = V K 2 fulfill ( 5 ) . Here we explain the fundamental scheme ofbeam splitting [8]. We define an isometric operator Va,p for coherent vectors such that V,,pI exp ( g ) ) = I exp (a&?)) €3 I exp ( B g ) )
+
with I a! l 2 I /Il2 = 1.This beam splitting is a useful mathematical expression for optical communication and quantummeasurements [2]. As one example, take a! = B = in the above formula and let K1 = K2 be the following operator of multiplication on L 2 ( G ) :
In this case, we put
then we obtain
Another example is given by taking K1 and K2 as the projections to the corresponding subspaces 311,312 of the orthogonal sum L 2 ( G ) = 311 @ 312. In [ 121 we used the first example in order to describe a teleportation model where Bob performs his experiments on the same ensemble of the systems as Alice. Further we used a special case of the second example in order to describe a teleportation model where Bob and Alice are spatially separated (cf. Sect. 5 of [ 121). 2. Previous Results on Teleportation
Let us review some results obtained in [12]. We fix an ONS {gl, . . . , g N } 2 L 2 ( G ) , operators K1, K2 on L 2 ( G ) with (9,a unitary operator T on L 2 ( G ) ,and d > 0. We assume
Using (I 1) and (I 2) we get
~
165 73
Quantum Teleportation and Beam Splitting
From ( 1 2) and ( 1 3 ) we get (Kzgk, Kzgj) = 0
( k f j ; k , j = 1 . . .. ,N ) .
(15)
The state of Alice asked to teleport is of the type
where
anda = &. One easily checks that (lexp (aKlgj) - exp (0)))y=land (lexp aK2gj) exp (0)))y=lare ONS in M . The set {as; s = 1 , . . . , N ] makes the N-dimensional Hilbert space 3cl defining an input state teleported by Alice. Although we may include the vaccum state lexp (0)) to define 3c 1 , here we take the N-dimensional Hilbert space ?f 1 as above because of computational simplicity. is still an ONS in M we assume In order to achieve that (I @,7)):=I
Denote cs = [csl,..., c , N ] E C N ,then (c,):=~ is an CONS in C N . Let (bn):=l be a sequence in CN,
with properties
(ba,b,)=O
( n f j ; n , j = 1 , . . . ,N ) .
(20)
Now, for each m, n (= 1 , . . . , N ) , we have unitary operators Urn,Bn on M given by
166 K.-H. Pichtner, M.Ohya
74
2.1. A perfect teleportation. Then Alice's measurements are performed with the projection
given by
One easily checks that (1Cn;2")):m=l is an ONS in M B 2 .Further, the state vector 14) of the entangled state o = 16) 1 is given by
(c
In [ 121 we proved the following theorem.
Theorem 2.1. For each n , m = 1, . . . , N , define a channel An, by
Then we havefor all states p on M with (1 6) and (I 7), Anm(p) = (r(T)UmB;) P (r(T>UmB,*)*.
(27)
Remark 2.2. Using the operators B , , U,, r ( T ) ,the projections F,,, are given by unitary transformations of the entangled state o : Fnm = (Bn €3 U m r ( T * )0 ) (Bn
8 umr(T*))*,
(28)
or Itnrn) = ( B n €3 U m r ( T * ) ) It).
If Alice performs a measurement according to the following selfadjoint operator N n.m=l
with {znmln,m = 1 , . . . , N ) 5 R - { 0 ) , then she will obtain the value znm with probability 1 / N 2 . The sum over all these probabilities is 1, so that the teleportation model works perfectly. Before stating some hndamental results of [12] for the non-perfect case, we note that our perfect teleportation is obviously treated in general finite Hilbert spaces %k ( k = 1,2,3) the same as the usual one [2]. Moreover, our teleportation scheme can be generalized a bit by introducing the entangled state 6 1 2 on "1 €3 Z2 defining the projections { Fnm}by the unitary operators B,, U,,, . We here discuss the perfect teleportation on general Hilbert spaces %k ( k = 1 , 2 , 3 ) . Let (5; j = 1, . . . , N be CONS of the
1
167 Quantum Teleportation and Beam Splitting
75
Hilbert space X k ( k = 1 , 2 , 3) . Define the entangled states and X 2 @ X3, respectively, such as
with
C12 :=
1
Cj=l N t 1j 8 4;
and
( 2 3 :=
1
012
and 023 on X I 8 X
2
Cj=, N C 2j 8 tj. By a sequence { b , =
[ b , ~., . , , b t 7 ~n ~= ] ; 1, . . . , N } in C N with the properties (19) and (20), we define the unitary operator B, and Urn such as
~,~ : =tbf, , j t f ( n , j = ~ , . . . , ~ ) a n d ~ , , ~ j : = t j ~ , ~ ( n . j, N = ~) , . . . w i t h j e m := j+m(modN).Thentheset(F,,,; n , m = I , . . . , N}oftheprojections of Alice is given by Ft7m = (Bn 8 urn)0 1 2 (Bn 8 urn)* 9
and the teleportation channels { AGm;1 2 , m = 1, . . . , N ) are defined as A n m ( P ) := tr12
(Fnm 8 1) (P 8 0 2 3 ) (F17m 8 1) trl23 (Fnm 8 1) (P 8 023) (Ftim 8 1)
Finally the unitary keys ( Wn,,; n, m = I , . . . , N } of Bob are given as Wtimtf = 5!7jt;em9( n ,m = 1,
. . . ,N ) ,
by which we obtain the perfect teleportation At,,
(PI= W n m P Wtt,
The above perfect teleportation is unique in the sense of unitary equivalence.
2.2. A non-perfect teleportation. We will review a non-perfect teleportation model in which the probability teleporting the state from Alice to Bob is less than 1 and it depends on the density parameter d (may be the energy of the beams) of the coherent vector. There, when d = a2 tends to infinity, the probability tends to 1. Thus the model can be considered as asymptotically perfect. Take the normalized vector
)'=( + I
with y :=
1
+ (N - I ) c d
and we replace in (26) the entangled state 0 by
1
( N - I)e-02
168 K.-H. Fichtner, M. Ohya
16
Then for each n , m = 1, . . . , N , we get the channels on any normal state p on M such as
where F+ = 1 - lexp (O))(exp (O)l, i.e., F+ is the projection onto the space M + of configurations having no vacuum part,
One easily checks that
that is, after receiving the state in,,, ( p ) from Alice, Bob has to omit the vacuum. From Theorem 2.1 it follows that for all p with (16) and (17),
This is not true if we replace A,,,,, by
h,,,, namely, in general it does not hold
In [ 121 we proved the following theorem. Theorem 2.3. ForallstatesponM with(16)and(17)andeachpairn,m (= 1 , . . . , N ) , we have
and
That is, the model works only asymptotically perfectly in the sense of condition (E2). In other words, the model works perfectly for the case of high density (or energy) of the considered beams.
169 Quantum Teleportation and Beam Splitting
71
3. Main Results The tools of the teleportation model considered in Sect. 2.1 are the entangled state cr and the family of projections ( F , l m ) ; , = I . In order to have a more handy model, in Sect. 2.2 we have replaced the entangled state cr by another entangled state 0 given by the splitting of a superposition of certain coherent states (30). In addition, we are going to replace the projectors Fnm by projectors Fnm defined as follows: Fnrn :=
(BIZ
8 u m r ( T ) * )6 ( B n CZI u,,r(T)*)*.
(36)
In order to make this definition precise we assume, in addition to (22), that it holds: U,,exp(O) = exp(0)
(m = 1.. .. ,N ) .
Together with (22) that implies
Formally we have the same relation between 0 and F,,,,, like the relation between cr and F,, (cf. Remark 2.2). Further for each pair n , m = 1, . . . , N we define channels on normal states on M such as
where P,L~(P>
:= tr123(Fnm 8 F+) ( P 8 0 ) (F,l,,, 8 F+)
(39)
(cf. (33), and (34)). In Sect. 4, we will prove the following theorem. Theorem 3.1. For each state p on M with (IS), and ( I 7)$each pair n , m(= 1, . . . , N ) and each bounded operator A on M it holds
From Theorem 2.1 and e-4 -0 (d + +m), Theorem 3.1 means that our modified teleportation model works asymptotically perfectly (the case of high density or energy) in the sense of conditions (El) and (E2). In order to obtain a deeper understanding of the whole procedure we are going to discuss another representation of the projectors F n m and of the channels &,,,. The starting point i s again the normalized vector Iq) given by (29). From (14) we obtain IlOJZKlgk1l2 = IlgkII
2 9
(42)
170 K.-H. Fichtner, M.Ohya
78
where O f denotes the operator of multiplication corresponding to the number (or function) f
Of1cI := f 1cI
(1cI
E L2(G)).
(43)
Furthermore (13) implies O f K l g j )= 0
(Oj’Klgk
tk # j ) .
(44)
From (42), and (44) follows that we have a normalized vector 1 i j ) given by
Remark 3.2. In the case of the example given by (1 1) of Sect. 1.2, we have
lii) = IV). Now let V be the unitary operator on M 8 M characterized by V (exp(fi) 8 exp(f2)) = exp
(2/21 (fl - f2))
1
8 exp (1/z ( f l
+ f2))
(fl,
L2(G)).
(46)
E L2(G)).
(47)
f2 E
One easily checks
v* (exp(f1) €3 exp(f2))
(h
= exp - ( f i
+ f2) ) 8 exp ($2 - (f2
- fd
1
(f1, f2
Remark 3.3. V describes a certain exchange procedure of particles (or energy) between two systems or beams (cf. [ 131). Now, using (12), (30), (49, and (47), (46) one gets I
t = V K , . K Z ( V ) = (1 8 r ( T ) ) V *tlexp(O>)8 16)) l = (1 8 ~ ( T ) ) (16) v 8 Iexp(o))>,
7
(48) (49)
and it follows 8 =I
ML
= (1 8 r ( T ) ) V *(lexp(O))(exp(O)l8 lii)(iil) ((1 8 W ) ) V * ) * .
(50)
171 ~
~
Quantum Telepnrtation and Beam Splitting
79
From the definition of Fnm(36) and (50)it follows
where
Xnm and consequently Xnm€3 1 are unitary operators. For that reason we get from (53h
and
Now from (38), (39), (55) and (56) it follows
According to (57,58) and (54), the procedure of the special teleportation model can be expressed in the following steps:
172 K.-€1. Fichtner, M. Ohya
80
Step 0 - initial state p - the unknown state Alice wants to teleport lexp(0)) (exp(0)I-vacuum state, Bob’s state at the beginning. Step 1 - Transformation according to that means: splitting of the state I f i ) (fi I. Step 2 - Transformation according to Step 3 - Transformation according to exchange of particles (or energy) between the first and the second part of the system. Step 4 - measurement according to checking for - first part in the vacuum? - in the third part is no vacuum? - second part reconstructed?
Final state sfin( p ) Now from (57) we get 6,,(p) = tr12 sfi,(p). Thus Theorem 3.1 means that in the case of high density (or energy) d we have approximately ( p with (1 6), and (1 7)) trl2 sfin(p) = ( ~ ( T ) u , B , *P) ( r ( w , , B , * ) * . The proof of Theorem 3.1 shows that we have even more, namely it holds (approximately) sfin(P) = lexp(0))(exp(0)l@~ i j ) ( f 8 i~
(~(T)u,BR)
Adding in our scheme the following step: Step 5 - Transformation (that means Bob uses the key provided to him)
P (~(T)u,B,*)*.
(59)
1 @ 1 @ ( T ( T ) U , B;)*
Then sfin(p)will change into the new final state lexp(0))(exp(0)l@lfi)(ijl 8 P . Summarizing one can describe the effect of the procedure (for large d!) as follows: At the beginning Alice has (e.g., can control) a state p , and Bob has the vacuum state (e.g., can control nothing). After the procedure Bob has the state p and Alice has the vacuum. Furthermore the teleportation mechanism is ready for the next teleportation (e.g. 16) (fi I is reproduced in the course of teleportation).
173 81
Quantum Teleportation and Beam Splitting
We have considered three different models (cf. Sects. 2.1, 2.2, 2.3). Each of them is a special case of a more general concept we are going to describe in the following: Let H I , H2 be N-dimensional subspaces of M + such that r ( T ) maps H I onto H2, and H I is invariant with respect to the unitary transformations B,, Urn ( n , rn = 1 , . . . , N). Further let c r 1 , 0 2 be projections of the type
where Mo is the orthogonal complement of M + , e.g., Mo is the one-dimensional subspace of M spanned by the vacuum vector lexp(0)). Now for each n , rn = 1, . . . , N and each pair q , qwe define a channel Qzha2from the set of all normal states p on H1 into the set of all normal states on M + ,
where
FZA := (Bn @ U,r(T*))
01
(B. @ U,r(T*))*.
In this paper we have considered the situation where H I is spanned by the ONS
and H2 is spanned by the ONS
Further the model discussed in Sect. 2.1 corresponds to the special case (TI = 0 2 = cr, e.g. Anm =
( n , m = 1, ... , N)
(perfect in the sense of conditions (El) and (E2)). The model discussed in Sect. 2.2 corresponds to the special case 0 1 = (T # e.g.
0 2 =6,
(perfect in the sense of (El), and only asymptotically perfect in the sense of (E2)). Finally the model from this section corresponds to the special case 01 = 0 2 = 5, e.g.
(non-perfect, neither (E2) nor (El) hold, but asymptotically perfect in the sense of both conditions).
174 K.-H. Fichtner, M.Ohya
82
4. Proof of Theorem 3.1
From ( 14) we get 02
IIexp (aK5gj) - exp(0)l12 = e T
( k .j = 1,.
-
1
(s = I , 2; j = 1 , . . . , N ) ,
(60)
.. ,N ) .
Lemma 4.1. Put for j , k = 1, . . . , N , aj/i := (Iexp(0)) 8
Iii) , v (texp ( U K i g j ) - exp(0)) 8 Iexp(aK1gk)))).
Theti it holds for-j , k = 1, . . . , N ,
Proof: We have (exp(0) , exP(f)) = 1 Using (62), (65), and (45) we get for j , k = 1 ,
(f E L2(G)). . .. ,N ,
(65)
175 83
Quantum Teleportation and Beam Splitting
From (61), (66), (67), and (68) it follows
Now (1 3) and (14) implies
1 ( K l g j Klgk) = i8jk. 9
For that reason (63), and (64) follow from (69). In the following we fix a pair n , m E [ 1 , . . . , N ) . Remark 4.2. Without loss of generality we can assume
(71)
Bn = 1,
which we can explain as follows: Using (57)-(59), and (54) we obtain in the case (71), 6km(P) = 6 n m (BlPBk)
( k = 1 , . . . ,N ) ,
$km(P) = $nm (BlPBk)
(k = 1 , . .. ,N ) .
On the other hand from Theorem 2.1 it follows that in the case (7 1 ) for all states p with (1 6) and (1 7) it holds
Akm(P) = Anm (BiPBk)
(k = 1 , . .. N ) . 7
Finally it is easy to show that B;pBk fulfills (16), and (17) if the state p fulfills (16) and (17). For those reasons Theorem 3.1 would be proved if we could prove (40), and (41) on the assumption that we have (71).
Lemma 4.3. Put for s = 1 , . . . , N
176 K.-H. Fichtner, M. Ohya
84
Pi-ooj From (1 7), (72), and (73) we get N
N
Further we have
That implies (74). Now we put .
N
Since
F+Iexp (UKrgk)) = (1 - e-+)'
( r = 1 , 2 ; k = 1, . . . ,m). Using (77), and (78) we obtain
Iexp(aK,gk) - exp(0)) (78)
177 Quantum Teleportation and Beam Splitting
Using the same arguments we get
Finally we have
For that reason we have the following lemma Lemma 4.4. For each bounded operator A on M and s = 1, . . . , N it holds
Now from (1 6) we get
85
178 K.-H. Fichtner, M.Ohya
86
On the other hand (IWs8 ij 8 exp(O)))y=, is an ONS because (q,7)r=l is an ONS. For that reason from (57,58), (84), and Lemma 4.4 with A = 1 it follows
AS (Iexp(0Klgj)
-
N
exp(0)))j=l is an ONS we can calculate easily
For that reason from (85) follows
Further we have
C h, = 1 and S
179 87
Quantum Teleportation and Beam Splitting
Lemma 4.5. We use the notation B,(A) from Lemma 4.4. Thenfor each bounded operator A on M ands = 1 , . . . , N it holds
2
2e--r
(N2
+N f i +
N ) IIAII
we get
i
Because of (86) it follows
Using (87) we get
180 K.-H. Fichtner, M. Ohya
88
and
That proves Lemma 4.5.
0
We have the representation (84) of p 63 / G ) ( f i l 63 lexp(O))(exp(O)I as a mixture of orthogonal projections. Thus from (56) and (57,58)we get with the notation O s ( A ) from Lemma 4.4,
For that reason (40) follows from Lemma 4.5, and
As = 1. S
That completes the proof of Theorem 3.1.
References 1, Accardi. L. and Ohya, M.: Teleportation ofgeneral quantum state.y.Voltera Center preprint, 1998 2. Accardi, L.,Ohya. M.: Compound channels. transition expectations and lifiings. Applied Mathematics & Optimization 39,33-59 (1999) 3. Bennett, C.H.. Brassard, G., Crkpeau, C., Jozsa, R., Peres, A. and Wootters, W.: Teleporting an unknown quantum state viaDual Classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895-1 899 ( I 993) 4. Bennett. C.H.. Brassard. G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722-725 (1996) 5. Ekert, A.K.: Quantum cryptography based on Bell's theorem. Phys. Rev. Lett. 67,661463 (1991) 6. Fichtner. K.-H. and Freudenberg, W.: Pointprocesses and the position distrubution of infinite boson systems. J. Stat. Phys. 47, 959-978 (1987) 7. Fichtner, K.-H. and Freudenberg, W.: Characterization of states of infinite Boson systems I. - On the construction of states. Commun. Math. Phys. 137, 315-357 (1991) 8. Fichtner. K.-H.. Freudenberg, W. and Liebscher, V.: Time evolution and invariance of Boson systems given by beam splittings. Infinite Dim. Anal. Quantum Prob. and Related Topics I, 51 1-533 (1998) 9. Lindsay. J.ht.: Quantum and Noncausal Stochastic Calculus. Prob. Th. Rel. Fields 97,65-80 (1993) 10. Fichtner, K.-H. and Winkler, G.: Generalized brownian motion, point processes and stochastic calculus for random fields. Math. Nachr. 161,291-307 (1993) 1 I . Inoue, K.. Ohya, M. and Suyari, H . : Characterization of quantum teleportation processes by nonlinear quantum mutual entropy. Physica D 120, 117-124 (1998)
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12. Fichtner, K.-H. and Ohya, M.: Quantum Teleportation with Entangled States given by Beam Splittings. Commun. Math. Phys. 222,229-247 (2001) 13. Fichtner, K.-H., Freudenberg, W. and Liebscher, V.: On Exchange Mechanisms for Bosons. Submitted to Infinite Dim. Anal., Quantum Prob. and Rel. Topics Communicated by H. Arab
182 Commun. Math. Phys. 222,229 - 247 (2001) With kind permission of Springer Science and Business Media
Communications in
Mathematical Physics
0 Springer-Verlag2001
Quantum Teleportation with Entangled States Given by Beam Splittings Karl-Heinz Fichtner', Masanori Ohya2
' Friedrich-Schiller-UniversitatJena. Fakultat f i r Mathematik und Infomlatik. lnstitut f i r Angewandte Mathematik. 07740 Jena, Germany. E-mail: fichtnerQininet.uni-jella.de Department o f Information Sciences, Science University o f Tokyo, Chiba 278-85 10. Japan. E-mail: ohyaQis.noda.sut.ac.jp Received: 21 January 2000 /Accepted: 23 April 2001
Abstract: Quantum teleportation is rigorously demonstrated with coherent entangled
states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication [2] and quantum stochastic [S]. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. ( 2 ) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity. We show that our quantum teleportation scheme with coherent entangled state is more stable than that with the EPR pairs which was previously discussed. It is in [ 3 ]that quantum teleportation was first studied as a part of quantum cryptography [5].Quantum teleportation is to send a quantum state itself containing all information of a certain system from one place to another. The problem of quantum teleportation is whether there exist a physical device and a key (or a set of keys) by which a quantum state is completely transmitted and a receiver can reconstruct the state sent. It has been considered that such a teleportation would not be realistic because the usual quantum state contains information which can not be observed simultaneously. In the above paper [ 3 ] ,Bennett et al showed such a teleportation is possible through a device (channel) made from proper (EPR) entangled states of Bell basis. The basic idea behind their discussion is to divide the information encoded in the state into two parts, classical and quantum, and send them through different channels, a classical channel and an EPR channel. The classical channel is nothing but a simple correspondence between sender and receiver, and the EPR channel is constructed by using a certain entangled state. However the EPR channel is not so stable due to decoherence. In this paper (1) we study the quantum teleportation by means of general beam splitting processes so that it contains the EPR channel, and (2) we construct a more stable teleportation process with coherent entangled states.
183 230
K.-H. Fichtner, M. Ohya
The quantum teleportation scheme can be mathematically expressed in the following steps [ l l ] : Step 0. A girl named Alice has an unknown quantum state p on (a N-dimensional) Hilbert space 7-11 and she was asked to teleport it to a boy named Bob. Step I. For this purpose, we need two other Hilbert spaces N2 and N 3 , 7-12 is attached to Alice and 7-13 is attached to Bob. Prearrange a so-called entangled state CJ on 7-12 @ 7-13 having certain correlations and prepare an ensemble of the combined system in the state p @ ~7on 7-11 @ 7-12 @ N 3 . Step 2. One then fixes a family of mutually orthogonal projections (F,fn,)zf,,=lon the Hilbert space 7-11 @ 7-12 corresponding to an observable F := C Z , , , ~ F , ~ ~ ~ , tr.m
and for a fixed pair of indices 1 2 , nz, Alice performs a first kind of incomplete measurement, involving only the 7-11 @ 7-12 part of the system in the state p @ C J , which filters the value z , ~ , ~that ~ , is, after measurement on the given ensemble p @ CJ of identically prepared systems, only those where F shows the value znfHare allowed to pass. According to the von Neumann rule, after Alice’s measurement, the state becomes (123)
Pnrn
._ .-
(Fnm @ 1 ) P @ CJ(F,,m‘8 1) tr123(Fnm ‘8 1 ) p @ a ( F n m @ 1)’
where tri23 is the full trace on the Hilbert space 7-11 ‘8 7-12 @ 7-13. Step 3. Bob is informed which measurement was done by Alice. This is equivalent to transmitting the information that the eigenvalue Z,xm was detected. This information is transmitted from Alice to Bob without disturbance and by means of classical tools. Step 4. Making only partial measurements on the third part on the system in the state p6!i3’ means that Bob will control a state A,,nl(p)on 7-13 given by the partial trace on 7-1 i @ N2 of the state pi::3’ (after Alice’s measurement) An,,,(P>= tr12 Pi;:3) -
(F,1,11
@ 1)P @ CJ(F,,,,,@ 1 )
tr12tr123(~,lm 8 1)p 8 a ( F n m
1)’
Thus the whole teleportation scheme given by the family ( F , , , , )and the entangled ) channels from the set of state CJ can be characterized by the family ( A f t mof states on 7-11 into the set of states on X3 and the family (P,~,,~)given by P , , ~ , ( P ):= tr123(Fflnz@ 1 ) @~ c ( F t l n Z @ 1)
of the probabilities that Alice’s measurement according to the observable F will show the value Z n m . The teleportation scheme works perfectly with respect to a certain class 6 of states p on 7-11 if the following conditions are fulfilled:
( E l ) For each 1 2 , m there exists a unitary operator u,,,,~: 7-11 + 7-13 such that A,,,,,(P) = U n t n P
U,Tnr
(P E 6).
184 23 1
Teleportation and Entangled States
( E l ) means that Bob can reconstruct the original state p by unitary keys (u,,,,,] provided to him. (E2) means that Bob will succeed to find a proper key with certainty. In [3,4], the authors used an EPR spin pair to construct a teleportation model. In order to have a more convenient model, we here use coherent states to construct a model. One of the main points for such a construction is how to prepare the entangled state. The EPR entangled state used in [3] can be identified with the splitting of one particle state, so that the teleportation model of Bennett et al. can be described in terms of Fock spaces and splittings, which makes us possible to work the whole teleportation process in general beam splitting scheme. Moreover to work with beams having a fixed number of particles seems to be not realistic, especially in the case of large distance between Alice and Bob, because we have to take into account that the beams will lose particles (or energy). For that reason one should use a class of beams being insensitive to this loss of particles. That and other arguments lead to superpositions of coherent beams. In Sect. 2 of this paper, we construct a teleportation model being perfect in the sense of conditions ( E l ) and (E2), where we take the Boson Fock space r ( L 2 ( G ) ):= Xi = ?'i2 = '?i3 with a certain class of states p on this Fock space. In Sect. 3 we consider a teleportation model where the entangled state D is given by the splitting of a superposition of certain coherent states. Unfortunately this model doesn't work perfectly, that is, neither (E2) nor ( E l ) hold. However this model is more realistic than that in Sect. 2, and we show that this model provides a nice approximation to be perfect. To estimate the difference between the perfect teleportation and non-perfect teleportation, we add a further step in the teleportation scheme: Step 5. Bob will perform a measurement on his part of the system according to the projection
F+ := 1 - lexp(0) > < exp(0)I. where lexp(0) > < exp(0)l denotes the vacuum state (the coherent state with density 0). Then our new teleportation channels (we denote it again by A,,,,,j have the form
and the corresponding probabilities are P,,,,,(P) := tr123(Fnm @ F + ) P @ ~ ( F l i l l @ t F+).
For this teleportation scheme, (El) is fulfilled. Furthermore we get
185 K.-H. Fichtner, M . Ohya
232
Here N denotes the dimension of the Hilbert space and d is the expectation value of the total number of particles (or energy) of the beam, so that in the case of high density (or energy) “ d + +oo” of the beam the model works perfectly. Specializing this model we consider in Sect. 4 the teleportation of all states on a finite dimensional Hilbert space (through the space RL). Further specialization leads to a teleportation model where Alice and Bob are spatially separated, that is, we have to teleport the information given by the state of our finite dimensional Hilbert space from one region X I C Rk into another region X2 C Rk with X In X2 = M, and Alice can only perform local measurements (inside of region X I ) as well as Bob (inside of X 2 ) .
1. Basic Notions and Notations
First we collect some basic facts concerning the (symmetric) Fock space. We will introduce the Fock space in a way adapted to the language of counting measures. For details we refer to [6-8,2,9] and other papers cited in [XI. Let G be an arbitrary complete separable metric space. Further, let p be a locally finite diffuse measure on G, i.e. p ( B ) < +oo for bounded measurable subsets of G and ,u(( x ) ) = 0 for all singletons x E G. In order to describe the teleportation of states on a finite dimensional Hilbert space through the k-dimensional space Rk,especially we are concerned with the case G = Rk x ( I , . . . , N } , ,u=lx##,
where 1 is the k-dimensional Lebesgue measure and # denotes the counting measure on ( 1 , . . . ,N). Now by M = M ( G ) we denote the set of all finite counting measures on G . Since I1
cp E M can be written in the form cp =
C 6,;
for some ri = 0, 1 , 2 , . . . and x ; E G
j=l
(where S., denotes the Dirac measures corresponding to x E G) the elements of M can be interpreted as finite (symmetric) point configurations in G. We equip M with its canonical a-algebra !D (cf. [6,7]) and we consider the measure F by setting
Hereby, XY denotes the indicator function of a set Y and 0 represents the empty configuration, i. e., O(G) = 0. Observe that F is a a-finite measure. Since p was assumed to be diffuse one easily checks that F is concentrated on the set of simple configurations (i.e., without multiple points)
Definition 1.1. M = M ( G ) := L 2 ( M ,m, F ) is called the (symmetric) Fock space over G.
186 Teleportation and Entangled States
233
In [6] it wasproved that M and the Boson Fock space r ( L 2 ( G ) in ) the usual definition are isomorphic. For each @ E M with @ f 0 we denote by I@ > the corresponding normalized vector
Further, I @ > < @I denotes the corresponding one-dimensional projection, describing the pure state given by the normalized vector I@ 1.Now, for each n 2 1 let M@"be the n-fold tensor product of the Hilbert space M . Obviously, M@"can be identified with L 2 ( M ' * ,F ' J ) . Definition 1.2. For a given function g : G bY
-+
C the function exp ( g ) : M
-+ C defined
is called an exponential vector generated by g . Observe that exp(g) E M if and only if g E L 2 ( G ) and one has in this case JJexp(g>1l2= ellRllZ and Jexp(8) >= e-tllnl12exp (g). The projection Jexp(g) > < exp(g)l is called the coherent state corresponding to g E L 2 ( G ) .In the special case g = 0 we get the vacuum state lexp(0) >= KIol. The linear span ofthe exponential vectors of M is dense in M , so that bounded operators and certain unbounded operators can be characterized by their actions on exponential vectors. Definition 1.3. The operator D : dom(D) -+ M g 2given on a dense domain dom(D) c M containing the exponential vectors from M by
is called a compound Hida-Malliavin derivative. On exponential vectors exp (8) with g
E
L 2 ( G ) ,one gets immediately
D exp ( 8 ) = exp ( g ) 63 exp ( g ) .
(1)
Definition 1.4. The operator S : dom(S) -+ M given on a dense domain dom (S) c M B 2containing tensor products of exponential vectors by @(@, p - @)
S @ ( y ):= (PSV
is called compound Skorohod integral.
(@ E dom(S). p E M )
187 234
K.-H. Fichtner,
M.Ohya
One gets
(03,@ ) M @= (@, S @ ) M ( 3 E dom(D), S(exp (8) 8 exp ( h ) )= exp (g
+ h)
@ E dom(S)),
(g, h E L 2 ( G ) ) .
(2)
(3)
For more details we refer to [lo].
Definition 1.5. Let T be a linear operator on L 2 ( G )with IIT I[ 5 1. Then the operator r ( T ) called second quantization of T is the (uniquely determined) bounded operator on M fuljilling r(T)exp (g) = exp (Tg) (g
E
L2(G)).
Clearly, it holds
It follows that r ( T ) i s an unitary operator on M if T is an unitary operator on L 2 ( G ) . Lemma 1.6. Let K I , K 2 be linear operators on L2 ( G ) with property KTK,
+ K;K~ = 1 .
(5)
Then there exists exactly one isometry U K , . K ~from M to M@' = M 8 M with v ~ , , ~ , e x p (= g )exp(Kig) 8 exp(K2g) (g
E
L2(G>>.
(6)
Further it holds VK,.K,
= (r(K1) '8 r ( K 2 ) ) D
(7)
(at least on dom(D) but one has the unique extension). The adjoint v ~ I ~of KV K2 , . K ? is characterized by v;,,~,(exp(h) '8 exp(g)) = exp(KTh
+ K;g)
(8, h E L 2 ( G ) )
(8)
and it holds viI,K2=
S(r(Kr) 8 r(K;)).
(9)
Remark 1.7. From K I , K2 we get a transition expectation ~ K , K: M ~ @ M -+ M , using v i , ,K , and the lifting K Z may be interpreted as a certain splitting (cf. [2]).
c:,
Proof of Lemma 1.6. We consider the operator B := S ~ U K T ) r(K;))(r(KI) 8 ~ ( K ~ ) ) D on the dense domain dom(B) C M spanned by the exponential vectors. Using (l), (3), (4) and (5) we get B exp (g) = exp (8)
(g E L 2 ( G ) ).
188 Teleportation and Entangled States
235
It follows that the bounded linear unique extension of B onto M coincides with the unity on M , B = 1. On the other hand, by Eq. (7) at least on dom ( D ) , an operator (2) and (4) we obtain II~KI.K21Cr1l2 = ( V K I . K 2 1 C , . V K I . K 2 1 C r )
=
($9
(1Cr
E
(10) UK,,K*
is defined. Using
dom ( D ) )
B1Cr),
which implies II~Kl.K21Cr1I2=
because of (10). It follows that on M with
II1Cr1I2 (1Cr
u K I . K 2 can
E dom ( D ) ) be uniquely extended to a bounded operator
I I ~ K ~ , K ~= ~ CII1Crl ~ I Il (1Cr E M I . Now from (7) we obtain (6) using (1) and the definition of the operators of second quantization. Further, (7), (3) and (4) imply (9) and from (9) we obtain (8) using the definition of the operators of second quantization and Eq. (3). Here we explain the fundamental scheme ofbeam splitting [8]. We define an isometric operator V,.p for coherent vectors such that Va.pl exp (g)) = I exp ( a g ) ) 63 I exp (Ps))
with 1 (Y l2 + I /3 12= 1. This beam splitting is a useful mathematical expression for optical communication and quantum measurements [ 2 ] . Example 1.8 (a = P = 1/&above). multiplication on L ~ ( G > :
Let K I = K2 be the following operator of
1 Klg = -g = K2g
a
(g E L 2 ( G ) ) .
We put
v := V K I . K 2 and obtain
Example 1.9. Let L 2 ( G )= 'HI 63 2 2 be the orthogonal sum of the subspaces 'Hl , 2 2 . K I and K2 denote the corresponding projections. We will use Example 1.8 in order to describe a teleportation model where Bob performs his experiments on the same ensemble of the systems like Alice. Further we will use a special case of Example 1.9 in order to describe a teleportation model where Bob and Alice are spatially separated (cf. Sect. 5 ) . Remark 1.10. The property (5) implies IIKig112
+ IIK2g1I2 = 11g1I2
(g E L 2 ( G ) ) .
( 1 1)
Remtrrk 1 . 1 1 . Let U , V be unitary operators on L2(G). If operators K1, K2 satisfy ( 5 ) , then the pair k1 = U K I , K 2 = VK2 fulfill (5).
189 K.-H. Fichtner, M. Ohya
236
2. A Perfect Model of Teleportation
Concerningthegeneralideawefollowthepapers[l,l I].WefixanONS(g1,. . . , g N ) C L2(G), operators K I , K2 on L 2 ( G )with (5), a unitary operator T on L 2 ( G ) ,a n d d > 0. We assume (k = 1 , . . . , N ) , ( k f j ; k , j = l ... , N ) .
TKIgk = K2gk (Klgk, K l g j ) = o
(12) (13)
Using (1 1j and (1 2) we get
11 KI gk 112 = 11 K2gk 11 2
=
1 y.
From (1 2) and (1 3 ) we get (k f j ; k, j = 1, ...
(Kzgk, K2g,j) = 0
The state of Alice asked to teleport is of the type N
P [email protected])(@s13 s=l
where N
(
~ ~ . ~ ) = C ~ , , i l e x p ( n ~ i g , i ) - e x p ( oClc,y,il )) 2 = 1;s= I,.. . , N
,j=l
.i
1
(17)
andn = &. One easily checks that (lexp(nK1g.j) - exp(0)))y=I and (lexpnK2g.j) exp (0)))y=lare ONS in M . Here we took the vacuum state exp(0) off, but it is just only for computational sympiicity. In order to achieve that is still an ONS in M we assume
([email protected]))y=l
N
E?,,.jck,i = 0
(j
#k;
j , k = 1, ... , N ) .
.i=1
Denote c , ~= [cSl,,.. , c , ~ N E] C N ,then (c,~):=~ is an CONS in CN. Now let (btl);=, be a sequence in C N , bri = [bnl .... , b n N ]
with properties 1
(12,
k = 1, . . . , N ) ,
( b I l ,b,,) = 0
(n
#
IBrrbI =
j ;
II,
j = I , . . . ,N ) .
Then Alice’s measurements are performed with projection
F,,,,, = It,,,,,) (hl,l I (n,in = 1, . . . , N )
(18)
190 Teleportation and Entangled States
231
given by
+
where j @ m := j m(mod N). One easily checks that (lt,l,,,))~m=, is an ONS in MB2.Further, the state vector 16) of the entangled state (T = 16) (41 is given by
Lemma 2.1. For each n , m = 1, . . . , N it holds
f r o o j From the fact that
On the other hand, we have
Using (26) and (27), we get (24). Now we have
s=l
191 K.-H. Fichtner, M.Ohya
238
From (30), (33) and Lemma 2.1 we obtain (34) It follows (35) Finally from (29), (34) and (35) we have 1 (Ftlrn'8 1) ( P '8 a)(F,,,,, '8 1) = N ~ F , , '8 ~ ~( r, ( T ) U n l B R P) (B,,U;,rV*))
(36)
That leads to the following solution of the teleportation problem. Theorem 2.2. For each n , rn = 1 , . . . , N , dejne a channel A,,,,, by
Then we have for all states p on M with (16) and ( I 7) A,,,,(P) =
(r(T)unlBi) P (r(~)u,,t~X)*.
(38)
192 Teleportation and Entangled States
239
.
Renzark 2.3. In case of Example 1.8 using the operators B,, , V,,, r(T ) . the projections F,,,,, are given by unitary transformations of the entangled state 0 : F,,,,, = (B,, 8 u,,m-*)) 0 (B,@ ~ u , , m T * ) ) * , or
~ b ,= ~ (& , ~ )@ U , , , w * ) ) it).
(39)
Remark 2.4. If Alice performs a measurement according to the following selfadjoint operator: N
F =
c
Zlltll
F,,,,,
~i,/ll=l
with { : , , , , , l r ~ ,YIZ = 1 , . . . , N ) C R - {O), then she will obtain the value z,~,,~with probability 1 / N 2 . The sum over all these probabilities is 1 , so that the teleportation model works perfectly.
3. A Non-Perfect Case of Teleportation In this section we will construct a model where we also have channels with property (38). But the probability that one of these channels will work in order to teleport the state from Alice to Bob is less than 1 depending on the density parameter ct (or cnergy of the beams, depending on the interpretation). If d = a 2 tends to infinity that probability tends to 1. That is, the model is asymptotically perfect in a certain sense. We consider the normalized vector
y :=
(1
+(N
-
1)ed
1
+(N
-
l)e-"?
and we replace in (37) the projector C-J by the projector
Then for each
PI, in
= 1 , . . . , N . we get the channels on a normal state p on
M such as
where F+ = 1 - lexp (0)) (exp (0)l e. g., F+ is the projection onto the space M + of configurations having no vacuum part, e. g., orthogonal to vacuum
M + := { $ E MI IIexp(O))(exp(O)I$II
= 01.
193 Fichtner. M . Ohya
240
One easily checks that @,,,(P)
=
F+
A
,1,11
( p ) F+
(44)
tr (F+A,,,,, @IF+)' that is, after receiving the state A,,,,, ( p ) from Alice, Bob has to omit the vacuum. From Theorem 2.2 it follows that for all p with (1 6) and (1 7 ) , An,,,(P) =
This is not true if we replace A,,, by
F+A,,m(p)F+
tr (F+ A,,,,, ( P ) F + )
A,,nl,namely, in general it does not hold
-
(~,,,,(P) = A,,,, ( P I .
But we will prove that for each p with ( I 6) and (1 7) it holds @,,,,,(P) = A,,,,,(P)>
which means
+),,,,m = (r(T)U,,B;r)p(r(T)U,,B,T)*
(45)
because of Theorem 2.2. Further we will show
(
Y2 (F,,,~,8 F+> ( p 8 6 ) (F,,,,, 8 F + ) = - e r N2 1'
-
1
)' ePf'
(46)
and the sum over 1 1 , m (= 1, . . . , N ) gives the probability
which means that the teleportation model works perfectly in the limit d + 00, e. g., Bob will receive one ofthe states O,,,, ( p ) given by (44). Thus we formulate the following theorem. Theorem 3.1. For all states p on M with (16) and (17) and each pair in (= 1, . . . , N ) , Eqs. (44) and (45) hold. Further; we have
N,
In order to prove Theorem 3. I , we fix p with (1 6) and (1 7) and start with a lemma. Lemma 3.2. For euch n , m , s (= 1, . . . , N ) , it holds
194 24 1
Teleportation and Entangled States
-
i f r = j andk = r @ m otherwise
and
195 K.-tl. Fichtner, M . Ohya
242
and
L,,(P) + ( r ( w & )P ( ~ ( T ) u , , , B : ) * . Now we have r(T)U,,B,t%
E
M + , lexp(0)) E M $ .
Hence, Lemma 3.2 implies (1 8 1 8 F + ) (F,I,,l 8 1 ) (%€4 l ) =
5
(I - e-’>
L,8
( ~ ( ~ ) ~ , l f ~ : % )
that is, we have the following lemma Lemma 3.3. For each 1 2 , m , s = I ,
. . . , N , it holds
(FtInL 8 F+) (%8e)=
Remark 3.4. Let
K2
$ (1 - e - 4 ) t,,,,, 8 (r(T)UnlB:@.s).
(48)
be a projection of the type K212 = h X x ;
h
E
L2(G),
where X E G is measurable. Then (48) also holds if we replace F+ by the projection F + , x onto the subspace M+.x of M given by
M+.x := {$
E
MI$(q) = 0 if p ( X ) = 0).
Observe that M+.G= M + .
(/@,s)).F=l
ProqfqfTheorem 3.1. We have assumed that is an ONS in M , which implies that (It,,,,,) 8 ( r ( ~ ) ~ , ~ f B ~ l ~is, san) ONS ) ) ~ ~in=Ml @ j .Hence we obtain Eqs. (45), (46 ) and (47) by Lemma 3 . 3 . This proves Theorem 3 . I . Remnr-k 3.5. In the special case of Remark 3.4, Eqs. (45), (46) and (47) hold ifwe replace F+ by F + . x in the definition ofthe channel On,,and in (46), (47 ),that is, Bob will only perform “local” measurement according to the region X , about which we will discuss more details in the next sections. 4. Teleportation of States Inside Rk
Let 3t be a finite-dimensional Hilbert space. We consider the case
% = C N =L2({I, . . . , N ) , # ) without loss of generality, where # denotes the counting measure on the set ( 1 , . . . , N ] . We want to teleport states on 3t with the aid of the constructed channels (An,fl)Z,n=,or (O,,,,,I:, -
=1
. w e fix
a CONS (lj))yZl of 3t,
f E L2 (Rk),’llfll = 1, d = a 2 > 0,
196 243
Teleportation and Entangled States
- K l , K 2 linear operators on L~ ( R ~ ) , unitary operator on ~2 ( R ~ )
f
with two properties KTKI f
= f?
tK$2f
F K - ,f = K 2 f .
(49) (50)
We put G = R~ x ( I , . . . , N J , p = i x #,
where 1 is the Lebesgue measure on R k . Then L 2 ( G ) = L 2 ( G ,p ) = L 2 ( R k )@ Ifl Further, put g / := f @ l j )
( j = I , . .. ,N ) .
Then (g,i)y=l is an ONS in L 2 ( G ) .We consider linear operators K I , K2 on L’(G) with (5) and K,.g,i = (k,.f)@ l j ) ( j = I , .
. . , N ; r = I , 2).
(51)
Remark 4.1. Equation (5 1 ) determines operators K I , K2 on the subspace of M spanned On the orthogonal complement, one can put for instance by the ONS (s~):=~. 1
K r @ = -@, Then K l , K2 are well defined and fulfill (5) because of (49). Further, one checks that (1 3) and ( 1 5) hold. Now let T be an unitary operator on L 2 ( G )with T(Klg,) =
( f K 2 f )
63 lj).
From ( 1 3) one can prove the existence of T using the arguments as in Remark 4.1. Further, we get (1 2) from (50). Summarizing, we obtain that (gl,. . . , g N ) , K I , K?, T fulfill all the assumptions @,,, given by (37) required in Sect. 2. Thus we have the corresponding channels AIl3,,,, and (43) respectively. It follows that we are able to teleport a state p on M = M ( G ) with (16) and (17 ) as it was stated in Theorem 2.2 and Theorem 3.1, respectively. In order to teleport states on 7-i through the space Rk using the above channels, we have to consider: a “lifting” E* ofthe states 6on 3-1 into the set of states on the bigger state space on M such that p = E* (6)can be described by ( 1 6), (1 7), (1 8). Second: a “reduction” R of (normal) states on M to states on Ifl such that for all states 6 on 3t it holds
First:
where (V,,llt)&,l=lare unitary operators on Ifl.
197 K.-H. Ficlitiier, M . Ohya
244
That we can obtain as follows: We have already stated in Sect. 2 that N
(Iexp(nK,.(g,)) - e x ~ ( O ) ) ) ~ = ~ ( r = 1,2) are ONS in M . We denote by M,. ( r = 1 , 2) the corresponding N - dimensional subspaces of M . Then for each I’ = I , 2, there exists exactly one unitary operator W ,. from U onto M,. C M with W,.lj) = lexp(aK,.g,) - exp(O))
( j = 1 , . . . ,N ) .
(53)
We put &* (6):= w I , ~ w ; ~ M , where
(6state on 3-1) ,
(54)
n , ~denotes , the projection onto M,. ( r = 1 , 2 ) . Describing the state 6 on U by
with N
I&\) = x c , / l j ) , /=I N
where ( c \ / ) \ I = I fulfills ( I @ , we obtain that p = &* Now, for each state p on M we put
Since
we get
and
As we have the equality
( b ) is given by (16) and ( I 7).
198 245
Teleportation and Entangled States
Put V,,,,, := W,*r(T)U,,B,TW, ( n , m = I ,
.. . , N),
(57)
then V,,,, ( n , m = 1, . . . , N ) is a unitary operator on 3t and (52) holds. One easily checks V,,, 1 j ) = bnj 1 j CB m ) ( j , m , n = 1, . . . , N )
Summarizing these, we have the following theorem: Theorem 4.2. Consider the channels on the set of states on 3t ( n , m = 1, . , . , N ) ,
(58)
O,,,, o &* ( n , rn = 1, . . . , N ) .
(59)
A,,,,, := R o A,,,, o € *
Or,, := R
o
where R,&*, A,,,, O,,, are given by (56), (54), (37). (43), respectively. Then for all states 6 on Z, it holds Awn
(b) = VnrnbV,~,= Grim (b) ( n ,nl
= 1,
... N ) , 3
(60)
where V,,, ( n , m = 1, . . . , N ) are the unitury operators on Z given by (57). Remark 4.3. Remember that the teleportation model according to works perfectly in the sense of Remark 2.4, and the model dealing with (C3r,r7z):rr,=, was only asymptotically perfect for large d (i.e., high density or high energy of the beams). They can transfer to (A,,.~,) ,
(e,,,,,).
Example 4.4. We specialize
Realizing the teleportation in this case means that Alice has to perform measurements (FrimI) in the whole space Rk and also Bob (concerning F+). 5. Alice and Bob are Spatially Separated
We specialize the situation in Sect. 4 as follows: We fix -
t E Rk,
-
X i , X 2 , X 3 G Rk are measurable decomposition of Rk such that Z(X1) f 0 and x2 =
x1 + t := ( x + t l x E X I } .
Put
? h ( x ) := h ( x - t )
( x E Rk , h E L 2 ( R k ) ) ,
K,.h := hXx,.
( r = 1 , 2 , h E L 2 ( R k ) ),
199 K.-H. Ficlitner, M . Ohya
246
and assume that the function .f E L2 (R k ) has the properties
fXx, =f
( f x x , )f,X X , = 0.
Then f is an unitary operator on L2 ( R k ) and (49 ), (50)hold. Using the assumption that X I , X2, X3 is a measurable decomposition of Rk we get immediately that
G, := X, x ( 1 . . . . , N )
(S
= 1,2.3)
is a measurable decomposition of G. It follows that M = M ( G ) is decomposed into the tensor product
M ( G )= M ( G I ) @ M ( G 2 ) 8 M ( G 3 ) [6,7,10]. According to this representation, the local algebras %(X,y corresponding to regions X,, E Rk (s = 1,2) are given by % ( X I ) := { A @ 1 8 1; A bounded operator on M ( G U(X2) := { 1 @ A @ 1 ; A bounded operator on M ( G One easily checks in our special case that
F,,,, E % ( X I ) @ % ( X I ) ( ~ z , m =1, . . . , N ) , CJ E %(XI) @ WX2) and &* (6) gives a state on %(XI) (the number of particles outside of G I i s 0 with probabiliy 1 ). That is, Alice has to perform only local measurements inside of the region X I in order to realize the teleportation processes described in Sect. 4 or measure the state &* (6). On the other hand, A,,,, (&* (6))and O,,, (&* (6))give local states on a ( X 2 ) such that by measuring these states Bob has to perform only local measurements inside of the region X2. The only problem could be that according to the definition (43) of the channels 8,,,,, Bob has to perform the measurement by F+ which is not local. However, as we have already stated in Remark 3.5, this problem can be avoided if we replace F+ by F+.x2 E U(X2). Therefore we can describe the special teleportation process as follows: We have a beam being in the pure state I q ) (171 (40). After splitting, one part of the beam is located in the region X I or will go to X 1 (cf. Remark 1.1 1) and the other part is located in the region X2 or will go to X2. Further, there is a state I *(6)localized in the region X I . Now Alice will perform the local measurement inside of X I according to F = 1 z,,,, F,,, involving 11 .in
the first part of the beam and the state &*(6).This leads to a preparation of the second part of the beam located in the region X2 which can be controlled by Bob, and the second part of the beam will show the behaviour of the state A,,,,, (&* (6))= O,,,,, (&* (6))if Alice’s measurement shows the value ~,,,,,. Thus we have teleported the state 6 on 3-1 from the region X I into the region X2.
200 24 7
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References I. Accardi. L. and Ohya. M.: Teleportation of general quantum states. Voltera Center preprint, 1998 2. Accardi L.. Ohya M.: Compound channels, transition expectations and liftings. Applied Mathematics & Optimization 39, 33-.59 ( I 999) 3 . Beni1ett.C. H., Brassard, G., 0epeau.C.. Jozsa, R., Peres.A. and Wootters, W.K.:Teleportingan unknown quantum state via Dual Classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895-1 899 (1993) 4. Bennett, C.H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722-725 ( 1996) 5 . Ekeit. A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661--663 (1991 ) 6. Fichtner, K.-ti. and Freudenberg, W.: Point processes and the position distrubution of infinite boson systems. J. Stat. Phys. 47, 959-978 ( I 987) 7. Fichtner, K.-H. and Freudenherg, W.: Characterization of states of infinite Bosoii systems I . On the construction ofstates. Coniniun. Math. Phys. 137. 315-357 (1991) 8. Fichtner, K.-14.. Freudenberg, W. and Liebscher, V.: Time evolution and invariance of Boson systems given by beam splittings. Infinite Dim. Anal., Quantum Prob. and Related Topics 1, 5 I 1-533 ( I 998) 9. Lindsay, J.M.: Quantum and Noncausal Stochastic Calculus. Prob. Th. Rel. Fields 97, 65--80 (1993) 10. Fichtner, K.-H. and Winkler, G.: Generalized brownian motion, point processes and stochastic calculus for random fields. Math. Nachr. 161,201-307 (1993) I 1. Inoue, K., Ohya, M . and Suyari, H.: Characterization of quantum teleportation processes by nonlinear quantum mutual entropy. Physica D 120, 117-124 (1998) -
Communicated by H . Araki
201 Open Sys. & Information Dyn. 7: 33-39, 2000 @ 2000 Kluwer Academic Publishers
33
NP Problem in Quantum Algorithm Masanori Ohya and Natsuki Masuda Science University of Tokyo, Noda City, Chiba 278-8510, JAPAN (Received: October 29, 1999)
Abstract. In complexity theory, a famous unsolved problem is whether N P is equal to P or not. In this paper, we discuss this aspect in SAT (satisfiability) problem, and it is shown that SAT can be solved in polynomial time by means of a quantum algorithm if the superposition of two orthogonal vectors 10) and 11) prepared is detected physically.
1. Introduction
Although the power of computers has highly progressed, there are several problems which still cannot be solved effectively, namely, in polynomial time. Among such problems, N P problems and NP-complete problems are fundamental. It is known [5] that all N P complete problems are equivalent and the essential question is whether there exists a n algorithm solving a n NP-complete problem in polynomial time. After pioneering papers of Feynman [4] and Deutsch [l],several important works have been done on quantum algorithms by Deutsch and Josa [2], Shor [7], Ekert and Jozsa [3] and many others [ll].Computation in a quantum computer is performed on a tensor product Hilbert space, and its fundamental point is t o use quantum coherence of states. All mathematical features of quantum computers and computations are summarized in [ll]. In this paper, we discuss the quantum algorithm of the SAT problem and we point out that this problem, hence all other NP problems, can be solved in polynomial time by a quantum computer if the superposition of two orthogonal vectors 10) and 11) is physically detected. 2. The SAT Problem
x
Let 5 { X I , . .. , xn} be a set. Then X k and its negation Tk,k = 1 , 2 , . . . , n, are called literals and the set of all such literals is denoted by X' = { X I ,T I , .. . , z, Tn}. The set of all subsets of X ' is denoted by F ( X ' ) and an element C E F(X')is called a clause. We consider a truth assignment t o all variables xk. If we can assign the truth value t o at least one element of C, then C is called satisfiable. When C is satisfiable, the truth value t(C) of C is regarded as true, otherwise, that of C is
202
34
M. Ohya and N. Masuda
false. We denote the truth values as “true + 1, false
+ 0”.
Then
C is satisfiable iff t ( C )= 1. Let L = (0, l} be a Boolean lattice with usual join V and meet A , and let t ( z ) be the truth value of a literal z in X . Then the truth value of a clause C is written as t ( C )f V,,ct(z). Moreover the set C of all clauses Cj, j = 1 , 2 , . . . , m, is called satisfiable iff the meet of all truth values of Cj is 1; t ( C ) A j ” l t ( C j ) = 1. Thus the SAT problem is written as follows: DEFINITION 1 (SAT Problem). Given a set X 5 ( 2 1 , . . . ,z,} and a set C ={C1, Cz, . . . , C,} of clauses, determine whether C is satisfiable or not. T h a t is, this problem is t o ask whether there exsits a truth assignment t o make C satisfiable. It is known [5] in usual algorithmic complexity theory that it takes polynomial time t o check the satisfiability when a specific truth assignment is given, but we cannot determine the satisfiability in polynomial time when a n assignment is not specified. 3. Quantum Algorithm of SAT
Let 0 and 1 of the Boolean lattice L be denoted by the vectors
in the Hilbert space C?, respectively. T h a t is, the vector 10) corresponds t o false and the vector 11) t o truth. As we explained in the previous section, an element x E X can be denoted by 0 or 1, and so by 10) or 11).In order t o describe a clause C with length at most n by a quantum state, we need the n-tuple tensor product Hilbert space 31 @’;@. For instance, in the case of n = 2, given C = { x ~ , Q }with an assignment 2 1 = 0 and x2 = 1, then the corresponding quantum state vector is 10) 8 Il),so that the quantum state vector describing C is generally written as IC) = 1x1) 8 1x2) E 31 with x k = 0 or 1, k = 1 , 2 . The quantum computation is performed by a unitary gate constructed from several fundamental gates such as Not gate, Controlled-Not gate, ControlledControlled Not gate [3, 111. Once X =_ { X I , . . . , xn} and C ={CI,C2,. . . ,C,} are given, the SAT is t o find the vector If(C)) Vc, t ( z ) ,where t ( z ) is 10) or 11) when z = Oor 1, respectively, and t(z)At(y) E t ( z A y ) , t ( z ) V t ( y ) t ( z V y ) . We consider a quantum algorithm for the SAT problem. Since we have n variables X k , k = 1,.. .n,and a quantum computation produces some dust bits, the
=
203
35
NP Problem in Quantum Algorithm
assignments of the n variables and the dusts are represented by n qubits and 1 qubits in t h e Hilbert space @yC2@iC!.' Moreover the resulting s t a t e vector If(C)) should be added, so t h a t the total Hilbert space is
7.l E @ y ? @ \ C 2 @ C 2 . Let us s t a r t t h e quantum computation of SAT problem from a n initial vector E @?lo) 8;10) @ 10) when C contains n Boolean variables 2 1 , . . .z,. We apply the discrete Fourier transform denoted by 1 0 .)
t o the part of t h e Boolean variables of t h e vector Iwo), then t h e resulting s t a t e vector becomes
).I
1
=
UF
+
8;I i v o ) = -8: (10) 11))8;10) 8 10)
fi
where I is the identity matrix in C2.This vector can be written as
).I
1 = -
fl
2
8:&j)8;
lo)@ 10).
...,x,=o
21,
Now, we use the quantum computer to check the satisfiability, which will be done by a unitary operator U f properly constructed by unitary gates. Then after t h e computation by U f , t h e vector ) . 1 goes to
1.f)
E
Ufl.) = -
k
@ XI,...,zn=o
-
1 -
2
f l x,,...,x,=o
@:=ll4
Uf
@;=l I Z j )
1 @;=1
IYi)
8; 10) @ 10)
8 lf(21, * ..,4),
where f(z1,. . . ,x,) K f(C) because C contains 2 1 , . . . 2 , , and Iy;) are t h e dust bits produced by the computation. As we will explain in a n example below, the unitary operator U f is concretely constructed. T h e final step to check the satisfiability of C is to apply t h e projection E @:+'I@ 11)(11t o the s t a t e Ivf)l mathematically equivalent t o compute t h e value ( v f I E l u f ) .If t h e vector E l v f ) exists or the value (vflElvf) is not 0, then we conclude t h a t C is satisfiable. T h e value of (vflElvf) corresponds to t h a t of a random algorithm as we will see in an example of the next section and it may or may not be obtained in polynomial time. Let us consider an operator Vg given by
vs = @ : + ' ( A ~ o ) (+o ~qi)(iI)8 Pf(%,
204 M. Ohya and N . Masuda
36
and apply it to t h e vector Iwr), where
and 6 is a certain constant describing the phase of the vector If(C)). T h e resulting vector is the superposition of two vectors with some constants a , 0,such as
one of which is polarized with 6 and t h e other one is non-polarized. T h e existence of the mixture of two vectors 10) and eiell) is the starting point of quantum computation which implies t h e satisfiability. 4. An Example of Computation
Let us explain the quantum computation for SAT in the case X = ( z 1 , x 2 , 2 3 } and C = ( ( ~ 1 ) {~2 2 1 2 3 } , { ~ l r T 3 }{T1,?F2,23)}. , T h e resulting s t a t e ( f ( z 1 , ~ , ~ 3 ) ) is written as l f ( ~ 1 ,Z2,23))
=
1. 1 )
A (1.2)
V 1x3)) A (1.)
V IF3)) A ( \ T i ) v IT2) V
1.3))
.
In t h e quantum computation, i t is not necessary to substitute all values of xj, j = 1 , 2 , 3as in the classical computation, we only have to use a unitary operator Uj for the computation of ] f ( ~ 1 , 2 2 , ~ ) )This . unitary operator Uj is constructed as follows: Let UNOT,UCN and UCCN be Not gate on Czl Controlled-Not gate on C2 '8 C2 and Controlled-Controlled-Not gate on C2 €4 C2@ Cz, respectively, which are given by
UNOT = l O ) ( l l + Il>(Ol UCN = UCCN
=
lo)(ol '8 + 11)(11'8 + I1)(ol) lo)(ol '8 lo)(ol '8 + 1°)(ol 8 I1)(l1@ + I1)(l1'8
+ 11)(11'8I l ) ( l l ( l O ) ( O l ' 8
@
Il)(OO.
Then the unitary operator U f is determined by the combination of the above three unitaries as
Uf 3
U36U35...U2Ul,
where, for instance, Ul = lO)(Ol '8y 1 + 11)(11€4; 1'8 ( l O ) ( l l + Il)(Ol) @yo 1 u2 n €4 lo)(ol 8t2n + 181i)(i1&18( I O ) ( ~+I Il)(ol) '8;' 1 u3 = 8 ; n €4 lo)(ol '8t1n +@mp)(ii'8; n '8 (lo)(li + IWI) d811
205 N P Problem in Quantum Algorithm
37
and other U4,. . . , u36 are similarly constructed (see the computation diagram 1). In this case, we need 20 dust bits (the number of the dust bits needed in a general case is counted in the next section), so that U J operates on the Hilbert space
@?4C2.
where we used the notation
Applying the operator Vs of the previous section to the vector Iv,), we obtain
which is t h e superposition of two vectors (0) and the polarized vector If(C)) = 11). Moreover, when we measure t h e operator E @!311 @ 11)(11in the s t a t e Vslvj)) we obtain 1 (vfVslEIVsvJ) = - . 10 This concludes t h e satisfiability of C 5 . Complexity of the Quantum Algorithm for SAT
Here we discuss the number of steps for the quantum algorithm solving the SAT problem. T h e size of input with n variables xk and their negations Tk, k = 1 , .. . n, and also C = {Cl).. . ,Cm}is N1 = l o g n 2 m n like for t h e classical algorithm because ICkl, the number of elements in Ck,is at most n,so that i t is t h e polynomial order ( O ( m n ) ) .T h e number of dust bits to compute f(C) is related t o that of the operations and substitutions of AND and OR, so t h a t the maximum number (complexity) N2 of the dust bits needed is the same as t h a t in t h e classical case,
+
)
d s: x
0
FZ"
\
207 39
N P Problem in Quantum Algorithm
namely, N Z = (the numbers of AND and OR operations )- (the steps to take the negation) = (5mn - 1 ) - mn = 4mn - 1. T h e number of steps N3 needed t o obtain the vector If(C)) is counted as follows: First take 1 step for the discrete Fourier transform t o get the entangled vector, next we need 3mn steps for t r u t h assignment and substitution t o n variables. Secondly t o compute the logical sum in each clause and take their logical product, the complexity corresponding to t h e logical sum, whose gate is made of four unitaries, is 4m(2n- 1 ) and that corresponding t o the logical product is m - 1. Thus N3 = 1 + 3mn+4m(2n- 1 ) + m - 1 = I l m n - 3m. Finally to check the satisfiability of C,we have t o look at t h e value of If(C)) registered at t h e last position of t h e tensor product state, such as the last position of physical registers (e.g., spins) lined up. This can be easily done by applying the operator Ve t o Ivf), and t h e resulting vector is a superposition of two vectors 10) and e i e l l ) . We can obtain in polynomial time (at most n I 1 steps) the vector IVguf) and t h e value of (Vevf,EVguf)l = 1,012for the projection E if needed. T h e existence of t h e above superposition is t h e starting point of quantum computation, so t h a t it should be physically detected being different from both 10) and Il),which implies the satisfiability. Thus, the quantum algorithm of the SAT problem is of polynomial order. In this paper, we assumed the physical detection of the vector &lo) PI1) t o prove t h e SAT, whose assumption can be omitted in [8].
+ +
+
Acknowledgment One of t h e authors (MO) appreciate Professor A. Ekert for his critical reading and valuable comments.
Bibliography 1. D. Deutsch, Proc. Roy. SOC.London series A 400, 97 (1985). 2. D. Deutsch and R. Jozsa, Proc. Roy. SOC.London series A, 439,553 (1992). 3 . A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996). 4. R . Feynman, Optics News 11, 11 (1985). 5. M. Garey and D. Johnson, Computers a n d Intractability - (I guide to the theory of NPcompleteness, Freeman, 1979. 6. M. Ohya, Mathematical Foundation of Quantum Computer, Mathematical Foundation of Quantum Information, Maruzen Publ. Company, 1998. 7. P. W. Shor, Algorithm for quantum computation: Discrete logarithm and factoring, in: Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, 124 ( 1994). 8. M. Ohya and I. V. Volvich, “Quantum computing, NP-complete problems and chaotic dynamics”, quant-ph/9912100.
208
CHAOS SOLITONS & FRACTALS PERGAMON
Chaos, Solitons and Fractals 1 I (2000) 1377-1385
~
www.elsevier.nl/locate/chaos
Application of chaos degree to some dynamical systems Kei Inoue a, Masanori Ohya
Keiko Sato
Department a/ Information Sciences, Science University of Tokyo, Noda City, Chiba 278-8510, Japan Department of Control and Computer Engineering. Numazu College of Technology, Numazu City. Shiruoka 410-8501, Japan a
Communicated by Prof. Y.H. Ichikawa Accepted 15 March 1999
Abstract Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models. Q 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction
There exist several approaches in the study of chaotic behavior of dynamical systems using the concepts such as entropy, complexity, chaos, fractal, stochasticity [1,3-6,131. In 1991, one of the authors proposed information dynamics (ID for short) [10,11] to treat such chaotic behavior of systems altogether. ID is a synthesis of the dynamics of state change and the complexity of systems, and it is applied to several different fields such as quantum physics, fractal theory, quantum information and genetics [7]. A quantity measuring chaos in dynamical systems was defined by means of two complexities in ID, and it is called chaos degree. In particular, among several chaos degrees, an entropic chaos degree was introduced in [8,12], and is applied to logistic map to study its chaotic behavior. This chaos degree has several merits compared with usual measures of chaos such as Lyapunov exponent. In Section 2, we review the complexity in ID and the chaos degree (CD for short). In Section 3, we remind the entropic chaos degree and the Lyapunov exponent (LE for short). In Section 4, the algorithm computing the entropic chaos degree is shown. In Section 5, we compute CD and LE for Bernoulli shift, Baker's transformation and Tinkerbell's map, then we discuss the merits of the entropic chaos degree. 2. Complexity of information dynamics and chaos degree
Information dynamics provides a frame to study the state change and the complexity associated with a dynamical system. We briefly explain the concept of the complexity of ID in a bit simplified version (see U11). Let (d, G,a(G)) be an input (or initial) system and (g,G,Z(c))be an output (or final) system. Here d is the set of all objects to be observed and G is the set of all means for measurement of d ,a(G) is a certain evolution of system. Often we have d = 2, G = G,a = a.
'Corresponding author. Tel.: +81-471241501 ext. 3358; fax: +81-471241532
0 2000 Elsevier Science Ltd. All rights reserved
209 K. Inoue et al I Chaos, Solitons and Fractals I 1 (2000) 1377-1385
1378
For instance, when d is the set M ( Q ) of all measurable functions on a measurable space ( Q , 9 ) and G ( d ) is the set P ( Q ) of all probability measures on Q, we have the usual probability theory, by which the
classical dynamical system is described. When d is the set B ( Z ) of all bounded linear operators on a Hilbert space Z and G ( d )is the set G ( X ) of all density operators on 2,we have a quantum dynamical system. Once an input and output system are set, the situation of the input system is described by a state, an called a channel. The element of 6, and the change of the state is expressed by a mapping from G to 8, concept of channel is fundamental both in physics and mathematics [7]. Moreover, there exist two complexities in ID, which are axiomatically given as follows: Let (d,,G,,uf(Gr))be the total system of ( d , G , u ) and (g,z,E), and let C(cp) be the complexity of a state cp and T(cp;A * ) be the transmitted complexity associated with the state change cp ---t A * q . These complexities C and T are the quantities satisfying the following conditions: 1. For any cp E 6, C(cp)
0,
>0
T(cp;A*)
2. For any orthogonal bijection j : e x 6
-+
ex6 (
the set of all extreme points in G ),
CCi(cp)) = C(cp),
T(j(cp);A') = T(cp;T).
3. For @ = cp 8 II, E G,, C ( @ )= C(cp)
+ C(*).
4. For any state cp and a channel A*, 0 < T ( p ;A * )< C(Cp). 5. For the identity map "id" from G to 6. T(q;id) = C(cp). When a state cp changes to the state A'cp, a chaos degree (CD) [ I l l w.r.t. cp and A' is given by D(cp;A * ) = C(A*cp)- T ( q ;A * ) .
Using the above CD, we observe chaos of a dynamical system as CD > 0 u chaotic, CD = 0 ++ stable.
3. Entropic chaos degree and Lyapunov exponent
Chaos degree in I D was applied to a smooth map on R and it is shown that this degree enables to describe the chaotic aspects of a logistic map as well [8,12]. Here we briefly review the chaos degree defined through classical entropies. For an input state described by a probability distributionp = (pi)and the joint distribution r = ( r i j )between p and the output state p = A'p = ('pi)through a channel A', the Shannon entropy S(p) = - x i p ilog pi and the mutual entropy I ( p ; N ) = Cijrij log rij/pipjsatisfy all conditions of the complexities C and T , then the entropic chaos degree is defined by D ( p ;X ) = C(A'p) - T ( p ;n')
0
=s p
- I ( p ;A*) ,
This entropic chaos degree is nothing but the conditional entropy of aposteriori state w.r.t. the channel A*. The characteristic point of the entropic chaos degree is easy to get the probability distribution of the orbit for a deterministic dynamics, which is discussed in the next section.
21 0 K. Inoue et al. I Chaos, Solitons and Fractals I 1 (2000) 1377-1385
1379
Lyapunov exponent (LE) is used to study chaotic behavior of a deterministic dynamics. The Lyapunov exponent A(j) for a smooth mapfon R is defined by [2]
where x@) = f ( x [ " - l ) )for any n E N. For a smooth map f = ( j l , . . . , f m ) on R", the vector version of LE is defined as follows: Let xo be an initial point of Rm and x(") = f ( x " - " ) for any n E N. After n times iterations off to xo, the Jacobi matrix ~ , ( x o )ofx(nJ= (xi"',. . . , x g ) ) w.r.t. x(o) is
Then the Lyapunov exponent
nu)of x@) is defined by
Here p; is the kth largest square root of the eigenvalues of the matrix J,(X('))J,,(X~~))~. An orbit of the dynamical system described byfis said to be chaotic when the exponent is positive, and to be stable when the exponent is negative. The positive exponent means that the orbit is very sensitive to the initial value, so that it describes a chaotic behavior. Lyapunov exponent is difficult to compute for some models (e.g., Tinkerbell map) and its negative value is not clearly explained.
nu)
4. Algorithm for computation of entropic chaos degree
It is proved [9] that if a piecewise monotone mappingffrom [u, b]" to [a,b]" has non-positive Schwarzian derivatives and does not have a stable and periodic orbit, then there exists an ergodic measure p on the Bore1 set b of [u, b]", absolutely continuous w.r.t. the Lebesgue measure. Take a finite partition {Ak} of I = [a,b]" such as
I = U A (~A , n A , = 0 ,
ifj).
k
Let IS1 be the number of the elements in a set S . Suppose that n is a sufficiently large natural number and rn is a fixed natural number. Let p @ '= (pi"') be the probability distribution of the orbit up to nth step, that is, how many X I k ) ( k = rn + 1,. . . , m+ n) are in A ,
1
{ k E N;x(k)E A , , m < k < m p,'"' = -
+ n}I
n
It is shown that the n + 00 limit of p,'"' exists and equals to p ( A L ) The . channel A* is a map given by P ( ~ + '= ' A*p(").Further, the joint distribution r("Xn+l) = (r:y+'))for a sufficient large n is approximated as I",~+= I) 'LJ
I { k E N; ( ~ ' ~ ) , f ( x ' E~ A' ), )x A,, rn < k < m + n} I n
Then the entropic chaos degree is computed as D(p"');A * ) = S(p("+'))- I ( P [ ~A') );
21 1 X Inoue el a/. / Chaos, Solitons and Fractals I 1
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(20110) 1377-1385
5. Entropic chaos degree for some deterministic dynamical models In this section, we study the chaotic behavior of several well-known deterministic maps by the entropic chaos degree. 5.1. Bernoulli shqt Let f be a map from [0, I] to itself such that 2ax(")
f(xn)
=
{ a(&(")
-
I)
(O<x(") <0.5), (0.5 < x(")< I ) ,
where x ( ~E) [0,1]and 0 Q a Q 1. Let us compute the Lyapunov exponent and the entropic chaos degree (ECD for short) for the above Bernoulli shift f. The orbit of Eq. (5.1) is shown in Fig. 1. The Lyapunov exponent &(f) is log 2a for the Bernoulli shift (Fig. 2). On the other hand, the entropic chaos degree of the Bernoulli shift is shown in Fig. 3. Here we took 740 different a's between 0 and 1 with
n = 100000.
5.2. Baker's transformation We apply the chaos degree to a smooth map on R 2 . Let us compute the Lyapunov exponent and the ECD for the following Baker's transformation fu:
where (xy',x!') E [0,1] x [0,1] and 0 < a
<1
$1 I
.s:.
.I.
AAf)
: 05
08
0 06
as I
04
I5 02
2
a
0 0
1
01
02
03
04
05
06
07
08
09
25
1
3
2
Fig 1 The bifurcation diagram for Bernoulli shlft Fig 2 Lyapunov exponent of Bernoulli shift
212 1381
K. Inoue et al. I Chaos, Solitons and Fractals I1 (2000) 1377-1385
The orbit for each a is shown in Figs. 4-9. These figures show that the larger a is, the more complicated the orbit is. The maximum Lyapunov exponent A;V)is log 2a for Baker's transformation (Fig. 10). On the other hand, the ECD of Baker's transformation is shown in Fig. 11. Here we took 740 different a's between 0 and I with
n = 100000.
5.3.Tinkerbell map Let us compute the CD for the following two type of Tinkerbell maps I = [-1.2,0.4] x [-0.7,0.3].
fa
and
fb
on
ECD Xi"'
-
I
-
09
08 07 06 05 04
I
r;l
03 02
01
0
08
0
~~
02
_ _ - _~
_
04
06
0.1
0.6
_
_
.___08
4
3
Fig. 3. ECD of Bernoulli shift. Fig. 4. Orbit off. w.r.t. 0 6 (I < 0.5.
X:"' 1 0.9
0.8
Xi"'
---
' I
j
0.7 1 0.6
1 05
04 I
0.3 0.2
03
I
0.2
I
0.1
0
~~
0
5
0.2
0.4
0.6
.. ..................
. _ . _ _ _ _ _ . ~ 0.8
1
0
0.2
6
Fig. 5 . Orbit off. w.r.t. a = 0.6. Fig. 6 . 0 r b i t o f f a w . r . t . a = 0 . 7 .
0.8
21 3 X Inoue er al. / Chaos, Solitons and Fractals
1382
0
7
0.2
0.4
0.6
0.8
0
I
11 (2#00) 1377-1385
0.2
0.4
0.6
0.8
I
8 Fig. 7 . Orbit offa w.r.t. u = 0.8. Fig. S . O r b i t o f f w . r . t . u = 0 . 9 .
where (xy',x?') E I ,-0.4
/
9
10
Fig. 9. Orbit off w.r.t. a = 1.0. Fig. 10. Lyapunov exponent of Baker's transformation.
I
214 X Inoue el al. I Chaos, Solitons and Fractals I 1 (2000) 1377-1385
1383
ECD 1.2
. .
. . . . . 0.4
* 0.5
.1.2
o.6 0.4
-
0.6
I
I
11
.1
-011
4.6
I
-0.4
-01
1.
-0.2
-0.1..
.on
.
-0.8
1 . -1.1
.
-1.4 ,
a
-1.0
.
Fig. 11. ECD of Baker's transformation. Fig. 12. Orbit offa w.r.t. a = 0.243.
. . ..
a3
0.6
0.1
L
0.t
a a11
-111
41.6
4 1
406
Q1 0.1
*
+
4.3 4.4 4.1
4.6
4.7
13
14
Fig. 13. Orbit offa w.r.t. b = 2.65. Fig. 14. Orbit off w.r.t. a = 0.670.
Xi"'
XI"'
4.7
1
Fig. 15. Orbit of fb w.r.t. b = 2.8
,
*
xi"' O.'
21 5 K. Inoue el a/. I Chaos. Solitons and Fractals I 1 (2000) 1377-1385
ECD
Fig.14
r
i
:::k O.’
P4
0.6
I-0.5
0.1
-0.1-04
-0.3
-0.2
-0.1
0.
- a 1.1
0.2 0.3
0.4
0.5
0.6
0.7
0.8
hig.12
Fig. 16. ECD for Tinkerbell map f..
CD 1.6
Fig.15
1.4
)r
1.2 1
0.8
7
0.6 0.4
0.2
b .
0
1
1.9 -0.2
2.0
2.1
2.2
2.3
2.3
2.4
2.5
2.6t 2.7 2.8
\ Fig.13
!.9 3.0
Fig. 17. ECD for Tinkerbell map fa.
Here we took 740 different a’s between -1.2 and 0.9 and 740 different b’s between 1.9 and 2.9 with A,.-
[-!-
%!I
x
[L1&!]
100’ 100 100’ 100 (i = - 120, - 119,. . . , - 1 , 0 , 1 , .. . ,38,39) ( j = -70, - 69,.. . , - 1,0, l , . . . ,28,29) n = 100000.
6. Conclusion
From our results, when the orbits for Bernoulli shift, Baker’s transformation or Tinkerbell map are chaotic, both Lyapunov exponent and chaos degree are positive. However, our chaos degree can resolve some inconvenient properties of the Lyapunov exponent in the following senses:
216 K.
Inoue
et al. / Chaos, Solitons nnd Fractnls I 1 (2000) 1377-1385
1385
1. Lyapunov exponent takes negative and sometimes -m. For instance, although the exponent of Bernoulli shift and Baker's transformation cannot be defined for a = 0, the ECD is always positive and defined for any a 0. 2. It is difficult to compute the Lyapunov exponent for the Tinkerbell mapsf, andfb because it is difficult to compute f,"and f; for large n. On the other hand, the ECD of f a and fb are easily computed. 3. Generally, the algorithm for CD is much easier than that for Lyapunov exponent. References [I] Akashi S. The asymptotic behavior of &-entropyof a compact positive operator. J Math Anal Appl 1990;153:250. [2] Alligood KT, Sauer TD, Yorke JA. Chaos - an introduction to dynamical systems. Textbooks in mathematical sciences. Berlin: Springer, 1996. [3] Alicki R. Quantum geometry of noncommutative Bernoulli shifts. Banach center publications. Mathematics Subject Classification 46L87, 1991. [4] Bennatti F. Deterministic chaos in infinite quantum systems. Berlin: Springer, 1993. [ 5 ] Devaney RL. An introduction to chaotic dynamical systems, New York Benjamin, 1986. [6] Hasegawa H. Dynamical formulation of quantum level statistics. Open Systems and Information Dynamics 1997;4350. [7] Ingarden RS, Kossakowski A, Ohya M. Information dynamics and open systems. Dordrecht: Kluwer Academic Publishers, 1997. [8] Kosaka M, Ohya M. A study of chaotic phenomena by information dynamics IEICE 1997; J80-A (12):2138 [in Japanese]. [9] Misiurewicz M. Absolutely continuous measntres for certain maps of interval. Pub1 Math IHES 1981;53:17. [lo] Ohya M. Information dynamics and its applications to optical communication processes. Lecture Notes in Physics 1991;378:81. [I I] Ohya M. Complexity and fractal dimensions for quantum states. Open Systems and Information Dynamics 1997;4:141. [I21 Ohya M. Complexities and their applications to characterization of chaos. Int J Theoret Phys 1998; 37(1): 495. [I31 Toda M. Crisis in chaotic scattering of a highly excited van der Waals complex. Phys Rev Lett 1995;74(14):2970.
217 Open Sys. & Information Dyn. 6 : 69-78, 1999 @ 1999 l(luwer Academic Publishers
69
Fundamentals of Quantum Mutual Entropy and Capacity Masanori Ohya Department of Information Sciences Science University of Tokyo Noda City, Chiba 278-8510, Japan (Received: March 3, 1998)
Abstract. The rigorous formulation of the quantum mutual entropy is reviewed. The capacities for various channels, such as quantum channel, classical-quantum channel and classical-quantumclassical channel are discussed exhaustively and some misuses of these capacities are indicated.
1. Introduction
The study of mutual entropy (information) and capacity in classical system was extensively done after Shannon by several authors like Kolmogorov [13] and Gelfand [8]. In quantum systems, there have been several definitions of the mutual entropy for classical input and quantum output [6, 9, 10, 151. In 1983, the author defined [22] the fully quantum mechanical mutual entropy by means of the relative entropy of Umegaki [36], and he extended i t [24] to general quantum systems by t h e relative entropy of Araki [4] and Uhlmann [35]. When introducing t h e quantum mutual entropy, t h e author did not indicate t h a t it contained other definitions of t h e mutual entropy including t h e classical one, so t h a t there still exists a misunderstanding concerning the use of the mutual entropy (information) to compute t h e capacity of quantum channels. In this note, we point out t h a t our quantum mutual entropy generalizes t h e other and where the misuse occurs. 2. Mutual Entropy The quantum mutual entropy was introduced in [22] for a quantum input and quantum output, namely, for a purely quantum channel, and it was later formulated for a general quantum system described in C*-algebraic terminology [24]. We review here the mutual entropy in usual quantum systems described Hilbert space language. Let 31 be the input Hilbert space, B ( U ) be t h e set of all bounded linear operators on U ,and S(U)be the set of all density operators on U.T h e output space is described by another Hilbert space %, but often 31 = ?i. A chancel from the input system t o t h e output system is a mapping A* from S(U)to S(31)[21]. A channel
218
70
M. Ohya
A* is said t o be completely positive if the dual map A satisfies the following ccndition: C;,j=lA;A(B;Bj)Aj 2 0 for any n E N and any Aj E B ( X ) , Bj E B(31). This condition is not strong at all because almost all physical transformations satisfy it [ 11, 241. An input state p E S(31)is sent t o the output system through the channel A*, so t h a t the output state is written as p" s A*p. Then it is important t o ask how much information of p is correctly sent t o the output state A'p. This amount of information transmitted from input t o output is expressed by the mutual entropy. In order t o define the quantum mutual entropy, we first mention the entropy of a quantum state introduced by von Neumann [20]. For a state p, there exists a unique spectral decomposition
where A k is an eigenvalue of p and Pk is the associated projection for each x k . T he is degenerated, so t hat the spectral projection Pk is not one-dimensional when decomposition can be further decomposed into one-dimensional projections. Such a decomposition is called a Schatten decomposition [34], namely,
where Ek is the one-dimensional projection associated with A k and the degenerated eigenvalue is repeated dim Pk times; for instance, if the eigenvalue A1 has the degeneracy 3, then X I = A 2 = A3 < A d . This Schatten decomposition is not unique unless every eigenvalue is non-degenerated. Then the entropy (von Neumann entropy) S(p) of a state p is defined by S(P) = - t r p l o g p ,
(2.3)
which equals t o the Shannon entropy of the probability distribution {A,}:
The quantum mutual entropy was introduced on the basis of the above von Neumann entropy for purely quantum communication processes. T he mutual entropy depends on input state p and channel A*, so it is denoted by I ( p ; A*). It should satisfy the following conditions: (1) Th e quantum mutual entropy is well-matched t o the von Neumann entropy. Furthermore, if the channel is trivial, i.e., A* = identity map, then the mutual entropy equals the von Neumann entropy: I ( p ; id) = S ( p ) .
(2) For a classical system, the quantum mutual entropy reduces to the classical one.
21 9 71
Fundamentals of Quantum Mutual Entropy and Capacity
(3) Shannon’s fundamental inequality, [33], 0
5 I ( p ; A*) 5 S ( p ) is satisfied.
Before turning t o the quantum m_utual entropy, we briefly review the classical mutual entropy [7]. Let (Q, F), (Q, F)be an input and output measurable spaces, respectively, and P ( R ) , P(fi)are the corresponding sets of all probability measures (states) on R and fi, respectively. A channel A* is a mapping from P ( R ) t o P ( 6 ) and its dual A is a map from the set B(R) of all Baire measurable functions on R t o B(fi). For an input state p E P ( Q ) ,the output state fi = A*p and t he joint state (probability measure) Q is given by
@(Q x where
nQ
a)
= /-A(~Q)+,
Q
Q E .T,
a
E
F,
(2.5)
is the characteristic function on R:
The classical entropy, relative entropy and mutual entropy are defined as follows:
s(P)= SUP { -
2 P(Ak)
1% p ( A k ) ;
iAk) E
p ( Q ) }>
(2.6)
k=l
where P ( R ) is the set of all finite partitions on R, t hat is, { A k } E P ( Q )iff Ak E F with Ak n Aj = 8 (lc # j ) and U&,& = 0. T h e quantum mutual entropy is defined as follows: In order t o define the quantum mutual entropy, we need the quantum relative entropy and the joint state (it is called “compound state” in the sequel) describing the correlation between an input state p and the output state A*p through the channel A*. A finite partition of R in classical case corresponds t o an orthogonal decomposition { E k } of t he identity operator I of 31 in quantum case because the set of all orthogonal projections is considered t o be the event system in a quantum system. I t is known [29] t hat the following equality holds sup{ - C t r p E k l o g t r p E k ; { ~ k } } = - t r p l o g p , k
and the supremum is attained when { E k } is a Schatten decomposition of p. Therefore, the Schatten decomposition is used t o define the compound state and the quantum mutual entropy.
72
M. Ohya
T h e compound state U E (corresponding t o joint state in CS) of p and A*p was introduced in [22, 231; it is given by
where E stands for a Schatten decomposition { E k } of p , so t hat t he compound state depends on how we decompose the state p into basic states (elementary events) or, in other words, on how we see the input state. The relative entropy for two states p and u is defined by Umegaki [36] and Lindblad [16], which is written as trp(1ogp
- log c)
when
c ran)
otherwise
(2.10)
Then we can define the mutual entropy by means of the compound state and the relative entropy [22], t hat is,
I ( P ; A*) = suP{s(oE, p @ A*P) ; E = { E k } }
i
(2.11)
where the supremum is taken over all Schatten decompositions because this decomposition is not unique in general. Some computations reduce it t o the following
This mutual entropy satisfies all conditions (1)-(3) mentioned above. When the input system is classical, an input state p is given by a probability distribution or a probability measure, in either case the Schatten decomposition of p is unique, namely, for the case of probability distribution, p = {&}, (2.13) where
hk
is the delta measure, t hat is, (2.14)
Therefore, for any channel A*, the mutual entropy becomes
XkS(A*hk,A*P) ,
I ( p ; A*) =
(2.15)
k
which equals the following usual expression of Shannon when the minus is welldefined : (2.16) I ( p ; h * ) = S(A*p) -
221 73
Fundamentals of Quantum Mutual Entropy and Capacity
The above equality has been taken as the definition of the mutual entropy for a classical-quantum channel [5, 6, 9, 10, 151. Note th at the definition (2.12) of the mutual entropy is written as
r ( f ;A*)
= SUP
{
A k S ( A * P k , A*p) ; p =
AkPk
E po(P)}
I
k
k
where Fo(p)is the set of all orthogonal finite decompositions of p . Here P k is orthogonal to p j (denoted by p k Ip j ) meaning t hat the range of p k is orthogonal t o th at of p j . Th e above equality is easily proved as follows. P u t
The inequality I ( p ; A*) 5 I j ( p ; A*) is obvious. Let us prove the converse. Each p k in an orthogonal decomposition of p is further decomposed into one-dimensional projections, pk = pi( k ) E j( k ), a Schatten decomposition of P k . From the following
cj
equalities for the relative entropy [4,291: (1) S(ap, ba) = aS(p, ~ 7 )- a l o g ( b / a ) ,for S(pl p 2 , a ) = S(p1,o) S(p2,a ) , we any positive numbers a, b (2) p1 Ip2 have
*
+
+
which implies the converse inequality I ( p ; A*) 2 If ( p ; A*) because c k , j X k p (j k ) E (j k ) is a Schatten decomposition of p. Thus I ( p ; A*) = I j ( p ; A*). More general formulation of the mutual entropy for general quantum systems was done [24, 111for C*-dynamical systems by using Araki's or Uhlmann's relative entropy [4,35, 291. This general mutual entropy contains all other cases including measure-theoretic definition of Gelfand and Yaglom [8]. The mutual entropy is not only a measure of information transmission but it also describes the change of state, so t hat this quantity can be applied in several fields [I, 2, 3, 17, 19, 24, 25, 28, 321. 3. Communication Processes
In this section, we discuss communication processes [7, 10, 291. Let { a l , u 2 , . . .,a,} be an alphabet and 0 be the infinite direct product of A : R = A" G nYrnA called a message space. In order t o send an information represented by a n element of this message space t o a receiver, we often need t o transform the message into a form proper for a communication channel. This transformation of message is
74
M. Ohya
called coding. Precisely, coding is a measurable one-to-one map [ from R t o a proper space X . For instance, we have the following codings: (1) When a message is expressed in binary symbols 0 and 1, such coding is a map from R t o ( 0 , l}N. (2) A message expressed by a {0,1} sequence in (1)is represented by an electric signal. (3) Instead of an electric signal, we use optical signal. Coding is a combination of several maps like the ones above. One of the main targets of the coding theory is to find the most efficient coding and also decoding for information transmission. Let (n,Fa,P ( R ) ) be an input probability space and X be the coded input space. This space X may be a classical object or a quantum object. For instance, X is a Hilbert space 71 of a quantum system, then the coded input system is described by (B(R),S(71)) of Sec. 2. An output system is similarly described as the input-system: T he coded output space is denoted by r? and the decoded output space is fi, possibly over a different alphabet. A transmission (map) from X t o r? is described by a channel reflecting all properties of a physical device, which is denoted by y here. With a decoding E", the whole information transmission process is written as
R
-LX
A
r? E, f i .
T h a t is, a message w E R is coded to c ( w ) and it is sent t o the output system through the channel y,then the output coded message becomes y o [ ( w ) and it is decoded t o c o y o [ ( w ) at the receiver. This transmission process is mathematically set as follows: 111 messages are sent t o the receiver, the k-th message w(lC)occuring with probability x k . Then the occurrence probability of each message in the sequence ( w ( l ) ,w('), . . . ,~ ( ~ of1 M) messages is denoted by p = {&}, which is a state in a classical system. If [ is a classical coding, then e ( w ) is a classical object such as a n electric pulse. If is a quantum coding, then [ ( w ) is a quantum object (state) such as a coherent state. Here we consider such a quantum coding, t hat is, [ ( w ( ~ )is) a quantum state, and we by) f f k . Thus the coded state for the sequence ( w ( l ) ,w ( ' ) , . . . ,d ' ) ) denote [ ( d k ) is written as ff = X X k C T k . (3.2) k
This state is transmitted through a channel y. This channel is expressed by a completely-positive mapping r*,in the sense of Sec. 1, from the state space of X t o that of X I hence the output coded quantum state 8 is r*a.Since the information transmission process can be understood as a process of state (probability) change, when and fi are classical and X and 2 are quantum, the process (3.1) is written as -
-
P(R)
5
S(71) r ; S(%)
-3P(fi),
(3.3)
where 3* (resp. E*) is the channel corresponding t o the coding (resp. ) and S(3c) (resp. S ( % ) )is the set of all density operators (states) on 3c (resp. %).
223 Fundamentals of Quantum Mutual Entropy and Capacity
75
We have t o study the objects in the above transmission process (3.1) or (3.3) carefully. Namely, we have t o make clear which object we are going t o study. For instance, if we want t o know the information capacity of a quantum channel y (= T'*), then we have t o take X describing a quantum system, i.e. a Hilbert space, and we need t o s ta r t the study from a quantum state in the quantum space X , not from a classical state associated t o a message. If we like t o know the capacity of the whole process including the coding and decoding, which means the capacity of the channel E o y o E = &* o T'* o E*, then we have t o st art from a classical state. In any case, when we are concerned with the capacity of a channel, we have t o take the supremum of the mutual entropy I ( p ; A*) over quantum or classical states p in a proper set determined by what we want t o study. We explain this more precisely in the next section.
-
4. Channel Capacity
We discuss two types of channel capacity in communication processes, namely, the capacity of a quantum channel r* and that of a classical (classical-quantumclassical) channel Z* o r*o E*.
-
4.1. CAPACITY O F QUANTUM CHANNEL The capacity of a quantum channel is the ability of the channel itself t o transmit information, so th at it does not depend on how a message (treated as a classical object) is coding; we have t o begin with an arbitrary quantum state and find t he supremum of the mutual entropy. One often makes a mistake at this point. For example, one begins with the coding of a message, and computes the supremum of the mutual entropy, identifying the supremum with the capacity of the quantum channel, which is not correct. Even when the coding is quantum and t he coded message is sent t o a receiver through a quantum channel, if one starts from a classical state, then the capacity is not the capacity of the quantum channel itself. In his case, the usual Shannon theory is applied because one can easily compute the conditional distribution in a usual (classical) way. Such a supremum is the capacity of a classical-quantum-classical channel, and it is in the second category discussed below. The capacity of a quantum channel I?* is defined as follows: Let SO C S(31)be the set of all states prepared for expression of information. Then the capacity of the channel T'* with respect t o $0 is defined by
cSo(r*) = s ~ p { i ( p r*) ; ; p E so}.
(4.1)
Here I ( p ; r*)is the mutual entropy given in (2.11) or (2.12) with A* = r*.When So = S(%),Cs(R)(I'*) is denoted by C(I'*) for simplicity. In [30, 181, we also considered the pseudo-quantum capacity Cp(I'*)defined by (4.1) with the pseudo-
224 76
M. Ohya
mutual entropy I p ( p ;r') where the supremum is taken over all finite decompositions instead of all orthogonal pure decompositions:
~ ~ (r*) p ;= sup
{ C X k S ( r * p k , r*p) ; p = C X k p k , finite decomposition 1. k
k
(4.2) However, the pseudo-mutual entropy is not well-matched with the conditions explained in Sec. 2, and it is difficult t o compute it numerically [31]. From the monotonicity of the mutual entropy [29], we have
o 5 cso(r*) 5 c,sO(r*)5 4.2. C A P A C I T Y
SUP{S(P) ; p E
O F CLASSICAL-QUANTUM-CLASSICAL
so]
CHANNEL
-
The capacity of C-Q-C channel Z* o r*o S* i s the capacity of t he information transmission process starting from the coding of messages, therefore it can be considered as the capacity including a coding (and a decoding). As is discussed in Sec. 3, the input state p is the probability distribution { A , } of messages, and its Schatten decomposition is unique as (2.13), so the mutual entropy is written by (2.15):
-
-
I ( p ; E*0 r*0 E*) =
Xks(E*
0
r*
0
E*6k
-
, E*0 r*0 2 * p ) .
(4.3)
k
If the coding E* is a quantum coding, then Let us denote the coded quantum state by the above mutual entropy is written as
is expressed by a quantum state. and put u = E * p = C k A k a k . Then
S*bk uk
-
-
-
I ( p ;=* 0 r*0 a*) =
Xks(E*
-
0
r*ok
, E*0 r.0) .
(4.4)
k
This is the expression of the mutual entropy of the whole information transmission process starting from a coding of classical messages. Hence the capacity of C-Q-C channel is cpo
(?* r* E*) =
sup
-
{ qp;E* r* a*); 0
E p0} ,
(4.5)
where Po c P ( R ) is the set of all probability distributions prepared for input (apriari) states (distributions or probability measures). Moreover, the capacity for coding free channel is found by taking the supremum of the mutual entropy (4.4) over all probability distributions and all codings E*:
c,P(?* r*) =
- r*
sup { ~ ( pE* ;
0
E*); p
E
pols*} .
(4.6)
225
77
Fundamentals of Quantum Mutual Entropy and Capacity
The last capacity is for both coding and decoding free channels and it is given by Po r*) = s u p { ~ ( p ~ ~ * o r * ~ Pz E * )P;, , E * , ~ * } .
ccd(
(4.7)
These capacities C?, C 2 do not measure the ability of the quantum channel itself, but measure the ability of r* through the coding and decoding. Note th at when x k AkS(r*ak) is finite, then (4.4)becomes
-
I ( ~z* ;
r* z*) = s(z* r*a)- C A ~ s ( So* r * O k ) .
r*
(4.8)
k
Further, if p is a probability measure having a density function f ( A ) and each X corresponds t o a quantum coded state .(A), then a = J f ( A ) a ( A ) d A and ~ ( pE* ; 0 r* which is less than
/
s*)= s(g* r*a)- f(x)s(g* r*c(x))d 0
~ , (4.9)
/
s(r*a)- f(x)s(r*a(x)) d~ , because of th e monotonicity of the quantum mutual entropy. T h e above bound is called the Holevo bound, and it is computed in several cases [30, 371. The above three capacities Cpo, satisfy the following inequalities
Cp, C 2
o 5
C ~ O ( ~ * ~ ~5 *c,Po(Sror*) ~ E * ) 5
c2(r*)5
sup{s(p);
E
poi,
where s ( p ) is not the von Neumann entropy but the Shannon one: - X k log X k . The capacities (4.1), (4.6), (4.7) and (4.8)are in general different. Some misunderstanding occurs due t o forgetting which channel is considered. That is, we have t o make clear whether the capacity of a quantum channel itself or t hat of a classical-quantum(-classical) channel or t hat of a coding free, is considered.
Bibliography 1. 2. 3. 4. 5.
L. Accardi, M. Ohya, and H. Suyari, Open Sys. Information Dyn. 2 , 337 (1994). S. Akashi, J. Math. Anal. Appl. 153, 250 (1990). R . Alicki, Open Sys. Information Dyn. 4, 53 (1997).
H. Araki, Publ. RIMS Kyoto Univ. 11, 809 (1976).
V.P. Belavkin,
Proc. of VIII-th Conference on Coding Theory a n d Information Transmission, Moscow-Kuibishev, 1982, p. 15. V. P. Belavkin and P. L. Stratonovich, Radio Eng. Electron. Phys. 18, 1839 (1973). P. Billingsley, Ergodic Theory a n d Information, Wiley, NY.,1965.
6. 7. 8. I. M. Gelfand and A . M . Yaglom, Amer. Math. SOC.Transl. 12, 199 (1959). 9. A. S. Holevo, Problemy Peredachi lnformacii 9, 3 (1973), in Russian. 10. R . S. Ingarden, Rep. Math. Phys. 10, 43 (1976). 11. R . S. Ingarden, A. Kossakowski and M. Ohya, Information Dynamics a n d Open Systems, Kluwer Academic Publisher, 1997.
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K. Inoue, M. Ohya, and H. Suyari, Physica D 120, 117 (1998). A. N. Kolmogorov, Amer. Math. SOC.Translation, Ser.2 33,291 (1963). S. Kullback and R. Leibler, Ann. Math. Stat. 22, 79 (1951). L. B. Levitin, Springer Lect. Note in Phys. 378,101 (1991). G.Lindblad, Commun. Math. Phys., 33,111 (1973). T . Matsuoka and M. Ohya, Rep. Math. Phys. 36,365 (1995). N . Muraki, M. Ohya, and D. Petz, Open Sys. Information Dyn. 1, 43 (1992). N. Muraki and M. Ohya, Lett. Math. Phys. 36,327 (1996). J. von Neumann, Die Mathernatischen Grundlagen der Quantenrnechanik, Springer-Berlin,
1932. 21. M. Ohya, J. Math. Anal. Appl. 84,318 (1981). 22. M. Ohya, IEEE Trans. Information Theory, 29, 770 (1983). 23. M. Ohya, L. Nuovo Cimento 38,402 (1983). 24. M.Ohya, Rep. Math. Phys. 27, 19 (1989). 25. M. Ohya, Quantum Probability and Related Topics 6, World Scientific, Singapore, 1991. 26. M. Ohya, Quantum Communications and Measurement, p. 309, 1995. 27. M. Ohya, Open Sys. Information Dyn. 4,141 (1997). 28. M. Ohya, Int. J. Theor. Phys. 37,495 (1998). 29. M.Ohya and D. Petz, Quantum Entropy and Its Use,Springer, 1993. 30. M. Ohya, D. Petz, and N. Watanabe, Probability and Mathematical Statistics, 17, 179 (1997). 31. M.Ohya, D. Petz, and N. Watanabe, Int. J . Theor. Phys. 38,507 (1998). 32. M. Ohya and N. Watanabe, Physica D 120, 206 (1998). 33. C. E. Shannon, Bell System Tech. J. 27, 379 (1948). 34. R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, 1970. 35. A. Uhlmann, Commun. Math. Phys. 54,21 (1977). 36. H.Umegaki, Kodai Math. Sem. Rep. 14,59 (1962). 37. H.P. Yuen and M. Ozawa, Phys. Rev. Lett. 7 0 , 363 (1993).
227 Appl Math Optim 39:33-59 (1999) With kind permission of Springer Science and Business Media
0 1999 Smineer-Verlag New York Inc
Compound Channels, Transition Expectations, and Liftings L. Accardi' and M. Ohya2
'
Dipartimento di Matematica, Centro Matematico V. Volterra, UniversitB di Roma 11, Rome, Italy 'Department of Information Sciences. Science University of Tokyo, Noda City, Chiba 278, Japan
Abstract. In Section I we introduce the notion of lifting as a generalization of the notion of compound state introduced in [21] and [22] and we show that this notion allows a unified approach to the problems of quantum measurement and of signal transmission through quantum channels. The dual of a linear lifting is a transition expectation in the sense of [ 3 ] and we characterize those transition expectations which arise from compound states in the sense of [22]. In Section 2 we characterize those liftings whose range is contained in the closed convex hull of product states and we prove that the corresponding quantum Markov chains [2] are uniquely determined by a classical generalization of both the quantum random walks of [4] and the locally diagonalizable states considered in [ 3 ] . In Section 4, as a first application of the above results, we prove that the attenuation (beam splitting) process for optical communication treated in [21] can be described in a simpler and more general way in terms of liftings and of transition expectations. The error probabilty of information transmission in the attenuation process is rederived from our new description. We also obtain some new results concerning the explicit computation of error probabilities in the squeezing case. Key Words. Compound state, Transition expectation, Lifting, Channel, Quantum probability, Quantum Markov chain, Beam splitting, Optical communication.
AMS Classification. 8 I Q99,94A40,60527.
228 L. Accardi and M. Ohya
34
Introduction The following situation is very common both in classical and quantum physics: one considers two systems, denoted respectively 1, 2, and their algebras of observables, dl , d 2 . One usually assumes that the interaction between the two systems is switched on at a sharp time to before which the two systems are considered to be independent. During the interaction the two systems merge into a larger system denoted (1,2) whose algebra of observables dcontains both d l and A2, in the sense that there are embeddings j l : d l +. A;
j 2 : d 2 +.
d
and that any physical information on the state of system I or of system 2 after the interaction can be obtained by choosing a state cp on A, i.e., a state of the composite ) jz(A2)). In system (1,2), and considering its restriction on the algebra j ~ ( A l (resp. most applications one chooses
d=Ai
@d2;
V=VI
@cp2,
j , ( a 1 ) = ~ 1 @ 1 2 ; jz(a2)=11@32,
ai
~ d l ,a 2 ~ d 2 ,
(0.2) (0.3)
i.e., the algebra of the compound system is described by a tensor product. In this paper we confine our analysis of such a situation. The choice of the state cp depends on the initial states of the two systems and on the interaction between them. In connection with this situation one studies several problems depending on the interpretation of systems 1 and 2. For example: (i) The state cp2 of system 2, after the interaction, is known (e.g., an output signal, a pointer in a measurement apparatus) and one wants to know the state q9 of system 1 before the interaction (e.g., an input signal, the state of a microsystem which has interacted with the apparatus). (ii) As in (i), exchanging the roles of 1 and 2, from the mathematical point of view, this exchange is trivial, but we want to underline that our approach avoids the separation of a macroworld, described by classical physics, from a microworld, described by quantum physics. (iii) The initial state of the composite system ( I , 2) is known and one wants to know the state of system 1 (system 2). (iv) The state cpl of system 1, before the interaction (e.g., the preparation of a microsystem) and the form of the interaction, is known and one wants to know the state of system 1 after the interaction. In all these cases the goal is to construct a map from the state space of a system to the state space of another system. In the literature on quantum information and communication systems, such a map is called a channel [20]. An important class of channels are those from the state space of an algebra AI into the state space of the algebra dl @ d 2 . These channels are called liftings; more generally, a lifting should be through as a channel from a subsystem to a compound system. An important example of liftings are the duals of transition expectations. Recall (see Definition I .3 below) that if d1,d 2 are C*-algebras, a transition expectation from dl @ d 2 to dl is a completely positive linear map &: d l @ A2 +. dl satisfying (1.6).
229 Compound Channels, Transition Expectations, and Liftings
35
Transition expectations play a crucial role in the construction of quantum Markov chains and they arise naturally within the framework of measurement theory in the following way: the composite system (1,2) undergoes an evolution ut : A + A ( t E R ) , which is a one-parameter group of *-automorphisms of A. This means that the state q of (1,2) evolves according to the law qt := q
0
(0.4)
Ut
and the state ql of the system 1 evolves according to the reduced evolution: q I , t ( a l ) : = ~ 1 ( E 2 o u ot ~ I ( ~ I ) ) ;
E
AI,
(0.5)
where j1 is given by (0.3) and E2: A = Al 8 A2 -+ A1 is the Umegaki conditional expectation characterized by E2(ai 8 a2> = aiq2(a2);
ai E A I , az E A2.
(0.6)
We fix a time T representing the moment when the experiment ends (ideally T = +co) and consider the linear map I TA :1 8 A2 -+A1 characterized by
8 a21 = E ~ ( U T @ ( ~~ 2I ) ) ; ai E -41, a2 E A2. (0.7) Then &T is a transition expectation. If one is ready to accept that the evolution U T does not take place inside the algbra A1 8 A2 but is a representation of A1 8 A2 into another algebra (usually much larger), then in some cases and in a certain technical sense, (0.7) represents the most general class of transition expectations (see Theorem 1.4 below). An instrument in the sense of the operational approach to quantum measurment is obtained by taking the restriction of a transition expectation & to a subalgebra CI 8 A2 of A1 8 A2 where CI is a a-finite abelian von Neumann sup-algebra of A ] .In this case it is known that if CI is a-finite, then there exists a probability space (Q, F,P ) such that CI is isomorphic to Loo(R,F ,P ) and the points w E Q are interpreted as macroscopic parameters of the apparatus. If Al = C I , i.e., if A1 is an abelian von Neumann algebra, the isomorphism (see [28]) implies that the elements of A, i.e., the observables of the composite system (1, 2 ) can be interpreted as functions (R, F,P ) -+ A2, i.e., as operator-valued random variables. Thus interpreting (Q, F ,P ) as the sample space of a classical stochastic process, the operational scheme becomes equivalent to the theory of operator-valued classical processes (see [7]). From this point of view an instrument in the operational sense is an object which is only half-quantum. The physical motivations for this choice go back to some ideas of Ludwig, according to which the measurement apparatus is usually a macroscopic body so that classical probability should be sufficient for its description. Several authors have introduced variations and modifications of Ludwig's ideas, however, since all the examples of physical interest of instruments in an operational sense produced up to now are the restrictions of liftings, we feel that the latter notion plays a more natural and fundamental role. In conclusion we show the theory of lifting includes the so-called operational approach.
230 L. Accardi and M. Ohya
36
1. Channels, Liftings, and Transition Expectations For a C*-algebra A, we denote by S(A)the convex set of its states. In this paper all C*and W*-algebras are realized on some separable Hilbert spaces and, unless explicitly stated, the tensor products are those induced by the tensor products of the corresponding Hilbert spaces. If A is a von Neumann algebra, S(A)denotes the set of its normal states and S(A)extr the set of extremal states. Both S(A)and S(A)extr are measurable spaces with their Bore1 structure and the set of probability measures on S(A) (resp. S(A)extr) is denoted by ProbS(A) (resp. Prob S(A),,,,).If A and B are C*-algebras, a channel from A to B is a map A*: S(A) -+ S ( B ) .If A * is affine we speak of a linear channel. If A* is w*-continuous and linear, then it can be extended by linearity to a linear map (still denoted A*) from A” to B*.Its dual A: B + A is apositive map. If it is completely positive, we call it a Markovian operator. Such channels have been studied with some applications by several authors (see [20], 1241, [25], and references quoted therein). Certain quantum channels are naturally associated to classical Markovian kernels on the measurable space S ( B )x S(A).In fact, given such a Markovian kernel, i.e., a measurable map
p: w
E
S(A) -+ p ( . I w )
Prob(S(B)),
E
one can define a channel in the following way: for any state cp decomposition
v=
s,,
E
S ( A )one fixes a convex
wdPv(w)
and defines the channel A*: S(A) + S ( B ) through the identity
s,,)
A*v :=
dPv(w)
s,,
w’p(dw’ I w ) .
The channel A* is usually nonlinear since the map + P~is affine if and only if S(d) is a simplex [8, Theorem (4.1.15) and Corollary (4. I . 17)J and this is the case if and only if A is Abelian [8, Example (4.1.6)]. On the other hand, given a linear channel A* one might try to associate to it a Markovian kernel p ( . lo) on the measurable space S ( B ) x S(A),by fixing, for each w E S(A),a convex decomposition A*w :=
L,,,
w’p(dw’ I w ) .
The possibility of choosing such a decomposition to assure the measurability of the map
p: w
E
S ( A ) + p ( . I w ) E Prob(S(B))
as well as study of the support of these measures give rise to some subtle measuretheoretic problems which will be discussed elsewhere. In many examples, however, these Markovian kernels can be explicitly constructed and, at least on a subset of the states, have good support and measurability properties.
Definition 1.1. Let At, A2 be C*-algebras and let A1 @A2be a fixed C*-tensor product of dl and A2. A lifting from A1 to dl @ A2 is a w*-continuous map E * : S ( A I )+ S(Al@A2).
(1.1)
231 Compound Channels, Transition Expectations, and Liftings
31
If &* is affine and its dual is a completely positive map, we call it a linear lifting; if it maps pure states into pure states, we call it pure.
Remark. Also in the nonlinear case some kind of complete positivity requirement should be included in the definition of lifting. However, the theory of nonlinear completely positive maps is still in its infancy but some is true for a satisfactory dudizution of it. Therefore we leave the general question open for further developments and we limit ourselves to presenting some examples of nonlinear liftings which are of some interest for the applications. To every lifting from dl to dl @ A2 we can associate two channels: one from dl to A1 , defined by ATpi(ai) := (&*pi)(ai @ 1 ) ;
di,
(1.2)
Va.2 E d2.
(1.3)
Vai E
and another from A1 to d 2 , defined by A;pi(ad := (I*pi)(l @ ~
2 ) ;
In general, a state p E S(A1 @ d 2 ) such that (otAlel = PI;
(PII@Az= p2
( 1.4)
has been called [21], [241 a compound state of the states pi E S(d1) and p2 E S(d2). In classical probability theory, the term coupling between pi and p2 is also used [ 131. The following problem is important in several applications: Given a state p1 E S(d1) and a channel A*: S ( A I )-+ S(d2), find a standard lifting I * :S(d1) --f S(A1 @ d2) such that &*pi is a compound state of P I and A*pl . Several particular solutions of this problem have been proposed in [9]-[ 1 11 and [21]-[24], however, an explicit description of all the possible solutions to this problem is still missing.
Definition 1.2. A lifting from dl to dl @ d 2 is called nondemolition for a state pi E S(d1)if p1 is invariant for At, i.e., if, for all a1 E dl, (E*pl)(al @ 1 ) = pl(al).
(1.5)
The idea of this definition being that the interaction with system 2 does not alter the state of system 1.
Definition 1.3. Let dl , d 2 be C*-algebras and let dl@A2be a fixed C*-tensor product of and d 2 . A transition expectation from dl @ d 2 to A1 is a completely positive linear map &: AI @ A2 .+ dl satisfying
Remark. The notion of nondemolition lifting discussed here is essentially (i.e., up to minor technicalities) included in the more abstract notion of state extension introduced by Cecchini and Petz [ I 11 (see also [lo]).
232 L. Accardi and M. Ohya
38
The two interpretations of the notion of standard lifting, which are used in this paper, are the following:
The measurement process. A1 (resp. d2)is interpreted as the algebra of observables of a system (resp. a measurement apparatus ) and &* describes the interaction between system and apparatus as well as the preparation of the apparatus. If P I E S(A1) is the preparation of the system, i.e., its state before the interaction with the apparatus, then AYpl E S(A1) (resp. A;pl E S(A2))is the state of the system (resp. of the apparatus) after the measurement. The signal transmission process. An input signal is transmitted and received by an apparatus which produces an output signal. Here A1 (resp. A2)is interpreted as the algebra of observables of the input (resp. output) signal and &* describes the interaction between the input signal and the receiver as well as the preparation of the receiver. If PI E S(A1)is the input signal, then the state A;pl E S(A2), defined by (1.3), is the state of the (observed) output signal. An important lifting related to this signal transmission is one due to aquantum communication process discussed below (Examples 1 a and 4). In several important applications the state P I of the system before the interaction (preparation, input signal) is not known and one would like to know this state knowing only h ; p l E S(Az),i.e., the state of the apparatus after the interaction (output signal). From a mathematical point of view this problem is not well posed, since usually the map A; is not invertible. The best one can do in such cases is to acquire a control on the description of those input states which have the same image under A; and then choose among them according to some statistical criterion. Another widely applied procedure is to postulate, on the basis of some experimental information, that the input state belongs to an a priori given restricted class of states and to choose among these ones by some statistical criterion. In the following we describe several examples of liftings which appear frequently in the applications.
Example 1 (Isometric Liftings).
Let V : 'HI + 'HI C3 'Fl2 be an isometry
v*v = lx,.
(1.7)
Then the map
is a transition expectation, and the associated lifting maps a density matrix w1 E T('FI1) into &*w1 = Vwl V * .Liftings of this type are called isometric. Every isometric lifting is a pure lifting. Isometric liftings have turned out to play a relevant role in some mathematical models for superconductivity [ 141.
Example l a (The Attenuation (or Beam Splitting) Lifting). ric lifting characterized by the properties = 'Ft2 =: r(C) = the Fock space over C.
It is the particular isomet-
233 Compound Channels, Transition Expectations, and Liftings
39
is characterized by the expression
vie) = iae) B we), where 10) is the normalized coherent vector parametrized by 8 such that
+ 18l2= 1.
1aI2
(1.10) E
C and a,B E C are (1.1 1)
Notice that this liftings maps coherent states into products of coherent states. So it maps the simplex of the so-called classical states (Le., the convex combinations of coherent vectors) into itself. Restricted to these states it is of convex product type in the sense of Definition 2.1 below, but it is not of convex product type on the set of all states. Denoting, for 8 E C, we as the state on B ( r ( C ) )defined by (1.12)
we see that, for any b E B ( r ( C ) ) , (E*ws)(b €3 1) = w,s(b),
(1.13)
hence this lifting is not nondemolition.
Interpretation. r(C) is the space of a one-mode EM field (signal). V represents the interaction, of the signal with an apparatus (e.g., a receiver or a semitransparent mirror). In r(C) 8 r(C) the second factor is the space of the apparatus. Equation (1.10) means that, by the effect of the interaction, a coherent signal (beam) 18)splits into two signals (beams) still coherent, but of lower intensity. Because of (1.1 I), the total intensity (energy) is preserved by the transformation.
Example l b (Superposition Beam Splitting). The only difference with Example 1a is the form of V , which in this case is
One easily checks that V extends linearly to an isometry of the form (1.9). This isometric lifting is not of convex product type in the sense of Definition 2.1 of the next section, neither it is a nondemolition lifting.
Example 2 (Compound Lifting). Let A*: S(AI) +. S ( & ) be a channel. For any P I E S(d2)in the closed convex hull of the external states, fix a decomposition of p I as a convex combination of extremal states in S(d1): (1.15)
234 L. Accardi and M. Ohya
40
where p ( . 1 P I ) is a Bore1 measure on define &*Pi := L , , , , W l
S(dI)with support in the extremal states, and
@ A*WP(dWl I P I ) .
(1.16)
Then &*: S(d1)+ S(d2 @ d2)is a lifting, nonlinear even if A* is linear, and nondemolition for P I . In Section 2 we shall see that the most general lifting, mapping S(d1)into the closed convex hull of the external product states on dl @ d2,is essentially of this type. Here “essentially” means that, in order to recover the most general case, we shall weaken, from the original definition of compound state in [22], the condition that p ( d o 1 I P I ) is concentrated on the extremal states. Therefore once a channel is given, a lifting of convex product type can be constructed by (1.16), and the converse is also true due to (1.3):
channel
tf
lifting.
For example, the von Neumann quantum measurement process is written, in our terminology, as follows: Having measured an observable A = C, a, P, (spectral decomposition with discrete spectrum and C, P, = I ) in a state p , the state after this measurement will be
and a lifting &*, of convex product type, associated to this channel A* and to a fixed decomposition of p as p = C,,h,p, (p, E S(d1)) is given by (1.17)
A more sophisticated example of lifting of this type is a reduction of a state associated with an open system dynamics. Namely, if a system C I described by a Hilbert space 7-1 interacts with an external system C2 described by another Hilbert space Ic and the initial states of C I and C2 are p and (T, respectively, then the combined state 0, of CI and C2 at time t after the interaction between the two systems is given by
where Ut = exp(itH) with the total Hamiltonian H of by taking the partial trace with respect to Ic, i.e.,
XI.and C2.A channel is obtained
p + ATp = trKQt.
A lifting associated to this channel is given by (1.17) with A: above. Example 3 (Canonical Form of a Lifting). Let dl = &‘HI), d2 = B(7-12).The most general linear lifting &*: S(dI)--+ S(dI@ d2)has the form WI E
T(7-11)+ C K i ( W I @ 1)K*
E
T(7-11 87-12)
(1.18)
235 Compound Channels, Transition Expectations, and Liftings
for some K ; E
41
B(7-118 7-12) such that (1.19)
This is a simple consequence of Krein's lemma.
Example 4 (The Lifting for a Quantum Communication Channel). K2 be Hilbert spaces. Denote by a the amplification a : b2 E
B(7-12)+ a(b2) = b2 @ IK,
E
Let 3-11,
B(7-12 8 K 2 ) .
7-12,
KI,
(1.20)
Let
be a completely positive identity preserving map and, for D I the conditional expectation
E
S ( B ( K l ) )denote , by
a ] 8 61 E B(3-11)@B(ICI)+ alal(bl) E B(3-11)Z B(3-11)8 l x , .
(1.21)
Then the lifting and the channel describing quantum communication processes are defined by & * = y * o o ,(2)* , (1.22) A * p = a* o & * ( p ) = trx2 y * ( p C3 01);
p E
S(B(WI)),
( I .23)
where p and 01 correspond to an input state and a noise state, respectively (see [24]). Moreover, the following remark, extending an unpublished result of A. Frigerio, shows that the above model of the quantum communication process is universal among the transition expectations, provided one chooses the space of the representation large enough.
Theorem 1.4. Let B = B(3-1)for a separable, injinite-dimensional, Hilbert space IFI and let &: B 8 B + B be a normal transition expectation. Then there exist a normal state cp on the W*-algebra (8B)' 8 M2 =: C (M2 is the algebra of 2 x 2 complex matrices), and a unitary element U of
(8B)48 M2 (1.24) such that, denoting 75 : B 8 B + A as the normal representation (amplification) A := C 8 B
(1.25) and q") := cp 8 Z B : A + B 2 I c 8 B as the conditional expectation (in is the identical map on B ) (1.26)
(1.27)
236 42
L. Accardi and M. Ohya
Proof. From Kraus' lemma [ 181 we know that, since & is normal, there exist a countable (since 1-I is separable family) a; E B @ B (i E N ) such that, identifying B with B @ 1, one has
i€N
If 1-I is infinite-dimensional in B @ 23 there exist isometries u ; (i E N ) such that, for each i, j , uru, = 6;j. Thus, defining d
(1.28)
u := x u ; @ a ; E (€4B)4, i=l
one finds u*(lg2 @ x ) u = l g 2 €4 & ( x ) ,
x
E
B @ B.
(1.29)
In particular, since u*u = 1 8 4 , u is apartial isometry with initial projection as the identity. Denote e = uu* as its final projection and define the unitary operator
u=
(;
;e)
Then for any vector (Vt
: (€4H)4 @
r] E
c22 (@H)4 @ (€4H)4 + (@W4@ C2.
H and for any fixed unit vector @
E(x)r])=(@@ V 7 (]a. @ & ( X I ) @ €3 r I ) = ( @
E
(1.30)
( & I H )one ~ finds
@ V, (u*(lgz @ x ) u ) @ @ v). (1.31)
Using the notation
t = @ @ r] E (@HI4
( 1.32)
the right-hand side of (1.3 1) can be rewritten as
(1.33) and, using Kraus' representation and the identifications (1.30) and (1.321,
The right-hand side of (1.33) then becomes equal to (1.34) Thus, defining the state cp on cp(.Y) =
((;), (Z)) Y
(B') €4 M(2, C) by ;
y E (B)3 @ c2,
(1.35)
237 43
Compound Channels, Transition Expectations, and Liftings
from ( 1.3I), (1.33), (1.34), and (1.35) we obtain that, for each q E H ,
and this is equivalent to (1.27).
0
Remark. If 'FI is finite-dimensional, then the identity (1.27) still holds, but on adifferent algebra and with a slightly less explicit form of the representation n.The argument which can be applied also to the infinite-dimensional case, if & is faithful, is the following: by Stinespring's theorem, & has the form &(x) =
v*ono(x)v~: x
E
a €3 B,
(1.36)
where KO is a (separable) Hilbert space and no is a faithful normal (since & is) representation of B @ B into B(Ko).Being faithful and normal, no must be (isomorphic to) a multiple of the identity representation (see Section 3 of [IS]). Therefore we may assume that KO has the form K I@ 'FI and
From (1.37) it follows that the restriction of no on
B @ 1 has the form
for some representation nl of B into B(K1).Define now acontraction W*: KI @%by
KO + KO =
where u E ICI is a unit vector fixed arbitrarily. Denoting W as the adjoint of W " , one easily verifies that
so W is a partial isometry with initial and final projections given respectively by
(1.41) ( 1.42)
u := (1
w ;"> p
I
(1.43)
the Riesy-Neagy unitary dialation of the partial isometry W, and using the identifications
238 L. Accardi and M. Ohya
44
denote, f o r b
E
B,
Then n is a representation of B in B(K1 @ K l ) and p is a state on K I@ K Isuch that, for any b , b’ E B and t ,rl E N,one has, using (1.40), (1.43), and the fact that any vector of the form u €3 6 is in the range of P ,
(6, p €3 ia(U*(n(b) €3 b’)U)rl) = ( u €3 6, U * ( n ( b )€3 b ’ w u €3 rl) =
((V i t ) (I1 ,
( b y b’ X I @ )O €3
b’
(v:rl)i
= (6, V*O(Tl(b) 8 b’)Vorl) = (6, &(b8 b’lrl)
and this proves ( I .27) with n given by (1.44).
Remark. The relevance of the above result for quantum probability is due to the fact that it allows us to prove the essential equivalence of different notions of quantum Markov chains.
2. Convex Combinations of Product States One of the main differences between classical and quantum probability is that while all the measures on a product space are in the closed convex hull (for the weak topology) of product measures, it is not true that all the states on the tensor product A1 8 A2 of two general C*-algebras are the limits (in some topology) of convex combinations of product states. In particular, the image under a general lifting &* of a state q will usually not be a convex combination of product states. However, the class of liftings with this property is particulary interesting because we expect that in this class some features of quantum probability will mix with some features of classical probability. This class is defined as follows:
Definition 2.1. Let dl and AZbe W*-algebras. A lifting &*: S(A1)+ S(AI 8 A2) is called of convex product type, or a convex product lifting, if any state w E S(A1)is mapped by &* into a convex combination of product states on A1 8 Az.If this property holds only for any state w in a subset F E S ( A l ) ,then &* is called a convex product lifting with respect to the family F. For any von Neumann algebra A, the set S(A)of all its states has a natural structure of measurable space with its Bore1 a-algebra. In the sequel, any probability measure on S(A)will be meant with respect to this a-algebra.
239 45
Compound Channels, Transition Expectations, and Liftings
Definition 2.2. A convex decomposition of cp S ( A )satisfying
E
S ( A )is a probability measure
p on
If p is pseudosupported, in the sense of [8], in the set of extremal states of S ( A ) ,we speak of an extremal convex decomposition of cp.
Proposition 2.1. To every lifting of convex product type &*: S ( A I one can associate a pair b p ( d w 1 ) . Pp(dW2
I
W1)J
+
S ( A I@ A 2 ) , (2.2)
with the following properties: (i) p p ( d w l )is aprobability measure on S ( A 1 ) . (ii) p,(d02 1 01) is a Markovian kernel from S(A1)to S(A2). Conversely every pair (2.2) satisfying (i) and (ii) above determines, via (2.4) and (2.5), a unique convex product lifting. Proof. For E* as in Definition 2. I , we fix a state pI E S ( A I )and also a decomposition of €*pl as a convex combination of product states (2.3) Denoting p p l(dw2 I W I ) as the conditional probability of p ( . 1 P I ) on the a-algebra of the first factor and dp,, ( W I ) as the marginal of p ( . I p l ) on the first factor, we obtain
(2.4)
Thus any lifting E': S(Al) + S(A1 @ A2),of convex product type, has the form (2.4), where p p l is a probability measure on S(A1) and the map A;]: S ( A I )+ S(A2) is given by (2.5). Notice that Azl is a channel in the sense of Section 1 and is usually nonlinear both in W I and P I . Conversely, given p p l and AZl as above, if we define &* by (2.3), then clearly €* is a convex product lifting from S(A1)to S(A1 @ A 2 ) . Finally, it is clear that the map
is a classical Markovian kernel on the Bore1 space S(A1)x S(A2).
0
240 L. Accardi and M. Ohya
46
Remark. If in (2.3) one conditions on the a-algebra of the second factor rather than on the first one, the resulting lifting is
where dq,, ( 0 2 ) is a probability measure on S(A2) and dq,, (dwl kernel from S(A2) to S ( A I ) .
I
w2) is a Markovian
We now consider the relation between the liftings of convex product type and Markov chains. The dual of a linear lifting is a transition expectation, therefore to any linear lifting one can associate a quantum Markov chain [ 2 ] in a standard way. If the lifting is of convex product type, then we can take advantage of this special structure to extend the construction of quantum Markov chains to the case of a not necessarily linear lifting &*. In what follows we describe this procedure. If &'*: S(d2) + S(A1 @ A2) is a lifting of convex product type, then it has the form
Notice that p ( d w ' , dw2 I p2) can be considered as a Markovian kernel on the space
which is constant on the first conditioning, i.e.,
Clearly, (2.6) is a state on A1 @ A2. If we apply &* to w2 in (2.6), we obtain the following state on ( A @ A2) @ A2:
where
241 Compound Channels, Transition Expectations, and Liftings
41
Applying again €* to w i we find
LIZLIZL12
p ( d w f ,d o : I P2)P(dw:, d l 4
@ w: @ w: €3w:.
At the nth iteration we obtain the state &,*Ip2on
This suggests introducing the classical Markov process
tn
:=
(6:, 6:)
: (Q, F ,P )
-+
S(AI) x S(A2) = S12
(2.9)
with the transition function given by (2.7) and initial distribution p ( . ) p 2 ) This . transition probability has a nice interpretation in terms of signal and noise: if system 1 represents the noise and system 2 the signal, then condition (2.7) means that the joint distribution at time (n 1 ) depends on the signal at time n , but not on the noise at time n : a natural assumption if we think of the noises at different times as generated by independent causes. Now let A := @N Al. The identification
+
a ~ 8 ~ ~ ~ @ a n ~ u ~ @ u ~ @ ~ ~ ~ @ a n @ l @ l @
induces a natural identification of (8dl)"with a subalgebra Af2,"lof A = @N A1 (the product of the first n-factors). In particular, if p2 E S(A2)is a state on A2, the restriction of &,*,p2on (8.Al)" is a state on (@" Al) and, with the above identification, we can consider it a state p [ l , non~ A. Following from above, in particular (2.8), we obtain
Proposition 2.2. For any p2
E
S(A2)the limit (2.10)
exists pointwise weakly on A. Moreover, if Ec denotes the mean with respect to the process [en],dejined by (2.9), then one has
3. Centralizer Liftings In this section we introduce an interesting class of nonlinear liftings generalizing the construction of [22]. It is shown that the Cecchini-Petz notion of state extension [ I I], introduced after [22] and for totally independent reasons, is a generalization of our construction hence, a fortiori, of the one in [22]. Recall that a linear map E from a C*-algebra A to a C*-algebra B is called anticompletely positive if the map E : A -+ B,defined by E ( a ) := E(a*);
a
E
A,
(3.1)
242 L. Accardi and M. Ohya
48
is completely positive antilinear, i.e., for any natural integer n , any a l , . . . , a, any bl, . . . , b, E t?, one has
E
A, and
Proposition 3.1. Let dl , A2 be W*-algebras, let Al @ ( O ) A2 denote their algebraic tensor product. For p E S(A1) let A: denote the centralizer of p and let E : A2 -+ A: be any anticompletely positive identity preserving linear map. Then there exists a unique ' , on A1 8"')Az such that state p
Proof. Let n be a natural integer and let bl, . . . , b,, E A, and a l , . . . , a , assumption the A:-valued n x n matrix B = (Bkj) defined by
E
A2. By
is of positive type, hence it has the form B = M * M for some A:-valued n x n matrix M = (Mkj).One has therefore
jkh
jkh
Remark. Clearly,
hence 'pp is continuous for the greatest cross norm on dl @(") A2. Cecchini and Petz [ 121 have proved that it is also continuous for the smallest C*-norm [28]. (This is clear if the centralizer of p, i.e., A:, is abelian because in that case all the C*-norms on Al @(") A2 coincide with the minimal C*-norm [28, Proposition 1.22.51. Moreover it is easy to check that the operator E , defined by (3.1) is an example of Cecchini's A-operator [9]. In this case in fact the Tomita involution J I acts as the identity on the cyclic space of A:, the centralizer of A!, therefore the identity (3.1) is precisely the defining relation of the A-operator. If A: is a discrete abelian algebra generated by a partition ( e j ) of the identity, then any positive map E , from A2 to A:, is also completely and anticompletelypositive and
243 Compound Channels, Transition Expectations, and Liftings
49
has the form
J
with pj E S(A2). In this case it is immediate to verify that
where
v,
is given by (3.2) and
p; := p(ej( . )e,).
In general, whenever the state bop, defined by (3.3), is continuous, the map p H pp defines a lifting &* in the sense of Definition 1.1. This lifting is in general nonlinear since the map E in (3.1) may depend on p. For example, if d1 is the algebra of all operators on some Hilbert space and p has the form p = tr(w . ) for some density matrix w with spectral decomposition given by (3.4) then the centralizer of the form
dy is the closed linear span of the (ej) and if the p, are chosen to be
for some channel A*: S(dI)+ S(A2).then (3.3) becomes of the same form as (1.17) giving an example of nonlinear compound lifting.
4. Error Probability for Optical Communication An optical communication process studied by several authors (see [2 1 ] for a mathematical analysis), the so-called attenuation process, can be described by the isometric lifting described in Example la. This description is simpler than the previous ones and allows quicker computations. This statement is illustrated with the computation of several error probabilities related to this model. Before introducing these computations, we briefly review some basic facts about the notions of quantum coding and of error probability in quantum control communication processes along the lines of [24]. Suppose that, by some procedure, we encode an information representing it by a . . ., where d k )is an element in a set C of symbols called sequence of letters d ' ) ,. . . , d"), the alphabet. A quantum code is a map which associates to each symbol (or sequence of symbols) in C a quantum state, representing an optical signal. Sometimes one uses a state as two codes: one for input and one for output.
244 L. Accardi and M.Ohya
50
In what follows we only consider a two-symbol alphabet: C = (0, 1).
(4.1)
One example of quantum code E = (to,61), where t;is the quantum state corresponding to the symbol i, E C, is obtained by choosing 60 as the vacuum state and as another state such as a coherent or a squeezed state of a one-mode field. Two states (quantum codes) 6:’) and t1(’)in the input system are transmitted to the output system through a channel A*. We here assume a Z-type signal transmission, namely, that the input signal “O’, represented by the state is error free in the sense that it always goes to the output signal “0’represented by while the input signal “I,” represented by the state {1(’), is not error free in the sense that its output can give rise to both states ti2)or .!f1(2) with different probabilities. The error probabilityqeis then the probability that the input signal “1” is recognized as the output signal “0,” so that it is given by
In the case of the quantum attenuation process, this error probability is written by using the attenuation operator V given in Example la with the construction (Example 4) of quantum lifting:
There are two main ways, called pulse modulation, to code the symbols of the alphabet C. We briefly explain them for completeness. A pulse is an optical signal, represented by a nonvacuum state of the EM field; its energy is here called the height of the pulse. To a single symbol of the alphabet C one associates one or more pulses. Time is discretized and each time interval between t k and tkkflhas length t.Each time interval corresponds to a single symbol of the alphabet. ( 1 ) PCM (Pulse Code Modulation) To the kth symbol Uk of the input sequence, one associates N pulses starting at a time t k . The ordered set of these pulses is denoted xk. For instance, for the alphabet (ao,a [ ) ,for N = 5 and choosing
the elementary pulses to be the vacuum (i.e., no pulse) denoted 0, and another fixed pulse, e.g., a coherent state, denoted 1, the code X k corresponding to Uk is determined by xo = ( I , 0, I , 0,0),X I = (0, I , I , 0, I ) , and so on. For this modulation, we need N slots (sites) in one time interval (e.g., between tk and t k + l ) to represent fully all M signals; 2 N - ’ < M 5 2N. (2) PPM (Pulse Position Modulation) In this case there is only one nonvacuum pulse in each time interval of length 5 . The code xk expressing a signal ak is determined by the position of the nonvacuum pulse, so that we need M slots (sites) in each time interval in order to express M signals. For instance, in the same notations as above, xo = (0, 0, 0, I , O), x2 = (0, I , 0.0,O). Given (4.2), the error probability of PCM with the to-tuple error correcting the
245 Compound Channels, Transition Expectations, and Liftings
51
following (4.3) and (4.4), respectively: (4.4a) j=ro+l
(4.4b)
PyM = qe,
where uCj = u ! / { ( u - j ) ! j ! ] . The most general case for the computation of qe is one where both t1(’) and ti2)are squeezed states, but in usual optical communication it is often enough to take a coherent or squeezed state as t,(’) and the vacuum state as ti2). Hence we first calculate the error probability qe for the latter two cases and compare them with the results previously obtained in [24]. Secondly, we show the computation for the most general case, 6;’) and t;2)squeezed, for mathematical interest and generality, although this somehow does not fit the assumption of our Z-type transmission. Case I: <,(I) = 10) (01 =coherent state and = 10)(01. The error probability (4.2) becomes
q e = t r x 2 ( t r ~ *v*IQ)(QIV)IO)(OI = trx2(trKc,IaQ)(UQl €3
IBQ)(molo)(ol
= trx*I.Q)(aQllO) (01 = I(0,
am2
= exp(-laO1’),
which is equal to the usual result (see [I61 and [19]), but our new derivation is much simpler than the old one.
Case 11: <,(‘I = squeezed state and = 10)(01. A squeezed state can be expressed by a unitary operator U ( z ) ( z E C), given in the Appendix, such that
t;’)= ~ ( z ) l v ) ( v l ~ ( z ) * . where I y ) is a certain coherent state. Therefore the error probability qe is
246 L. Accardi and M. Ohya
52
which implies
Therefore qe is qe = =
1 ; /d2s!(U(Z)Y.
I ) V w ) ( w ,U(Z>Y)
V*(lO)(Ol
11 d2wexp(i(1al2lp121w12)} IT x(U(z)y, IBI2w)(O, a w ) ( w , U(z)y).
This can be computed by expression (A.27) given in the Appendix and a Gaussian type integration:
+ Iy12))(coshr)-’/2
= exp{-i(lw12
x exp{G@(coshr)-l
+ tanhr(i(exp(-icp)O2
- exp(icp)G2))),
The result is (coshr)-l exp(i(y2 9e =
+ p2)(tanhr) - lyI2)
- ( I - la12)2(tanhr)2
J1
-;(I
-
v2)
la12)2(coshr)-2(tanhr)(y2 + ( 1 - la12)ly12(COShr)-2
+
xexp(
1
-
( 1 - la12)2(tanhr)2
1
-
J(cosh r ) 2 - ( 1
{(
-
la12)2(sinhr)’ 1 - 1011’
(coshr)2 - ( 1
+(I
-
la12)2(sinhr)2 (1
-
-
1)
IY 1’
la12)2 (coshr)’ - ( 1 - la/2)2(sinhr)’ -
which is same as the result obtained in [24]:
(i(v’ + p’)(tanhr))
247 Compound Channels, Transition Expectations, and Liftings
Case 111:
51’)
= squeezedstute U ( p ) J y ) ( y J U ( p )and *
U ( q ) l a ) ( a l U ( q ) * . By a similar way as in case 11,
53
eA2’
= squeezed srute
248 L. Accardi and M. Ohya
54
Appendix. Squeezed States In this section we recall the definition and the basic properties of a class of states which in the past have been the object of several studies in the field of quantum optics [ 171, [29], [30]. We mainly follow [30]. Let C denote the set of all complex numbers. A Fock representation of the Canonical Commutation Relations (CCR) over C is a triple
where 7-1 is a Hilbert space and W: z E C H W(z) unitary operators on 7-1 such that W(0) = id and W(u)W(u) = exp{i Im Uu)W(u and @
E
(@,
+ u);
E
u, u E
Un(7-1)is a map from C to the C,
(A. 1 )
7-1 is a unit vector, called the Fock vacuum, satisfying
w ( z ) @= ) exp{-;lz~~];
z
E
C.
(A.2)
It is moreover assumed that the weak closure of the complex vector space generated by the { W ( z ) : z E C) coincides with the algebra of all bounded operators on 7-1. This property is called irreducibility. Clearly, any two Fock representations are canonically isomorphic. The Stone-von Neumann theorem asserts that if ( W ( z ) : z E C) is any irreducible family of unitary operators on a Hilbert space 7-1 satisfying (A.l), then it is isomorphic to the Fock representation. In particular, for any such a family, there will exist a (necessarily unique) vector @ satisfying (A.2), i.e., a Fock vacuum for this family. A corollary of the Stone-von Neumann theorem is the following: let T : C + C be any real linear transformation such that V u , u E C,
Im(Tu)-(Tu) = Im Uu;
(A.3)
where the overbar denotes the complex conjugate, and define WT(Z) = W ( T z ) ;
z
E
c.
04.4)
Then the set {WT(z): z E C) (because any T satisfying (A.3) must be invertible) is irreducible. Moreover, it satisfies (A.1) because of (A.3). Hence by the Stone-von Neumann theorem, there exists a vector @ T E 7-1 and a unitary operator U T : 7-1 + 31, characterized by the property UTW(Z)@ = WT(z)@T;
z
E
c.
(A.5)
The vector Q T = U T @(i.e., the vacuum for the WT-representation) is called a squeezed vector for the W-representation. The most general operator T , satisfying (A.3). is given by the following:
Proposition A.l. terized by
Let V : C -+ R2 be the isomorphism ?f real linear spaces charac-
249 Compound Channels, Transition Expectations, and Liftings
55
Then a real 2 x 2 matrix T induces on C a transformation satisbing (A.3) ifand only ifdet T = I . Proof. A direct calculation: The identity (A.2) implies that for each z E C the oneparameter unitary group { W ( t z ) )( t E R) is strongly continuous, hence
for some self-adjoint operator B ( z ) . Moreover, the map z E C H B ( z ) is real linear. The operators
,I B ( I ) = q ;
iB(i)= p
(A.8)
are called momentum and position operators, respectively. Condition (A. 1 ) implies that [ B ( u ) ,B ( u ) ] = 2i Im Uu
64.9)
so that, in particular
Finally, denoting a=p-iq;
a*=p+iq
(A. 10)
one has
[ a ,a * ] = 1 , iB(z) = za*
-
z
ia;
E
c.
(A.11)
The vectors
18) = W(O)@;
8
E
c,
(A. 12)
are called coherent vectors. Now let T : C -+ C be a real linear map satisfying (A.3) and let W r , U T ,@r be characterized respectively by (A.4) and (AS). Then one has, for
z
E
c,
On the other hand, by definition (A.4) of W T ,one also has W T ( z )= W ( T z )= exp((Tz)a* - (Tz)-a)
(A.14)
250 L. Accardi and M. Ohya
56
Proposition A.2. In the above notations, if T : C + C is represented, in the iden@cation of Proposition A. 1, by the matrix (A. 15) then,for each z
(&-)
C,
E
=
(s ;") (n)
(A.16)
with
+ + i[c
C = +([a d ]
-
b]);
= + ( [ a- d ]
-S
+i[c+b]).
(A.17)
Remark. Notice that any c, s given by (A. 17) satisfy (A. 18)
~ 1, we can define a , b , c , d by Conversely, given c, s E C such that IcI2 - 1 . ~ 1 = (A.17) and the resulting matrix is in SL(2; R). Proof. Denote by C2(R) the real vector space
and by Vo: R2 + C2(R) the isomorphism of real vector spaces characterized by
vo
(1) (1); =
vo
(:> (!i). =
Then, if V: C + R2 is the isomorphism of Proposition A. 1 and z = x
VoVTz = vo
(p i)(;) ( =
(ax (ax
+ i y , one has
+ b y ) + i(cx + d y ) + b y ) - i(cx + d y )
Expressing x , y in terms of z , 2, one finds (A. 16), (A.17). Putting together Proposition (A.2) and the identity (A. 14), we obtain
+
= cxp{z~(a*c a s ) - Z(ac
+ a*s))
(A. 19)
Comparing (A. 19) with (A. 13) we finally find a.7' = ac
+a*$
(A.20)
251 Compound Channels, Transition Expectations, and Liftings
51
or equivalently
(:;) (6 :) (:*) =
(A.21)
4+
However, from (AS), (A.7), and (A. 1 l), it follows that the operator UT is characterized by the property
or, in view of (A.20), by U;aUT = ca
(A.22)
$a*.
Our goal is to find the operator UT satisfying (A.22) for given c and s satisfying (A. 18). To this goal first notice that, in view of (A.18), there exist real numbers r, q , cp such that
c = exp(iq)coshr = exp(iq)c,;
s = exp(-icp) sinhr = exp(-icp)s,.
(A.23)
Moreover, due to the identities exp(xa*a)aexp(- x a * a ) = exp(- x ) a ; expIxa*u}u*exp(-xa'a) = exp(x)a*;
x E C,
(A.24)
by replacing the representation W(z) with the equivalent representation W (exp{iq)z), we can always suppose that c, in (A.22), is real, i.e., q = 0 in (A.23). 0
Proposition A.3. Let c , s be given by (A.23) with q = 0; r > 0. Then the operator U T ,characterized by (A.22), is given by (A.25) (A.26) Proof.
Denote Dz = $(za2 - ?a*') and define
f ( t ) = exp(tD7)aexp(-tD7}.
Then, due to the easily verified commutation relations
one deduces the equation
with initial condition
252 L. Accardi and M. Ohya
58
whence ( ; : i t ) ) =) exp (t
(: 6) (a".))
cosh t r = (exp{icp) sinhtr
exp{-icp} sinh t r ) cosh t r
For t = 1 , using (A.23) and the assumption
Q = 0, one
(ny.) finds
+sa*,
exp{D,)a exp{- D z ) = ca
Remark. Let z = r exp(icp) and denote by V, the one-parameter unitary group V, = exp(ita*a}. Then one easily checks, using (A.24), that
So we can limit ourselves to studying the operator D; in the case of real z . In several applications it is useful to know the matrix elements of the operator exp(D,) = U(r) with respect to the coherent states in the W-representation. Proposition A.4. In the notation (A.12), (A.23), (A.25) one has (A.28) (A.29) Proof.
Denote f ( & ) = (a,U(r)B). Then
Solving this equation, we find (A.27) Now put f ( r ) = (0, U(r)B). Then f ( 0 ) = 1 and -df ( r ) dr'
=-
2
( P2 -' r ) c,'
c,
f(r).
The solution of this equation is (A.30)
Taking into account (A.23), the integral in (A.30) is easily evaluated and leads to (A.29).
253 Compound Channels, Transition Expectations, and Liftings
59
References 1.
2. 3. 4.
5.
Accardi, L.; Quantum Kalman filtering. Contribution to the memorial volume for the 60th birthday of R.E. Kalman. Accardi, L.; Noncommutative Markov chains with preassigned evolution: an application to the quantum theory measurement, Advances in Mathematics, 29 (1978), 226-243. Accardi, L.; An outline of quantum probability. Unpublished manuscript ( 1990). Accardi, L., Watson, G.S.; Quantum random walks and coherent quantum chains. Quantum Probability and Applications, IV. LNM 1396. Springer-Verlag. Berlin (1987). pp. 73-88. Alicki, R., Frigerio, A.; Quantum Poisson noise and linear quantum Boltzmann equation. Preprint (March
1989). 6. Bach, A,; The simplex structure of the classical states of the quantum harmonic oscillator, Communications in Mathematical Physics, 107 (1986), 553-560. 7. Barchielli. A,; Stochastic processes and continual measurements in quantum mechanics. Preprint. 8. Bratteli, 0.. Robinson, D.V.; Operator algebras and quantum statistical mechanics. I and 11. SpringerVerlag, Berlin (1981). 9. Cecchini. C.; Stochastic couplings for von Neumann algebras. Quantum Probability and Applications, 111. LNM 1303. Springer-Verlag, Berlin (1988). pp. 1-5. 10. Cecchini. C., Petz. D.; Classes of conditional expectations over von Neumann algebras, Journal of Functional Analysis 91 (1990). I I . Cecchini, C., Petz, D.; State extention and a Radon-Nikodym theorem for conditional expectations on von Neumann algebras, Pacific Journal of Mathematics, 138 (1989),9-24. 12. Cecchini. C.. Kiimmerer. B.: Stochastic transitions on preduals of von Neumann algebras. Quantum Probability and Applications, V. LNM 1442. Springer-Verlag, Berlin (1990). pp. 126-130. 13. Chen Mu-Fa. Selected topics in probability theory. Preprint, Volterra N.94, Universita di Roma. (1992). 14. Fannes, M., Ndchtergaele, B., Werner, R.F.; Construction and study of exact ground states for a class of quantum antiferromagnets. Preprint ( I 989). 15. Haagerup, U., A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space, Journal of Functional Analysis 62 (1985). 160-201. 16. Hirota. 0..Yamazaki, K., Nakagawa, M., Ohya. M.; Properties of error correcting code using photon pulse, Transaction of IECE Japan, E69 (1986), 917-919. 17. Hollenhorst, J.N.: Quantum limits on resonant-mass gravitational-radiation detectors, Physical Review, D19 (1979), 1669-1679. 18. Kraus, K.; States, Effects and Operations. LNP 190 Springer-Verlag, Berlin (1983). 19. McEliece, R.J.; Practical code for photon communication, IEEE Transaction of Information Theory, 27 (1981), 393-398. 20. Ohya, M.; Quantum ergodic channels in operator algebras, Journal of Mathematical Analysis and Applications. 84 (1981), 318-327. 21. Ohya. M.; On compound state and mutual information in quantum information theory, IEEE Transactions of Information Theory, 29 (1983), 770-774. 22. Ohya, M.; Note on quantum probability, Lettere al Nuovo Cimento, 38 ( 1 983), 402-404. 23. Ohya, M.; State change and entropies in quantum dynamical systems. Accardi. L.. von Waldenfels, W. (eds.). LNM 1136, Springer-Verlag, Berlin (1985), pp. 397-408. 24. Ohya. M.; Some aspects of quantum information theory and their applications to irreversible processes, Reports on Mathematical Physics, 27 (1989), 19-47. 25. Petz, D.; Characterization of sufficient observation channels. Preprint. 26. Petz, D.; Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Communications in Mathematical Physics, 105 (1986), 123-131. 27. Rondoni, L.; Nonlinear Boltzman maps in classical and quantum probability. Quantum Probability and Related Topics, VIII. World Scientific, Singapore (1993). and W*-Algebras. Springer-Verlag. Berlin (1971). 28. Sakai, S.; (?Algebras 29. Stoler, D.: Equivalence class of minimum uncertainty packets, Physical Review, DI (1970). 3217-3219 andD4 (1971), 1925. 30. Yuen, H.P.; Two-photon coherent states of the radiation field, Physical Review, A13 ( I 976), 2226-2243. Accepted 11 Februaty 1997
254 Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 2, NO. 2 (1999) 267-282 @ World Scientific Publishing Company
QUANTUM DYNAMICAL ENTROPY FOR COMPLETELY POSITIVE MAP
A. KOSSAKOWSKI Institute of Physics, Nicholas Copernicus University, Grudziadzka 5, 87-100 Torun, Poland M. OHYA and N. WATANABE' Department of Information Sciences, Science University of Toko, Noda City, Chiba 278-8510,Japan Received 5 November 1998 A dynamical entropy for not only shift but also completely positive (CP) map is defined by generalizing the AOW entropy1 defined through quantum Markov chain and AF entropy defined by a finite operational partition. Our dynamical entropy is numerically computed for several models.
0. Introduction
Quantum dynamical entropy was first introduced by Emch' and Connes-Stormer3 around 1975. In 1987, Connes, Narnhoffer and Thirring4 defined a dynamical entropy (CNT entropy) in C*-dynamical systems. Park computed the CNT entropy for several model^.^ In 1994, Alicki and Fannes6 defined a quantum dynamical entropy (AF entropy) by means of a finite operational partition of unity. In 1994, Hudetz7 discussed the dynamical entropy in terms of topological entropy. In 1995, Ohya' introduced a quantum dynamical entropy and a quantum dynamical mutual entropy based on the C*-mixing entropyg and the complexity in information dynamics." In 1995, Voiculescul' introduced a dynamical approximation entropy for C*- and W*-algebra automorphisms based on a general approximation approach. In 1997, Accardi, Ohya and Watanabel defined a quantum dynamical entropy (AOW entropy) through quantum Markov chain. The difference among some definitions of the dynamical entropy were discussed in Refs. 12-14. Some computations of the dynamical entropy are done in several papers such as Refs. 15 and 16. We introduce a dynamical entropy for a completely positive (CP) map in Sec. 1. The Stinespring type expression of transition expectation associated to CP maps is *E-mail: [email protected] 267
255 268
A . Kossakowski, M. Ohya d N . Watanabe
discussed in Sec. 2. It is shown that the dynamical entropy of Sec. 1 is a generalization of both A F and AOW dynamical entropies in Sec. 3. Several computations of our dynamical entropy giving a finite value are carried for several models in Sec. 4. 1. Quantum Dynamical Entropy for CP Map Let B(K) (resp. B(X)) be the set of all bounded linear operators on separable Hilbert space K (resp. 3-1). We denote the set of all density operators on K (resp. X)by G(K)(resp. B ( X ) ) .Let
r : B(K)
@ B(X)
+ B(K)
8 B(X)
(1.1)
be a normal, unital CP linear map, i.e. l? satisfies
B,
t B + r(B,) ?r(B), (Ix(resp. I K ) is the unity in X(resp.
r ( I K @ 1 ~ =)I K 8 I n for any increasing net
K))
{ g , } c B ( K ) @ B(X) converging to B E B(K) 8 B(X) and n
C B;r(A;Ai)Bi 2 o i,j=l
hold for any n E N and any Ai,B j E B(K) @ B(3C). For a normal state w in B(K), there exists a density operator i;l E B(K)associated to w (i.e. w(A) = trGA, b' A E B ( K ) ) . Then a map
Erl" : B ( K ) 8 B(X)
B(X)
(1.2)
defined as
E ~ > " (= A )w(r(A)) = trK
i;lr(A), v A E B(K)
@ B(X)
(1.3)
is a transition expectation in the sense of Ref. 14 (i.e. Er?" is a linear unital CP map from B(K) 8 B(X) to B(X)), whose dual is a map
E*r'"(p) : B ( X ) + 6(K @ X)
(1.4)
E * ~ + (=~r )* ( G 8 p ) .
(1.5)
given by
The dual map E*r@is a lifting in the sense of Ref. 17; i.e. it is a continuous map from B ( X ) to B(K 8 X). For a normal, unital CP map A : B(X) + B(X), id @ A : B ( K ) 8 B(X) + B ( K ) 8 B(X) is a normal, unital CP map, where id is the identity map on B ( K ) . Then one defines the transition expectation
Ef;'"(A)= w((id 18A)r(A)),
V A E B ( K ) 8 B(X)
(1.6)
V p E G(7-1).
(1.7)
and the lifting
Eir'"(p) = r * ( G @ A * ( p ) ) ,
256 Quantum Dynamical Entropy f o r Completely Positive Map
269
The above A* has been called a quantum channells from G(7-1)to G(7-1),in which p is regarded as an input, signal state and (;I is as a noise state. The equality tr(gq~ ) ~ . t l @ i ~ ~ " ( @ p ). (. A . @i A,, 8 B )
= tr.tl p(E,r'W(Al@ E:Iw(Az
8.. . @ An-l 8 Ei'"(An @ B ) . . .)))
(1.8)
for all Al, A l , . . . ,A, E B(K), B E B(7-1)and any p E G(7-1)defines
(a) a lifting
and (b) marginal states (1.10) (1.11)
(1.12) is a compound state for -r ,w is not equal to p. pA,n
and p?,:
in the sense of Ref. 19. Note that generally
Definition 1.1. The quantum dynamical entropy with respect to A, p, defined by 1 S(A;p, r,w ) = limsup pi;:), n+w
r and w is
n
where S(.) is von Neumann entropy2'; i.e. S(a) = - t r a l o g a , a E G(@; dynamical entropy with respect to A and p is defined as
(1.13)
K).The
S(A;p) = s ~ p { S (p,~ r;, w ) ; r,w } . 2. Transition Expectation In this section, we discuss the transition expectation Eiiwassociated to the C P maps I?, A and a state w . For a complete orthonormal system (CONS) { e i } in K ,put Eij = lei)(ej I. There exist operators ua E B(7-1)and complex numbers Xkla,mnp E C such that the unital C P map r of (1.1) can be written in the form (p. 145 of Ref. 16)
r(A)=
C C h a , m n p ( E i l @u:)A(Emn kl,m,n.a,P
@ up),
A
E
B ( K ) 8 B(7-1). (2.1)
257 270
A . Kossakowska, M. Ohya B N . Watanabe
Let A E B(K) 8 B(7-I)be
From (2.1) and (2.2), we have
The equality (2.4) implies
As the matrix
0
= (Akla,mnp)
is positive definite, it can always be diagonalized as
ikL
where the positive number T~ is the eigenvalue of (T and the complex number is the component of the orthogonal eigenvectors I@*)) associated to T ~ From . the positivity of W ,we have
Quantum Dynamical Entropy for Completely Positive Map
271
where { f , } is a CONS of K. From the above equality, the transition expectation Er,w is expressed as (2.6) below.
k,in
P
4
Since r(IK €3 I%)= IK €3 1% holds, one obtains
P 4 h
For a channel A* : 6(31) -+ 6(31),the transition expectation (1.6) and the lifting (1.7) are expressed as follows:
El
(x
Eij
i,j
@Aij
)
=
A(u;qk AkmUpqm) k,m P
4
(2.10)
259 272
A . Kossakowski, M . Ohya t3 N . Watanabe
and
3. Dynamical Entropy
In this section, we generalize both the AOW entropy and the AF entropy. Then we compare the generalized AF entropy with the generalized AOW entropy. Let 6 be an automorphism of B(X), p be a density operator on 3c and E," be the transition expectation on B(K) @ B(N) with A = 6 defined in Sec. 2. One introduces a transition expectation E," from B(K) @B(X)to B(3c) such as
C
=
(3.1)
e(u~qk)6(Akm)6(UPqm)~
k,m,p,q
The quantum Markov state {P;,~} expectation E," by
on
B(K) is defined through this transition
t r g y Klp;,,(Al 8 . .. @ An)]
= tr.tl[pE,"(Al@ E,"(Az @ . . . @ An-l
@ E,"(A, @ I ) . .
.))I
(3.2)
for all A l , . . . , A , E B ( K ) and any p E G(3c). Let us consider another transition expectation e& such that
One can define the quantum Markov state tr@T K"pZI,,(Al 3
@
{bg,,}
in terms of e;
. . . c3 An)]
tr.tl[pe:(Al @eg2(Az@...@AA,-1 @e&(A,@I)...))]
(3.4)
for all A l , . . . , A , E B(K) and any p E e(X). Then we have the following theorem.
Theorem 3.1. p:,n
=
p;,,.
Proof. It is easily seen that e;l,(A@ B ) = BieU(A@ F i ( B ) )= EZ{(A@ K i ( B ) ) for any A E B(K) and any B E B(X), where e" (3.5), we have
=
(3.5)
e&,entity. From the equality
eg(A1 @ I ) = 6e"(A1@ I ) = E,"(A1c3 I )
(3.6)
260 Quantum Dynamical Entropy for Completely Positive Map
273
and
Generally, we observe
ei(A1 8 e& (A2 8 . . . @ e& ( A , @ I ) ) ) =Et(Ai 8 E ~ ( A 2 ~ . . . 8 E t ( A n ~ I ) ) ) for any n
2. 1. Hence pt,n
=
Pt,, holds.
0
Let ,130 be a subalgebra of B(K). Taking the restriction of a transition expectation E : B(K) 8 B(X) + B(X), to 130 8 B(X), i.e. EO = E ( B ~ ~ BEO ( ~ is ) , the transition expectation from ,130 @ B(R) to B(3t). The QMC (quantum Markov chain) defines the state pi,(:) on ,130 through (3.4), which is P;,:)
= PtJL 1 8; Bo
.
(3.8)
The subalgebra ,130 of B(K) can be constructed as follows: Let P I , .. . , P, be projection operators on mutually orthogonal subspaces of K such that Czl Pi = I K . Putting Ki = PiK, the subalgebra ,130 is generated by 7n
CPiAPi,
A E B(K).
(3.9)
i=1
One observes that in the case of n = 1 m
~i(l"' E pi,1
=
C pipt,,pi
(3.10)
i=l
and one has for any n E N
c
P0,n "(O) = il
(Pi,8 . .. @ Pi,)p;,n(Pz,
8 . .. @ Pzn)
(3.11)
,...,i,,
from which the following theorem is proved (cf., see Ref. 16).
Theorem 3.2.
Proof. Let {Fi,,,,,,i,,} be PVM (projection valued measure) on 8TK. If &,(A) is a map defined by n
261
274
A . Kossakowski, M . Ohya & N . Watanabe
then Ei, satisfies (a) E p ( l @ ; ~= ) I E 87,130, (b) &p o Ep(A) = Ep(A), V A E ,130. Therefore Ep is B ( K ) , (c) trEp(A)B = tr AB, V A E B(K), V 8 E a conditional expectation from B(K)to ,130, so that 40)
-
S(P,,, )
s (EP- ( P;,n))
= trV(Ep(P;,n))
2 trfp(V(P;,n))
= t r ~ ( ~ ; , n=) S(Pzl,n) 7
(3.12)
where ~ ( t5) -tlogt (t 2 0). We used the concavity: v ( ~ P ( P ; , ~ )2) Ep(V(P;,n)).
Taking into account the construction of subalgebra Bo of B(K), one can construct a transition expectation in the case that B(K)is a finite subalgebra of B(X). Let B ( K ) be the d x d matrix algebra h f d (d 5 dimX) in B(X) and Eij = lei)(ejl with normalized vectors ei E X ( i = 1 , 2 , . . . , d ) . Let 71,. , . ,yd E B(X) be a finite operational partition of unity, i.e. d $yi = I , then a transition expectation
EY : h f d C3 B(7-1)+ B(X)
(3.13)
is defined by
(3.14) Remark that the above type complete positive map EY is also discussed in Ref. 21. Let M: be a subalgebra of h f d consisting of diagonal elements of Md. Since an d element of M j has the form CiZl blEii(bi E C),one can see that the restriction EY(O)of EY to M j is defined as
($g1
EY(0)
Eij €3
d
Ail)
=p 4 i i 7 i .
(3.15)
When A : B(X) + B(3C) is a normal unital CP map, the transition expectations EZ and El(') are defined by
262 Quantum Dynamical Entropy for Completely Positive M a p
275
(3.17) where we, put
Wij(A)
EyfA~j,
W,+j(P) ?E Tjp"Yz', pil, ...,i n
A E B('H),
(3.18)
P E G(3-1) 1
(3.19)
= t r x pA(Wilil (A(Wiziz(. . . "inin = trx
(1~))))))
WCii,(A* ... A * ( W ~ i z ( A * ( W ~ i , ( A * ( p ) ) ) ) ) (3.20) ).
The above px,,, px::) become the special cases of ,of;:: defined in Sec. 2 by taking l? and w in (2.10). Therefore the dynamical entropy (1.13) becomes
(3.21) (3.22) S(')(A; p , {ri}) = limsup - S ( Pm * , ~) . n-+w n The dynamical entropies of A with respect to a finite-dimensional subalgebra B c B(X) and the transition expectations E i and Ex(') are given by
SdA;P ) = SUPmA; P , { T i } ) , {TiK a1 , ,!?:'(A;
p ) E sup{s(')(A; p , {yi}), {-yi}c B} .
(3.23) (3.24)
We call (3.23) and (3.24) a generalized AF entropy and a generalized AOW entropy, respectively. When {ri} is PVM (projection valued measure) and A is an automorphism 8, Sf'(8; p ) is equal to the AOW entr0py.l When { ~ f y i is } POV (positive operater valued measure) and A = 8, s B ( 8 ; p ) is equal to the AF entropy.6 From Theorem 3.2, one obtains an inequality
Theorem 3.3.
< s, -('I
sB(A;P)
(A; p ) .
(3.25)
That is, the generalized AOW entropy is greater than the generalized A F entropy. Moreover the dynamical entropy S,(A; p ) is rather difficult to compute because there exist off-diagonal parts in (3.16), so we mainly consider the dynamical entropy $$)(A; p ) in the next section. Here, we note that the dynamical entropy defined in terms of p:,, on 8:B(K) is related to that of flows by Emch,2 which was defined in terms of the conditional expectation, provided B(K) is a subalgebra of B(7-L).
263 276
A . Kossakowski, M. Ohya €4 N . Watanabe
4. Some Models We numerically compute the generalized AOW entropy for several models. Let yi = y,t be projection operators on one-dimensional mutually orthogonal subspaces of 3t such that yi = I N holds. For unitary operators Vi and V on 'Hi = UiyiV satisfies
xi
Let us consider a transition expectation E Y ( 0 ) : A(31) @ B(31)
+ B(31)
(44
defined by
where A(X) is an Abelian subalgebla of B(31) generated by A : B(X) + B(X) be a normal unital CP map given by
X k l , k n = 61,
(@ A(I'+l) == 174) and
Xkl,kn =
k
c
E C).
Akj,kjAjl,jn.
Let
(4.5)
j
We will compute the dynamical entropy of A based on
Theorem 4.1. When yi p and {yi = Eii} is
= Eii,
if {Aij,ij}
E7(0)
above.
the quantum dynamical entropy with respect to A,
S(O)(A;pi { y i = Eii}) = - C C A i j , i j A j l , j k where
xibiEii(bi
tr'+l(PElk) lOgAij,ij
i
1,k
are the coeficients of (4.4) associated to the CP map A .
(4.6)
264 Quantum D y n a m i c a l E n t r o p y for C o m p l e t e l y Positive M a p
277
and
(4.10) which implies Pn = { p j l , , , , , j n } has the Markov property. Therefore the quantum dynamical entropy with respect to A, p and {yi = Eii) is computed as
S(')(R;P, {ri = Eii)) = -
Xij,ijXjl,jntr.tl(pEln)
log ~ i j , i .j
i , j Z,n
Theorem 4.2. W h e n ~i = U * E i i U ( = Ifi)(fil = Fii) for a unitarg operator U o n Z, the quantum dynamical entropy with respect to A, p and { ~ = i Fii) is
S(')(A; p, {Ti
=
Ei))= i,j
where
{Xij,ij}
Cp,q,r,s Apq,rs
x$:jjAj[,jk Z,k
tr.tl(pU*ElkU)
1OgA:fjj
,
are the coeficients of (4.4) associated to the CP map A and tr.tl E q p u * E i i u E r s u * E j j u .
(4.11) =
265 278
A . Kossakowski, M . Ohya 63 N . Watanabe
Inserting (4.13) into (4.4), one obtains
A(A) =
C
Xpq,rsE;qAErs
p,q,r,s
(4.14)
(4.15)
= A(.F) 3 n J n - 1 >. 3 . n .3 n - l
3231 i32.71 . .
CXj,l,j,k
tr%(pU*ElkU)
(4.16)
l,n
because of yi = Fii. Since PiF’ = { P(~F,), , , , , ~has , } the Markov property, we obtain the quantum dynamical entropy with respect to A, p and {Ti = Fii} as
S ( O ) ( A ; p, {ri = F ~ ~=} )
7,x ! : ~ ~ x ~tr%(pU*ElkU) ~,~~ log ill, . i,j
l,k
0
Theorem 4.3. Take Au(A) = U*AU for a unitary operator U on ?.!I When U has a simple point spectral { e i p k } and its eigenvector fk, the dynamical entropy with respect to AlJ, P and {Yk = Ifk)(fkl} i s S(O)(AlJ;P , {Yk = Ifk)(fkl))
=0.
(4.17)
Proof. For a CONS e = { e m } of %, U is written by
U=
CX:LEmn
,
m,n
where X mn (4
= -
(ern,u e n ) 7
(4.18) (4.19)
266 Q u a n t u m Dynamical Entropy f o r Completely Positive Map
279
From the definition of Au, one obtains h(A)
=
A;!,,ElkAErnn
7
k,l,m,n
where AEimn s
ik)Azk. Therefore, one has 4k
,,,A;:
f
I (em,Ue,) I2 .
P, = {pjl,,.,,jn} has Markov property. Moreover, for the eigenvectors simple point spectral {ezvk} of U such that
uf k
= eivkf k
,
{fk}
of the (4.20)
since =
I(fm, Ufn)I2 =
&Tm
,
(4.21)
pjl (f) = - ~ A ~ [ ~trN(plfl)(fkl) , j , ~ = ( f j i ~p
fji)
7
(4.22)
l,k
the dynamical entropy with respect to Av, p and {yk
= Ifk)(fkl}
is
We remark here that for another choice of base {gi} c N ,one has
S ( o ) ( A U ; P ,{Yk
=
1gk)(gk1}) > 0
Now we study the dynamical entropy for a quantum communication process, in particular, the attenuation process. That is, A* is the attenuation channel" defined as follows: Let 3t = L 2 ( R ) ,(0) be a coherent state vector in 3t and y = { ~ j } ; = ~ , where yj = Izj) ( z jI and
b j ) = %lo) + bjl0) ,
Tj =
-(1- 2A) - (-1)jJl- 4A(1- A)(1 - exp(-1012)) 2(1- ~)exp(-+18(2)
267 280
A . Kossalcouiski, M. Ohya & N . Watanabe
The attenuation channel
A* with
a transmission rate 77 is defined by
A*(lO)(Ol)
= lfi~)(ml.
Theorem 4.4. W h e n p is given by p = XlO)(Ol+ (1- X)lO)(O( and A* is the attenuation channel with a transmission rate 77 satisfying C jp k , j p j = p k , the quantum dynamical entropy with respect t o A, p and {yj} is obtained by
S(’)(A; p, { y j ) ) = - c P k , j p j l o g P k , j ,
(4.23)
j,k
whe.repj = Xl(0,zj))’
+ (1- X ) I ( J i j 6 , ~ j ) 1 ~and
Pk,j = v:I(Zk,Zf)l2
IzT) = aT10)
at 3
=
Eta.
+ (1 - v j ’ ) l ( z k , z Y ) 1 ’
+ b T I f i O ) , 1x7) = a i l 0 ) + b ; I f i O ) , a T = &;a.
3 3 , 3
bt
3 3 , 3
= &?b. bT = E - 6 . 3 . 1 ’ 3
v + = -1 ( l + e x P ( - 2 ( 11- ~ ) 1 6 1 ~ ) ) 3
7
3 3 ’
1
2
Proof. The formula (3.17) can be rewritten in the form ( n 3 3).
gb
pjl,,,,>jn
PA,n ’(O) j1,...,jn=l
where
k=l
lzjk)(zjkl
(4.24)
268 Quantum Dynamical Entropy for Completely Positive Map 281
cj
When p k , j p j = p,+ is hold, we obtain the dynamical entropy with respect to A, p and { y j } such as
Acknowledgment We thank Prof. Petz for his useful comments to our present work. We also thank Prof. Accardi for his encouragement. References 1. L. Accardi, M. Ohya and N. Watanabe, Dynamical entropy through quantum Markov chain, Open System Infor. Dynamics 4 (1997) 71-87. 2. G. G. Emch, Positivity of the K-entropy on non-Abelian K-flows, Z. Wahrscheinlichkeitstheory Gebiete 29 (1974) 241. 3. A. Connes and E. Stormer, Entropy for automorphisms of II1 von Neumann algebras, Acta Math. 134 (1975) 289-306. 4. A. Connes, H. Narnhoffer and W. Thirring, Dynamical entropy of C*-algebras and von Neumann algebras, Commun. Math. Phys. 112 (1987) 691-719. 5 . Y. M. Park, Dynamical entropy of generalized quantum Markov chains, Lett. Math. Phys. 32 (1994) 63-74. 6. R. Alicki and M. Fannes, Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994) 75-82. 7. T.Hudetz, Topological entropy for appropriately approximated C*-algebras, J . Math. Phys. 35 (1994) 4303-4333. 8. M. Ohya, State change, complexity and fractal in quantum systems, Quantum Commun. Measurement 2 (1995) 309-320. 9. M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys. 27 (1989) 19-47. 10. R. S. Ingarden, A. Kossakowski and M. Ohya, Information Dynamics and Open Systems (Kluwer, 1997). 11. D. Voiculescu, Dynamical approximation entropies and topological entropy i n operator algebras, Commun. Math. Phys. 170 (1995) 249-281. 12. F. Benatti, Deterministic Chaos in Infinite Quantum Systems (Springer, 1993). 13. N.Muraki and M. Ohya, Entropy functionals of Kolmogorov Sinai type and their limit theorems, Lett. Math. Phys. 36 (1996) 327-335. 14. L. Accardi, M. Ohya and N. Watanabe, Note on quantum dynamical entropies, Rep. Math. Phys. 38 (1996) 457-469. 15. M. Choda, Entropy for extensions of Bernoulli shifts, Ergodic Theory Dynamic Systems 16 (1996) 1197-1206. 16. M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, 1993). 17. L. Accardi and M. Ohya, Compound channels, transition expectations and liftings, to appear in J. Multivariate Anal. 18. M. Ohya, Quantum ergodic channels in operator algebras, J . Math. Anal. Appl. 84 (1981) 318-328. 19. M.Ohya, Note on quantum probability, Lett. Nuovo Cimento 38 (1983) 402-404. 20. J. von Neumann, Die Mathematischen Grundlagen der Quantenmechanik (S pringer-Verlag, 1932).
269 282
A . Kossakowska, M. Ohya & N . Watanabe
21. P. Tuyls, Comparing quantum dynamical entropies, Banach Centre Publication 43 (1998) 411-420. 22. M. Ohya, O n compound state and mutual information i n quantum information theory, IEEE Trans. Information Theory 29 (1983) 770-774. 23. M. Ohya, D. Petz and N. Watanabe, Numerical computation of quantum capacity, Internat. J . Theor. Phys. 37 (1998) 507-510. 24. M. Ohya, D. Petz and N. Watanabe, Capacity of a noisy quantum channel, SUT
preprint.
270 International Journal of Theoretical Physics, Vol. 37, No. I , 1998
Complexities and Their Applications to Characterization of Chaos2 Masanori Ohya' Received July 4, 1997
The concept of complexity in Information Dynamics is discussed. The chaos degree defined by the complexities is applied to examine chaotic behavior of logistic map.
1. INTRODUCTION There are several tools to describe chaotic aspects of natural or nonnatural phenomena such as entropy. The concept of complexity is one such tool. In 1991 the author proposed Information Dynamics (ID, for short) to synthesize the dynamics of state change and the complexity of a system. In this paper, I briefly review the concept of ID and discuss some applications of the entropic complexities in ID to the characterization of chaos.
2. INFORMATION DYNAMICS Information Dynamics is an attempt to provide a new view for the study of chaotic behavior of systems (Ohya, 1995). _-Let (d, G, a (G)) be an input (or initial) system and (d, G, a(G))be an output (or final) system. Here d is the set of all objects to be observed and (5is the set of all means of getting the observed value, a(G) is a certain a = E. evolution of the system. Often we have d = 2, G = G,
' Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278, Japan. 'Dedicated to Professor GCrard G. Emch on his 60th birthday. 495 0 1998 Plenum Publishing Corporauon
271 Ohya
496
Therefore we claim Giving a mathematical structure to input and output triples
= Having a theory For instance, when d is the set M(R) of all measurable functions on a measurable space (R, 9) and G(d) is the set P of all probability measures we have usual probability theory, by which the classical dynamical on system is described. When d = B ( X ) ,the set of all bounded linear operators the set of all density operators on a Hilbert space X , and G(d)=‘G(X), on X , we have a quantum dynamical system. The dynamics of state change is described by a channel A*: G -+ (sometimes G + G). The fundamental point of ID is that there exist two complexities in ID itself. - - Gi, at(G,))be the total system of (d,(5, a) and (d, 6 a), Let (di, and Y be a subset of G from which we measure the observables and we call this subset a reference system [e.g., Y = Z(a),the set of all invariant elements of a]. G’(cp) is the complexity of a state cp measured from Y and T’(cp; A*) is the transmitted complexity associated with the state change cp + A*q, which satisfy the following properties:
(a)
a,
(i) For any cp
E
Y
C
a,
C”(cp)
2
0,
T’(cp; A*)
(ii) For any orthogonal bijectionj : ex Y points in Y ) ,
2
0
+ ex Y (the set of all extreme
C””’(j(cp)) = C”p((p)
Tj(’)(j(cp); A*) = T”(cp; A*) (iii) For @ = cp @ Ji
E
Y , C G,,
C”r(@) = C’(cp)
+ C9(*)
(iv) For any state cp and a channel A*,
0
IT’(cp;
A*)
ICy(cp)
(v) For the identity map id from (5 to GS,
T’(cp; id) = CYp(cp)
CP (i.e., the Instead of (iii), when ‘‘(iii’) CP E Y , C G,, put cp restriction of CP to d),Ji = CP , f C’f (@) 5 C’(cp) + CY(Ji)”is satisfied,
272 497
Complexities and the Characterization of Chaos
C and Tare called a pair of strong complexity. Therefore ID can be considered as follows.
Dejnition 1. Information Dynamics (ID) is defined by
(d, 6, a(G);3,G, E ( c ) ;A*; C”(cp), T’,
(cp;
A*))
and some relations R among them.
Thus, in the framework of ID, we have to: (i) Determine mathematically
d,EJ, a(G);3,F, E(C) (ii) Choose A* and R. (iii) Define Cs(cp), T s (cp; A*). Information Dynamics can be applied to the study of chaos in the following ways: (a) $ is more chaotic than cp as seen from the reference system Y if CY(*> 5 CY(cp). (b) When cp changes to A*cp, a degree of chaos associated to this state change is given by
-
DY(cp;A*) = C’(A*cp) - T’(cp; A*)
In ID, several different topics can be treated from a common standpoint (Matsuoka and Ohya, 1995; Ohya, 1991a, n.d.-a, c; Ohya and Watanabe, 1993). Although there exist several complexities (Ohya, 1997), one of the most fundamental pairs of C and T in quantum system is the von Neumann entropy and the mutual entropy, whose C and T are modified to formulate the entropic complexities such as €-entropy (e-entropic complexity) (Ohya, 1989, 1991b, 1995) and Kolmogorov-Sinai type dynamical entropy (entropic complexity) (Accardi et al., 1996; Muraki and Ohya, 1996). In this paper, we discuss some applications of entropic complexities to the study of chaos.
3. CHANNEL The concept of channel or channeling transformation is fundamental in ID and it is a convenient mathematical tool to treat several physical dynamics in a unified way (Ohya, 1981). In classical systems, an input (or initial) system is described by the set of all random variables SQ = M ( n ) and its state space 6 = P ( n ) , and an output (or final) system by M and P
(a)
(a).
273 Ohya
498
A quantum system is described on a Hilbert space X.That is, an input
d is the set B ( X ) of all bounded linear operators on X,and G is the set T
(X)of all density operators on X.An output system is 3 = B (X)and G = T (%). A more general quantum system is described by a C*-algebra
and its space, but this general frame is not used in this paper. - In any case, a channel is a mapping from G(P(R)) or T (X), resp.) to G (P or T (%), resp.). Almost all physical transformations are described by this mapping.
(a)
Definition 2. Let A* be a channel from G to (1) A* is linear if A*(Aq + (1 - A)$) = AA*q + (1 - h) A*$ holds for all cp, 4 E G and any A E [O, 11. (2) A* is completely positive (C. P.) if A* is linear and its dual A: 3 + d satisfies
ATA(ATAj)AjL 0 i,j = 1
for any n
E
N and any
(3;) C 3, { A ; ) C d.
Most channels appearing in physical processes are C.P. channels. We here list a few examples of such channels (Ohya, 1989). Take a density operator p as an input (initial) state. (1) Time evolution: Let { U, ; t E R+)be one-parameter group or semigroup on X.We have p
4
AFp = U,pU:'
(2) Quantum measurement: When a measuring apparatus is described by a positive operator-valued measure Q , ) and the measurement is carried out in a state p, the state p changes to a state A*p by this measurement such that p
+ A*p =
~ f i ~ ~ p ~ ; ~ ~ n
( 3 ) Reduction: If a system C, interacts with an external system C2 described by another Hilbert space 3%and the initial states of Cl and Z2 are p and u, respectively, then the combined state 9, of C, and CC2 at time t after the interaction between two systems is given by
e, = uf(p8 U)U: where U, = exp( - i t H ) with the total Hamiltonian H of Zl and 2,. A channel is obtained by taking the partial trace w.r.t. 3%such as p
+ ATp = tr&
274 Complexities and the Characterization of Chaos
499
4. QUANTUM ENTROPY AS COMPLEXITY The concept of entropy was introduced and developed to study the following topics: irreversible behavior, symmetry breaking, amount of information transmission, chaotic properties of states, etc. Here we review quantum entropies as an example of our complexities C and T A state in quantum systems is described by a density operator on a Hilbert space X.The entropy of a state p was introduced by von Neumann (1932; Ohya and Petz, 1993) as S(p) = -tr p log p
If p = &pkEkis the Schatten decomposition (i.e., p k is the eigenvalue of p and Ek is the one-dimensional projection associated with Pk, this decomposition is not unique unless every eigenvalue is nondegenerate of p, then
because { Pk] is a probability distribution. Therefore the von Neumann entropy contains the Shannon entropy as a special case. For two states p. (T E 6 ( X ) , the relative entropy (Umegalu, 1962) is defined by S(P9 a) =
{
tr p(1og p - log a), p << (T +m, otherwise
where p << a means _ that_tr u A = 0 + tr pA = 0 for any A 2 0. Let A*: G(X)--;r G(X)be a channel and define the compound state by 0,5
=
c PkEk @ A*Ek k
which expresses the correlation between the initial state p and the final state A*p (Ohya, 1983a, b). The mutual entropy (Ohya, 1983a) for a state p E
G ( X ) and a channel A* is given by A*)
=
SUP{S(~E, p 8 A*p); E = {Ek]1
where the supremum is taken over all Schatten decompositions. The above entropy and mutual entropy become a pair of our two complexities according to the following facts:
275 Ohya
500
(1) The fundamental inequality of Shannon type (Shannon 1948; Ohya, 1983a)
0
5
Z ( p ; A*)
5
min(S(p), S(A*p))
because of s(A*Ek,A*p) = S(A*p) - &J k s(A* Ek) 5 S(h*p) and the monotonicity (Uhlmann, 1977; Ohya and Petz, 1993) of the relative entropy: s(A*&, h*p) 5 s(&,p ) . (2) Z ( p ; id) = S ( p ) , which is proved as follows:
because of S ( E k ) = 0. In Shannon's communication theory in classical systems, p is a probability distribution p = ( p k )and A* is a transition probability (t& so that the Schatten decomposition of p is unique and the compound state of p and its output p [- p = (p,)]is the joint distribution Y = (rV)with rij = tijpj.Then the above complexities C and T become the Shannon entropy and mutual entropy, respectively, s(P) =
-x
Pk
k
log P k
We can construct several other types of entropic complexities and they are used to define the quantum dynamical entropy (Muralu and Ohya, 1996; Ohya, n.d.-a; Accardi et al., 1996), which is one of fundamental tools to describe chaotic aspects of a dynamical system (Billingsley, 1965; Benatti, 1993; Connes et al., 1987; Connes and Stormer, 1975; Emch, 1975). For instance, one pair of the complexities is
where p = CkPkPk is a finite decomposition of p and the supremum is taken over all such finite decompositions. The mutual entropy given above contains other definitions of the mutual information (Holevo, 1973; Ingarden, 1976; Levitin, 1991). Moreover it is not only a fundamental quantity to study quantum communication processes
276 Complexities and the Characterization of Chaos
501
such as the capacity of a quantum channel (Ohya, n.d.-b; Ohya et al., 1997), but also can be used to study irreversible processes (Ohya, 1989). 5. CHAOS DEGREE
We apply the chaos degree in ID to a deterministic dynamical system and discuss its usefulness. The degree of chaos for a state p (density operator or probability distribution) and a channel A* is defined as D(p; A*)
E
C(A*p) - T(p; A*)
E
S(A*p) - Z(p; A*)
We shall see how this degree works to describe the chaotic aspects of a logistic map. The logistic map is given by the following equation: x,+I
=fa(x,,) = ~ , , ( l x,,),
x,,
E
[0, 11,
0
5 u 5
4
(5.1)
The solution of this equation bifurcates as shown in Fig. 1. The Lyapunov exponent of this map has been calculated by Shaw (Shaw, 1981) (Fig. 2). The Lyapunov exponent A is defined as
Figure 2 is the result of computing A,, for 1000 a's from 3.0 to 4.0 with
1
0.8
-
0.6
0.1
-
0.2
3.2
3.4
3.6
Fig. 1. Bifurcation diagram of logistic map.
3.8
4
277 Ohya
502
a 3
3.4 3.6 3.8 Fig. 2. Lyapunov exponent of logistic map.
3.2
4
A positive exponent means that the trajectory is very sensitive to the initial value and is called chaotic; a negative exponent means that the trajectory is stable. If a logistic map fa does not have a stable and periodic trajectory, then there exists an ergodic probability measure p on the Bore1 set of [0,1], absolutely continuous with respect to the Lebesgue measure (Misiurewicz, 1981). Take a finite partition (Ak) of I = [0,1] such as n = 100,000 steps.
m
I = U
Ak
(A;nAj=@,ifj)
k= 1
Let lQl be the number of the elements in a set Q and p(") = (pp)) be the probability distribution of the trajectory up the nth step, that is, how many x . (1' = 1, . . . , n - l ) a r e i n A k :
,
This probability distribution is obviously from the difference equation of (5. l), hence it depends on the initial value x1 and fa. The channel A* is a map given by p ( " + l )= A* p'"). It can be shown (Misiurewicz, 1981) that the n + 00 limit of p p ) exists and is equal to p(Ak). Further, the joint distribution r("."+l)= (rp"")) for a sufficiently large n is approximated as
278 503
Complexities and the Characterization of Chaos ( 0.7
0.6
Y
0.3 0.4
0.3 0.2
0. I
0
L
-
.
-
.
'
- -
- .
.
3.2
3
3.4
1 .
3.6
3.8
4
Fig. 3. Chaos degree.
r(n.n+l)
0
=
I(k; (
~ k ~, k + l ) E
Ai X Aj, 1 n
5
k
5
n)l
Then the chaos degree (CD) is
D@("); A*)
E
C(A*p("))- T(p(");A*)
=
,'j(A*p("))- I@("); A*)
=
,@("+I))
-
I@("); A*)
For a computer simulation, we take 1000 a's from 3.0 to 4.0 and
i = 0 , . . . , 1999 n = 100,000 The choice of these quantities does not alter the results so much if we take large n. The result is shown in Fig. 3 , from which we conclude that our chaos degree describes the chaotic aspects of the logistic map, namely,
D > 0 w chaotic D
=
0 e non-chaotic
Although the Lyapunov exponent becomes negative, sometimes-m, our degree
279
504
Ohya
is always nonnegative, so that it might be useful to make the chaotic domain clear-cut. More rigorous study of the chaos degree and its use for other dynamical channels including quantum systems is now in progress.
REFERENCES Accardi, L., Ohya, M., and Watanabe, N. (1996). Note on quantum dynamical entropy, Reports on Mathematical Physics, 38, 457469. Benatti, F. (1993). Deterministic Chaos in Infinite Quantum Systems, Springer-Verlag, Berlin. Billingsley, P. (1965). Ergodic Theory and Information, Wiley, New York. Connes, A., and Sterner, E. (1975). Entropy for automorphisms of 11, von Neumann algebras, Acta Mathematica, 134, 289-306. Connes, A,, Narnhofer, H., and Thimng, W. (1987). Dynamical entropy of C*-algebras and von Neumann algebras, Communications in Mathematical Physics, 112, 69 1-7 19. Emch, G . G . (1974). Positivity of the K-entropy on non-abelian K-flows, Zeitschriff fur Wahrscheinlichkeits theorie Verwandte Gebiete, 29, 241-252. Holevo, A. S . (1973). Some estimates for the amount of information transmittable by a quantum communication channel, Problemy Peredachi Informacii, 9, 3-1 1 [in Russian]. Ingarden, R. S. (1976). Quantum information theory, Reports on Mathematical Physics, 10, 43-73. Ingarden, R. S., Kossakowski, A,, and Ohya, M. (1997). Information Dynamics and Open System, Kluwer Academic Publishers. Levitin, L. B. (1991). Physical information theory for 30 years: basic concepts and results, in Springer Lecture Notes in Physics, Vol. 378, pp. 101-1 10. Matsuoka. T., and Ohya, M. (1995). Fractal dimensions of states and its application to Ising model Reports on Mathematical Physics. 36, 365-379. Misiurewicz, M. (198 I). Absolutely continuous measures for certain maps of interval, Publications Mathematiques IHES, 53, 17-5 1. Muraki, N., and Ohya, M. (1996). Entropy functionals of Kolmogorov Sinai type and their limit theorems, Letters in Mathematical Physics, 36, 327-335. Ohya, M. (1981). Quantum ergodic channels in operator algebras, Journal of Mathematical Analysis and Applications, 84, 3 18-327. Ohya, M. (1983a). On compound state and mutual information in quantum information theory, IEEE Transaction of Information Theory, 29, 770-774. Ohya, M. (1983b). Note on quantum probability, Lettere a1 Nuovo Cimento, 38, 402406. Ohya, M. (1989). Some aspects of quantum information theory and their applications to irreversible processes, Reports on Mathematical Physics, 27, 1 9 4 7 . Ohya, M. (1991 a). Information dynamics and its application to optical communication processes, Springer Lecture Notes in Physics, Vol. 378, pp. 81-92. Ohya, M. (I991 b). Fractal dimension of states, Quantum Probability and Related Topics, 6, 359-369. Ohya, M. (1995). State change, complexity and fractal in quantum systems, Quantum Communications and Measurement, 2, 309-320. Ohya, M. (n.d.-a). Foundation of entropy, complexity and fractal in quantum systems, in International Congress of Probability toward 2000, to appear. Ohya, M. (n.d.-b). Fundamentals of quantum mutual entropy and capacity, submitted. Ohya, M. (n.d.-c). Applications of information dynamics to genome sequences, SUT preprint. Ohya, M. (1997). Complexity and fractal dimensions for quantum states, Open Systems and Information Dynamics, 4, 141-157.
280 Complexities and the Characterization of Chaos
505
Ohya, M., and Petz, D. (1993). Quantum Entropy and Its Use, Springer-Verlag, Berlin. Ohya M., and Watanabe N. (1993). Information dynamics and its application to Gaussian communication process, Maximum Entropy and Bayesian Method, 12, 195-203. Ohya, M., Petz, D., and Watanabe, N. (1997). On capacities of quantum channels, Probability and Mathematical Statistics, 17, 179-196. Shannon, C. E. (1948). Mathematical theory of communication, Bell System Technical Journal, 27, 379423. Shaw, R. (1981). Strange attractors, chaotic behavior and information flow, Bitschrifl fur Naturforschung, 36a, 80-1 12. Uhlmann, A. (1977). Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory, communication in Mathematical Physics, 54, 2 1-32. Umegalu, H. (1962). Conditional expectations in an operator algebra IV (entropy and information), Kodai Mathematical Seminar Reports, 14, 59-85. Von Neumann, J. (1932). Mathematischen Grundlagen Quantenmechanik, Springer, Berlin.
281
Amino Acids (1998) 14:343-352 With kind permission of Springer Science and Business Media
Amino Acids 0 Springer-Verlag 1998
Analysis of HIV by entropy evolution rate K. Sato', S. Miyazaki* and M. Ohya' Department of Information Sciences, Science University of Tokyo, Noda City, Chiba, Japan Center for Information Biology, National Institute of Genetics, Mishima City, Shizuoka, Japan Accepted October 29,1997
Summary. We analyze the variation of HIV after infection by means of an information measure, called the entropy evolution rate. In our analysis, we use a part of the external glycoprotein gp120 including the V3 region observed from six patients. Then we could make the following two aspects clear; (1) the relation between the change of the entropy evolution rate and the appearance of symptoms of disease, and (2) the relation between the change of the entropy evolution rate and that of the CD4 count of the patients. Keywords: Amino acids - HIV - Entropy evolution rate
1. Introduction The main purpose of this study is to find a new criterion grasping the processes of the change of CD4 count and the immunity of patients from the gene level after HIV infection. In Section 2, we summarize the data of HlV genes of the six patients used in this paper. The entropy evolution rate and the method of how to use it for our analysis are discussed in this section. In Section 3, we present our results by the graphs. We discuss our results and the usefulness of our method in Section 4.
2. Material and methods We consider two aligned amino acid (resp. base) sequencesA and 3,which are composed of 20 (resp. 4) kinds of amino acids (resp. bases) and the gap *. The complete event system &,p) ofA is determined by the occurrence probabilityp; of each amino acid (resp. base) aiand the gap * (0 5 i 5 20) (resp. 0 5 i 5 4) with a, = *;
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In the same way, the complete event system (&q) of
is
We can construct the compound event system (Ax 3,r ) for two sequencesd and 3.
where rjj represents the joint probability of the event i of A and the event j of g. These event systems define various entropies, among which the following two are important: (1) Shannon entropy
which expresses the amount of information carried by (2) The mutual entropy
(As).
which expresses the amount of information transmitted from A(resp. 3)to B(resp. A). Using the above information measures, a measure indicating the difference between two amino acid sequences was introduced in (Ohya, 1989). This measure is called the entropy evolution rate and defined as follows: Put
which is the rate how much information is transmitted fromA to $, and it is symmetrized as
The entropy evolution rate @(Ag)is defined by
@(A&) = 1 -
3)
In this paper, we use this entropy evolution rate to examine the variation of HIV sequences of six patients. The entropy evolution rate takes the value in [0,1]; @(A,3) =0 ifA and 3 are completely same and @(A,z)= 1if they are completely different. Therefore the variation of HIV becomes larger, the entropy evolution rate is getting larger. Data used in our analysis are the base sequences of HIV for the six patients reported in (Wolfs et al., 1991; Holmes et al., 1992; McNearney et al., 1992). We obtained the data from the International Nucleotide Sequence Database (DDBJ/EMBL/GenBank). Here, the six patients are designated as patient A to patient F. The facts reported for the six patients are summarized in Table 1.
Table 1. Data used in our analysis Designation in our analysis
Patient A
Patient B
Patient C
Patient D
Patient E
Patient F
Designation in our analysis
Patient 1
Patient 495
Patient 82
sl
s2
s4
Presumed trans mission mode
homosexual contact
homosexual contact
a single batch of factor VIII
no information
no information
no information
Clinical status
p24antigenemia (1988)
AIDS (1989) p24antigenemia
asymptomatic
no information
no information
no information
CD4 counts Antiviral therapy
decreasing None
decreasing AZT (1989)
decreasing None
fluctuating None
decreasing None
decreasing None
Term
1985- (about 5 years period)
1985- (about 5 years period)
1984-1991 (7 years period)
1985.11-89.5 (4.5 years period)
1985.5-87.10 (2.5 years period)
1985.1-89.6 (4.5 years period)
Length
183-276nt
183-276nt
234nt
332-335nt
332-335nt
332-335nt
Tissue
serum
serum
plasma
peripheral blood leucocyte
peripheral blood leucocyte
peripheral blood leucocyte
Molecular type
RNA
RNA
RNA
DNA
DNA
DNA
N
03
w
284
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K. Sat0 et al.
We use the sequence, a part of the gp120 region including the third variable (V3) region whose mutation rate is particularly high in HIV (Watson et al., 1993; de Jong et al., 1992). The V3 region is composed of disulfide bounds of cysteine residues located in the amino acids 296 and 330 of the gp120, as shown in Fig. 1. Although it has been called the principal neutralization domain (PND) as the antibody for the segment is able to block the HIV infection, the antibody for a specific virus is gradually losing its effect because of the variation (mutation) of the virus. It is reported (Wolfs et al., 1991) that patient B was diagnosed as having AIDS in about 5 years after the primary infection and that patient A remained healthy although the p24-antigenemia reappeared at 3 years after his infection. The p24-antigenemia in blood reflects the amount of the virus, it is reappeared when the patient has AIDS, so that it is used as a value measuring the condition of the patient. The CD4 count has been used as a measure to know having AIDS by several researcher and medical doctors. The number of CD4 for patient D fluctuates, and that for other patients gradually decreases. This CD4 count represents the number of immunocyte destroyed by HIV. The immunocyte for healthy people is around from 800 to 1,000 per lpl-blood. When the CD4 count of a patient decreases and it becomes less than 200, various infections are considered to appear. Therefore, according to the diagnosis standard of CDC (Centers for Disease Control), when the CD4 become less than 200, the patient is recognized to have AIDS. The CD4 count is reported only for patient D, E, and F. For patient D, it changes as 470,826,273, and 515 from the primary stage (called 1 year) up to the fourth stage (called 4 year). For the patient E and F, it changes as 1225,756,368 and 943,575, 187, respectively. The number of the sequence data observed from the six patients are listed in Table 2. We used the base sequences having the same length of bases for each patient. For example, in the primary stage of patient A, we used 6 data out of 8 data because the length of six data is 276 and that of other two is 183. Moreover, in order to carry out our analysis, first we translate the base sequences of HIV collected from the six patients into the amino acid sequences, and secondly, we directly used the base sequences. Our analysis is done in the following two cases (I) and (11). (I) In order to compare the genome sequences of HIV in successive years, the entropy evolution rate is computed for the sequences obtained at one year with respect to those obtained at the next year (we call it the entropy evolution rate for each year), and we examine the variation (mutation) rate with the mean of the entropy evolution rates for each year and their standard deviation for each year. (11) In order to check the variation of HIV from the primary stage, we compute the entropy evolution rate for the sequences of each year w.r.t. the primary year (we call it the entropy evolution rate for the primary year). Similarly as the case (I), we examine the variation by means of the entropy evolution rates for the primary year and their standard deviations. As an example, we explain how to compute the entropy evolution rate and others mentioned above in (I) and (11) for the patient A. From Table 2, the number of genome sequences for the patient A are as follows: n = 6 (year 0), n = 7 (year l ) , n = 7 (year 2), n = 5 (year 3 ) , n = 6 (year 4), n = 6 (year 5). For the case (I), we compute every entropy evolution rate for the aligned sequences in successive years, for instance, @(A:,A;) (i = 1,
IVIRSDNITDNAKTIIVQLKEAVQIN CTRPNNNTRKSIHIGPGKAFYATGEIIGDIRQAIICNLSRVDWEDTLKQIAEKLREQFRNKTIVFNQ IVIRSDNITDNSKTIIVQLKEAVQIN CTRPNNNTRKSIHIGPGKAFYATGEIIGDIRQAHCNLSRVDWEDTLKQIAEKLREQFRNKTIVFNQ IVVRSDNITDNAKTIIVQLKKAVQIN CIRPNNNTRKSIHIGPGKAFYATGETIGDIRQAHCNLSGGDWENTLKQIAEKLREQFRNKTIVFNQ
Fig. 1. 3 sequence data collected from patient A in primary stage of infection. It's V3 region is underlined
285
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Analysis of HIV by entropy evolution rate
Table 2. The number of sequences used in this paper Patient A Number of data collected from GenBank Number of data used
year 0 8
6
Patient B Number of data collected from GenBank Number of data used
year 0 11
Patient C Number of data collected from GenBank Number of data used
year 0 1
Patient D Number of data collected from GenBank Number of data used
year 0 5
Patient E Number of data collected from GenBank Number of data used
year 0 5
Patient F Number of data collected from GenBank Number of data used
year 0 5
7
1
5
5
5
year 1 7
year 2 9
7
7
year 1 6
year 2 6
3
4
year 3 9
5 year 3 6
4
year 4 9
6 year 4 7
4
year 5 8
6 year 5 8
4
year 3 15
year 4 11
year 5 23
year 6 15
year 7 13
15
11
23
15
13
year 2 2
year 3 4
year 4 3
2
4
year 2 5
3
year 2.5 6
5
6 year 4.5 6
year 4 6
6
6
. . . , 5. j = 1, . . . , 6 ) for the sequence A: of the third year and the sequence A! of the fourth year. Then we compute their mean value given by
which enables us to examine the variation of HIV. In the same way, we compute Q(Ao,A1), e(A1,AA2), Q(A2,A3),@(A3,A4), Q(A4,A5).The standard deviation of the entropy evolution rate for year 3 and year 4 is defined as follows:
Y 30 For the case (11), we compute the mean entropy evolution rates for every sequence of each year with respect to that of the primary year. For instance, the mean entropy evolution rate for the fifth year w.r.t. the primary year is given by
f:
&(A!
Q(AO,AS) =
,A;)
;=li=1
36 We similarly compute e(Ao,A1),Q(Ao,AA2), @(A0,A3), @(AoJ4),Q(A0,A5),and their standard deviations.
286 348
K. Sat0 et al.
All six patients are examined with these quantities, and our results are shown in the next section. Here we note that we should align the sequences to compute the entropy evolution rate, and the alignment is done by the method in (Ohya and Uesaka, 1970; Neeleman and Wunsch, 1970).
3. Results
The following figure (Fig. 2) is the results of the mean entropy evolution rates for each year and the standard deviations obtained from the amino acid
(0.11
(1.2)
(2.3)
(3.4)
(4.5)
patient A
patient B
I211
10.21
patient C
(3.11
patient D OP 01
om 016
Om
;
012
!om
i
om :
om
on1
0
0 IW
patient E
1
on a
14,451
patient F
Fig. 2. Mean entropy evolution rate (bars) and standard deviation (lines) for each year
287
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Analysis of HIV by entropy evolution rate
I
01
ow
OW
...... ........ ...... ...
I, 2
llXM ‘UYI
. 001
0 Ib
I I6
10.01
10.11
10.21
10.31
10.41
10.51
patient B
patient A
10.01
10.41
10.31
10.51
10.61
patient C
1ODI
Ul
patient E
10.71
10.01
10.21
10.31
10.41
patient D
10lIl
patient F
Fig. 3. Mean entropy evolution rate (bars) and standard deviation (lines) measured from primary year
sequences. Here (i, i + 1) denotes the (i + 1)-th year w.r.t. i-th year and the mean value means the mean entropy evolution rate. Fig. 3 shows the results of the mean entropy evolution rates for the primary year and their standard deviations, so that (0, i) denotes the i-th year w.r.t. the primary year. The following figures (Fig. 4,Fig. 5 ) are the results obtained from the base sequences.
288 350
K. Sat0 et al.
1O.Il
11.2)
patient A
(0.31
15.6)
(4.9
0.4)
16.7)
13.4)
(4.5)
1UI
111)
patient D
patient C
i021
12.3)
patient B
N)l
1441
patient E
iiin
patient F
Fig. 4. Mean entropy evolution rate (bars) and standard deviation (lines) for each year
4. Discussion
Patient B was diagnosed as having AIDS at about 5 years after the primary infection. According to the result of the mean entropy evolution rate (m-EER for short) for each year, the change of the m-EER for the patient B is met the second extreme increase at that time. The change of the m-EER (Fig. 2) for patient B is considered as a fundamental pattern of the outbreak of AIDS. Based on this pattern, we may say the following conclusions for other patients. Patient A will be diagnosed as having AIDS in a few years, because the second extreme increase seems starting. Similarly, patient C will have an attack of AIDS soon, because the second moderate increase is occurred.
289
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Analysis of HIV by entropy evolution rate
patient A
patient B
,401
patient E
10.4)
lDAJl
patient F
Fig. 5. Mean entropy evolution rate (bars) and standard deviation (lines) measured from primary year
Patient D, patient E and patient F have few number of data, so that we merely say a few comments. Patient D may possibly increase here after. Patient E may be far from the outbreak of AIDS. Since the data of patient F are lacked a first few years, it is very difficult to judge the variation of his HIV. According to the result of the mean entropy evolution rate measured from the primary stage, the m-EERs of the patients except the patient D increase as shown in Fig. 3. This consequence agrees with a report that the CD4 counts of the patients except patient D decrease. That is, the gradual decrease of
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K. Sat0 et al.: Analysis of HIV by entropy evolution rate
the CD4 count is equal to the increase of the m-EER for the primary year. Further, the CD4 count of patient D fluctuates and his m-EER also fluctuates. This result means that there exists a positive correlation between the m-EER for the primary year and the CD4. We merely note that the standard deviation shows how many different HIV exist in each stage (year). The results obtained by using the whole base sequences are shown in Fig. 4 for the case (I) and Fig. 5 for the case (11). These results show that patient B, like the case of the amino acid sequences, have a characteristic variation and other patients are also similar to the case of the amino acid sequences. From our analysis, we may conclude that the mean entropy evolution rate can be a measure of the variation of HIV and the outbreak of AIDS as the CD4 count. References Holmes EC, Zhang LQ, Simmonds P, Ludlam CA, Brown AJL (1992) Convergent and divergent sequence evolution in the surface envelope glycoprotein of human immunodeficiency virus type 1 within a single infected patient. Evolution 89: 48354839 de Jong JJ, Goudsmit J, Keulen W, Klaver B, Krone W, Tersmette M, de Ronde A (1992) Human immunodeficiency virus type 1 clones chimeric for the envelope V3 domain differ in syncytium formation and replication capacity. J Virol66: 757-765 McNearney T, Hornickova Z, Markham R, Birdwell A, Arens M, Saah A, Ratner L (1992) Relationship of human immunodeficiency virus type 1 sequence heterogeneity to stage of disease. Medical Sciences 89: 10247-10251 Needleman SB, Wunsch CD (1970) A general method applicable to search for similarities in the amino acid sequence of two proteins. J Mol Biol48: 443-453 Ohya M (1989) Information theoretical treatment of genes. Trans IEICE E 725: 556-560 Ohya M, Uesaka Y (1992) Amino acid sequences and DP matching: new method for alignment. Information Sciences 63: 139-151 Watson JD, Gilman M, Witkowski J, Zoller M (1993) Recombinant DNA, 2nd edn. Freeman and Company Wolfs TW, Zwart G, Bakker M, Valk M, Kuiken C, Goudsmit J (1991) Naturally occurring mutations within HIV-1 V3 genomic RNA lead to antigenic variation dependent on a single amino acid substitution. Virology 185: 195-205 Authors’ address: Dr. Keiko Sat0 and Prof. Dr. M. Ohya, Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278, Japan. Received May 29, 1997
291 PROBABILITY AND MATHEMATICAL STATISTICS Vol. 17, Fax. 1 (1997), pp. 17%1%
ON CAPACITIES OF QUANTUM CHANNELS BY
MASANORI OHYA, DENES PETZ*
AND
NOBORU W A T A N A B E (TOKYO)
Abstract. Capacities of quantum mechanical channels are defined in terms of mutual information quantities. Geometry of the relative entropy is used to express capacity as a divergence radius. The symmetric quantum spin 1/2 channel and the attenuation channel of Boson fields are discussed as examples.
1. Introduction. A discrete communication system - as modeled by Shannon - is capable of transmitting successively symbols of a finite input alphabet {xl,x2, . .. , x,,,}. In the stochastic approach to the communication model it is assumed that the input symbols show up with certain probability. Let pjibe the probability that a symbol xi is sent over the channel and the output symbol y j appears at the destination. The joint distribution pji yields marginal distributions (pl,p 2 , ..., p,) and (ql, q2, ..., qk) on the set of input symbols and output symbols, respectively. Shannon introduced the mutual information
I = c p j i log- Pji i,i Pi% in order to measure the amount of information going through the channel. The interest in quantum communication channels arose in the late 1960’. The scheme of a quantum communication system is not different from a classical one, however, zero point fluctuation (noise) cannot be avoided in quantum systems. Important recent devices for communication are based on optical fiber which is a quantum object. Hence we may assume that the actual signal transmission is over a quantum mechanical medium which is described in the usual Hilbert space formalism of quantum theory. Coding, actual signal transmission and decoding (or measurement) are the main components of the communication chain. The splitting of the communication chain into these three parts C-T-M is somewhat arbitrary. The parts can be investigated individually and their capacity can be defined by means of mutual information.
* On leave from the Technical University Budapest and from the Mathematical Institute of the Hungarian Academy of Sciences. The author was partially supported by JSPS and by OTKA TO 16924.
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et al.
Our approach to capacity is based on quantum mutual information which is defined in terms of relative entropy (or informational divergence). Therefore, relative entropy is the basic tool in the paper. The capacity is not compared with performance bounds of classical coding (as in [5] and [13]) because we are mainly interested in the purely quantum part of the channel. Section 2 contains some generalities of quantum communication channels, mutual information and relative entropy. Kholevo’s bound is also discussed and we show that it is rarely achievable. In Section 3 our capacities are introduced and the quantum mechanical counterpart of Csisziir’s information geometry is used to realize the pure quantum capacity as the divergence radius of the range. The toy example of symmetric quantum spin 1/2 channel is used to demonstrate our ideas. We verify that the pure quantum capacity of this channeling transformation is the same as the performance bound from coding (see [S]). Section 4 treats an infinite-dimensional example, the attenuation channel of Boson fields. The channeling transformation is treated in the abstract Weyl algebra setting as well as in Fock representation. It is proved that the capacity of the attenuation channel is infinite, however, the transmission of arbitrarily much information requires infinite energy. 2. Generalities of quantum mechanical channels. To each input symbol xi there corresponds a signal state (pi of the quantum communication system, ‘pi functions as the codeword of xi. The signal states ‘pi are mostly pure but they can be non-orthogonal. However, we do not make any assumption on them at this level of generality. The channel state is a convex combination
whose coefficients are the corresponding probabilities, p i is the probability that the letter x i should be transmitted over the channel. In the mathematical sense the quantum channeling transformation A* is an afine transformation of the state space of the input quantum system into the state space of the output quantum system. (The notation A* is used here because very often A* is the dual mapping of a linear transformation of observables.) At the output some sort of detection scheme retrieves the transmitted information. To each output symbol y j there corresponds a non-negative observable A j , that is a self-adjoint operator A j on the Hilbert space 2,such that C.Aj= I. (Some people speak about effects, or call ( A j )a generalized measurement). In terms of the quantum states the transition probabilities are (A*qi)(Aj) and the probability that x i was sent and y j is read is (2.2)
Pji
= Pi(A*cPi)(Aj). \
On the basis of these joint probability distributions the classical mutual information (1.1) is given. Kholevo’s theorem provides a fundamental bound for the mutual information in terms of the quantum von Neumann entropy. Before stating
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Kholevo’s result, we review the simplest entropy quantities used in quantum information theory, for details see [lo]. The relative entropy of two states is defined (following Umegaki [14], Lindblad [S] and Araki [l]) as (2.3)
S(Cp1, 432) = TrD1 (logD1 -logD,),
where D 1 and D 2 are the corresponding density operators. (Ths formula extends the Kullback-Leibler information measure.) We shall use a kind of algebraic language and view the states as linear functionals on (operator) algebras. The basic property of relative entropy is its monotonicity under channeling transformation. More precisely, if a : d 93 is a unitial (completely) positive mapping between the algebras d and 3,that is, the dual a* is a channeling transformation from the state space of 23 into that of d,then --f
(2-4)
S(cp,oa, C p z 0 4 d
S(Cp1,
cpzh
Relative entropy (or information gain) is the fundamental information quantity, many other information quantities are expressed by it. For example, the von Neumann entropy is
(2.5)
S ( q ) = -Tr(DlogD) = sup{~lljS(cpj,40): C;ljqi= cp, llj 3 0). i
i
Let a: d -+ 93 be positive unitial mapping and cp be a state of 3.So cp is an initial state of the channel a*. The quantum mutual entropy is defined after [9] as (2.6)
C ~ j=~C Pj } ,
~ ( c Pa) ; = sup{CljS(qjoa, q~oa): i
i
where the least upper bound is over all orthogonal extremal decompositions. (One checks easily that this formula reduces to (1.1) when d = Ck and ~49 = Cmsince in this case the orthogonal extremal decomposition is unique and (PI, ~ 2 3
pm)
=CiPi8i-I
THEOREM 2.1. With the above notation the inequality I
= cpji10g* i,i
< S(A* (43))-
Pi4j
C P i S ( A *((pi)) i
holds true.
Kholevo [ 6 ] proved this inequality in 1973 when the concept of quantum relative entropy was not well understood yet. Kholevo’s upper bound is (2.7)
S(A*q)- C p i S ( n * q i ) = CPiS(A*cpi, A*CP)* i
I
Let C”, B ( X ) , B(%) and C“ be operator algebras and consider the mappings a: C”+B(%),
(c1,
c2,
..., c k ) w x c j A j , i
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A : B ( X )+ B ( X ) ,
B:
BW)-+C",
HH(cpl(H), Cpl(H), -.. qm(H)). Y
So the duals of the positive unitial mappings CL, A and /3 correspond to the measurement, quantum state transmission and coding procedures, respectively. In this terminology the upper bound is the mutual information I ( p , BOA) of the quantum channel a with input quantum input state p (where p ((cl, c2, .. ., cm)) = p i ci) and the classical mutual information I becomes the quantum mutual information I ( p , BoAoa) of a composite quantum channel. So Kholevo's theorem is read as
xi
I(P, BOA04
(2.8)
< 1(p,P o 4
which follows from the monotonicity of the quantum mutual information. For the details see [lo], in particular pp. 139-140. It is noteworthy that these ideas work in the continuous case as well as it was observed also in [lS]. Yuen and Ozawa [lS] propose to call Theorem 1 thefundamental theorem of quantum communication. The theorem bounds the performance of the detecting scheme. We see that in most cases the bound cannot be achieved. Namely, the bound may be achieved in the only case when the output states A* (pi)have commuting densities.
PROPOSITION 2.2. If the states A* (pi), 1 < i < m, do not commute, then I=
1p j i l oPiclj g E < S(A*(p))i,i
&qA*(qli)) I
is a strict inequality.
In the terminology of Chapter 8 of [lo] the equality in Kholevo's theorem means that the measurement channel CI is sunicient for the states A* (pi),1 < i < m, and the sufficiency has several characterizations, for example, the existence of states mi of the output quantum system such that
1 pi(Aj)wj = pi i
for every i.
In particular, if the bound is achieved, the states A*(pi) have to commute. Suppose that the state A* (pi)has a density Diand let D = C i p i D i (which is the density of A* (40)). Then the generalized measurement Ai = DiD-' achieves Kholevo's bound. (The operator DiD-' is well defined even if D is not invertible, because the kernel of D is contained in that of Di.) The technicalities of the detailed proof depend very much on the level of generality. For finite dimension one has to investigate the equality case in the Jensen inequality and this was carried out by Kholevo [ 6 ] . We consider now the infinite-dimensional case but under the restrictive assumption of faithfulness. Since A* does not play any role, we skip it from the notation. The proof of the next theorem uses the
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Capacities of quantum channels
idea of the paper [12] and the proof presented here is a bit sketchy. (The interested reader may consult [12] for the more detailed justification of the steps.)
THEOREM 2.3. Assume that cp = ralized measurements such that
xi
picpi and there is a sequence (a,) of gene-
CPiS(a,*(qi),~;(cP))+ C p i S ( q i , CP). i
i
If the limit isfinite and the states
‘pi are faithful,
then the family (cpi) must commute.
P r o o f . Since S(a,*(cpi),a,*(cp))f S(cpi, cp), the assumption implies that for every i-
S(a,*( q i ) , a,*(r~)) S ( c ~ iq) , +
Let Di, D be the statistical operators of cpiy cp, respectively. In the sequel we shall use the relative modular operator technique and we work on the Hilbert space .fof Hilbert-Schmidt operators. There exists a positive operator Ai such that (AD!/’: AEB(&’)} is a core for A!12, and 11d!blD;/2112 = cp(AA*)
(AEB(3P)).
I n fact, the relative modular operator di is the extension of the linear operator AD!/* t--,D1l2A defined on a dense linear subspace of X . In terms of the relative modular operator, we have S ( ~ p i40), = - (D;I2, (IOgA)D!’2)
(2.9)
00
=
5 (D!/’,
( ~ l ~ + t ) - ’ D ! / ~ ()l-+ t ) - ’ d t .
0
Similarly, m
S(a*(cpi),a*(cp)) =
J (d;!?,
(s,,i+t)-ld,l!iz)-(l+t)-ldt,
0
where the probability vector d,l!; (d,) corresponds to the state a; qi (a,*q) and 6 n . i = dn/&,i-
(Note that is the relative modular operator of a:q with respect to cmz(cpi) but, due to the simple situation coming from finite dimension and commutativity, we may just regard it as a vector.) Since
{a,:!,
(6,,i+t)-1d;/:) 6 ( D f / 2 , (Ai+t)-’ D‘/’)
for every t > 0 and i,
our assumption implies that
(d;::, (d,,i+t)-ldi,’:)
+
(D’/”, (Ai+t)-’D/j2)
From this we infer that
(2.10)
a,((t+6,,i)-’
a:):-’+,
(t+Ai)-’D!l2.
as n-, co.
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Now we consider the function Fi(z) = Af Dill2. We know that DtI2 is in the domain of A t / 2 . The range of At/' contains the operators D1I2A for a bounded A . Since (1 Dti2 11 d p ; ' I 2 , we may choose A = D - ' I 2 Dti2, and infer that DfI2is in the domain of As a consequence, the function F i ( z )is analytic on the strip { z E C : -1/2 < Rez < 1/2}. We should not gather so much withf,,i(z) = ~5;,~d;!',because it is analytic on the whole complex plain. Our next aim is to show that (2.11)
K.i ( f n , i ( 2 ) )
--+
if -1/2 < Rez < 1/2 for the contraction K,i (a, di!?)
Fi (2)
K,i defined
by
= a, (a,)Dti2
Since we have an analytic function at our disposal, it suffices to prove (2.11)for 0 < s < 1/2 in place of z. For 0 < s < 1/2 we obtain
sinsn a, -j" t"-'Ai(t+Ai)-' D;l2dt = P i ( s ) . n o So we may consider in (2.11) a pure imaginary z = it:
K,i(J', (it))= K,i (d:,i d;!?) +
= Dit
l)I2
and c(,
(&)
Dillz --+
(D"D,ri')
in the Hilbert-Schmidt norm. The strong operator convergence follows from the Hilbert-Schmidt norm convergence and we arrive at (2.12)
a, (S:,i)
-,(DitD r i t ) (strongly).
In particular, D"D;" is a unitary group for fixed i, and D and D i must commute.
3. Capacity of channels. Let A? and 3- be the input and output Hilbert spaces of a quantum communication system. The channeling transformation A*: C ( 2 ) --+ C ( X ) sends density operators acting on Z into those acting on X . A pseudo-quantum code is a probability distribution on C ( Z )with finite support. So {(pi), (cpi)) is a pseudo-quantum code if (pi) is a probability vector and cpi are states of B(Z).The quantum states (pi are sent over the quantum mechanical media, for example, optical fiber, and yield the output quantum states A*rpi. The performance of coding and transmission is measured by the mutual information (3.1)
I((Pi), (qi),
"*)
= CpiS(A*yi, A*CP)i
297
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Capacities of quantum channels
Taking the supremum over certain classes of pseudo-quantum codes, we obtain various capacities of the channel. Here we consider one subclass of pseudo-quantum codes. A quantum code is defined by the additional requirement that {cpi} is a set of pairwise orthogonal pure states. Correspondingly, we arrive at two alternative concepts of capacity: (3.2) and
Cp,(A*) = sup {I((pi),(cpi), A): ((pi),(cpi)) is a pseudo-quantum code)
(3.3)
C,(A*) = sup{I((pi), (cpi), A*): ((pi),(cpi)) is a quantum code}.
We can write C,(A*) in a slightly different form by using the notation (2.6): C, (A*) = sup {I(cp, A*):cp is an input state}.
(3.4)
The capacity C, may be viewed as the characteristic of the purely quantum mechanical signal transmission. It follows from the definition that
c, (A*>d c,, (A*> holds for every channel.
EXAMPLE 3.1. Let A* be a channel on the 2 x 2 density matrices such that A*:
(x
:)t-+("
o
O). c
Consider the input density matrix
For I # 1/2 the orthogonal extremal decomposition is unique; in fact,
1 D*=Z(-l
1-1 1 1
1
Y1)+T-(l
1)
and we have I ( D A 7A*) = 0
for I # 1/2.
However, I(D,,,,A*) = log2. Since C,(A*) ,< C,,(A*) < log2, we conclude that C,(A*) = C,,(A*) = log2. The example shows that the quantity I(cp, A*) may be discontinuous at cp when cp has some degeneracy in the spectrum. In order to estimate the quantum mutual information, we introduce the concept of divergence center. Let {ai: i ~ l be } a family of states and R > 0. We say that the state w is a divergence center f o r {ai:~EI} with
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if
S ( o i , o)< R
for every i E I .
In the following discussion about the geometry of relative entropy (or divergence as it is called in information theory) the ideas of [3] can be recognized very well. 3.2. Let ((pi), (cpi)) be a pseudo-quantum code for the channel A* and LEMMA o be a divergence center with radius < R for {A*cp,}. Then
P r o o f . We assume that the states A*cpi, A*cp = CipiA*cpi and o have finite entropy and their densities are denoted by Di, D and D’, respectively. We have - S (A*cpi)-TrDi
logD’ d R,
and hence (3.5)
c p i S ( A * c p i ,A*cp) = - cpiS(A*cpi)-TrDlogD i
i
< R-TrD(1ogD-1ogD‘)
= R-S(A*cp, 0).
The extra assumption we made holds always in finite dimension. When the entropies are not finite but the relative entropies are so, one has to use more sophisticated methods for the proof. It is quite clear that inequality (3.5) is close to equality if S(A*cpi,a)is about R and CipiA*cpi is about o. m Let {mi: i E I } be a family of states. We say that the state o is an exact divergence center with radius R if R = infsup { S ( o i , cp)} ‘
p
i
and o is a minimizer for the right-hand side. (When R is finite, then there exists a minimizer, because cp H sup { S (ai,cp): i E I > is lower semicontinuous with compact level sets; cf. Proposition 5.27 in [lo].) LEMMA3.3. Let $o, $1 and o be states of B ( X )such that the Hilbert space X is finite dimensional and set
(0d I
$ A = (l-A)$o+A$1
< 1).
If S ( $ o , o)and S ( $ , , o)are finite and SWAY4 2 S($1, 4
(0< 1 6 11,
then SWlY 4 + S ( $ o ,
$1)
< S($o, 4.
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P r o of. Let the densities of $ A and w be D land D,respectively. Due to the assumption S($17 w )< +a, the kernel of D is smaller than that of Dl.The function f(n) = S(cpA,w )is convex on [O, 11 andf(L) 3f(l)(cf. Proposition 3.1 in [lo]). It follows that f’(1) < 0. Hence we h3ve
f‘(1) = Tr(D, -Do)(I+logD,)-Tr(Dl-Do)logD = ‘($1,
w)-S($o,
0)+S($07
$1)
< O.
This is the inequality we had to obtain. We note that in the differentiation of the functionf(1) the well-known formula
a
+
-Tr F ( A tB)I,=, = Tr (F‘ ( A )B) at can be used. LEMMA 3.4. Let (mi:i e l } be a finite set of states of B ( Y ) such that the Hilbert space X isfinite dimensional. Then the exact divergence center is unique and it is in the convex hull on the states m i .
Proof. Let K be the (closed) convex hull of the states wl,w 2 ,. . . w, and let w be an arbitrary state such that S(oiy a)< CQ. There is a unique state w E X such that S (a’, w ) is minimal (where w’ runs over K ) , see Theorem 5.25 in [lo]. Then
+
S(koi+(l-L)o’, w) 2 S(w’, w )
for every 0 < A
< 1 and 1 < i < n .
It follows from the previous lemma that S ( O i , 0)3 S ( W i , 0’).
Hence the divergence center of 0:s must be in K. The uniqueness of the exact divergence center follows from the fact that the relative entropy functional is strictly convex in the second variable.
THEOREM 3.5. Let A* : C (2) 4 Z , ( X )be a channel with finite-dimensiona1 X . Then the capacity C,,(A*) is the divergence radius of the range of A*. P r o o f . Let ((pi), (qi))be a pseudo-quantum code. Then I((&), (qi),A*) is at most the divergence radius of {A*cpi>(according to Lemma 3.2), which is obviously majorized by the divergence radius of the range of A*. Therefore, the capacity does not exceed the divergence radius of the range. To prove the converse inequality we assume that the exact divergence radius of A* (C (2)) is larger than t E R. Then we can find cpl c p 2 , ..., q,,E C (2) such that the exact divergence radius R of A* (ql), ..., A* (q,)is larger than t. Lemma 3.4 states that the divergence center w of A*(cpl),..., A*(q,,) lies in their convex hull K. By possible reordering of the states ‘piwe can achieve y
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that
; Let K‘ be the convex hull of A * ( q l ) , . .., A*(qn).We claim that ~ E K ‘we choose o’EK’ such that S ( d , w) is minimal (w’ is running over K‘). Then S(A*cpi, EO’+ (1-E)W) < R
for every 1 < i < k and 0 < E < 1, due to Lemma 3.3. However, S(A*cpi, EO’+ (I-E)o)< R
for k < i d n and for a small E by a continuity argument. In this way, we conclude that there exists a probability distribution (pl, p z , ... , Pk) such that k
C piA*’pi = O ,
S ( A *q i, 0) = R .
i=l
Consider now the pseudo-quantum code ((pi), (pi)) such that k
k
C p i S ( A * V i , A * ( 11
i= 1
k pjqj))
=
j=
1 piS(A*ipi, w )= R. i= 1
So we have found a pseudo-quantum code which has quantum mutual information larger than t. The channel capacity must exceed the entropy radius of the range. a Up to now our discussion has concerned the capacities of coding and transmission, which are bounds for the performance of quantum coding and quantum transmission. After a measurement is performed, the quantum channel becomes classical and Shannon’s theory applied. The total capacity (or classical capacity) of a quantum channel A* is (3.6)
Cct (A*) = SUP (1 ((Pi),
(qi),
Y*oA*)),
where the supremum is taken over both all pseudo-quantum codes (pi),(pi)and all measurements y*. Due to the monotonicity of the mutual information we have (3.7)
EXAMPLE 3.6. Consider the Stokes parametrization of 2 x 2 density matrices: D , = $(I+xlol + x z o z + x 3 o 3 ) , where cl,02,o3are the Pauli matrices and (xl,x2, x3)E R3 with x i +x; f xi:d 1. For a positive semidefinite (3 x 3)-matrix A the application r*:D, I-+ DAx gives a channeling transformation when (IAII d 1. This channel was introduced in [ S ]
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under the name of symmetric binary quantum channel. We want to compute the capacities of r*.Since a unitary conjugation does not obviously change capacity, we may assume that A is diagonal with eigenvalues 1 2 A1 3 I 2 > ,I3 2 0. The range of r*is visualized as an ellipsoid with (Euclidean) diameter 21,. It is not difficult to see that the trace state z is the exact divergence center of the segment connected the states ( 1 ~ 1 , ~ , ) / 2and , hence z must be the divergence center of the whole range. The divergence radius is
s(l(l
O)+-( I 1 2 0 0 2 0
0 -1
= 10g2-s(5( 1
),.)
1+1
O 1-1
))
= log2-q((l+A.)/2)-q((1-1)/2).
This gives the capacity Cpq(r*)according to Theorem 3.5. Inequality (3.7) states that the capacity C,(r*) cannot exceed this value. On the other hand, I ( z , I-*) = log2-q ((1+1)/2)-q ((1- 1)/2),
and we have Cp,(r*) = C,(r*).
H
Shannon’s communication theory is largely of asymptotic character, the message length N is supposed to be very large. So we consider the N-fold tensor product of the input and output Hilbert spaces X and N
X N
=
N
@ 2,
Y N =
i=l
@ x. i= 1
Note that N
N
B ( X N ) = @ B(c8)t
B ( x N ) = @ B(%)i= 1
i= 1
The (multi-) channeling transformation is a mapping A::
c(XN)
z(&7).
The main example is the memoryless channel, which is the tensor product of the same single site channels: A:
=
r*0 .. . @ r* (N-fold).
The sequences C,, (A:) and C , (A:) of capacities are defined as above for a single channel. For a memoryless channel the sequences C,, (A;) and C, (A;) are superadditive. Indeed, if ((pi), (cpi)) and ((qj),( $ j ) ) are (pseudo-) quantum codes of order N and M , then ((pi, qj), ( ~ p ~ @ $ ~ ) ) is a (pseudo-) quantum code of order N + M and
(3.8)
I((pi,
qj), (CPiO$j), A%+M)= l((Pi), (Vi),A:)+I((qj)y
($j)7
A$)
follows from the additivity of relative entropy under taking tensor product.
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One can check that if the initial codes are (pseudo-) quantum, then the product code is (pseudo-) quantum as well. After taking the supremum, the additivity (3.8) yields the superadditivity of the sequences C,,(A;) and C,(A;). So the following limits exist and they are well known to coincide with the suprema: 1 (3.9) CG = lim-C,,(A$),
N
CT
1
= lim-C,(AA),
N
Cz
1 N
= lim-Ccl(Ag).
(For multiple channels with some memory effect, one may take the limsup in (3.9) to get a good concept of capacity per single use.) We have CZ (4 <) CPmp(4) d
(3.10)
cm , (A$)
for the capacities per single use.
EXAMPLE 3.7. In the case of the memoryless symmetric binary channel we have c;(r*)
=
C,m(i-*) = log2-q((1+1)/2)-q((1-1)/2),
that is the capacity of the single channel coincides with the capacity per single use for the multiple channel. The proof consists in checking that the trace state remains the divergence center of certain states in the range. Since 7 = (cpl + rp2)/2 for certain output states pl, rpz such that S(cp,, z) is the capacity, we have
Due to symmetry, the trace state is the divergence center, the exact divergence radius is n times S ( q i , z) according to the additivity of the relative entropy. This implies that the entropy C,; equals the single site one. The argument for C: is similar to the single site case. The work [ S ] deals with Ccl(A*) in detail, and, among other things, a coding theorem relates C$ to the code rate of a sequence of pseudo-quantum codes and measurements with asymptotically vanishing average error probability. The picture looks rather similar to Shannon’s coding theorem. (Note that in [S] our capacity C,, was called pseudo-capacity because the authors were interested in the classical capacity.) The relations among C,,, C, and CC1form an important problem, worthy of study. For a noiseless channel, CcI= logn was obtained in [SJ,where n is the dimension of the output Hilbert space (actually identical to the input one). Since the trace state is the exact divergence center of all density matrices, we have C,, = logn and also C, = logn. We expect that C,, < Cclfor “truly quantum mechanical channels” but Cz = ,C : = CT must hold for a large class of memoryless channels. In the case of the binary symmetric channel, all the three capacities coincide as computed in Example 3.7 and in [ S ] .
303 Caoacities o f auantum channels
191
4. The attenuation channel. First we discuss the attenuation channel in the context of the Weyl algebra in a representation free way. It will turn out somewhat later that what we are describing is identical to the attenuation channel defined in terms of the bosonic Fock space in [9]. Let o be a non-degenerate symplectic form on a linear space A?.Typically, A? is a complex Hilbert space and o ( f , g ) = -Im (f,g). The Weyl algebra C C R ( 8 ) is generated by unitaries { W ( f ) :fc A?} satisfying the Weyl form of the canonical commutation relation:
(4.1)
~ ( fW ) ( g )= eia(f,g) W(f+g)
(f,g E A?).
Since the linear hull of the unitaries W ( f )is dense in CCR(A?), any state is determined uniquely by its values taken on the Weyl unitaries. The most important state of the Weyl algebra is the Fock state which is given as
cp(W(f))= exP{-llfl12/2) (fEW. The GNS Hilbert space corresponding to the Fock state is called the (bosonic) Fock space r ( Z )and the cyclic vector @ is said to be a vacuum. The states
(44
(4.3)
cpf
(9 = cp (W(f)*. Wf))
are called coherent states and they are induced by the coherent vectors rF
( W ( f ) @) = @f
in the Fock representation xF. We have
(4.4)
(@f>
@g)
=
cp(W(f)*W(9)) = exP{-+11s-fI12) e x p { - i d f , 9))
= exp { -3(11fl12
+ llSllZ)+ >,
and (4.5)
911 = exp{-+11gtIZ+2iIm(f, s>> (f, gEx).
(Pf(W(g))=exP{-~tI91l2--i~(f,
The field operators are obtained as the generators of the unitary groups t-nF(W(tf)) in the Fock representation. In other words, B(f)is an unbounded self-adjoint operator on r ( S )such that
.a
B(f)= -l-xF(W(tf))(t=O dt with an appropriate domain. The creation and annihilation operators are defined as a*(f) = f ( B ( i f ) - i W ) ) ,
a(f) = + ( W f ) + i B ( f ) ) .
The positive self-adjoint operator N (f) = a* (f) a (f)has spectrum Z f and it is called the particle number operator (for the “$mode”).
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Let T be a symplectic transformation of A? to 3? @ X , i.e., a(f, g) Tg). Then there is a homomorphism
= c(Tf,
aT: CCR (A?)+ CCR (20Z) such that
(4.6)
ET(W(f))=
W(Tf).
We may regard the Weyl algebra CCR ( 20X ) as CCR ( X )0 CCR ( X )and, given a state I) on CCR(Z),a channeling transformation arises as (4.7)
( A * o ) ( A )= (00 $ ) ( a m ) ,
where the input state o is an arbitrary state of CCR (A?)and A E CCR (2). (In the language of optical communication, $ is called a noise state.) To see a concrete example discussed in [9], we choose % = X , I) = cp and
(4.8)
S ( 5 ) = a5 @ b5.
If la(2+lb12= 1 holds for the numbers a and b, this S is an isometry and a symplectic transformation, and we arrive at the channeling transformation (4.9)
(A*w) W(g) = o(W(ag))exP{-511bgll2}
(9EX).
In order to have an alternative description of A* in terms of density operators acting on r(&)we introduce the linear operator F r ( 2 )+ r ( 2 ) @ r ( 2 ) defined by 1/XF(A)Q) = ZF(aT(A))Q) @ @ -
We have VnF(W(f))@ =(Z~(w(clf))OXF(W(bf)))@o @>
and hence (4.10)
V i f = !DUf @ G b f .
LEMMA4.1. Let o be a state of CCR(X)which has density D in the Fock representation. Then the output state A * o of the attenuation channel has density Trz VDV* in the Fock representation.
P r o o f . Since we work only in the Fock representation, we skip xFin the formulas. First we show that (4.11)
v* (Yf) 0 1)I/=
W(af)exp { -3 IIbf1I2)
for every f E P.(This can be done by computing the quadratic form of both operators on coherent vectors.) Now we proceed as follows:
305 Capacities of quantum channels
193
Tr (TrzVDV*)W (f ) = Tr VDV* ( W ( f )@ I ) = Tr D V* (W(f) 0 I)V = TrDW(af)exp{-tIIbf
\Iz}
= o(w(af))exp{-tIIbf112},
which is nothing else but (A*w)(W(f)) due to (4.9). a The lemma states that A* is really the same (attenuation) channel discussed in [9] or [lo], p. 305. We note that ’4 is a so-called quasi-free completely positive mapping of C C R ( 2 ) given as (4.12)
A(W(f)) = W(af)exp { -t IIbf1I2}
(cf. [4] or Chapter 8 of [ll]).
PROPOSITION 4.2. If $ is a regular state of CCR ( X ) ,that is t t+ $ (W(tf)) is a continuous function on R for every f € 2 ,then (A*)”(JI) + q pointwise. (q denotes the Fock state.) P r o o f . It is enough to look at the formula n- 1
and the statement is concluded.
a
It is worth noting that the singular state
iff # 0, if f = O is an invariant state of CCR (Z)On the other hand, the proposition applies to states with density operator in the Fock representation. Therefore, we have
COROLLARY 4.3. A* regarded as a channel of B ( r ( 3 ) )has a unique invariant state, the Fock state, and correspondingly A is ergodic. A is not only ergodic but it is completely dissipative in the sense that
(4.13)
A @ * A ) = ’4 (A*)A ( A )
may happen only in the trivial case when A is a multiple of the identity. The authors are grateful to M. Fannes and A. Verbeure for this information (private communication). In fact, (4.14)
A = (id 0 o)o as,
where as is given by (4.6) and (4.8), and o ( W ( f ) )= exp {- IIbf112} is a quasi-free state. Here id @ o is just a conditional expectation which leaves invariant a separating product state.
LEMMA4.4. Let A* be the attenuation channel. Then SUPI((PA (qi),A*) = logn
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when the supremum is taken over all pseudo-quantum codes ((pi);=1, ( q f ( i $ = applying n coherent states.
I)
P r o o f . We know that A * q f = q a f 7so the output {A*V,-(~), . ..) A*qf(.,) consists of n pure states. The corresponding vectors of r ( X )span a Hilbert space of dimension k < n. Since the trace state on that Hilbert space is a divergence center with radius < log k < log n, log n is always a bound for the mutual information according to Lemma 3.2. In order to show that the bound logn is really achieved we choose the vectors f (k) such that (1 d k d n), f ( k ) = Akf where fE Xis a fixed non-zero vector. Then in the limit I q f ( k ) become orthogonal, since
(4.15)
I(@lkf
I @lmf)l
2
= exp {
I l f 112/2)
-I2
-+
--*
00
the states
0
whenever k # m. In the limit A + 00 the trace state (of a subspace) becomes the exact divergence center and we have
This proves the lemma. The next theorem follows directly from the previous lemma. THEOREM 4.5.T h e capacity C,, of the attenuation channel is infinite.
Some remarks are in order. Since the argument of the proof of Lemma 4.4 works for any quasi-free channel, we can conclude C,, = 00 also in that more general case. Another remark concerns the classical capacity C,,. Since the states qfcn,used in the proof of Lemma 4.4 commute in the limit A -+ 00, the total capacity Cclis infinite as well. CC1= 00 follows also from the proof of the next theorem.
4.6. T h e capacity C, of the attenuation channel is in$nite. THEOREM Proof. We follow the strategy of the proof of the previous theorem, but we use the number states in place of the coherent ones. The attenuation channel sends the number state In) (nl into the binomial mixture of the number states 10) (01 = 11) ( 1 1 7 In> (nlHence the commuting family of convex combination of number states is invariant under the attenuation channel, and the channel restricted to those states is classical with obviously infinite capacity. Since C , (as well as CCJ cannot have a smaller value, the claim follows. a @7
. - - ?
Let us make some comments on the previous results. The theorems mean that arbitrarily large amount of information can go through the attenuation
307 Capacities of quantum channels
195
channel, however the theorems do not say anything about the price for it. The expectation value of the number of particles needed in the pseudo-quantum code of Lemma 4.4 tends to infinity. Indeed, 1
i
n
which increases rapidly with n (here N denotes the number operator). Hence the good question is to ask for the capacity of the attenuation channel when some energy constraint is posed: (4.16)
c(E0) = SUP {I(bi), (qi), A*): C ~ i q i ( N ) E0)i
(To be more precise, we have posed a bound on the average energy, different constraints are also possible, cf. [Z].) Since A ( N ) = a2N for the number operator N, we have
The solution of this problem is the same as that of and the well-known maximizer of this problem is a so-called Gibbs state. Therefore, we have (4.18)
C(Eo) < a2Eo+log(a2Eo+1).
This value can be realized as a classical capacity if the number states can be output states of the attenuation channel.
REFERENCES [l] H. Araki, Relative entropy for states of von Neumann algebras, Publ. Res. Inst. Math. Sci, Kyoto Univ., 11 (1976), pp. 809-833. [Z] C. M. Caves and P. D. Drummond, Quantum limits on bosonic communication rates, Rev. Modem Phys. 66 (1994), pp. 481-537. [3] I. Csiszir, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), pp. 146158. [4] B. Demoen, P. Vanheuverzwijn and A. Verbeure, Completely positive maps on the CCR-algebra, Lett. Math. Phys. 2 (1977), pp. 161-166. [5] A. Fujiwara and H. Nagaoka, Capacity of memoryless quantum communication channels, Math. Eng. Tech. Rep. 94-22, University of Tokyo, 1994. [6] A. S. Kholevo, Some estimates for the amount of information transmittable by a quantum communication channel, Problemy Peredachi Informatsii 9 (1973), pp. 3-1 1. - Capacity of a quantum communication channel, Problems Inform. Transmission 15 (1979), pp. 247-253.
[n
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[S] G. Lindblad, Completely positive maps and entropy inequazities, Comm. Math. Phys. 40
(1975), pp. 147-151. [9] M. Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Inform. Theory 29, pp. 770-777, [lo] - and D. P e t & Quantum Entropy and Its Use, Springer, 1993. [l 11 D. P e t z, The Algebra of the Canonical Commutation Relation, Leuven University Press, 1990. [12] - Discrimination between states of a quantum system by observations, J. Funct. Anal. 120 (1994), pp. 82-97. [13] B. Schumacher, Quantum coding, Phys. Rev. A (1995), pp. 273S2747. [14] H. Umegaki, Conditional expectations in an operator algebra I Y (entropy and information), Kodai Math. Sem. Rep. 14 (1.962), pp. 59-85. cl5l-H. P. Yuen and M.Ozawa, Ultimate information carrying limit of quantum systems, Phys. Rev. Lett. 70 (1993), pp. 363-366. Department of Information Sciences Science University of Tokyo Noda City, Chiba 278, Japan Received on 9.5.1996
309 Open Sys. & Information Dyn. 4: 141-157, 1997 @ 1997 Iiluwer Academic Publishers
141
Complexity, Fractal Dimension for Quantum States Masanori Ohya Department of Information Sciences Science University of Tokyo Noda City, Chiba 278, Japan (Received November 28, 1996)
Abstract. The complexities in information dynamics are reviewed and their examples are given. The fractal dimensions of a quantum state are discussed from a general point of view of complexity. It is shown trough a model that the fractal dimensions of a state provide measures for order structure of chaotic systems.
Introduction There exists several approaches in the study of chaotic behavior of systems using of concepts such as entropy, complexity, chaos, fractality, stochasticity. In 1991, the author proposed Information Dynamics (ID) t o find a common frame t o treat such chaotic behavior of systems altogether. That is, ID is an attempt t o synthesize the dynamics of state changes and the complexity of systems [32]. Since then, the author and his coworkers attempted to refine this concept and apply it t o several topics. In particular, the fractal dimensions of states are defined [33] not only for geometrical sets but also for general states, so that we can examine whether a complicated (chaotic) object obeys a certain rule (i.g., the fractal structure like self-similarity) or not by means of the fractal dimensions of states. This means t h a t this and other complexities might provide measures for some order structures of chaotic systems. In Section 1, we briefly review ID and an axiomatic approach to the complexity. In Section 2 , various examples of the state changes (channels) are presented, some of which are new expressions of physical processes. In Section 3, some examples of the complexities are discussed. In Section 4, fractal dimensions of states are discussed on the basis of two complexities. In Section 5 , the use of the fractal dimensions is discussed to characterize chaotic aspects of physical phenomena. 1. I n f o r m a t i o n Dynamics
Information dynamics (ID) is a synthesis of the dynamics of the state change and the complexity of states [15, 32, 361. It is an attempt t o provide a new view for the study of chaotic behavior of systems. We briefly review what ID is.
310
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Masanori Ohya
-_
Let (A,6,a ( G ) )be an input (or initial) system and ( A ,6,E(G))be an output (or final) system. Here A is the set of some objects t o be observed and 6 is the set of some means t o get the observed value, a(G) describes certain evolution of the system. We often have A = 2,6 = GI a = Z. Therefore, we claim [Giving a mathematical structure t o input and output triples E Having a theory]. T h e dynamics of the state change is described by a channel, which will be explained in the next section, A*: 6 -+ (sometimes 6 3 6). T h e fundamental point of ID is t h a t ID contains two complexities in itself. Let (At,Bt,at(Gt))be t h e total system of (A,6,a) and (x,Z,Z),and S be a subset of 6 in which we are measuring observables (e.g., S is the set of all KMS or stationary states in a C*-system). Two complexities are denoted by C and T . C is the complexity of a state y measured from the reference system S, in which we actually observe the objects in A, and T is the transmitted complexity associated with the state change y -+ A*y, both of which should satisfy the following properties:
Axiom of complexity (i) For any p E S c 6 ,
(ii) For any orthogonal bijection j: e x 6 -+ e x 6 , where e x 6 is the set of all extremal points of 6, ci(s)(j(P)) = CS(d,
Ti(‘) ( j (p) ; A*)
=
TS(9;A*) .
(iii) For @ f p @ i, E St C 6t,
+
CSt(@)= CS(9) C“(qj). (iv)
0
5 T ” ( y ;A*) 5 CS(p).
(v) TS(p;id) = CS(p),where “id” is the identity map from 6 to 6. If instead of (iii), the following is satisfied (iii’) @ E St c B t , put p E @ I A, II, 3 @ I 2 (i.e., the restriction of @ t o CSt(@)5 CS(p)+C“(qj), C and T is called a pair of strong complexity. Therefore, ID is defined as follows:
x),
DEFINITION 1.1. Information Dynamics is described by
( A ,6 , a ( G ) ; ~ , 8 , ~ ( ~ ) ; h * ; C S ( p ) , T S ( p ; A * ) ) and some relations R among them.
31 1 Complexity, Fractal Dimension for Quantum States
143
Therefore, in the framework of ID, we have t o (i) mathematically determine
d,6,a(G);Z,B1Z(C), (ii) choose A* and R , and (iii) define Cs(p), T S ( y ; A * ) . Information Dynamics can be applied t o the study of chaos in the following meaning (a) 11, is more chaotic than p seen from the reference system
S if CS($) 2 Cs(p).
(b) When p changes to A*v, the degree of chaos associated t o this state change is given by D s ( y ; A * ) = Cs(A*v) - T S ( v ; A * ) .
This degree of chaos plays simi1a.r r d e as several other expressions of chaos in classical dynamical systems (CDS) such as Lyapunov’s number or topological entropy. In ID several different topics can be trea.ted on common grounds so t h a t we can find a new clue bridging several different fields [3, 16, 21, 22, 32, 33, 38, 391.
2. State Changes
ID contains the dynamics of the sta.te change as its part. A state change is mathematically described by a channeling transformation (it is called “channel” [26]) or a bit restricted notion of “lifting” [2]. I n this section, we discuss the notions of channel and lifting, and we show that several dynamics encountered in physics can be expressed by these notions. Before defining the above notions, we set the notation used throughout this paper. Let (0,F)be a measurable space and P ( R ) be the set of all probability measures, A4(R) be the set of all random variables on (R, F). The usual quantum system is described on a Hilbert space denoted by 3t and B ( X ) is the set of all bounded linear operators on X ,and 6(3t) is the set of all density operators (normal states) on B(3t).A general quantum system is described in a C*-algebraic or von Neumann algebraic framework by a C*-algebra or von Neumann algebra A, and the set 6(d)of all states on A. The descriptions of classical dynamical systems (CDS), quantum dynamical systems (QDS) and general quantum dynamical systems (GQDS) are given in Table 2.1.
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real r.v. in
observable
Hermitian operator A on 31 (self adjoint operator in B(')t))
M(R)
self-adjoint A in C*-algebra A p.l.fna1 'p E B with 'p(Z) = 1
expectation
I
s, f dw
I
I
trpA
4'4)
Tab. 2.1 Descriptions of CDS, QDS and GQDS
-_
The input and output triple ( A ,6,a ( G ) )and ( A ,G , E ( C ) )are the above sets, t h a t is, A is M ( 0 ) or B(31)or A (C*-algebra), and 6 corresponds t o the state space in each case, and a ( G )is a n inner evolution of A with a parameter group G (or semigroup) and so is the output system. A channel is a mapping from G ( d ) to e(2).Almost all physical transformations are described by this mapping. We first give the mathematical definition of various types of channels. DEFINITION 2.2. Let ( A ,6 ( A ) , a )be an input system and ( X , G ( X ) , abe ) an output system. Take any cp, 4 E G ( A ) . For A*: 6 ( A )+ B ( 2 ) we have: (1) A' is linear if A*(Acp+ (1 -A)+)
= XA*cp+ (1 - X)A*+ holds for any X E [0,1].
(2) A* is completely positive (C.P.) if A* is linear and its dual map A : X satisfies
-+A
n
ij=l
for any n E N and any
Zi E 2,A; E A.
(3) A* is of Schwarz type if A ( T ) = A@)* and
A(Z)*A(z)5 A ( Z 2 ) .
(4) A* is stationary if A o at = Zt o A for any t E Iw.
(5) A' is ergodic if A' is stationary and A*(exI(a)) c e x I ( a ) , where I ( @ )(resp. I @ ) ) is the set of all a (resp. Z) invariant states in 6 (resp. E ) . (6) A* is orthogonal if for any two orthogonal states cp11cp2) one has A * c p l l A * ~ 2 .
cp1, cp2
E 6 ( A ) (denoted by
313 Complexity, Fractal Dimension for Q u a n t u m States
145
(7) A* is deterministic if A* is orthogonal and bijective. (8) For a subset S of
6(d),A* is chaotic for S if A*cpl = A*p2 for any c p l , c p 2 E S.
(9) A* is chaotic if A* is chaotic for 6(d). When we take 2 = A g B , B is another algebra, the channel is called a “lifting”. This special channel is useful t o arrange several processes.
DEFINITION 2.3. (1) A continuous map &*: 6 ( A )-+ e(d@Ba) is called a lifting. (2) A lifting L* is nondemolition for a state cp E 6 ( A ) if &*cp ( A 8 I ) = cp ( A ) for any A E A. Lifting is not necessary linear. An important example of nonlinear lifting is a compound state, which will be discussed in Section 3. A linear lifting is the dual map of a transition expectation of Accardi [l]from A @ B t o A. We show several examples of channels and liftings which appear in physics and qua,ntum communication [15,30]. 2.1. UNITARYEVOLUTION For any density operator p E
6(X)
2.2. SEMICROUP EVOLUTION p
-+ A;p
= V,pVt+,t E R+,where V, ( t E R+) is a one-parameter semigroup
on 31. 2.3. Q U A N T U M MEASUREMENT When we measure an observable A = CnanPn (spectral decomposition with C , Pn = I ) in a state p, then p changes to a state A*p by t h i s measurement, such as p -+ h * p = c P n p P n . n
2.4. REDUCTION (OPEN SYSTEMDYNAMICS) If a system C1 intemcts with an external system E2 described by another Hilbert space K and the initial states of C1 and E2 are p and 0,respectively, then the
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Masanori Ohya
combined state Bt of C1 and C2 a t time t after the interaction between two systems is given by &;p = ut(p8 q; ,
et
where Ut = exp(itH) with the total Hamiltonian H of C1 and C2. A channel is obtained by taking the partial trace w.r.t. Ic, such as
If C1 is an observed system and C2 is a measuring apparatus, then &; exhibits the interaction between C1 and C2. Namely, when an initial state of C1 is p then A;p is the final state after the interaction between C1 and C2 and R'ip E trx&;(p) is the final state of Cz. Therefore, a measuring process can be described by a lifting.
2.5.
O P T I C A L COMMUNICATION PROCESSES
Quantum communication process is described by the following scheme.
4 Loss
G(3-1)
1;
G(3-1)
Y* -1 1' a* G ( X @ K )- ; t G ( R @ h ' ) The above maps y*,a* are given as
where D is a noise coming from the outside of the system. T h e m a p x* is a certain channel determined by physical properties of the combined system. Hence the lifting and the channel for the above process are given as
=
f*p x* (p €3 0) , h * p s (a* o x * o y*)(p) = trnx* ( p @ 0).
31 5 Complexity, Fractal Dimension for Quantum S t a t e s
147
2.6. BEAM SPLITTING [a] V is an isometry from
31 t o 31 @ 72 and a lifting defined by f*p =
vpv* ,
p E 6 (31) ,
is called an isometric lifting. One of this type is the beam splitting (attenuation) process, where the isometry Vap(o,P E C with laI2 Ipl2 = 1) is defined on 31 (the usual Fock space) as
+
va4 18) =
b e ) 8 IPO)
for a coherent state vector 16) E 31. Then the lifting associated with Vap is GctpP = VactpPV,;,,
P E 6 (31)*
This beam splitting lifting was used to construct a new quantum Markov processes [14]. The attenuation process, a special channel of the type (2.5), is written by = trT&&p.
2.7. AMPLIFIER PROCESS In quantum optics, a linear amplifier is usually expressed by means of annihilation operators a and 6 on 31 and K ,respectively: c = Ga@Z+dmI86t,
where G ( 2 1) is a constant and c satisfies CCR (i.e., [c,ct] = I) on 31 @I K. This expression is not convenient to compute several measures of information like entropy. The lifting expression of the amplifier is as follows [35]: Let c = pa @ Z v Z @ 6 t with (pI2- (vI2= 1 and 17) be the eigenvector of c: clr) = 717). For two coherent vectors 18) on 31 and 18‘) on K , 17) can be written by the squeezing expression: 17) = l8@8’; p , v ) and the lifting is defined by an isometry Vet p) = 18 @ 8’; p , v)
+
such t h a t &&p = v @ l p v ; , The channel of the amplifier is
p E 6 (31)
.
A:tp = trK&i,P.
3. Examples of Complexities
In this section, we give several examples of complexities C and T related mainly t o information.
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Masanori Ohya
1. The first examples of C and T are the entropy S and the mutual entropy I, respectively. Both classical and quantum S and I satisfied the conditions of the complexities. Here we only discuss the quantum case. For a density operator p in a Hilbert space and a channel A*, the entropy S ( p ) and the mutual entropy I ( p ; A * )are defined in [28] as S(P) = - t r p l o g p ,
where the supremum is taken over all Schatten decompositions { E k } of p ; p = C kX k E k . From fundamental properties [34] of the entropy S ( p ) , it satisfies (i) S ( p ) 2 0, (ii) S ( j ( p ) ) = S ( p ) for an orthogonal bijection j , t h a t is, it is a map from a set of orthogonal pure states to another set of orthogonal pure states, (iii) S ( p l @p 2 ) = S(p1) S(p2 ), so that S ( p ) is a complexity C of ID.
+
T h e mutual entropy I ( p ; A * ) satisfies the conditions (i), (ii), (iv) from the fundamental inequality of mutual entropy [as]:
0
5 I(p;A*) 5
min{S(p),S(A*p)}.
Further, for the identity channel A* = id,
k
= sup
{
c
X,,trEl,(log
Ek -
logp);
Ek}}
k
= -trplogp
because of S ( E k ) = 0, hence it satisfies the condition (v). Thus S and I become a pair of the complexity. Moreover, S satisfies the condition of the strong complexity (subadditivity).
2. Fuzzy entropy has been defined by several authors like Zadeh [43], DeLuca and Termini Ill] and Ebanks [13]. Here we take Ebanks's fuzzy entropy and we show t h a t we can use it to construct the complexity C. Let X (this is A of ID) be a countable set (21,. . . ,xn} and f~ be a membership function from X to [0,1] associated with a subset A c X. If f~ = l ~then , A is a usual set, which is called a sharp set, and if f~ # l ~then , A is called a fuzzy set. Therefore, the correspondence between a fuzzy set and a membership function is one t o one. Take a membership function f and let u s denote fi = f (xi)for each xi E X. Then Ebanks's fuzzy entropy S (f) for a membership function f is defined by n
~ ( f =) -"Jlogf;, ,i=l
(v = 1og2e) .
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Complexity, Fractal Dimelision for Quantum States
4
When f is sharp, t h a t is, fi = 0 or 1 for any xi E X , S (f)= 0. When f; = for any i, S (f)attains t h e maximum value. Moreover, any two membership functions (or equivalently fuzzy sets) f and f’ have t h e following order < :
Ifl If[
f
< f’@
If;
-
f (x) L f’(x) f (x)5
1
-
(when f’(x) L 7 (x) (when f’(x) 5
fl,
which implies
S(f) 5 This fuzzy entropy S
+) +) .
W’) .
(f)defines t h e complexity C (f)
T h e positivity of C ( f ) is proved as follows: from Klein’s inequality, log $ 1 - x for any x > 0, we have
n
2
n
T h e invariance under a permutation K of indices i of directly from t h e invariance of S under K .
2;
(i.e, i + 7r (i)), comes
This C (f)satisfies not only t h e additivity but also t h e subadditivity. Let Y be another set { y / ~ ,. . , ym} and 9 be a membership function from Y to [0, 13. Moreover, let h be a membership function on X x Y to [O, 13 satisfying m
Ch
n (xi1
~ j )=
f (xi)
1
C h (xi1 yj)
= Y (yj) .
i=l
j=1
W h a t we have t o show is the inequality
C ( h ) 5 C ( f )+ C ( g ) . Without loss of generality, we assume n
~ ( t =) - t U l o g t , 7 ( t ) is monotone increasing in 0
2 m 2 2. P u t (v = l o g 2 e ) .
5 t 5 f, so t h a t we
have
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because of h;j for any i , j .
G
h (z;,
yj)
5 f; G f (xi),and
Thus we have
hence 0
5 h;j/nm 5 fi/nm 5
m
mT(”) nm
2
~ q (n mk ) . j=1
Now
which is positive since n
2 m 2 2.
Hence
which implies 1
C(f) 2 s C ( h )* Similarly we can prove C(g)
1
L ;zC ( h )
’
Therefore, we have the subadditivity
3. Kolmogorov [17] and Chaitin [S] discussed the complexity of sequences. Consider e.g. the following two sequences a and 6 composed of 0-s and 1-s: a : 010101010101, 6 : 011010000110. In both u and 6, the occurrence probabilities p ( 0 ) and p(1) are the same, p ( 0 ) = p(1) = 1/2. However, the sequence 6 seems to be more complicated than u. It suffices to know the first two letters t o guess the whole a, but one may need the whole sequence of letters to know b. In general, we consider a computer (an automaton) transforming binary input sequences into output ones. Formalizing the description of such a machine it is possible to introduce the notion of the minimum programme as the “simplest” algorithm which produces a given output sequence. The amount of information contained in
319
Complexity, Fractal Dimension for Quantum States
151
t h e specification of t h e minimum programme (measured e.g. in “bits”) is called t h e complexity of a given (output) sequence. Let A be t h e set of all finite sequences over a n alphabet, say { O , l } , and d be another set. Further, let f be a partial function from A to A (i.e., f is not necessarily defined on t h e whole A ) . T h e triple ( A ,d, f ) can be regarded as a language describing certain objects. For an element a E A, t h e length of a is denoted by l ( a ) . T h e minimum length of a E A describing ii E d ( t h a t is f ( a ) = ii) is called t h e complexity of description. If there is no a E A such t h a t f ( a ) = ii, then we set t h e complexity of u t o co. When both A and d are sets of binary sequences, we consider partial computable functions f : A -+ A (i.e., such that there exists a programme which, given an input a E A , terminates (halts) with t h e o u t p u t f ( a ) E d whenever a E Domain ( f ) ,but it need not halt at all for inputs a $! Domain (f)).T h e complexity H,(U) determined by ( A ,A, f ) is defined as min {!(a); a E A, f(a) = U} (when 3a E A s.t. f ( a ) = ii) oc) (otherwise) . For a E A let 0’1a denote the sequence obtained from a by appending k symbols 0 a n d a single 1 on t h e left, i.e. 0 . . .Ola. Then it is shown [8] t h a t there exist k E N and a computable partial function fu such t h a t for any f , f~(O‘1a) = f ( a ) . This f ~ isi called t h e universal partial function. W i t h this function certain universal computer U is associated. In particular, U is capable of computing f ( a ) for a E Domain ( f ) .Some important consequences of t h e above are: (1) there exists a constant E such t h a t H j u ( i i ) 5 H j ( i i ) + E for any f , and ( 2 ) there exists a constant E’ for two universal partial functions fv,fufsatisfying I H f , ( U ) - H f u f( a ) / 5 E’. T h e above facts imply t h a t H j L rgives t h e minimum value for H f if we neglect t h e constant E . Kolmogorov and Chaitin introduced t h e following complexity
H(ii) = H f u ( a ) which does not depend on a choice of fu because of t h e statement (2). Moreover, Chaitin introduced t h e mutual entropy type complexity in t h e s a m e framework as above. This complexity and mutual entropy type complexity can be associated with our complexities C and T , respectively. 4. Generalizing t h e entropy S a.nd the mutual entropy I , we can construct complexities of entropy type: Let (d,G(A),ol(G)), ( ~ , ~ ( ~ ) , be ZC (’ ~systems ) ) as before. Let S be a weak *-compact convex subset of 6 ( A ) and M,(S) be t h e set of all maximal measures p on S with t h e fixed barycenter p [9]
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Masanori Ohya
Moreover, let F,(S) be the set of all measures of finite support with the fixed barycenter p. The following three pairs C and T satisfy all conditions of the complexities: TS(p;A*)
= sup {
S(A*w,h * p ) d p ; p E M,(S)}
(L
C$(p) E Ts(y;id) I S (v;A*) E SUP S w €3 A*wdp, y @ A*.> ; p E M,(S)}
{
C,s(p) E IS(p;id) JS(p;A*> E s u p { ~ . S ( A * w , A * p ) d p ;p E F,(S)}
Cf(p)
E Js(p;i d ) .
Here, the state Jsw @ A * w d p is the compound state exhibiting the correlation between the initial state and the final state A*p. This compound state was introduced in [27] as a quantum generalization of the joint probability measure in CDS because it does not exist in QDS [42]. Moreover, it is a nondemolition nonlinear lifting from G(A) t o G ( d @ 2). These complexities with the mixing S-entropy SS(p)and the CNT (ConnesNarnhofer-Thirring) entropy H,(d) satisfy the relations of Theorem 3.1 [24, 361. Before stating a theorem, we review the definition of the S-entropy and the CNT entropy: For a state p E S C G ( d ) , put
where 6(p) is the delta measure concentrated on {p}, and put
for a measure p E D,(S). Then the S-entropy of a state p E S is defined in [29,30] as
Connes, Narnhofer and Thirring introduced the entropy of a subalgebra !Bl of (EN) is defined as follows:
A [lo]. The CNT-entropy H ,
321 Complexity, Fractal Dimension for Quantum States
153
For a s t a t e y and a subalgebra M
( y j lm, y 1 m ) ; y = X p j y j ( f i n i t e decomposition o f y ) j
where S(., . ) is t h e relative entropy for C*-algebra according to t h e definition of Araki [4] or Uhlmann [41] and y l m is t h e restriction of y to M.
THEOREM 3.1. (I) 0 <: I S ( y ; A * )5 T S ( y ; A * 5 ) JS(y;A*). (2) CF(y) = C,G(y)= CF(p) = S " ( q ) = H v ( A ) . (3) W h e n A = 3 = B(?l), for a n y density operator p 0
5
I s ( p ; A*) = T S ( p ;A*)
5 J s ( p ; A*) .
These complexities provide u s with a new definition of t h e quantum dynamical entropy [Xi],which can be used t o characterize quantum communication processes [37]. I t is possible t o construct other complexities, not of t h e entropy type, in several fields like genetics [38], economics and computer sciences [39].
4. Fractal Dimension of States
Usual fractal theory t r e a t s mostly geometrical sets. I t is desirable to extend t h e fractal theory so as to be applicable to some other objects. For this purpose, we introduce t h e fractal dimension of general states. First, we recall two fractal dimensions of geometrical sets [20].
4.1. SCALING DIMENSION We observe a complex set F built from a fundamental pattern. If t h e number of t h e patterns observed is N ( 1 ) when t h e sca,le is very rough, sa.y 1, and t h e number is N ( r ) when t h e scale is T , then we call the dimension defined through
the scaling dimension of the set F
322
154 4.2.
Masanori Oliya
CAPACITY DIMENSION
Let us cover a set F in t h e n-dimensional Euclidean space En by copies of a certain convex set with t h e diameter E . If t h e smallest number of t h e convex sets needed to cover t h e set F is N ( E ) , then we call t h e dimension given by d , ( F ) = lim 1% N E+O log (1/&)
t h e capacity dimension (or t h e &-entropy dimension) of t h e set F . These two fractal dimensions become equal for almost all sets for which they can be computed. T h e &-entropy was extensively studied by Kolmogorov [18]and his &-entropy was defined for a probability measure, which gives u s an idea t o define t h e &-entropy for a general state. Kolmogorov introduced t h e notion of &-entropy in probability space (0,F ,p). His formulation is as follows: for two random variables f,g E M (Q), t h e mutual entropy I (f,g) is defined by t h e joint probability measure p ~ / and , ~ t h e direct product measure p~ @ pg such that
1(flu) =
s
(CLf,Sl
Pf @ Pg)
I
where S ( . , .) is t h e relative entropy [19]. T h e &-entropy for a random variable f is given by s I < (f,&) inf (1 (f,9) ; 9 E Md (f1&)} I ~~
JJQ
where Md (fl&)’= (9 E M ( Q ) ; Ilf - 911 5 &) with Ilf-gll If - 91’dPfg. For a general probability measure p on (RIFT), t h e Kolmogorov &-entropy &(p; E ) is given by SI<(p;E)
= i n f ( S ( P c o , P @ p ) ; p E PO(Q)}I
where pco is t h e joint (compound) probability measure of p and ii and Po(Q) is t h e set of all probability measures p satisfying IIp - fill 5 E . We introduced the &-entropy of a general quantum s t a t e y and t h e fractal dimensions of states in [31, 33, 361. Let C be a set of some physically interesting channels a n d define two sets:
c1(A*; y ) = {r*E c; r*y= A*Y> , c2( w ) = {r*E c; I I -~r*vii L. E ) . Then t h e &-entropy of a s t a t e of y w.r.t. S is defined by means of t h e transmitted complexity T S ( y ;A*) as
s:,~(9; &)
{
&I} ,
= inf J: (9; A*) ; A* E C’ (9;
323 Complexity, Fractal Dimension for Quantum States
where
J; (9; A*)
= s u p { T (9; ~ r*);
155
r* E c1(A*; 9)}.
When S = 6 a n d C is t h e set of all channels on 6, t h e above &-entropy is denoted by SO ( 9 ;for ~ )simplicity. T h e capacity dimension of a s t a t e 9 w.r.t. S and C is defined by
where
T h e above d: (9; E ) is called t h e capacity dimension of &-order. T h e information dimension of a state 9 for S of E order is defined by
T h e above new €-entropy and fractal dimensions of states provide measures to classify chaotic phenomena in both classical and quantum systems. We will discuss some applications of such uses i n the next section.
5. Some Applications of Fractal Dimensions of States These &-entropy and fractal dimensions are applied to several physical phenomena and mathematical objects. For instance, we can classify the shapes of river basins or seas on t h e moon [22], as well as consider symmetry breaking in Ising systems [21]. We s t a t e some results concerning the classification of Gaussian measures [15,16]. For a random variable f = ( f i , . . . , fn) from R t o Rn and t h e measure p ~ f associated with f,t h e random variable norm 1 1 . I[R,v of p ~isf defined by
THEOREM 5.2. If the distance of two states is defined through the above random variable norm o n R = Rn and the transmitted complexity T is the mutual entropy in CDS, then (1)
324 156
Masanori Ohya
where A l l . . . , An are the eigenvalues of the covariance operator R for pf and O2 is the constant uniquely determined b y the equation n
i=l
(2) do (Pf)= dK (Pf) = 7LAccording to this theorem, t h e Kolmogorov &-entropy coincides with our &-entropy. T h e difference between SO and Sr< comes from t h e norm taken for measures. For instance, when we take t h e norm of measures by t h e total variation, namely, Ilvl(~v= Ivl(s2) for any finite measure v. We show the difference for t h e case R = R. If a n input s t a t e p is described by t h e mean 0 and t h e covariance u2 and a noise of a channel is exhibited by one-dimensional Gaussian measure PO = [0, ug] E P ( R ) , then t h e o u t p u t s t a t e A*p is given by [O, d2a2 a;] with some constant 6. Since t h e channel A* depends on d and we denote A* =
0-02,
T H E O R E M 5.3. Let
+
be a channel sutisfying 111-1 - A i ( h , b ; ) p I I 1 ~ ~6 for
any 6 E M ( E ) .If a Gaussian ch.anne1 Ai(6,ug)satisfies the condition then we have
where
O(E)
is the order of& : lim
O(E)
= 0 and M ( S ) = ( 6 E R;O
E+O
(w2
5
Ca - 6 u2
’
5 6 5 E}.
This theorem tells us t h e difference between the Kolmogorov e-entropy and our €-entropy. It concludes t h a t our fractal dimension shows the fractal structure of Gaussian measures. Moreover, our fra.cta1 dimensions enable t o characterize chaos in t h e following sense: when a system is random, our fractal dimension is infinite, but when it has some order like self-similar structure, it will yield a finite value.
Bibliography 1. L. Accardi, Noncommutative Markov Chains, International School of Mathematical Physics, Camerino, pp. 268-295, 1974. 2. L. Accardi and M. Ohya, Compound Channels, Transition Expectations and Liftings, to appear in J. Multivariate Analysis. 3. L. Accardi, M. Ohya and N. Watanabe, Rep. Math. Phys. 38, 1 (1996). 4. H. Araki, Publ. RIMS, Kyoto Univ. 11, 809 (1976).
325 Complexity, Fractal Dimension for Q u a n t u m States
5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 34. 25.
157
F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Trieste Notes in Physics, Springer-Verlag, 1993. P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965. 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 11, Springer, New York, Berlin, Heidelberg, 1981. G. J. Chaitin, Algorithmic Information Theory, Cambridge Uni. Press, 1987. G. Choquet, Lecture Analysis I, 11, 111, Bengamin, New York, 1969. A. Connes, H. Narnhofer, and W. Thirring, Commun. Math. Phys. 112, 691 (1987). A . DeLuca and S. Termini, Inform. Control. 20, 301 (1972). G. G. Emch, Z. WahrscheinlichkeitstheorieVerw. Gebiete 29, 241 (1974). B. R. Ebanks, J. Math. Anal. Appl. 94, 24 (1983). K.H. Fichtner, W. Freudenberg, and V. Liebscher, Beam Splitting and T i m e Evolutions of Boson S y s t e m s , preprint. R. S . Ingarden, A. Kossakowski, and M. Ohya, Open Systems and Information Dynamics, to be published in Kluwer. I<. Inoue, T. Matsuoka, and M. Ohya, New Approach to &-Entropy and Its Comparison with l ~ o l m o g o l o v ’ €-entropy, s SUT preprint. A . N . Kolmogorov, Dokl. Akad. Nauk SSSR 119, 861 (1958). A . N. Kolmogorov, Amer. Math. SOC.Translation, Ser. 2 33, 291 (1963). S. Kullback and R. Leibler, Ann. Math. Stat. 22, 79 (1951). B.B. Mandelbrot, T h e Fractal Geometry of Nature, W.H. Freemann and Company, San Francisco, 1982. T . Matsuoka and M. Ohya, Rep. Math. Phys. 3G, 27 (1995). T . Matsuoka and M. Ohya, Fractal Dimension of Stutes and Its Application to Shape Analysis Problem, S UT preprint. N. Muraki, M. Ohya, and D. Petz, Open Sys. Information Dyn. 1, 43 (1992). N. Muraki and M. Ohya, Lett. Math. Phys. 3G, 327 (1996). J. von Neumann, Die Mathenmtischen Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.
Ohya, J . Math. Anal. Appl. 84, 318 (1981). Ohya, L. Nuovo Cimento 38, 403 (1983). Ohya, IEEE Trans. Information Theory 20, 770 (1983). Ohya, J. Math. Anal. Appl. 100, 222 (1984). Ohya, Rep. Math. Phys. 27, 19 (1989). Ohya, Proc. Symp. Appl. Func. Anal. 11, 45 (1989). Ohya, Lecture Notes in Physics 378, Springer, 81 (1991). M. Ohya, Quantum Probability and Related Topics 6 , 359 (1991). M. Ohya and D. Petz, Q u a n t u m Entropy and Its U s e , Springer-Verlag, 1993. M. Ohya and H. Suyari, Rep. Math. Phys. 36, 403 (1995). M. Ohya, Quantum Communications and Measurement 2, Plenum, 309 (1995). N. Ohya and N. Watanabe, Note on Irreversible Dynamics and Quantum Information, to appear in the Alberto Frigerio Conference Proceedings. M. Ohya, Analysis of Geiieonie Sequences b y Complexity, SUT preprint. N. Ohya and N. Watanabe, O n Alatheniatical Treatment of Fredkin-Toffoli-Milburn Gate, S UT preprint. C . E . Shannon, Bell System Tech. J. 27, 379 (1948). A. Uhlniann, Commun. Math. Phys. 54, 21 (1977). I<. Urbanik, Stud. Math. 21, 119 (1961). L. A. Zadeh, J. Math. Anal. Appl. 23, 421 (1968). M. M. M. M. M. M. M.
326 Studia Scientzarum Mathematicarum Hungarica 31 ( 1 996), 423-430
NOTES ON QUANTUM ENTROPY M. OHYA
and D.
PETZ
The present paper consists of two parts. In the first one it will be proved that the von Neumann entropy governs the size of rather sure projections in the course of independent trials. The second part is devoted t o the extension of the von Neumann entropy t o states of arbitrary unital C*-algebras. By a finite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a finite quantum system is a finite dimensional C*-algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is “tracial” in the sense that
Tr ab = Trba
( a , b E d).
Every functional w on d is determined by a density operator D , E A in the form ~ ( a=)Tr D,a (aE A). The entropy S ( w ) of a functional w is defined by means of its density operator as S ( w ) =Trq(D,). This notion was introduced by von Neumann in 1927 and we term it von Neumann’s entropy or shortly entropy (cf. [S]). It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behaviour of certain probabilities in the course of independent trials (see, for example [l]or [ll]).Now we will discuss this phenomenon for finite quantum systems. Let A be a finite quantum system with a faithful state w . The n-fold algebraic tensor product A, = A 8 . . .@IA is again a finite quantum system and the product functional w, = w 8 . .. @ w is a state of A,. Using the -m obvious identifications the inclusion (A,,w,) c ( A m , w m ) holds for n 5 and we set (A,, urn)= u{(A,, w,) : n E N}. 1991 Mathematics Subject Classification. Primary 46L30; Secondary 82B10, 94A17. Key words and phrases. Density matrix, s t a t e space, von Neumann entropy, relative entropy. 0 1 9 9 6 Akade‘miai Kiadd, Budapest
327 424
M . OHYA and D . PETZ
On the *-algebra A, the right shift endomorphism y is defined for a1 @$ a2 @ @ . . . @ u , E A , as ~ ( a l a2 @ 8 . ..@ an) = I @a1 @$ a2 @ . . . @ a , E An+1
and w, is invariant under y. Now perform the GNS-constructions with the state w, and arrive at the triplet ( r , X , R ) . We identify A, through its faithful representation T with a subalgebra of the generated von Neumann algebra M = 7r(A,)” C B ( X ) . The normal state W(.)
= (Q, aR)
(aE M )
is an extension of w, and the endomorphism y extends t o M such that the relation w o y = w is preserved. (For the sake of simpler notation we do not use a new letter for the extension.) T h e following result may be called the weak law of large numbers (for independent finite quantum systems). Since it is well known its proof will be omitted.
PROPOSITION 1. f n the above described situation the following statements hold. (i) ff a E M and y ( a ) = a then a E C I . (ii) For every a E M the sequence s,(a) = 71-l ( a y ( a ) . . . y”-’(a)) converges to w ( a ) I in the strong operator topology. (iii) ff a E M”” and J c R is closed interval such that W ( U ) @ J and p , is the spectral projection of s,(u) corresponding to the interval J then p , -+ 0 in the strong operator topology.
+
+ +
Let us fix a positive number E < 1. For a while we say that a projection Q , E A, is rather sure if w,(Q,) 2 1 - E . On the other hand, the size of Q,, the cardinality of a maximal pairwise orthogonal family of projections contained in Q,, is given by Tr,Q,. (The subscript n in Tr, indicates that the algebraic trace functional on A, is meant here.) The theorem below says t h a t the von Neumann’s entropy of w governs asymptotically the size of rather sure projections: A rather sure projection in A, contains at least exp(nS(w)) pairwise orthogonal minimal projection.
THEOREM 2. Under the above conditions and with the above notation the limit relation 1 lim - inf {log TrnQn: Q , E A, is a projection, w,(Q,) 2 1 - E } = S ( w ) n+cc n holds. P R O O F . If D, denotes the density of w, then one can see easily t h a t
c
n-1
- log D, =
i=O
yz (- log 01)
328 425
NOTES ON QUANTUM E N T R O P Y
where y stands for the right shift. The sequence (rZ(- logD1)) behaves as independent identically distributed random variables with respect t o the More precisely, the previous proposition applies for a = - log D1 state w,. and tells 1 -log D, + S ( w ) l n strongly. Let P ( n ,6) be the spectral projection of the selfadjoint operator -n-l log D, corresponding t o the interval ( S ( w ) - 6, S ( w ) 6). According t o (iii) of Proposition 1 one has
+
P ( n ,6)+ I
(1)
strongly for every 6 > 0. In particular,
w ( P ( n ,6)) = ( P ( n ,6)Q, 0)-+ 1 as n + OCJ and P ( n ,6) is a rather sure projection if n is large enough. It follows from the definition of P ( n ,6) that
+
D,P(n, b)exp ( n S ( w )- n b ) 5 P ( n ,6) 5 D,exp ( n S ( w ) n6) (2) which gives 1 - log Tr,P(n, 6) 2 S ( w ) 6. n Since 6 > 0 was arbitrary we establish
+
(3)
1 limsup -inf{logTr,Q,:Q,}$S(w). 71-m
To prove that S ( w ) is actually the limit we shall argue by contradiction. Assume that there exist a sequence n(1) < n(2)< . . . of integers, a number t > 0 and projections Q ( n ( k ) )E & ( k ) (k = 1 , 2 , . . .) such that (i) ~Co(Q(.(W2 1- E , (ii) logTrn(k)Q(4k)) 5 n(k)(S(w) - t). The bounded sequence ( Q ( n ( k ) ) ) khas a weak limit point in the von Neumann algebra M , say T E M . Instead of selecting a subsequence we suppose that Q ( n ( k ) )+ T weakly. It is straightforward t o show that from (1) the weak limit Q ( n ( w + ( k ) , 6) T follows. Consequently, -+
(4)
lim inf w , ( Q ( n ( k ) ) P ( n ( k ) , 6)) k+w
2 w ( T ) 2 1- E .
Using the first part of (2) we estimate T r Q ( n ( k ) ) Z T r Q ( n ( W ' ( n ( k ) , 6) 1TrD,(k)Q(n(k))P(n(k), 6 ) e x (~n S ( w )=exp ( n S ( 4 - ~ ~ ) ~ , ( Q ( n ( k ) ) P ( n6)()k ) ,
4
329 426
M . OHYA and D. PET2
and
The limit term on the right-hand side vanishes due t o (4)and we arrive at a contradiction with (ii) if 0 < 6 < t. This proves the theorem. Opposite t o the commutative case the state space of a quantum system is not a Choquet simplex in the sense that states admit several extremal decompositions. For example, for A = M2 ( C ) the general form of a density matrix is (5)
D = -1( l f a b + i c 2
b-ic
1-a
where a , b , c are real numbers and u2 + b2 + c2 5 1. Thanks t o the affine correspondence D +) ( a ,6 , c ) we can visualize the state space as a ball (of radius 1) and surface points correspond t o pure states. A;$; be an Let cp be a state of a finite quantum system and p = extremal decomposition (that is, every $; is pure). Approaching from information theory one might think t h a t the entropy of cp is A; log Xi. This, however, would not be satisfactory because the Xi’s are not in general the probabilities of mutually exclusive events. In fact,
xi
S(cp) -
c
A; log A;
a
and the equality holds if and only if the extremal decomposition C A;$; is orthogonal. This was obtained in [4]a long time ago and here it will be deduced by means of the relative entropy. The inequality (6) is interpreted as follows. In the sense of information content, the most economical extremal decomposition is the orthogonal one, which is implemented by the density matrix. The entropy of w with respect t o cp is defined by Tr D, (log D, - log D,) if supp D, 2 supp D, otherwise. Here supp Di denotes the smallest projection p such that $ ( p ) = $(I). In particular, S(w, cp) is always finite if the density of p has strictly positive eigenvalues. (Such a cp is called faithful.) When D, commutes with D, and their eigenvalue lists are ( A l , Aa,. . . , A), and ( ~ 1 , .. . , K,), respectively, then S(.7.) reduces t o the classical expression due t o Kullback and Leibler.
330 427
NOTES ON QUANTUM E N T R O P Y
Although we mostly speak of the relative entropy of states it is convenient t o allow w and cp in the definition of S ( w , cp) t o be arbitrary positive functionals. The relative entropy may be defined for linear functionals of an arbitrary C*-algebra. Now we do not give the details of the rather technical chain of definitions going through von Neumann algebras and normal functionals t o an arbitrary C*-algebra. We just mention a possible extension of (7), the so-called Kosaki's formula ([ 13, [ 5 ] ) .
7
+
~ ( wcp), =supsup { w ( ~logn ) - [ w ( y ( t ) * y ( t ) ) t-'cp(x(t)x(t)*]t-'dt) l/n
where the first sup is taken over all natural numbers n, the second one is over all step functions z : ( l / n ,co)+ N with finite range and y ( t ) zI - x ( t ) . The relative entropy of positive functionals of a C*-algebra shares the following properties (see [ l ] [5] , and [9]). (i) ( w , cp) r-) S(w, cp) is convex and weakly lower semicontinuous. IIcp - w1I2 5 2S(w, cp) if p(I) = w ( I ) = 1. (111) S(w, cpl) 1 S ( w , $92) if cp1 5 9 2 . (iv) For a unital Schwarz map a : A0 I-+A1 the relation S(w o a , cp o a ) 5 - S ( w , cp) holds. 5 (v) For w = wi we have S ( w , q) S(w;,w)= S(wi7 cp). Below the properties (ii) and (v) will not be required.
(!!)
xi
+
zr=l
PROPOSITION 3. Let cp be a state of a finite quantum system A. If cp is a convex combination C j pjcpj of states then
W h e n all the
cpj
's are pure then the equality sign applies.
PROOF. It suffices t o prove the equality because the inequality follows by convexity of the relative entropy. By simple computation we have
j
j
and the first term on the right-hand side vanishes when all the D V J ' s are projections. Now let cp = C Xi+* be an extremal decomposition. Combining Proposition 3 with the monotonicity (iii) of the relative entropy in the first variable we infer (6) as follows.
331 428
M . OHYA and D . PETZ
We note t ha t Proposition 3 and its consequence (6) remain valid if A is a von Neumann algebra which is the direct sum of type I factors and y is an
arbitrary normal state. (In this case the functional Tr in the proof should be understood as the faithful normal semifinite trace which takes the value 1 at each minimal projection.) Proposition 3 allowed the following definition of the entropy of states of arbitrary C*-algebras in [7].
i
i
Here the supremum is taken over all decompositions of p into finite (or equivalently countable) convex combinations of other states. Apparently the background uniform distribution provided by the trace functional in finite quantum systems is not present in this definition. Some properties of S(p) are immediate from those of the relative entropy. The quantity S(p) is nonnegative and vanishes when and only when, y is a pure state. Moreover, the entropy is lower semicontinuous because it is the supremum of lower semicontinuous relative entropy functionals (see (i) above). The invariance of S(p) under automorphisms is obvious as well.
PROPOSITION 4. Let do c A be C*-algebras and assume that there exists a conditional expectation of A onto do preserving a given state y of A. Then (9) = p I do for some states $i of A0 then C Ai$io E was written for the conditional expectation in the statement. The rest follows from S(p I Ao, $i) 5 S(p,$i o E ) .
PROOF. Indeed, if
C Ai$i
o E is a decomposition of p where
The existence of the conditional expectation preserving the given state is an essential hypothesis in Proposition 4. The monotonicity property (9) does not hold in general. The simplest counterexample is due t o the fact that in the quantum case a pure state of the algebra can yield a restriction which is a mixed state on the subalgebra. The next observation is obvious.
PROPOSITION 5 . Let A = A1 $A2 be C*-algebras and p = Ap1@(1- X ) p 2 a state of A ( 0 < X < 1). Then Now let p be a normal state on a von Neumann algebra M . Then a decomposition p = C X i p i is necessarily built from normal states pi if A; # # 0. Hence, if we wish, in t he definition (8) we may restrict ourselves to normal states pi. When p is the support projection of p then S($,p) = = S ( $ I p M p , p I p M p ) whenever $ ( p ) = 1 for the state $. Consequently
(10) S(v)= S ( v l P M P ) . The following result is due ot Hiai ([3]) and its proof uses the structure theory of von Neumann factors.
332 429
NOTES ON QUANTUM ENTROPY
THEOREM 6. Let p be normal state of a von N e u m a n n algebra M and let p be the support projection of p. If p M p is a countable direct s u m of type I factors then S(p) = Tr q(D,) where D, is the density of p with respect t o the canonical semifinite normal trace Tr o n p M p . Otherwise, S(p)= 00. Now we turn t o entropy of states of C*-algebras. Let ?,h be a state of a C*-algebra A. We write $ for the vector state induced by the cyclic vector 9 on the von Neumann algebra r+(A)”when ( H Q ,9,r+) is the GNS-triple corresponding t o ?,h.
L E M M A7. W i t h the above notation we have S(?,h)= S($). PROOF.The key t o the proof is the fact that a finite relative entropy may be computed in the GNS representation of the reference state. This yields Ai?,hi is a convex readily that S(?,h)2 S($). On the other hand, if ?,h = decomposition in the state space of A then one can find (normal) states -of r$(d)” such t h a t ?,hi o ir$ and Since S(?,h,?,hi) = S(?,h,?,hi) the converse inequality S(?,h)5 S($) follows.
Ci
=qi
T = C i&Ti.
qi
T H E O R E8.MLet $ be a state of a C*-algebra A. T h e n
where the infimum is taken over all possible decompositions $ = CiA;$; into pure states.If $ is not a countable convex combination of pure states then S ( $ ) = 00.
PROOF.Lemma 7 allows us t o reduce the theorem to the von Neumann algebra version. We have to consider two cases. If ?,h is not a countable convex combination of pure states, then the von Neumann algebra pr+(A)”pis not a countable direct sum of type I factors. ( p is the support projection of 3.) Theorem 6 tells us that S($) = m in this case. If $ is a countable convex combination of pure states then p r + ( A ) ” p is a countable direct sum of type I factors and we may refer to Theorem 6 again. The expression (11) appeared as the definition of entropy in [6]. It is noteworthy t h a t in [9] this formula was generalized t o define the entropy of a state with respect to a compact convex subset. Theorem 6 and Lemma 7 with the additivity of the von Neumann entropy under tensor product gives that (12)
S(98 4)= S(P) + S ( $ )
if 9 and ?,h are arbitrary states of C*-algebras. It is quite remarkable that this property does not follow readily from the definition (8).
333 430
M . OHYA and D . PETZ: NOTES O N Q U A N T U M ENTROPY
REFERENCES [I] ARAKI,H., Recent progress on entropy and relative entropy, I/lllth h t . Congr. on
[2]
[3] [4] [5] [6]
[7] [8]
[9]
[lo] [11] [12]
Math. Phys. (Marseille, 1986), World Sci. Publishing, Singapore, 1987, 354365. M R 88j:81004 CSISZ~R I., and K ORNER, J., Information Theory. Coding theorems for discrete memoryless systems, Probability and Mathematical Statistics, Academic Press, New York-London, 1981. M R 84e:94007 HIAI, F., Minimum index for subfactors and entropy 11, J . Math. SOC.J a p a n 4 3 (1991), 347-379. JAYNES, E. T., Information theory and statistical mechanics. 11, Phys. Rev. (2) 108 (1957), 171-190. M R 20 #2898 KOSAKI, H., Relative entropy of states: a variational expression, J . Operator Theory 16 (1986), 335-348. M R 87j:46110 M A N U C E A U , J . , N A U D T S , J . and VERBEURE, A , , Entropy of normal states, Comm. Math. Phys. 27 (1972), 327-338. NARNHOFER, H. and THIRRING, W . , From relative entropy to entropy, Fitika 17 (1985), 257-265. NEUMANN, J. VON, Mathematische Grundlagen der Quantenmechanik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 38, Springer, Berlin, 1932. Jb.Fortschritte Math. 58, 929-930. Dover Publications, New York, N . Y., 1943. M R 5-165 O H Y A , M . , Entropy transmission in C*-dynamical systems, J . Math. Anal. Appl. 100 (1984), 222-235. P E T Z , D . , Properties of quantum entropy, Quantum probability and applications, I1 (Heidelberg, 1984), ed. by L. Accardi and W. von Waldenfels, Lecture Notes in Math., No. 1136, Springer, Berlin-New York, 1985, 428-441. M R 873:46144 R E N Y I , A , , O n the foundations of information theory, Rev. I n s t . Internat. Statist. 33 (1965), 1-14. M R 31 #5712 S H A N N O N , C. E., A mathematical theory of communication, Bell System Tech. J . 27 (1948), 379-423. M R 10-133 (Received April 12, 1992)
DEPARTMENT OF INFORMATION SCIENCES SCIENCE UNIVERSITY OF TOKYO NODA CITY CHIBA 278 JAPAN MTA MATEMATIKAI K U T A T ~ I N T ~ Z E T E POSTAFIOK 1 2 7 H - 1 3 6 4 B UDAP EST HUNGARY
334 Vol. 38 (1996)
R E P O R T S ON M A T H E M A T I C A L P H Y S I C S
No. 3
NOTE ON QUANTUM DYNAMICAL ENTROPIES
LUIGIACCARDI Centro Matematico Vito Volterra, Universith di Roma 11, via di Tor Vergata, Roma, Italy
MASANORIOHYAand NOBORUWATANABE Department of Information Sciences, Science University of Tokyo, Noda, Chiba 278, Japan (Received September 12, 1996)
Classical dynamical entropy is an important tool to analyse the efficiency of information transmission in communication processes. Quantum dynamical entropy was first studied by Connes, Stmmer and Emch. Since then, there have been many attempts to formulate or compute the dynamical entropy for some models. Here we review four formulations due to (a) Connes, Narnhofer and Thirring, (b) Ohya, (c) Accardi, Ohya and Watanabe, (d) Alicki and Fannes. We consider mutual relations between these formulations and we show some concrete computations for a model.
Introduction The classical dynamical (or Kolmogorov-Sinai) entropy S ( T ) [13, 2G] for a measure preserving transformation T was defined on a message space through finite partitions of the measurable space. The classical coding theorems of Shannon are important tools to analyse communication processes which have been formulated by the mean dynamical entropy and the mean dynamical mutual entropy. The mean dynamical entropy represents the amount of information per one letter of a signal sequence sent from an input source, and the mean dynamical mutual entropy does the amount of information per one letter of the signal received in an output system. The quantum dynamical entropy (QDE) has been studied by Connes and S t ~ r m e r [lo], Emch [ll],Connes, Narnhofer and Thirring [9], Alicki and Fannes [5], and others [7, 221. Their dynamical entropies were defined in the observable spaces. Recently, the quantum dynamical entropy and the quantum dynamical mutual entropy were studied by one of the present authors [14, 231. They are formulated in the state spaces through the complexity of Information Dynamics [ 2 l , 231. Furthermore, another formulation of the dynamical entropy through the quantum Markov chain (QMC) was done in [4]. In Section 1 we briefly review the formulation by Connes, Narnhofer and Thirring (CNT). In Section 2 we explain the formulation by complexity [23, 241. In Section 3 we
335
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L. ACCARDI, M. OHYA and N. WATANABE
review the formulation by QMC, and in Section 4 we briefly explain the formulation by Alicki and Fannes (AF). Mutual relations among these four formulations are discussed in Section 4, and in Section 6 we compute the mean dynamical entropies in quantum communication processes.
1.
The CNT formulation
Let A be a unital C*-algebra, 0 be an automorphism of A, and cp be a stationary state over A with respect to 0; p o 0 = p. Let B be a finite dimensional C*-subalgebra of A. The CNT entropy [9] for a subalgebra B is given by
where plB is the restriction of the state cp to B and S ( . , .) is the relative entropy for C*-algebra [6, 27, 281. The CNT dynamical entropy with respect to 0 and B is given by
1
H,(B, B) = limsup -H,(B v 8 8 v . . . v W ' " ~ B ) , N
N+m
and the dynamical entropy for 0 is defined by
H,((e) 2.
= supH,(B, B
a).
Formulation by complexity
In this section, we first review the concepts of channel and complexity, which are the key concepts of ID (Information Dynamics) introduced by Ohya [2l, 231. , (G)), (A, :(A),?i (G)) be an input (initial) and an output (final) C*Let (A,C ( A ) cr _ _ systems, respectively, where A (resp. A) is a unital C*-algebra, C(A)(resp. C ( A ) )is the set of all states on A (resp. A) and cy (G) (resp. 5 ( E ) )is the group of automorphisms of A (resp. A) indexed by a group G (resp. G). The channel is a map A* from C (A) to (d).If the dual map A from d to A of A* satisfies the complete positivity, the channel A* is called a complete positive channel (CP channel). For a weak * compact convex subset S of C , there exists a maximum measure p with the barycenter p such that Ip=
S,4.
The compound state, introduced in [17, 181, which exhibits the correlation between the initial p and final A * p states is given by
336 459
N O T E ON QUANTUM DYNAMICAL ENTROPIES
In the sequel, we use a CP channel A* and the compound state to formulate the dynamical entropy. There are two types of complexities in ID. One of them is the complexity C$ ('p) of the system itself, the other is the transmitted complexity T S ('p ; A * ) from an initial system to a final system. These complexities should satisfy the following conditions: (i) For any
'p
ES
c C, cS('p) 10,
~; A *() 1 0 ~.
T
(ii) If there exists an orthogonal bijection j : exC + exC; the set of all extremal points in C , then
Ci%('p))
Ti(')( j ( ' p ) ; A * ) (iii) For a state !P = 'p 8
+ E S,, with
'p
E
S,
= CS('p)1 = T S ('p ; A * ) .
+ E s;
CSt(9) = C S ( ' p )
+ C"(+).
(iv) 0 5 T S ('p ; A * ) 5 Cs(p). (v) TS ('p ;id) = Cs(p). Next we review the definitions of the three types of entropic complexities introduced in [23]. Let (A, C ( A ) ,a (G)), (2, E(x),5 and S be as above. Let Mv(S) be the set of all maximal measures p on S with the fixed barycenter 'p, and let Fv(S)be the set of all measures having finite support with the fixed barycenter cp. Then the three pairs of complexities are
(c))
{ s, s (A*w,
TS ( ' p ; A * )
=
SUP
C$('p)
=
T S ( ' p ;id)
A*P) dp ;
,
1
(S) ,
pE
1
I S ( ' p ; A * ) = s u p { S ( ~ ~ 8 A * ~ d p , ' p & A *; ' pp t h i , ( S ) } , Cf (9) J S ('p
;A*)
Cf('p)
3
= =
Is
('p;
SUP
id)
{
,
s( A * w ,
dPf ;
pLf E
Fv ( S ) }
1
J s ( ' p ; id) .
Based on the above complexities, we explain the quantum dynamical complexity (QDC)
~71.
e)
be a stationary automorphism of A (resp. A); 'p o 19 = cp, and A be a Let 6' (resp. covariant CP map (Le., A o B = o A ) from A to A. & (resp. a k ) is a finite subalgebra
e
337 460
L. ACCARDI, M. OHYA and N. WATANABE
of A (resp. A). Moreover, let ak (resp. 6 ! k ) be a CP unital map from d (resp. A) and a M and are given by
(resp. &) t o
67
. . ., a M ) ,
aM =
(a1,a 2 ,
67
( ~ ' I o 6 l , A O 6 ! 2 ,. . . , A
=
Two compound states for a M and ,&;
o6!~).
with respect t o p E M v ( S ) , are defined as M
Using the above compound states, the three transmitted complexities [23] are defined by
When B k = 81, = B, A = A, O = 0, a k = OkP1 o a = from dot o A, the mean transmitted complexities are
T : (e, a , A * ) =
?: (0, A * ) =
limsup -TZ 1 N-03
sup?:
6kr
( aN
where a is a unital CP map
, -N ) ,
N (O,a,A * ) ,
oi
and the same for f$ and j $ . These quantities have properties similar t o those of the CNT entropy [14, 231. 3. Formulation by QMC A construction of dynamical entropy is due t o the quantum Markov chain [4]. Let A be a von Neumann algebra acting on a Hilbert space 'Ft and let cp be a state on A and A0 = Md ( d x d matrix algebra). Take the transition expectation &,. : Ao@d+ A of Accardi 11. 21 such that i
where A =
eij @
Aij E
A0
@
A and y = {yj} is a finite partition of unity I E A . 03
Quantum Markov chain is defined by $ E {cp,&.,,e}
$ ~ ( j i ( A i ). . . j n ( A n ) )
CP (&,,@(A1 8 &,,e(Az 8
E C(@&) such that 1
.. . @ A - i & , , s ( A ,
@ I ) . . .)))
,
338
46 1
NOTE ON QUANTUM DYNAMICAL ENTROPIES 03
where Ey,e
=
0 o E-,, 0 E Aut ( A ) ,and j k is an embedding of do into @ do such that 1
j k ( A ) = I @ . . . @ I @A @ I . . . k-th
Suppose that for cp there exists a unique density operator p such that cp(A) = Tr pA n
for any A E A. Let us define a state &, on @ do expressed as 1
+n(A1@ . . ' 8 An) = q ( j l ( A 1 ). . j n ( A n ) ). '
The density operator
en for $ J ~is given by
Put
Pi,...i= l w 4 ( @ n ( y i n )..'yalP-Yil . . . O " ( ? i " ) ) ) . The dynamical entropy through QMC is defined by
If Pi, ...i, satisfies the Markov property, then the above equality is written by
S,(O;y) = -
x
P(iZlil)P(il)log P(i2lil).
il ,iz
The dynamical entropy through QMC with respect to 8 and a von Neumann subalgebra B of A is given by S,(O; B) = sup 3,(0; y) ; y c 8) .
{
4.
Formulation by AF
Let A be a G*-algebra, 0 be an automorphism on A and cp be a stationary state with respect to 0 and B be a unital *-subalgebra of A. A set y = {yl,y2,. . . ,yk} of elements of B is called a finite operational partition of unity of size k if y satisfies the following condition: k
cy,ryi =I . i=l The operation o is defined by y ~ E = { y i E j ; i = l , 2 ,..., k ,
j = 1 , 2 ,..., 1 )
for any partitions y = {yl,yz,. . . ,yk} and [ = {[I, ( 2 , . . . ,El}. For any partition y of size k , a k x k density matrix p[y] = ( p [ y ] i , j )is given by P[Yli,j = cp(^lj-Yi).
339 462
L. ACCARDI, M. OHYA and N. WATANABE
Then the dynamical entropy fi,(O,B,y) with respect to the partition y and shift 6 is defined by von Neumann entropy S( . );
The dynamical entropy fiv(8, B) is given by taking the supremum over operational partition of unity in B as
fiJ& B) = SUP{ &(el a, 7 ) ;7 c B}. 5.
(4.3)
Relations among the four formulations
In this section we discuss relations among the above four formulations. The S-mixing entropy in GQS introduced in [20] is
ss('P) = inf { H (PI ;
P E M , ( S ) )I
where H ( p ) is given by
and P ( S ) is the set of all finite partitions of S. The following theorem [14, 231 shows the relation between the formulation by CNT and that by complexity.
THEOREM 5 . 1 . Under the above settings, we have the following relations: (1) 0 Is ('P ;A*) I TS('P ; A * ) I Js ('P ; A * ) , (2) ('PI = CT"(9)= CJ"('PI = sz ('PI = 4 (4 , (3) A = A = B(?L),for any density operator p, and
2
c,
0 5 Is ( p ; A * ) = TS ( p ; A*) 5 J s ( p ; A*) .
Since there exists a model showing that S'(")(cp) 2 H,(d,),Ss((p) distinguishes states more sharply than H,(d), where A, = { A E A; a ( A ) = A } . Furthermore, we have the following results [24]. (1) When An,A are abelian C*-algebras and
(Yk
is an embedding map, then M
-
T C ( p ; a M ) = Silassical( V A m) , m=l
1 . ? 2 ( ~ ;a M , & N )
=
Iclassical p
M
(
v
m=l
-
Am,
N
-
v Bn)
n=l
are satisfied for any finite partitions Am, fin on the probability space ( 0= spec(A), T ,P I .
340 N O T E ON QUANTUM DYNAMICAL ENTROPIES
463
(2) When A is the restriction of A to a subalgebra M of A; A = IM,
H,(M) = J c ( p ; IM) = JF(id; JM). Moreover, when IN N C & ,
A=@&,
@EAut(A);
( a ,e a , . . . ; e N - l a ) ; a =6 ; --+ A an embedding; aN E
N
we have
We show the relation between the formulation by complexity and that by QMC. Under the same settings in Section 3, we define a map from C ( H ) ,the set of all
&Tn,?)
density operators in 7-1, to
C((61 C d )8 7-1) by
8 e"-1(Yi,)e("-2)(Yi,-1) . ..
Y . . .e(n-2)(yi,-l ~ ~ ) e n~- l ( T i n )~,
~
for any density operator p E C(7-I). Let us take a map ETn) from C ( ( $ C d )8 'FI) to C ( 6 C d )such that 1
ET")(0)= TrR0 ,
v0 E c ( ( @ e d )
8 'FI) .
1
From Theorem 5.1, we have C?(I'Tn,?)(p))= S(TTn,,,(p)). Hence
341 464
L . ACCARDI. M. OHYA and N. WATANABE
Now we briefly show the relation between the formulation by complexity and that by
AF . Under the same settings in Section 4, for any partitions y = the k x k density matrix p [ y ] = ( p [ y ] i , j )is defined by
..
(71,. , yk}
of size k ,
P[Yli,j = cp(YjYi)
which is acting on the k-dimensional Hilbert space ?&. We define a map Z;m,r) from C(Zk) to C ( a k m ) by
Z;,,,)(P[YI) = P[Qm-l(y)0 . . oQ(Y) O 71 for any partitions y = ( 7 1 , . . . ,yk} of size k and any density matrices '
p [ y ] E C ( ' F l k ) . The
dynamical entropy by AF is given by
Since c,c(qm,,)(PIYI)) = s(Z;m,,)(PIYl)), we have &(Q,a,y) =
Wq,)(P[m
In any case, the formulation by the entropic complexities contains other formulations. Moreover it opens other possibilities to classify dynamical systems more fine [ 5 ] .
6. Computation of quantum dynamical complexity for a model Let X = ( a l , . . . ,U M }be an alphabet used to construct the input signals and let S = { E l ,. . . , E M } be a set of one-dimensional projections on a Hilbert space satisfying (1) E n I E , ( n # m) and (2) En corresponds to the character a,. By COwe denote the set of density operators generated by S
n=1
n=l
Suppose that the input quantum state is an element of CO. To send the information effectively, the state is first transmitted through a quantum modulator; the transmitted state is called the quantum modulated state. Let yTM) be a map from COto CAM)such that Y ( M ) is a completely positive unital map from A to A induced by the modulator
( M ) . For any En E S, the modulated state Ei'' ELM)
is given by
-
- yTM)(En)'
By ELM' we denote the set of modulated states
n=l
n=l
In this paper, we consider the modulated states constructed by the photon number states.
342
465
NOTE ON QUANTUM DYNAMICAL ENTROPIES
(1) For any En E CO,the modulated state ELpA”) for PAM (Pulse Amplitude Modulator) is defined by
EkpAM) = Y ; P A M ) where In)(./ is the n photon number state on
=
In)
1
H.
(2) For any En E CO,the modulated state E i p p M )for PPM (Pulse Position Modulator) is defined by
M
where EAPAM)is the vacuum state and E Y A M )= Id)(dl ( d is fixed). The transmission efficiency using the MER (mutual entropy - entropy ratio) [29] is calculated for some modulators. Now we compute the mean dynamical mutual entropy for PAM and PPM expressed by the photon number state (as above). Let B(X0)(resp. B(’Fl0))be the set of all bounded linear operators on a Hilbert space KO (resp. Ro),and let BO (resp. &) be a finite subset in B(Ko)_(resp. B(R0)). Let A (resp. A) be an infinite tensor product space of B(Ko) (resp. B ( K 0 ) )denoted by
Moreover, let 6’ (resp.
e) be a shift transformations on A (resp. A) defined by
Let a (resp. 6) be the embedding map from BO into A (resp.
to
A) given by
The set of all density operators on Ho (resp. Ro)we denote by CO(resp. C (resp. be the set of all density operators on A (resp. p on d).
x)
Eo),and let
343 466
L. ACCARDI, M. OHYA and N. WATANABE
where we took a special channel and modulator such that
A=
05 ,
8 A and
2=--m
T(M) =
-m
.
2=-M
Y(M).
(I) PAM.
Let us take a stationary initial state p E C
When A* is an attenuation channel, we obtain [17]
j ; =O
where
F,(PAM) =
Iji)(jil is the ji-photon number state in the output space TOand
where 77 is the transmission rate of the channel [17, 191. The compound states related to the channel d* become @E((Y&AM)
u &;(PAM))
344 NOTE ON QUANTUM DYNAMICAL ENTROPIES
467
Having the above equality, we obtain the following theorem.
THEOREM 6.1. (1) For an initial state p in (6.1), we have the lower bound of that
a ( p ~ ~such ) )
cp(e,a(pAM))
L
cp(e,
c
pmS(pg)).
m
(2) Let PO = 1,pk = 0 (b’k 1) and a = 6 , we obtain the following equalities:
= An i n (6.1). When A = d,0 =
e and
M n=l
(11) PPM. For an initial state p in (6.1), we obtain the following compound states:
345 468
L. ACCARDI, M. OHYA and N. WATANABE
346 NOTE ON QUANTUM DYNAMICAL ENTROPIES
469
REFERENCES [l] L. Accardi: Noncommutative Markov chains, Internatinal School of Mathematical Physics, Camerino (1974), 268. [2] L. Accardi, A. Frigerio and J. Lewis: Quantum stochastic processes, Publications of the Research institute for Mathematical Sciences Kyoto University, 18 (1982), 97. [3] L. Accardi and M. Ohya: Compound channels, transition expectations and liftings, to appear in J. Multivariate Analysis. [4] L. Accardi, M. Ohya and N. Watanabe: Dynamical entropy through quantum Markov chain, to appear in Open System and Information Dynamics. [5] R. Alicki and M. Fannes: Lett. Math. Phys. 32 (1994), 75. [6] H. Araki: Relative entropy for states of von Neumann algebras, Publications of the Research institute for Mathematical Sciences Kyoto University 11 (1976), 809. [7] F . Benatti: Determin&tic Chaos in Infinite Quantum Systems, Springer, Berlin 1993. [8] L. Bilingsley: Ergodic Theory and Information, Wiley, New York 1965. [9] A. Connes, H. Narnhoffer and W. Thirring: Commun. Math. Phys. 112 (1987), 691. [lo] A. Connes and E. Stormer: Acta Math. 134 (1975), 289. [ll] G. G. Emch: 2. Wahrscheinlichkeitstheorie v e m . Gebiete 29 (1974), 241. [12] L. Feinstein: Foundations of Information Theory, MacGrow-Hill, 1965. [13] A. N. Kolmogorov: Dolcl. Akad. Nauk SSSR 119 (1958), 861. [14] N. Muraki and M. Ohya: Lett. Math. Phys. 36 (1996), 327. [15] J. von Neumann: Die Mathematischen Grundlagen der Quantenmechanik, Springer, Berlin 1932. I161 M. Ohya: J. Math. Anal. Appl. 84 (1981), 318. [17] M. Ohya: IEEE Duns. Information Theory 29 (1983), 770. [18] M. Ohya: L. Nuovo Cimento 38 (1983), 402. [19] M. Ohya: J. Math. Anal. Appl. 100 (1984), 222. [20] M. Ohya: Rep. Math. Phys. 27 (1989), 19. [21] M. Ohya: Information dynamics and its application to optical communication processes, Springer Lecture Notes in Physics, 378,p. 81, Springer, Berlin 1991. [22] M. Ohya and D. Petz: Quantum Entropy and its Use, Springer, Berlin 1993. [23] M. Ohya: Quantum Communications and Measurement 2 (1995), 309. [24] M. Ohya: Foundation of entropy, complexity and fractal in quantum systems, to appear in International Congress of Probability Towards 2000, 1996. [25] M. Ohya and N. Watanabe: Note on irreversible dynamics and quantum information, to appear in the Albert0 fi-igerio Conference Proceedings. [26] J. G. Sinai: Dokl. Akad. Nauk SSSR 124 (1959), 768. [27] A. Uhlmann: Commun. Math. Phys. 54 (1977), 21. [28] H. Umegaki: Kodai Math. Sem. Rep. 14 (1962), 59. [29] N. Watanabe: Quantum Probability and Related Topics 6 (1991), 489.
347 Letters in Mathematical Physics 36: 321-335, 1996 0 1996 Kluwer Academic Publishers.
327
Entropy Functionals of Kolmogorov-Sinai Type and Their Limit Theorems N A O H U M I M U R A K I ' and M A S A N O R I O H Y A Z 'Department of Applied Science, Yamaguchi University, Tokiwadai 2557, Ube City, Yamaguchi 755, Japan 'Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278, Japan (Received: 14 March 1995) Abstract. Two functionals s and 7 are introduced for C*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals s" and I: Our functionals s and I" are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya. Mathematics Subject Classification (1991). 94A17 Key words: Kolmogorov-Sinai theorem, entropy functionals, quantum information theory.
1. Introduction Quantum information theory was formulated in the operator algebraic setting [6]. In general quantum information theory, a stationary information source is described by a C*-dynamical system (d, O,, 4), where d is a unital C*-algebra, Qd is an automorphism of d,and 4 is an invariant state over d with respect to O d , that is, 4 8, = 4. A stationary information channel from an input C *-dynamical system (d,8,) to an output C*-dynamical system ( 8 , 0 B ) is here the transpose A*: d *+B* of a completely positive unital (c.P.u.) map A: 8-d being covariant with 8, and O#, i.e., A 0 Oa = Od 0 A. Such A* is called a stationary channel from an input system (d, 0,) to an output system (8,Oa), and maps an input state 4 over d to the output state A* 4 over 8. In [7,8], Ohya introduced the concept of the compound state of an input state 4 and the output state A * 4 , and, using them, he defined the mutual entropy I(4;A*) which expresses the amount of information transmitted from the input state 4 to the output state A* 4 through the channel A*. In this Letter, in order to formulate the entropy of KS Type [4] in a new direction, we extend the construction of mutual entropy in [6,7] to the situation involving automorphisms 8,, 8, of C *-algebras d,8, certain evolutions of systems. That is, we introduce two functionals s"(4)and r(4;A*) for an invariant state 4 over d and a stationary channel A* from (d, 8,) to (8,QB). The functional s" (resp. is interpreted as a kind of quantum analogue of entropy rate (resp. mutual entropy rate) in information theory. The contents of this Letter are as follows. In Section2, using the concept of the compound state (or compound lifting [l]), we define the functionals S,(aM) and 0
r)
348
328
NAOHUMI MURAKI AND MASANORI OHYA
la(aM,8") for a pair of finite sequences aM=(a1,a2, ...,aM),
BN=(B17flz?. . . , P N )
of completely positive unital maps a,: A , -+ d,fin:B , -+ d from finite-dimensional unital C*-algebras A , (m= 1, ..., M ) , B, (n= 1, ...,N ) to d,and for a decomposition measure p of 4. In Section 3, we introduce the functionals T(4)and r(4;A*) based on the functionals S,(aM) and I,,(a', BN), and we show that the KolmogorovSinai-type theorems hold for the functionals $(+) and T(4;A*).
2. Functionals S and Z for Completely Positive Unital Maps Let d be a unital C*-algebra with a state 4. In this section, we introduce functionals S,(aM), S,(a"), I,(aM,p"), and 19(aM,BN), for a pair of finite sequences of BN=(P1,P2, ..., B N )
aM=(a1,a2, ...,UM),
of completely positive unital maps a,: A , unital C *-algebras A m ( m = 1, ...)M ) ,
8.: B,
+ d,
-+
d from finite-dimensional
B,(n= 1, ..., N )
to d.At the end of this section, these functionals S and I will be interpreted as some kinds of 'joint entropy' and 'joint mutual entropy', respectively. Let M ( 4 ) be the set of all regular Bore1 probability measures p on the state space S ( d ) of d such that p is maximal in the Choquet ordering and p represents 4 : fS(&) o dp(o)= 4. We refer to such measures as extremal decomposition measure for 4. For any state 4 over d,such measures exist from Choquet's theorem. For a given finite sequence of completely positive unital maps a, from finite-dimensional unital C*-algebras A , (m= 1, ..., M ) to d and a given extremal decomposition measure p of 4, we define on the tensor product algebra 8=: A , a state m p ( a M by ) , l
The state @,,(aM)is a compound state of a:$, a ; 4 , ..., a L 4 in the sense of [7,S], so that @,,(aM)(x @0=44(x), is satisfied for any x E A , , I E C3,,,m'ZmA,.. Furthermore, @,,(aMu/31V) is a compound state of @,(aM) and @,(/I") with a M u g N3 ( a 1 , a 2 ,..., a M ,B1, fi2,..., B N ) constructed as (2.1). This compound state satisfies the following marginal conditions @,(CIMuP1V)(x~')=@,(aM)(x),
for x E C 3 g = 1 A mI, E C 3 f = 1 B , ,and @,(@
u"B
@ Y ) = @J
B") ( Y )>
349 ENTROPY FUNCTIONALS OF KOLMOGOROV-SINAI TYPE
329
for Z E @ $ = , A , , Y E @:=lBn. For any pair (aM,P") of finite sequences
a M = ( a l , ..., a M ) and
PN=(P1,
...,P N )
of completely positive unital maps (c.P.u. maps for short) such that a,: A , -+ d, P,: B, + d and for any extremal decomposition measure p of 4, we define two functionals; the entropy functional S, and the mutual entropy functional I,, by
,
P
s,(aM)= J
s(@$=,a:wl@,(a') S(.4
PN)= s(@,(a"uP")I
j
dp(w),
@,(aM) @ @,(P",
respectively, where S(.l.)stands for the relative entropy [2, 111 S ( 0 , Iw2)=Tr{Po,(logPo1-log Po,)}
where pol (resp. po,) is the density operator of a state o1(resp. w2) over some finitedimensional unital C *-algebra. The following is a fundamental relation of functionals S, and I , . PROPOSITION 2.1. S , ( a M ) + S,( PN)= S , ( a M u P N )
+
I,(aM,
P").
Proof. Let pw (resp. o m )be the density operator in @ $ = , A , (resp. @ ; = ' = , B , ) corresponding to a state SF=,a : o (resp. @=: P ; w ) in the fixed trace Tr, then S,(aM u B N ) + I p ( a MP," )
n
r
n
omlogoo-aolog
350
330
N A O H U M I MURAKI A N D MASANORI OHYA
Note that this relation between S, and I , is analogous to the classical relation H,(A")
+ H , ( 8 ) =H,(A" v 8)+ I,(&
8)
of the entropy H and the mutual entropy I for measurable finite partitions
-
B = { Y , , ...) Y N z }
A " = { X 1 ,...) XN1},
of a probability space (QP, P ) (see [2,9]). If d, B , ( n = 1, ...)N )
A m ( m =1, ...)M ) ,
are Abelian C*-algebras and each a,: A , + d (resp. fin: B, + d )equals to embed(resp. jB,,), of A , (resp. B,) into d,then there exists the unique extremal ding jAm decomposition measure p for each state 4 over A , and our functional S, coincides with the classical entropy
A",),
S,(aM)=Hp(V:=l
and I, coincides with the classical mutual entropy, Z,(aM,
p") = I,( v=:
1
A",, v=;
1
I&),
where A", (resp. 8,) is a finite partition of the compact Hausdorff space R = Spec(&) of all pure states over d. For a given pair of finite sequences of C.P.U.maps aM=(M1,
b N = ( p l ?. . . ? P N ) ,
...,aM)?
we define the functional S, (resp. Z,) by taking the supremum of S , ( a M ) (resp. M , p N ) )for all possible extremal decompositions p's of 4:
Z,(a
sup
S,(aM)=
S,(aM),
, E M : (@) Z@(UM,
p")=
sup
Z,(aM,
8").
,EM:(@)
Recall that the original mutual entropy
Z(4;A * ) in [ 6 ] was defined by
I ( 4 ; A * ) = sup Z,(&A*), ,€M:($)
where Z,(4;A*)=S(OPI463A*4) with 0 , = ~ s c s ) ~ 6 3 A * ~ d pand ( ~ )S(.l.) , denotes Uhlmann's relative entropy [101 for infinite-dimensional C *-algebras. If we drop the assumption in the definition of Z,(a M , p") that A's and B's are finite-dimensional, and we take Uhlmann's relative entropy to set S(.l.), then the corresponding functional Z,(a M , 8") can be formally viewed as a natural generalization of Ohya's
351
331 mutual entropy Z(4;A*) to the cases involving many C.P.U.maps. In fact, one clearly has ENTROPY FUNCTIONALS O F KOLMOGOROV-SINAI TYPE
~(4;A*)=~,Wd>N> where id, stands for the identity map of d. Using the well-known properties of relative entropy such as joint convexity, monotonicity, scaling property, and others [S, 91, the following proposition is easily proved due to the finite-dimensionality of { A , } and { B , } . PROPOSITION 2.2. Let y,: A h + A , (resp. 6,: BL + B,) be C.P.U.maps from a finite-dimensional C*-algebra A ; to A , (m= 1, ..., M ) (resp. BL to B , (n= 1, ...,N ) ) , and let 8 : d -+ d is a C.P.U. map with 408=4. Then Fundamental inequality:
0 < z 4 ( a M ,P " )
< min{S,(a'),
Forfinite sequences of
C.P.U.
S,( P")}.
maps
a M ' = ( a 1 , a 2 ,..., a L ) and ~ ~ ~ = ( a ~ +...,~a M, )a ~ + ~ , with L < M and M = M ' v M ,
max{S4(aM'),S,(aw)) For jinite sequences of a M y M =(a 0
0
< &(aM) < s,(a'')+s,(aM").
C.P.U.
maps
y l , a 2 y z , ...,aM y M ) and 0
0
P"0S" -(Pi 061, P z 0 8 2 , ..., P N o h N ) , S , ( a M 0 y M )d S,(a"), Z,#,(aM
0
yM,
B" 8 " ) d Z&M, D"). 0
Equality holds if the y, and 6 , are conditional expectations for all m and n, associated to o a , and o P,, respectively. For finite sequences of C.P.U. maps 0
0
0 o a M - ( 8 ~ a l , 0 0 a ..., 2 , e 0 a M ) and
8 0 ~ N ~ ( e o p l , 8 0..., p zeopN), ,
s,(e
M , G s,(aM), l , ( e o a M ,e o p N ) G Z , ( ~ M , ~ N ) .
3. Functionals $ and f for Invariant States and Stationary Channels In this section, we introduce two functionals s"(4)and r(4,A*) for a stationary channel A * with an invariant input state 4, and we prove the KolmogorovSinai-type theorems for these functionals. Let d (resp. 9i?)be a unital C*-algebra with a fixed automorphism 8, (resp. 8#), A be a covariant C.P.U.map from to d , and 4 be an invariant state over d,that
352
332
NAOHUMI MURAKI AND MASANORI OHYA
is, 4 0, = 4. For each C.P.U. map a: A -+ d (resp. p : B sional unital C*-algebra A (resp. B ) to d (resp. B), put 0
-+
B ) from a finite-dimen-
1 s"b(a)= Iim sup - s b ( a n ) n-rm n
where an=(a, 8,
o
a, .. ., 85- a) 0
and
fi; =(A /3, A 0
0
Qg 0
B, ..., A O$-' 0
0
B).
Note that indeed $+(a) (resp. r"b,A*(a,/3))is the limit of S,(a")/n (resp. l / n Z,(a", A p i ) ) from the subadditivity (3) in Proposition 2.2. Then we define the functionals s"(4)and r(4,A*) by taking the supremum for all possible A's, a's, B's, and ps: 0
s"(4)=sup s"&), OL
r"(4,A*)=sup r"4,Aa(a3 B). a. B
Note that the above functionals s"(4) and T(4,A*) depend on 8, and (8&,Og), respectively, but we abbreviate them for simplicity here. The fundamental inequality in information theory certainly holds for our s" and I" from Proposition 2.2: PROPOSITION 3.1. 0 < r"(4,A*) < min {s"(4),$(A*
4)},
In ergodic theory and information theory, the Kolmogorov-Sinai theorem for the entropy of automorphisms is very important because it makes possible to compute the mean entropy for concrete examples. In [4], Connes-Narnhofer-Thirring proved the non-Abelian Kolmogorov-Sinai theorem for their dynamical entropy. For the case of our functionals s"(4)and r(4,A*), the Kolmogorov-Sinai-type theorem still holds (Theorems 3.5, 3.6). The proof of this Kolmogorov-Sinai-type theorem for functionals s" and r" is similarly as that in [3], because our functionals obey similar estimates to that of the Connes-Narnhofer-Thirring joint entropy. The points we stress are that our functionals s"(4)and r"(4,A*) are constructed from the functionals S , ( a M ) and I,(aM, B") which are related in information theory and our functionals are obtained by using a channel transformation so that ours contains the dynamical entropy in some cases. For the proof of the Kolmogorov-Sinai-type theorem for functionals s" and 7, we need a Lemma from [4] LEMMA 3.2 [4]. Let d , B be unital C*-algebras with B finite-dimensional and d = dim B. Let 4 = J w dp(w) be any decomposition measure of a state 4 over d.Then
353 ENTROPY FUNCTIONALS OF KOLMOGOROV-SINAI TYPE
333
for any C.P.U.maps a, a:B + d with (Ia - E (I 9 E one has
d 6&(;
+ log (1 +
%)).
Using this, we can show the continuity of the functionals S b ( a M )and I b ( a M P, N ) with respect to a’sand p’s.
PROPOSITION 3.3. Let d be a unital C*-algebra with a state 4 and A,, m= 1, ..., M be finite-dimensional unital C*-algebras, a,, E m be C.P.U. mapsfrom A , to sd. Let d be the max of the dimensions of the Am’sand max, 1) a , -a, 11 < E , then
PROPOSITION 3.4. Let d be a unital C*-algebra with a state 4, and A,, m = 1, .. ., M , and B,, n = 1, ..., N , be jinite-dimensional C*-algebras, a,, E m be C.P.U. maps from A , to d, and P,, fl, be C.P.U. maps from B , to d.Let d be the max of the dimensions of the Am’sand B,’s, and max{ 11 a,
-a, 11, llPn -p, 11 I m= 1, ..., M ; n= 1, ..., N } d E,
then
Under the conditions of the above propositions with an invariant state the following Kolmogorov-Sinai-type theorems (Theorems 3.5 and 3.6).
4, we get
THEOREM 3.5. Let a , be a sequence of C.P.U. maps a,: A , + d such that for suitable C.P.U. maps ah:d + A,, one has a , ah + idd in the pointwise topology. Then 0
s”($)= ~ i ms”b(a,). m+ m
THEOREM 3.6. Let a , and /jm be sequences of C.P.U. maps a,: A , + d and Pm:B , + such that for suitable C.P.U. maps M A :d + A , and P&:9i? B,, one has a , ah + idd and Prn0 P& -+ ida in the pointwise topology. Then 0
f(4,A*)=
lim f b , A * ( a m P,)., m+m
Proof. Put y, =a, o a; y for any C.P.U.map y : A + d with a finite C*-algebra A . Then ( y,(x)) converges to y(x) for each x E A and, hence, 0
lim IIa,-aII=O, rn- m
354
334
NAOHUMI MURAKI A N D MASANORI OHYA
since y m and y are bounded linear maps from a finite-dimensional C*-algebra A . In the same way, lim 116,-611=0,
S m = p m o p ~ o S
m- m
for any
C.P.U.
6: B
z4((e>
--$
A? with a finite C*-algebra B. From Proposition 3.4, we have
(es
,;::
m)
A 6m)
::;) - z4((e>
N
))
< 2 4 N ~ ( i+ log( 1 + for IIym-yII d s and 116,-611
7 ); : ;), (0% A 6 ) : ~ ; N
d & and, hence,
and I
lim
m- w
T4,A*(Ym,
dm)=I$,A*(y, 6).
From the monotonicity of the functional I (Proposition 2.3 (3)), we have I
Ir$,A*(Ym,
6,)
T+,A*(am? P m )
and, hence,
-
Z4,A.(y,6)~liminfT4,A.(am,~m)~T(~,A*). m- m
Since y and 6 are arbitrary completely positive unital maps from A to d, we have lim m- m
?4,A*(~m,Pm)=?(4, A*).
0
The assumption of the above two theorems are always satisfied if we take C*-algebras to be nuclear. If we apply these theorems for AF algebras (the norm closure of an increasing union u A , of sequences of finite-dimensional subalgebras) we can get the followings. ~
COROLLARY 3.7. Let d = u A , be an AF algebra with an automorphism Od, then for any invariant state 4 over d,
,T(4,e)= lim ,T4,@(~,,). n+m
-
-
COROLLARY 3.8. Let d = u A , (resp. A?=uB,) be an AP algebra with an automorphism Od (resp. %#). Then for each covariant c.p.u. map A : A? +. d and an invariant state 4 over d,
T(4,
eB))=
lim
n+
m
T4,A*(Am,
Bn).
355
ENTROPY FUNCTIONALS OF KOLMOGOROV-SINAI TYPE
335
In the above corollaries, the subalgebra A , c d stands for the inclusion map j A n :A , + d for every n. Some extensions and verification of two functionals s"(4) and I"(4,A*) are discussed in [lZ], in order to study communication processes and complexities of various physical systems.
References 1. Accardi, L. and Ohya, M., Compound channels, transition expectations and liftings, to appear in J . Multivariate Anal. 2. Araki, H., Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Uniu. 11, 809-833 (1976). 3. Billingsky, P., Ergodic Theory and Information, Wiley, New York, 1965. 4. Connes, A., Narnhofer, H., and Thirring, W., Dynamical entropy of C * algebras and von Neumann algebras, Comm. Math. Phys. 112, 691-719 (1987). 5. Muraki, N., Ohya, M., and Petz, D., Entropies of general quantum states, Open Systems Inform. Dynam. l(l), 43-56 (1992). 6. Ohya. M., Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys. 27, 1 9 4 7 (1989). 7. Ohya, M., On compound state and mutual information in quantum information theory, IEEE Trans. Inform. Theory 29, 770-774 (1983). 8. Ohya, M., Note on quantum probability, Lett. Nuouo Cim. 38, 402406 (1983). 9. Ohya, M. and Petz, D., Quantum Entropy and its Use, Springer, New York, 1993. 10. Uhlmann, A., Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Comm. Math. Phys. 54, 21-32 (1977). 11. Umegaki, H., Conditional expectation in an operator algebra IV (entropy and information), Kddai Math. Sem. Rep. 14, 59-85 (1962). 12. Ohya, M., State change, complexity and fractal in quantum systems, to appear in Quantum Communications and Measurement, Plenum, New York.
356
Informat ion Dynamics and Its Applications t o Optical Communication Processes
Masanori OHYA Department of Information Sciences Science University of Tokyo Noda City, Chiba 278, Japan
Abstract Various physical or nonphysical systems can b e described by states, so that the dynamics of a system is described by the state change. One of essential characters of a state is expressed by its complexity. Complexity such as entropy is a key concept in Information theory. We call the study of the state change together with such complexities “Information dynamics”, which is a kind of synthesis of dynamics of state change and information theory. Here we explain what information dynamics is and indicate how it can be used in optical communication. Some of concrete applications of information dynamics are discussed in the papers [9,10] of this volume. f l .What is Information Dynamics 1
Let an input _ -dynamical system and an output dynamical system be described by
(A,6 , a)and (A,G, Z), respectively. Here A is a set of all objects to be observed and G is a set of all means getting the observed value for each element A in A, and a describes an inner evolution of the input system. We call G a “state space” here. Same for the -output system (A,G,E). Thus we may say [Giving a mathematical structure to input and output triples z Having a theory].
A map providing a bridge between two systems is called a “channel” if it sends a state of the input system to that of the output system ; A’ : G --t ???.There exist several channels, whose properties specify the character of two systems. Mathematical structure of almost all systems can be expressed by the following charts. ~ Input ( I ~ ~ Transformation ~ ( I I ) - Output j 4 1 1 ) ~ system
system channel
system
(chart 1) Lecture Notes in Physics, Vol 378 C Bendjaballah, 0 Hirota, S Reynaud (Eds ), Quantum Aspects of Optical Communications 8 Springer-Verlag BerlinHeidelberg 1991 With kind permission of Springer Science and Business Media
357 a2
Input state
-
processing, Control
1 4
Artificial processing, Control (D)
-
(III)---
output state
(El
(Chart 2)
(B)
In the above chart 2, (I) ( = (A) + (B)) corresponds t o an input system, (11)( = + (C) + (D)) does t o a transformation (channel) system (111)( = (D) + (E)) to an
output system. More complex systems are constructed from this first structure. A “naked” state (A) is artificially processed or controlled t o a “dressed” state, and it is suffered t o change by a physical (natural) transformation, and it is again artificially processed and controlled. The fundamental part of the process for this state change is obviously “A -+ C + E” Let us give some examples of an input system (I),a transformation system (channel) (11) and an output system (111).
.
(1) Causal System : (A) z E R”; (B) i = f ( z ) ; (C) ~ ( t = ) @:(a!), evolution (semi)group generated by f.
where Gt is an
Remark: If z ( t ) can not be obtained directly, then examines the properties of f and find some proper approximation to obtain at. We may have a chaotic system. (2) Signal Transmission : (A) causal signal z ( t ) ; (B) coding causal signal z ( t ) to (2”) by e.g., sampling theorem with cut off; (C) some transformation yn = f(z,) for any n ; (111) interpolating or decoding {y,} t o y(t).
(3) Discrete systems : (A) input probability distribution p = { p l , . . . , p , } of events X = ( 2 1 , . ,2,) ; (C) transition probability ( p ( i 1 j ) ) ; (E) output probability distribution q = (qi)iqi = C j p ( i l j ) p j .
..
(4) Continuous systems : (A) probability measure p on measurable space (0,x) ; (C) Markov kernel X : s2 x 5 -+ [0,1]s.t. X(z,.) E P(n)and ;\(-,A) E M ( 0 ) ; (E) CC f sn -)+.
358 03
( 5 ) Quantum system 1 :(A) z E (Hilbert space) ; (C) unitary operator U or isometric operator V ; (E) y = U z or Vz E 'He (6) Quantum system 2 : (A) density operator p E T ( X ) + , 1 ; ( C ) A* (E) = A'p.
AdU or AdV.
f
(7) C*-system containing all above: (A) (A,6 , a ( G ) )C*-triple; (C) A' : 6 ( A )4 G(Z) -a dual map of a completely positive map A : Z --t A, where (A,6,a(E)) C*-triple; (E) = A'p.
v
Once input and output systems are mathematically fixed and a transformation rule (channel) is given, we next consider some complexities of the state associated with the systems, which are a corner stone of information dynamics. The first complexity is one for a state itself : For a state p, the complexity seen from a reference system S, a certain subset of G , is dented by C"(p). The second complexity is determined by both input and output states p, (p or an input state p and a channel A*, so that it is denoted by Ts(p;jij) or P ( p ; A * ) , which is called a transmitted complexity from p to or f l p . Typical examples of these complexities are entropy and mutual entropy, which play essential roles in several fields. These complexities should satisfy the following properties: Let (All GI, a l ( G ) ) , (A2, 6 2 , a 2 ( G ) )be two systems and (A, 6, a ( G ) )be the compound system such that A = A 1 8 d2, and A' be a channel from 61 to ( 5 2 . (i) For any state 'p E S c 61, Cs((p)2 0 and TS(p;A*) 2 0. (ii) For any bijectionj from 6 1 t o 61,
Cj(S)(j(p))= C"(p) Tj(")(j(p);A*) = TS(p;A*).
(iv) 0 5 P ( p ; A * )5 Cs(p). (v) Ts(p;id) = C"(p), where "id" is an identity channel from p, vp E 6 1 .
6 1
to 6,; id(p) =
Definition 1.1: Information dynamics is a dynamics described b y a set
where R i s a certain relation among above quantities. Therefore, for a system of interest, we have t o - - _ (1) mathematically determine A, G ,S,(Y ; A, G,S,(Y (2) choose and R ; (3) fix C"(p) and T"(p;r).
359 84
By setting the above (1) (3) in general quantum systems (GQS),we can apply this general frames t o several topics such as optical communication[l,5,6],fiactd theory [8], molecular genetics We here mainly discuss a n application of information dynamics to optical communication processes. N
"73.
52. Channel and Lifting
In order to consider an application of information dynamics to optical communication processes, we have to mathematically - - set all quantities needed. Let both input system (A, 6, a) and output system (A, 6, E ) be described by certain C*-algebraic triples. Very often A = 6 = E, a = isi. A map A' : 6 + E is called a channel. In particular, A' is called completely positive channel if the dual map A : 2 + A satisfies
II,
BfA(AfAj)Bj 2 0 for VBj E A and VA; E 51. i,i
A certain class of channels may be a lifting &' from A t o A @ z[l]; that is, a continuous map such as E' : 6(d)+ 6 ( d @ Z ) .
A lifting &* is said to be nondemolition for a state p if ( E ' p ) ( A @ I )= p ( A ) for VA E A. Given a lifting &', we can construct channels A* and
fi such that
A':6+E by A'p(Z) = ( E ' p ) ( l @
z),VzE 2, and ----.
A :6+6
by Ti'p(A) = (E'p)(A @ I ) , V z E A. Conversely, given a channel A' : 6 + p E S c 6 and a decomposition of p ; p = w d p
Ss
w @ A'wdp
E'p =
(compound state
and
[3])
is a nondemolition lifting. Hence we have
[Channels e, Liftings ]
< Examples of channel and lifting > Here we give some examples of channel and we construct a quantum communication channel. Let p = C X,p, be a certain decomposition of a density operator when A = B(31) and 6 = T(X)+,l on a Hilbert space 3t.
<1> Unitary evolution : p + A;p = AdUt(p)
U,'pUt, t E R
+ E'p
=
C Xnpn @ A'p,
360
where Ut is a unitary operator Ut = exp(itH).
<2> Semigroup evolution : p
+ Ayp
= Ad&(p)
3
v,'p&, t E R+
E'p =
C Xnpn @ A'pn
where {& ; t E R+) is a one parameter semigroup on 'H.
<3> Measurement : Measure A = EnanPn (spectral decomposition of A) in a state p,
<4> Reduction : When two systems described by Hilbert spaces 'H and 1c interacts and we look at the state change of the first system.
w
W
Then
-, nyp= tT,et. More generally for <3> and <4>, A (resp. 71) is interpreted as the algebra of observables of a system (resp. a measuring apparatus) and E' describes an interaction between the system and the apparatus as well as the preparation of the apparatus. If p E S(d) is the preparation of the system, i.e., the state before the interaction with the appazatus, then Tcp E S(d) (resp. A'cp E S ( 2 ) ) is the state of the system (resp. of the apparatus) after the measurement.
<5> Isometric lifting : For an isometry V(V'V = IN)from 'H to 'Ha&a lifting defined by &*p = VpV', trp E 6(311)
is called an isometric lifting [I]. <6> Quantum Communication Channel : Let us construct a quantum (communication) channel.
361 86
Let Y E 6(&)be a state representing the noise and a, r,7 be the following maps: (1) a : B('H2) + B('Hz@Xz) given by a(A) = A@Ifor any A E B('Hz), (2) n : B('H,@Xz) + B('H1 @&) completely positive with n(1) = I, (3) 7 : B('H1 @XI) + B('H1) by 7 ( Q ) = t r K I Y Q for m y Q E B ('Hi @ X i ) . A=yonoa.
Then
A* = a* o r* o 7 . or equivalently, A*p = t 7 K a r * ( p @ v), Vp E 6(7fi). We call this channel "quantum communication channel". When 'HI= 'HZ= 'H,
E' : p E Cq3-1) + 7 r ' ( p @ v ) E 6 ( ' H @ K ) is a lifting, and A*p = t T K E * p .
In <5> and <6>, a n input signal is transmitted and received by an apparatus which produces a n output signal. Here A (resp. 2)is interpreted as the algebra of observables of the input (resp. output) signal and €* describes an interaction between the input signal and the receiver as well as the surroundings of the receiver. If p1 E S(d) is the input signal, then the state A*p E S ( 2 ) is the state of the (observed) output signal.
f 3. Entropies describing complexities in GQS Next we discuss two types of entropy for general quantum states as the complexities of information dynamics which are needed for optical communication. Let (A,6,a ( R ) ) be a C*-dynamical system and S be a weak* compact and convex subset of 6, exS be the set of all extreme points of S. Every state 'p E S has a maximal measure p pseudosupported on ex S such that
The measure p giving the above decomposition is not unique unless S is a Choquet simplex, so that we denote the set of all such measures by M,(S).For the probability measure p, define
H(p) = SUP{-
p(Ak)logP(Ak);
2 E p(s)),
AbEZ
where P ( S ) is the set of all finite partitions of S. Then the entropy of a state p E S w.r.t. S is defined by
362 a7
This entropy does depend on the set S chosen, and we call it "S-entropy". Even when S(p) = +oo, Ss(p) < +oo for some S, which is a remarkable property of S- entropy [4]. A compound state (lifting) +:(= €;p) of p and (p =)A*p with respect t o S and p was introduced as
+;
=
J, w 8 A'wdp.
The mutual entropy w.r.t. a n initial state p E S, the decomposition measure p and a channel A* is defined by [2,6] I;b;A*) = s(*:l+o)l where S(.l.) is the relative entropy for two states. I n some cases, this mutual entropy L can be written as
The mutual entropy w.r.t. an initial state
'p
E S and a channel A* is then defined by
IS('p;A*) = limsup{It('p;A*);p E F,(S;E)}, r-0
where F,(S) is the subset of the set M,(S) such that F,(S;c) = { p E M,(S);Ss(p) 5 H(p) < SS('p)+c < +oo} or F,(S; e) = M,(S) when Ss(p) = 00. Note that the mutual entropy should be used when the decomposition measure is fixed. In the sequel we use the simple notations S(p), G P , IP('p; A*) and I(p; A*) when S = (5. Let us write the mutual entropy in usual quantum system, namely, when A is the full algebra B(3-I) and any normal state 'p is described by a density operator p such as p(*)= t r p -. Then our entropy S ( ' p ) is shown to be equal to that of von Neumann: S(p) = S(p) = -trplogp. Every Schatten decomposition p = AnEn , En = 12, >< 2-1 (i.e., An is the eigenvalue of p and 2, is its associated eigenvector) provides every orthogonal measure in IM,(G) defining the entropy S(p). Since the Schatten decomposition of p is not unique unless every eigenvalue An is nondegenerate, the compound state 9 is expressed as
En
UE =
C XnEn 8 A*Eni n
where E represents a Schatten decomposition and the channel A* is given by
{En}.Then the mutual entropy for p
363
where uo = p @ A’p. This form of the mutual entropy was introduced in [2] to study optical communication processes. Fundamental properties of S’(97) and 15(p;A*) :
Theorem 3.1: W h e n A = B(E) and at = Ad(Ut) with a unitary operator Ut, for a n y state p given by p(-) = t r p with a density operator p, w e have t h e followings: (1) S ( y ) = -t7p log p. ( 2 ) If p i s a n a-invariant faithful state and every eigenvalue of p i s nondegenerate, t h e n Sr(p) = S(p). ( 3 ) If p E K ( a ) , t h e n SK(p) = 0.
-
Two states p1 and are said to be orthogonal each other (denoted by p1 I p2) if their supports s(p1) and s(p2) are orthogonal, where the support s(p) of p means the smallest projection E satisfying p(1-E) = 0. The measure p E M,(S)is said to be orthogonal if (JQw d p ) l ( J n l Q w d p ) is satisfied for every Bore1 set Q in S. A channel is called normal if it sends a normal state to a normal state. Theorem 3.2: For a n o r m a l C.P. channel A* and a normal state p, i f a measure p has a discrete support and i s orthogonal, t h e n I,(p;A*) =
S(A*wJA*p)dp< S(p)
+
E.
Theorem 3.3: For a density operator p
In the notatins of Section 1, two complexities are
c5
(P)= ss(97) T5(p; A*) = Is(p;A*). Without seriously taking the original meaning of the entropy S (i.e., forgetting the correspondence between each element of p and that of A*p), and as far as the complexity is concerned, Cs(p) and ‘ l f ( p ;A*) can be given as follows : For any p E S and any decomposition of p
the complexities are
‘I?(p;A*) --=
/
S(A*wlA*p)dp
6
C5(p)
E
TS(p;id).
364 89
Watanabe [lo] discusses the efficiency of a modulation M by
where p~ is a modulated input state. Remark: The channel capacity is given by
c"(A*)
= S U P { I ~ (am) ~~; E
s},
which is also useful to study the efficiency of channel for communication of information. 54. Applications of Auantum Channel to Optical Communication Processes
As applications of a mathematical expression of quantum channel given in <6> of 52, we here discuss a n attenuation optical communication process and derivation of error probability. 4.1 Attenuation process
4.1.1 Conventional expression: A quantum system composed of photons is described by the Hamiltonian H = a*a+1/2, where a* and a are creation and annihilation operators of photon, respectively. The Schrodinger equation H z ( q ) = Ez(q) is easily solved, whose eigenvalue is E,, = n+1/2 (n 2 0) and the eigenvector z,,(q) for E,, is ( l / ( d ~ z n ! ) l / z ) H , , ( & )exp(-q2/2), where H,,(q) is the n-th Hermite function. The Hilbert space of this system is a certain closed linear span of linear combinations of z R ( q )(n=0,1,2 .). The model is considered as follows: When n1 photons are transmitted from the input system, ml photons from the noise system add t o the signal. Then m2 photons are lost to the loss system through the channel, and nz photons are detected in the output system. In this model, the Hilbert spaces are denoted by 'HI, 'Hz,XI, XZ and their coordinates represented wave functions are respectively denoted by
..
+
+
According to the conservation of energy (nl ml = nz mz), we take a following linear transformation among the coordinates ql,tl, q z , t z of the input, noise, output and loss systems, respectively : qz = %?l
+ Ptl,
t z = -pq1
+ crtl.
( 2+ p2 = 1)
365 90
7r"
= V(.)U*.
Thus from the expression A" of quantum channel of <6> in 52, the attenuation channel A" is given by A'p = trx,U'(p 8 Y)U, (44 with the noise Y = Iyc) >< yc)l E 6(K1)due t o the "zero point fluctuation" of electromagnetic field (yc) is a vacuum state vector in XI). Note that we may take 7 i = 312 = K2.
4.1.2 A simple expression [l]: The above attenuation process (4.1) can be written by a little simple way. Let 3-1 = X: = r(C) (Fock space) and let
denote the coherent vector, where In r(3-1)to r(3-1)8 r(n)85
> is the number state.
Define a mapping V from
vie >= Iae > 8lpe >
+
= 1. with a l p E C, lala V represents the interaction of the signal with an apparatus or a receiver and it means that by the effect of the interaction a coherent signal (beam) lO > splits into two signals (beams) still coherent but of lower intensity although the total intensity (energy) is preserved by the transformation. Now, let us show the equivalence of the above operator V and the operator U in the conventional expression.
366 91
which implies, for any nonnegative integer N,
Thus U equals to V by replacing p with -p. Therefore the attenuation channel can be written as
4.2 Error probability
Let
(i
be the quantum code corresponding t o a symbol q E C: For simplicity, take
c = { O , l } e E = {(ol&}. One expression of quantum code is due to a state of photon; for instance, ( 0 is the vacuum state and €1 is another state such as a coherent or a squeesed state. and ($'I in the input system are transTwo states (quantum mechanical codes) mitted to the output system through a channel A*. Consider a Z-type signal transmission, namely, the signal "0" represented by the state to (1) goes always to "0" represented
(c)
(r)
(c'
(c)
and the signal "1" represented by the state goes t o or other states. by Then the error probability qc comes from that the signal "1" is recognized as the signal "O", so that it is given by (1)
Qe
(2)
= t W € 1 )Eo = t ~ ~ a ( t r ~ a ~ * ( €v))€o j l )(2) (" (1)) Z : : e r
1
states
Based on this error probability qe, the error probability of PCM modulation with the to-tuple error correcting code with the weight N and that for PPM modulation are given by N
NCjqi(1 - qe)N-j,
p y f= j=to+l
pTPM
= Qe 1
where N C = ~ N ! / { ( N-j)!j!}. Concrete computation and physical discussion of error probability for some optical processes are given in the paper [9] of this volume, by which we obtain an interesting observation. A certain input squeezed state gives us a better error probability than the input coherent state; that is, the error probability very much depends on the way of squeezing the coherent state.
367 92
References: I here simply give references of mine and see [2] and [6] for reference.
a
complete
[l]L. Accardi and M. Ohya : “Compoud channels, transition expectations and lifti n g ~,”preprint. [2] M. Ohya : “On compound state and mutual information in quantum information theory”, IEEE.Trans.Inf.Theory, 29, pp.770-774 (1983). : uNote on quantum probability”, L. Nuovo Cimento, 38,pp.402-404 131 (1983). [41 : “Entropy transmission in C*-dynamical systems”, J. Math. Anal. Apple, 100, pp.222-235 (1984). [5] M. Ohya, H. Yoshimi and 0. Hirota : “Rigorous derivation of error probability in quantum control communication processes”, IEICE of Japan, J71-B, N0.4~533-539 (1988). [6] M. Ohya : “Some aspects of quantum information theory and their applications to irreversible processes”, Rep. on Math. Phys., 27, pp.19-47 (1989). : “Information theoretical treatment of genes”, Trans. IEICE, E70, No.5, [71 pp.556-560 (1989). : “Fractal dimensions of states”, to appear in Quantum Probability and PI Applications (edited by L. Accardi and W. von Waldenfels), KLUWER Publishing Company. [9] M. Ohya and H. Suyari : “zligoxous derivation of error probability in coherent optical communication”, in this volume. [lo] N. Watanabe : “Efficiency of optical modulations with coherent state”, in this volume.
T H E TRANSACTIONS OF T H E IEICE, VOL. E 7 2 , NO. 5 MAY 1989
556
c
0 1989 IEICE
LETTER
)
(Special Issue on Information Theory and Its Applications)
Information Theoretical Treatments of Genes Masanori OHYAt, Member SUMMARY Some concepts in information theory are tried t o apply to the study of genes. The mutual entropy is used to define a measure indicating the similarity between two genetic sequences. The alignment of sequences is briefly discussed. Some phylogenetic trees are written by using the entropy measure. According to this results, usefulness of information theory is discussed in the study of genes such as molecular evolution.
1. Introduction
Recently it becomes possible to study the biological evolution from genes, more precisely, from information carried by DNA. For rather long time, the evolution has been studied through the forms of several species existing now and of fossils found in stratums. When the evolutionary process is discussed, the form of organism is considered to be an important clue, but there are several different interpretations of the form. Thus nowadays, DNA or amino acid sequences are often used to establish more objective interpretation of biological evolution. A genetic information preserved on DNA is regarded as a message made by a sequence of four bases (adenine ( A ) , guanine ( G I , thymine (T)and cytosine ( C ) ) . This information is transmitted to RNA and is used to make twenty amino acids and proteins. If we regard biological replication or multiplication as a communication of the message carried by DNA, then information theory is applicable in analysis of base sequences (DNA) and amino acid sequences (proteins). This letter proceeds as follows : In the first place, we briefly mention how to compare genetic sequences. Namely, we explain the idea of the alignment of sequences for this purpose. Secondly, we construct complete event systems for genetic sequences. Thirdly, a measure indicating the similarity of two sequences is formulated by using the mutual entropy (information), Then the UPG method writing phylogenetic trees is briefly reviewed. Finally, we write phylogenetic trees for biological evolution by using the genetic matrix obtained through our measure of similarity, and we compare our results with those derived by other kinds of genetic matrices. Manuscript received December 5, 1988. Manuscript revised February 20, 1989. t The author is with the Faculty of Science and Technology, Science University of Tokyo, Noda-shi. 278 Japan.
2.
DNA Sequences and Amino Acid Sequences
We briefly review fundamental facts of DNA and animo acids for self-consistency of this letter. Self-replication and multiplication of a living system are caused by a gene having the information of the form and functions of the system. In 1944. Avery, MacLeod and McCarty found that a gene is a part of DNA itself. In 1953, by the X-ray structure analysis, Watson and Crick showed that the shape of DNA is a double helix. The main chain of DNA consists of a regular sequence of alternating deoxyribose-phosphate. Each of four bases, ( A ) , ( G ) , ( T ) , ( C ) , joins with a part of deoxyribose. The bases on two main chains are joined according to the constraint that ( A ) only bonds to ( T ) and ( C ) only bonds to ( G ), which is called the Watson-Crick base-pairing rule. The character of a protein is determined by three of four bases. The mechanism for the production of a protein from DNA is the following : The information stored in the base sequence of a main chain of DNA is copied by messenger-RNA (mRNA). The four bases in DNA are transformed to the following four bases on m R N A ; ( A ) , ( G ) , ( C ) , ( U ) (uracil corresponding to thymine ( T ) in DNA), respectively. The amino acid sequence of RNA is translated into a protein in ribosome. A triplet of mRNA bases, called a codon, specifies one of twenty amino acids. A protein is synthesized by a combination of several amino acids starting from the codon named f-methionine to a stop codon. Here we list the names and symbols of twenty amino acids for the later use: alanine ( A ) , cysteine ( C ) , aspartic acid ( D ) , glutamic acid ( E l , phenylalanine ( F ) , glutamine ( G ) , histidine ( H ) , isoleucine ( I ) , lysine ( K ) , leucine ( L ) , methionine (M), asparagine ( N ) , proline ( P ) , glutamine ( Q ) , arginine ( R ) , serine ( S ) , threonine (TI, valine ( V ) , tryptophan (W) and tyrosine ( Y ) . 3. Alignment of Genetic Sequences
Let Sa and W be amino acid sequences two organisms specifying an identical protein. These sequences are generally considered to be close each other, but there might exist some difference betweeen them
369 LETTER (Special Issue on Information Theory and Its Applications)
because of the biological evolution. For instance, suppose that these sequences are given as A : A C D A C D E
8 : A E D E A C D, where each alphabet A, C, D, ... represents each amino acid existing in the sequences. The above two amino acid sequences look not so close each other, whose difference might come from the fact that some amino acids in A (resp. B ) change, delete or insert in 8 (resp. A ) during the course of the biological evolution. Therefore we have to align two sequences by taking account of this fact. When an amino acid in A ( r e s p . 8 ) is considered to be lacking in 8 (resp.A), we insert the gap (dummy) " in the corresponding place of B (resp. A ) , and when an amino acid, say A, in A is considered to change to another amino acid, say B, in 8 , we correspond A with B. This arrangement makes us possible to take the matching (alignment) of two amino acid sequences, whose result will be
*"
d : A C D
*
551
4. Event Systems and Entropy Ratio
For a set &=(Al, Az,..., A,} and the occurrence probability p=(p,, pz, ...,p,,]of each event A k (i. e. ph= P(Al), Z p , = l ) , a pair (A, P) is called a complete event system. For two complete event systems (A, p) and (3, 4 ) the compound event system is denoted by (4.8,r ) w h e r e A B = ( ( A , B ; A E A , B E 8 1 and 7ij=g(Ai,E,) with ZL7ii=qj3 Xjyjj=pi, Shannon introduced several information measures to study the communication of inf~rmation'~'. One of them is the information (entropy) carried by a system (d, p), which is given by
S ( d ) = - 2 ' s z 1ogp2 The most fundamental information measure in Shannon' s communication theory is the mutual entropy expressing the amount of information correctly transmitted from an input system (4,p) to an output system (8,q ) , which is described as
A C D E
$ : A E D E A C D * . After this alignment, we can see the similarity of two sequences A and 8 as expected. The alignment can been done by using a computer on the basis of mathematical formulation of the distance between two sequence^""^'. The fundamental points of such a mathematical formulation are to define the distance d ( X , Y) of two amino acids X E A, Y 6 8 and to minimize the total distance between A and 8 such that dtOt,,= Z ( d ( X , Y); X E A , YE 81. For instance, the alignment of the amino acid sequences of Hemoglobin (I for human and carp becomes Human : V L S P A D K T N V K A A W G K V G A Carp : SLSDKDKAAVKIAWAKI S P HAGEYGAEALERMFLSFPT KADDIGAEALGRMLTVYPQ
In order to apply these two information measures to the study of amino acid (or DNA) sequences, we have to set the complete event systems of amino acid (DNA) sequences. For an amino acid (DNA) sequence A, the complete event system associated to A is nothing but (A, p) with the occurrence probability ph=p(AJ for each event Aa in A , where A& represents each amino acid (base). When we consider two amino acid sequences A and 8, it is not so easy to set a proper compound event system of two sequences. However, once we know the correspondence between A and 8,we can construct the compound event system ( A 8 , 7 ) , so that the information transmitted from (A, p) to ( 3 ,4 ) can be calculated. P) here contains Remark that the event system (A, as an event. Indeed "
*
"
TKTYFPH*FDLSHGSAQVK TKTYFAHWADLSPGSGPVK GHGKKV*ADALTNAVAHVD *HGKKVIMGAVGDAVSKID DMPNALSALSDLHAHKLRV DLVGGLASLSELHASKLRV DPVNFKLLSHCLLVTLAAH DPANFKILANHIVVGIMFY LPAEFTPAVHASLDKFLAS LPGDFPPEVHMSVDKFFQN VSTVLTSKYR LALALSEKYR
By constructing these event systems, a measure indicating the difference between two amino acid sequences can be introduced. This measure is called the entropy ratio"' defined by
370 T H E TRANSACTIONS OF T H E IEICE, VOL. E 7 2 , NO. 5
558
This quantity is regarded a s the ratio of the information transmitted from A into W to the information carried by A . By symmetrizing the entropy ratio. we here introduce a more suitable measure
Table I Genetic distance matrix.
r ( A ,W)=+ir(wlA)+ r ( AIS)), which may be called the symmetrized entropy ratio or the evolution entropy rate. 5. A Method Constructing Phylogenetic Trees
In this section, we briefly review the UPG (unweighted pair group clustering) method writing a phylogenetic tree. This method is initiated by P. H. A. Sneath and R. R. Sokai and by Nei’” it is now understood a s a way to divide organisms into several groups. The pair having the smallest difference makes the first group. Then we try to find a next group (pair or triple) giving the second smallest difference calculated for any pair out of organisms and the first group. Moreover we consider the difference between two groups, that is. the averaged difference of all pairs of organisms in two groups. We repeat this procedure and make a final relation among all organisms. Let us show this procedure by an example. Let the difference between an organism A and an which is an element organism W be denoted by p ( d , a), of the properly defined genetic (distance) matrix given in advance. The averaged difference between an organism A and a group ( ~ , 6 is denoted ) by p (90, ( 3 ,a)). and it is computed by
-: Fig. 1 Example of phylogenetic tree
Since the pair having the smallest difference forms a group, the group (..4,3) and the organism 6 are combined. We next compute the following three differences : p(((,d,3),6), .)=P(A,8)+Q(-RJJ)+p(~,
3
p(((n,8), 6 , 8)=P(A, & ) + d W ,
3)
B ) + P ( 6 , 8)
3
-~ 8 + 8+ 10 =8.3,
-
The averaged difference between a group (90.( 3 ,6 )) and a group (XI,& ) is computed a s
(a,& )) =
P ( ( d , ( 3 ,6)),
dA, Z))+P(A,& ) + p ( W , a)+P(w,G )+p( 6, dg)+P(d,8 )) 6 Now, suppose that the genetic distance matrix is given a s in Table 1. Then an organism A and an organism 3 are first combined together because the difference between A and W is the smallest. Secondly, we compute the differeces for a group (d,W) and one of three organisms d , a ,& : /$(A,B), 6)=P ( d , 6 ) + d - 8 a , ) =5+6,5.5, 2 2
3
( P ( B ) , R )=7).
Accordingly, we have a group
(a,G ). Finally,
d ( ( d . W ) , ), 6 (a,8 ) ) = 8 . 3 . On the basis of the above results, we can write a phylogenetic tree of these organisms as Fig. 1. In the next section, we write phylogenetic trees by using this UPG method with the genetic matrix constructed from our symmetrized entropy ratio r ( A , a ) .
6. Phylogenetic Tree by Hemoglobin a
In this section, we write phylogenetic trees for hemoglobin 0.We here consider the following species : Monodelphia (human. horse), Marsupialia (gray kangaroo), Monotremata (platypus), Aves (ducks, greylag goose), Crocodilia (alligator, nile crocodile), viper, bullfrog tadpole, Osteinthyes (carp, goldfish), port jackson shark. All data are taken from@’.Let the degree of difference between two organisms A and W be given as PdA, W)=l-
r(A,8)
371 LETTER <Special Issue on Information Theory and Its Applications) 559 Monodeiphia Marsupialia Monotremata
Aves
i
Tree X
Crocodiiia viper
1
bul I f r o g .
'
OSteichthyeS
Tree Y
P. j. shark
Fig. 5 RF operations
Fig. 2 Tree constructed by entropy ratio. Table 2 RF-distance between trees
1
Monodeiphia Marsup i a I i a
Fig.i
Aves
Fig.2
Fig.1
I
Fig.2
1
Fig.3
2
2
2 4
Crocodiiia Monotremata viper
bul I f r o g . Osteichthyes P. j. S h a r k
Fig. 3 Tree constructed by substitution rate
Aves
Crocodilia
~
viper bullfrog.
I
p. j. s h a r k
Fig. 4 Tree constructed by fossils.
or pz(I,
m=$,
where n is the number of replacing amino acids between the sequences j4 and W and N is the number of amino acids in I or 59 both after the alignment. Then we can make the genetic matrix pa from p d I , W ) ( k = l , 2) such that p a = ( p a ( I , W ) ) . by which we can construct the phylogenetic tree for the above species. The difference p, is new, but the difference pz (or its modification) is one often used in several occasions. The phylogenetic trees written by P I and PZ are shown in Fig. 2 and Fig. 3, respectively. The Fig. 4 is a result estimated from the fossils of
species. At first glance, Fig. 2 is closer to Fig. 4 than Fig. 3. For more scientific judgement among the resulted phylogenetic trees, Robinson and Foulds considered'7' a certain operation which expresses the movement of a branch between two phylogenetic trees. For two phylogenetic trees, say X and Y , if we can overlap X with Y by moving n branches in X,then the difference between X and Y is said to be n. There are two such operations a and /3 : CI is the operation adding a branch in a tree and B is that eliminating a branch, as shown in Fig.5 for an example. The difference of three phylogenetic trees Fig. 2, Fig. 3, and Fig. 4 are shown in the Table 2. Therefore if we believe the phylogenetic tree written by the data of fossils and if the UPG method is a plausible way to write the phylogenetic tree, then the genetic matrix pl constructed by the entropy ratio will be better than the genetic matrix pz. Even so, we have a little difference between Fig. 2 and Fig. 4, so that we might need to refine both UPG method and genetic matrix in order to write more accurate phylogenetic trees, about which we are now on the working bench. 7 . Phylogenetic Tree by Cytochrome C
A phylogenetic tree written by using Cytochrome C
is shown in Fig, 6. The genetic matrix for this phylogenetic tree is due to the distance pi. We here consider the following species : Vertebrate (human, horse), Invertebrate (locust, garden snail), Higher plats (wheat, ginkgo biloba) , Algae (enteromorpha intestinalis), Fungi (yeast, debaryomyces kloeckeri) , Protozoa (euglena gracilis, tetrahymena pyriformis) . Cytochrome C is suited for estimating phyletic lines among organisms being far from each other because the substi-
372 T H E TRANSACTIONS OF THE IEICE, VOL. E 72, NO. 5 560 Vertebrate Invertebrate Higher p l a n t s Algae Fungi Protozoa (euglena) Protozoa (tetrahymena) Fig. 6 T r e e constructed by entropy ratio with C
tution of amino acids in Cytochrome C may take a longer time than that in other protein. This phylogenetic tree well matches to some known results, for example, by Whittaker's'''.
rate. Moreover, the phylogenetic tree (Fig. 6) for cytochrome C by our measure turns out to be well-matched to an experimental result as explained in Sect. 7. In the case when the fossils of some organisms have not been found, we can use the entropy ratio and the UPG method to construct the phylogenetic tree for these organisms. Therefore we can conclude that the information theoretical approach to genetics might give us a clue to understand some aspects of biological evolution. Further development of the alignment and more detail examinations for phylogenetic trees with an extension of the UPG method will he discussed elsewhere. References
8. Consequences
In this letter, we try to show that the information theoretical treatments will be important for the study of genes such as molecular evolution. We introduced the symmetrized entropy ratio as an application of the mutual entropy, and we wrote phylogenetic trees of some species by the following genetic matrix (GM for short) with the UPG method : ( 1) GM constructed from the symmetrized entropy ratio for hemoglobin LI ; ( 2 ) GM constructed by a conventional method, namely, the substitution rate of amino acids, for hemoglobin LI ; ( 3 ) ( 1 ) for cytochrome C. We compared the phylogenetic trees written by the above methods with that by the fossils of the same species. As a consequence based on the RF-criterion, the tree by ( 1j is better than that by ( 2 j. Namely, the information theoretical treatment gives us better description for biological evolution than the conventional treatment with the substitution
S.B. Needleman and C. D. Wunsch: "A general method applicable t o search for similarities in the amino acid sequences of two proteins", J. Mol. Biol., 48, 443-453 (1970). P. H. Sellers : "On the theory and computation of evolutionary distances", SlAM J. Math., 26. pp. 787-793 (1974). M. Ohya and Y . Uesaka: "Amino acid sequences and DP matching", Amino Acid Sequences and DP Matching", Sci. Univ. of Tokyo, Res. Rep. (1986). M. Ohya and Y . K i t a g a w a : "A mathematical analysis of DNA sequences", Symp. Appl. Fuct. Anal., 8. pp.36-47 (1985) P. H. A. Sneath and R. R. Sokal : "Numerical taxonomy", W H Freeman, S a n Francisco (1973) W. C. Barker , et al. : "Protein sequence database of the protein identification resource (PIR)", N B R F (1985). D. F. Robinson and L. R. Foulds : "Comparison of phylogenetic trees'', Math. Biosci., 53, pp. 131-147 (1981). L. Margulis and K. V. Schwartz : "Five Kingdoms". W. H. Freeman (1982) H. Umegaki and M. Ohya : "Entropies in Prohablistic Systems", Kyoritsu Pub. Company (1983).
373 Vol. 27 (1989)
R E P O R T S ON M A T H E M A T I C A L P H Y S I C S
No. 2
SOME ASPECTS OF QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS TO IRREVERSIBLE PROCESSES* MASANORI OHYA Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278, Japan (Received October 12, 1986
-
Revised M a y 16, 1988)
Several quantum entropies are systematically studied and the mathematical structure of a channel in optical communication processes is presented. As applications of these entropies and channel, general formulas of error probability in some communication processes using, for instance, coherent or squeezed states, are obtained and the irreversibility for some dynamical processes is discussed.
Introduction The notion of entropy was introduced by Clausius around 1865 in order to discuss the thermal behaviour of physical systems on the basis of Carnot’s work. Since then, the irreversibility of physical systems such as the second law of thermodynamics has been understood in terms of the entropy increase. It was Boltzmann who first tried to explain the entropy increase from the microscopic dynamics, that is, from the dynamics of large numbers of atoms. As every fundamental equation of motion such as Newton’s equation, Schrodinger equation or Liouville equation, is invariant under the time reflection, it is almost impossible to show the entropy increase, hence to explain the irreversibility, by a direct application of such a fundamental equation of motion. In this paper we consider quantum mechanical systems but our formulation mathematically contains any classical system as a special case. A quantum physical system is usually described by a density operator e, and the entropy for the state 4 is given by S ( 4 ) = - treloge
according to von Neumann [l]. For a Hamiltonian dynamics, the state e changes in time due to the unitary time evolution U , generated by the Hamiltonian H of the
*
An invited review paper.
374
20 system: U , = exp(itH) and
MASANORI OHYA
e, = Ut.QU,,
so that we have
S(eJ = S ( e ) , that is, the entropy is invariant under the time evolution of the system. Therefore, in order to explain the irreversible phenomena rigorously, we have to (1) modify the fundamental equation of motion in quantum mechanics, for instance, by adding some external effects (noise, fluctuation, etc.) to the reversible equation, (2) introduce new concept or criteria, besides the entropy, interpreting the irreversibility, or (3) construct a new theory containing quantum mechanics as a special case. There are many trials along the line of (l), but most of such trials are not so satisfactory, namely, some modifications do not involve the entropy increase in themselves and others are not mathematically well controlled. As for (2), there are a few different directions introducing new criteria, one of which is to develop von Neumann’s quantum mechanical entropy and formulate the so-called quantum information theory along the ideas of Shannon [2], in which we might be able to find a useful expression for an irreversible process. Apart from comprehension of the irreversibility, rigorous formulation of quantum information (communication) theory is very important from both mathematical and physical points of view because of the following reasons: (i) Optical communication of information is an indispensable technique today, and photon is a typical object of quantum mechanics, so that we like to have a rigorous theory describing quantum communication processes. Concerning this in terms of quantum control theory and quantum statistics, there have been several interesting investigations initiated by Helstrom, Liu, Gordon, Holevo and others [68, 69, 72, 73, 74, 75, 76, 771, although I will not review them here but present them as the Part I1 of this paper on some other occasion. (ii) An aspect of Shannon’s information theory is the theory of entropy, so the formulation of quantum information theory agrees to some extent with the development of the theory of quantum mechanical entropy initiated by von Neumann. And such a development will be useful in physics, for example, to study some cooperative behaviours [3, 41 of physical systems. In this paper, we review and systematically reformulate, with a few new concepts and extensions, some of our works concerning (1) and (2) above. More precisely, the following topics are considered here: (1) Formulation of several entropies in general quantum systems. (2) Mathematical construction of a communication channel and its application to optical communication processes. (3) Reconsideration of irreversibility with some entropies.
$1. Preliminaries In this section, we briefly review the gist of information theory in classical (commutative) dynamical systems (CDS for short) and fix the notations used throughout this paper both for CDS and quantum (noncommutative) dynamical systems (QDS for short).
375
21
QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
Once we make clear all mechanisms of the state change, we can know almost all properties of a physical system because each aspect of the physical system is described by a certain state. One of the most general descriptions of the state change for CDS is suggested in communication theory of Shannon and its measure theoretic extension by Kullback, Leibler, Kolmogorov, Gelfand, Yaglom and others [ S , 6, 71. Therefore it is interesting to study the state change for QDS following Shannon's philosophy, which is a motivation of the present work. Every CDS is generally described by measure theoretic terminology, that is, a state in CDS is expressed by a probability measure ,u on a measurable space (52, 9) and an observable of the system corresponds to a real random variable on 52. The state change in CDS is described by a mapping from P(Q), the set of all probability 9)). Generally, measures on 52, into itself (or P ( 0 ) on another measurable space the state change in CDS is given by a mapping between two dynamical systems, P(Q)) and an output system 9, P(a)). The namely, an input system (52, 9, following linear mapping A* from P(52) to P ( 0 ) is important and called a channel (or channeling transformation):
(a, (a,
@(Q) = A*cp(Q)= S4o, Q)dcp(o),
cp~P(%
(1.1)
12
where II is a mapping from 5 2 x 9 to [0, 11 satisfying the following conditions: (i) A(., Q) is a measurable function on 52 for each Q E and~ (ii) A(o,. ) E P ( Q )for each w E 52. This A is often called the Markov kernel on 52 x 9. It was shown by Umegaki [S, 91 that for the above A, there exists a unique bounded linear map A from B ( a ) , the set of all bounded measurable functions on to B(52) satisfying (i)f 2 0 +A(f) 3 0; (ii) f,.lO+A(f,)lO; (iii) @ ( f ) (= Sfd@) = J A ( f ) d c p .
a,
ii
R
Shannon's definition of the entropy of a state (probability distribution) p
'(PI
=
-x!?klogpk'
=
(pk}is (1.2)
k
When 52 is a discrete set, say 52 = {ol, 02, 03,. . . , on}, the occurrence probability of an event W E Q is denoted by pk = p(ok) and the probability distribution of 52 is denoted simply by p , namely, p = (pl, p 2 , p s ,... , p , } . For two states p , q on 52, the relative uncertainty between p and q is expressed by
which is called the relative entropy of p and q, and it was extended in a general P(52)) by Kullback, Leibler and others. Their definition of probability space (52, 9, the relative entropy is
{9
flog(dcpld*)dcp
'(cpl*)
for cp and
Ic/ in P(52).
=
oo
if cp 6
*>
otherwise
( 1.4)
376
22
MASANORI OHY A
When a state cp of an input system dynamically changes to a state Cp(= A*cp) of an output system under a channel A*, we ask how much information carried by cp can be transmitted to the output system. It is the mutual entropy (information) that represents this amount of information transmitted from cp to Cp, which is defined in terms of a compound state of cp and Cp and the relative entropy. The compound state @ of cp and (p is a measure (state) expressing the correlation between cp and Cp, and it is given by
for any Q, e . 9 , Qz t.F.The mutual entropy in CDS is defined by the relative entropy S(.(.) such as A*) = S(@I@J,
(1.6)
where Qi,, is the direct product state (measure) cp@Cp. In particular, if cp is a probability distribution p = {pk} and the channel A* is a transition probability (pij), then the compound state ds is the joint probability distribution: Qi = { p ( i , j ) ) with p(i, j ) = pijpj and the mutual entropy becomes
I@; A*) =
j)log(p(i, jIlPj4i)
(1.7)
i,j
with qi = c p i j p j . The following inequality is a fundamental inequality for comj
municatjon theory. 1.1 (Shannon). 0 < Z(p; A*) < S(p). THEOREM
See [9-121 for the classical information theory. Around 1950 von Neumann reformulated quantum mechanics on Hilbert space and demonstrated the theory of operator algebras [1] and Haag and Kastler [13] found that the C*-algebraic method is important for studying some physical systems with infinite degrees of freedom. That is, the study of such a physical system without using a Hilbert space is essential when the system involves, for instance, a kind of symmetry breaking. The operator algebraic method essentially starts from 2 set d containing all physical observables of interest and the set 6 (or G(d)) of all states The most fundamental set d is a C*-algebra. A symmetry of a physical system on d. is defined by a * preserving automorphism a of d and the action of a symmetry group G is a homomorphism a: G-+Aut(d), the set of all automorphisms of d satisfying (i) ag(a,(A))= agk(A)for any g, k E G and A ~ d (ii); a,(.) is continuous in some topology. A concrete and important C*-algebra is a von Neumann algebra although it is defined on a Hilbert space 2.A subset % of B ( X )is said to be a von = %, where a’’ = (’3’)’ with %’ Neumann algebra acting on a Hilbert space JF if a’’ = { A d ? ( . * ) ; AB = B.4, B E % ) .
377 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
23
Before closing this section, we show the correspondences between CDS and QDS in Table 1. Table 1
I
QDS
CDS
Observable
real random variable on measurable space
self-adjoint element of d (C*-a'lg.) or %(v.N. alg.)
State
probability measure
p.l. fnal cpeG(d) with 4 m=1
cp E
Consult Refs. [14--211 for the details of operator algebra and noncommutative probability theory and their physical applications. $2. Entropy in C*-systems Let us discuss the entropy in C*-dynamical systems introduced in [22]. The formulation of quantum mechanical entropy was presented by von Neumann about 1930, 20 years ahead of Shannon, and it now becomes a fundamental tool in analysing physical phenomena. His entropy is mentioned in Introduction; namely, for a density operator e E G(X'),the set of all density operators in a Hilbert space 2, the entropy is
S ( e ) = --reloge, which, in terms of any CONS {xk} in Z,equals
(2.1)
This does not depend on the choice of the CONS {xk}. Now, the spectral set of e is discrete, so that we write the spectral decomposition of e as n
where I , is an eigenvalue of e and P, is the projection from % onto the eigenspace associated with A,. Therefore, if every eigenvalue 1, is non-degenerate, then the dimension of the range of P , is one (we denote this by dim P, = 1). If a certain eigenvalue, say A,, is degenerate, then P, can be further decomposed into onedimensional projections: dimP,
P,
=
c Ey).
j= 1
(2.3)
378
24
MASANORI OHYA
where EY) is a one-dimensional projection expressed by EY) = Ix$'")(x$'"Iwith the eigenvector x$")( j = 1, 2,. . . , dim P,) for A,. By relabelling the indices j , n of { E Y ) } , we write
with
A, 3 A, 3 .. . 2 An 2 .. . ,
(2.5)
E,IE, ( n # m). (2.6) We call this decomposition the Schatten decomposition [23]. Now, in (2.5), the eigenvalue of multiplicity n is repeated precisely n times. For example, if the multiplicity of A, is 2, then A, = A,. Moreover, this decomposition is unique if and only if no eigenvalue is degenerate. In the sequel, when we write e = xA,E,, it is the n
Schatten decomposition of e, otherwise stated. For two Hilbert spaces %l and %, let % = %l@A"2 be the tensor product and If2and let us denote the tensor product of two operators Hiblert space of S1 A and B acting on and A?,,respectively, by A O B . The reduced states el in 8, and Q 2 in %, for a state e in % are given by the partial traces, which are denoted by ek = t r x j e ( j # k ; j , k = 1, 2). The properties of S(e) are summarized in
THEOREM 2.1. For any density operator e E G ( X ) , the followings hold: (1) Positivity: S(e) 2 0. (2) Symmetry: Let e' = U - ' e U for an invertible operator U . Then
S(e') = S ( d . (3) Concavity: S(Ae,+(l-A)e2) 3 AS(el)+(l-A)S(~,) for any el, Q,EG(%) and any AECO, 11. (4) Additivity: S(Q1@e2)= s(e,)+s(@,) for any e k E G ( 8 k ) . (5) Subadditiuity: For the reduced states el, e2 of @EG(%10%2), S(e)
(6) Lower Semicontinuity: Zf
S(@l)+S(@,).
lie, - e 11 , (= tr le, -el)
-+0,
then
S(e) d lim inf S(e,). (7) Continuity: Let e,, e be elements in G ( X ) which satisfy the following conditions: (i) e, -+Q weakly as n -+ co,(ii) en d A (Vn)for some compact operator A, and (iii) -Ia,loga, < co for the eigenualues {ak} of A. Then S(e,)+S(e).
+
k
( 8 ) Strong Subadditivity: Let 8 = X1@%,@X3 and denote the reduced states trJPkeand trXiBxj @ by eij and @ k , respectively. Then S(e) +S(e2) d S(e12)+s(e23) and s(@l) + s(@Z) d s(@13) + s(@23).
379 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
25
The proofs of the propositions of this theorem can be seen in [24-291. There exists some difference between in CDS and in QDS, for instance, the monotonicity is satisfied in CDS but not in QDS. In order to discuss some physical phenomena, for instance, phase transitions, we had better start without Hilbert space. Therefore we here formulate the entropy of a state in a C*-dynamical system. Let ( d ,6 , a(R))be a C*-dynamical system and Y be a weak* compact and convex subset of 6, e x 9 be the set of all extreme points From the Krein-Milman theorem, Y is equal to the weak* closure of convex of 9. hull of ex 9. We are interested in the following three cases for the set 9: (1) Y = 6, (2) Y = I(a),the set of all a-invariant states (i.e., q(at(.))= cp(.)), (3) Y = K(a),the set of all KMS states at an inverse temperature fi (i.e., for any A , B in d , there exists a bounded function FA,B(z)of a complex value z continuous on and holomorphic in the strip - b d Imz d 0 with boundary values: FA,B(t)= cp(a,(A)B) and FA,B(t-ifi)= cp(Ba,(A)) for any t e R [30]). Note that K ( a ) c I(a). Every state cp E Y has a maximal measure p pseudosupported [19, 201 on ex Y such that cp =
J adp. Y
The measure p giving the above decomposition is not unique unless Y is a Choquet simplex [20, 311, so that we denote the set of all such measures by M , ( Y ) . Take
where 6 ( q ) is the Dirac measure concentrated on {cp}, and put H(pL)= -xpklogpk k
for a measure p e D , ( 9 ) . Then the entropy of a state cp~9w.r.t. Y is defined by SY(cp) =
{+oo
inf(H(d; p E D , ( Y ) ) if D , ( 9 ) = 0. 7
This entropy is an extension of von Neumann's entropy as shown below, and it deDends on the set Y chosen. Hence it represents the uncertainty of the state Three interesting entropies S"(cp) ( = S(cp) for measured in the reference system 9. short), Srca)(cp)( = Sr(cp) for short) and SKc"'(cp) (= SK(cp) for short) are generally different even for c p ~ K ( a ) . THEOREM 2.2. When d = B(&) and a, = Ad(U,) with a unitary operator U,, for any state cp given by cp(.) = tre. with a density operator Q, the followings hold:
380
26
MASANORI OHYA
(1) S(cp) = -trgloge. (2) If cp is an a-invariantfaithful state and every eigenvalue o f g is non-degenerate, then S'(cp) = S(cp). (3) I f c p ~ K ( a )then , SK(cp) = 0. Sketch of proof: ( 1 ) Let g
=
2Akgk be a decomposition of g into extremal states k
mg, (i.e., g:
= gk). It
is easily seen that -
1Aklogl, attains the minimum value when k
the above extremal decomposition is the Schatten decomposition of e. Hence S(cp) = -trglogg. (2) Since cp is a-invariant, the equality [U,, g ] = 0 holds for all t E R. From the Ek] = 0 for each E , of the Schatten decomposition assumptions, we have [U,, g = x l k E k .Thus E , is a-invariant for every k, by which we obtain S(cp) 3 S'(qD). The k
converse inequality is shown by using the ergodic decomposition of cp. (3) The KMS state is unique for d = B ( . X ) , so SK(cp) = 0, Q.E.D. There are some relations among S(cp), S'(cp) and S"(cp), for instance, we have
THEOREM 2.3. For any c p ~ K ( a )the , followings hold: (1)SK(44G S'(cp). (2) S K ( d d S ( d (3) If our dynamical system (d, a@)) is G-abelian on cp, then SK(44d S ' ( d d S(44.
((a, a@))
is called G-Abelian if E , x , ( d ) " E , is an Abelian von Neumann algebra, where E , is the projection from 2 onto the set of all U,(t)-invariant vectors.) (4) If our dynamical system (d, a(R)) is q-Abelian, then mcp) S'(cp).
( ( da(R)) , is called q-Abelian i j the equality
lim T+co
l T -
1cp(C*[a,(A),B ] C)dt = 0
To
holds for any A , 3 , C E ~ . ) Sketch of proof: I t is enough to prove the case when the decomposition of cp is discrete: cp = c I k c p k .The set of such states is denoted by Y,. k
(1) The extremal decomposition cp = ~ l k c p ofkc p ~ K ( ainto ) ~ ex'K(a) is unique k
and orthogonal (i.e., cp,Icp, for n # m; see Section 5 for the definition of .the orthogonality), so the set inclusion ex K ( a ) c I ( @ )implies that each cp, can be further decomposed into the ergodic states. We denote this ergodic decomposition by
381 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS cpn
= xP;$k,
27
$kEexI(a). Then
k
S’(cp) = inf{ - ~ ~ n P Y o g ~ n {PPi; );) = C ~ , S ’ ( c p n ) + S K ( 4 42 S K ( d . k,n
n
(2) is proved similarly as (1) and (3) is a consequence of the uniqueness of the ergodic decomposition of cp by the G-Abelianness, and (4)is due to the set inclusion ex K ( a ) c ex I ( a ) obtained by the q-Abelianness, Q.E.D. This theorem tells us that even if the entropy S(cp) of cp is infinite, the entropy measuring in a proper reference system Y becomes finite. Therefore our entropy can be applied in continuous systems for CDS. Moreover, when a physical system has a symmetry breaking, the entropy SK(cp) might change w.r.t. some parameters such as temperature, so that our entropy can be used to study some phase transitions in physical systems, which will be discussed elsewhere [32]. By the way, most properties of von Neumann entropy S ( Q ) also hold for our entropy Sy(cp) under some conditions, for instance: S”(cp)
S”(cp) , 2 0 and = 0 iff c p ~ e x Y . THEOREM 2.4. (1) Positivity: For any ~ € 9 (2) Symmetry: For the dual map E* from ex .4c to ex Y of a *-automorphism &from a2 to d,put cp‘ = E * ( ( P ) . Then S”(cp’) = S”(cp). (3) Concavity: For any cp, $ E G and AE[O, 11, put w=Acp+(l-A)$. Then (i) S ( w ) 2 AS(cp)+(l-A)S($); (ii) when cp, $ € K ( a ) , SK(w)2 ASK(cp)+(l-A)SK($) and S’(w) >, /zS’(cp)+ +(1 -A)S’($). (iii) when cp, $ ~ l ( a and ) if one of the following conditions is satisfied (a) every element of the centres Z , for cp and 2, for $ is invariant under a; (t E R), the canonical extension of at; (b) (d, a(R))is G-Abelian for cp and $, then we have S’(o) 2 AS‘(cp)+(l -A)Sr($). (4) Additivity: Let B = &@J? and yt = a,@E,. For a weak* compact convex subset Y(y),if the extremal decomposition of any state in Y(y) is unique, then SY(Y)((p@$)= sY(=) (cp) S”q$). (5) Lower Semicontinuity: Suppose that there exists a unique maximal orthogonal decomposition measure defining (2.7)for each cp in Y and any two states o,$ E ex Y are orthogonal. When a sequence {cp,} c Y converges to a state cp~Yin norm ~ ~ c p n - c p+~O~ as n-tco, we have S”(cp) Q liminfS”(cp,). We omit the proof of this theorem, which is essentially same as those given in [22, 331.
+
53. Quantum mechanical channels By a direct extension of the classical channel A* given by (1.3) and its dual expression, we define quantum mechanical channels in this section on the basis of [34, 35, 72, 751. In order to define a channel in QDS, we need two dynamical
382
28
MASANORI OHYA
systems, an input system and an output system denoted by C*-triples (d, 6, a)and sd satisfying
(2,G,i),respectively. A mapping A from d to
c BiA(AT A j ) B j3 0 n
i,j= 1
for any Bi E d , A j E d and every n E N is called a completely positive map. Remark that the usual completely positive map is a linear map with (3.1). The dual map A* of A from Y, the set of all positive functionals on d ,to 9 on d is called a quasichannel, and the map A* from 6to is called a channel. Namely, a channel is the dual map of a linear complete positive map. In this paper, we mainly deal with channels, but a quasichannel is indispensable when we consider Gaussian measures and their transformation [36]. Most physical state changes are described by such channels and we here give some examples encountered in usual discussions in physics [28, 37401. Let e be a density operator in a Hilbert space 2 of a physical system.
e
(1) Unitary evolution: Q +A:@ = AdU,(e) = U : e U , , t E R, where U , is a unitary operator on 2 generated by the Hamiltonian of the system, i.e., U , = exp(itH). (2) Semigroup evolution: e + A : @ = V:e V,, t E R+, where { V,; t E R + } is a one-parameter semigroup on 2. (3) Measurement: When we measure an observable A = x a n P n(spectral decomn
position) in a state e, the state e changes to a state A * @ by this measurement according to the rule Q + A * @ = P n e P , .
c n
(4) Reduction: If a system C , interacts with an external system C, described by another Hilbert space X and the initial states of C , and C, are e and cr, respectively, then the combined state Bt of C , and C, at time t after the interaction between two systems is given by
et = u : ( m w , , where U , = exp(itH) with the total Hamiltonian H of C, and C,. A channel is obtained by taking the partial trace w.r.t. X , viz.,
e +A:@ = tr,Q,. ( 5 ) Conditional expectation [41]: Let % be a von Neumann algebra, YJl be its von Neumann subalgebra and € be the conditional expectation (norm one projection [42]) if it exists, or the generalized conditional expectation [43] from % to YJl. Then the dual map €* of € is a channel. The following channels have been introduced in [35] to study in quantum 6 ,a) and G,5) be an input communication processes. Let two C*-triples (d, and an output system, respectively, and A* be a channel from 6 to 6. (Cl) A* is said to be stationary if A o i , = a,oA for any t E R . (C2) A* is said to be ergodic if it is stationary and A*(exZ.(a)) c exI(i) holds.
(a,
383 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
29
(C3) A* is said to be orthogonal if it maps any pair of orthogonal states into orthogonal states (i.e., cp I$+ A * cp IA* cp). (C4) A* is said to be deterministic if it is orthogonal and bijective. (C5) A* is said to be chaoticfor Y (c6 )if A*cp = A*$ for any pair cp, * ~ e x Y , and it is said to be chaotic if Y = 6. 94. Relative entropy
The relative entropy of two states was first introduced by Umegaki [44] for a-finite and semifinite von Neumann algebras. For two density operators Q and a it is defined as S(elo) = tre(1oge-logo).
(4.1)
Lindblad [45, 461 studied some fundamental properties of this relative entropy corresponding to those of Shannon's type relative entropy in CDS. There were several trials to extend the relative entropy to more general quantum systems and to apply it to some other fields [35, 47-61]. Here we review Araki's [49, 501 and Uhlmann's [52] definitions of the relative entropy and state the fundamental properties of the relative entropy. [Araki's definition] Let % be a-finite (this condition is easily removed [SO]) von Neumann algebra acting on a Hilbert space %' and q,$ normal states on % given by q(.)= (x, .x) and $(.) = ( y , . y ) with x, EX (a positive natural cone). The operator Sx,, is defined by S,,,(Ay+z)
= s"(y)A*x,
A€%,
s"'(y)z
= 0,
(4.2)
on the domain %y+(I-s"'(y))%, where s"(y) is the projection from % to { % ' y } - , the %-support of y. Using this SX,,, the relative modular operator Ax,, is defined as m
= (S,,,)*S,,,
with spectral decomposition denoted by
i 'Adex,,(A). Then
the
0
relative entropy is given by
otherwise, where $ 6 cp means that cp(A*A)= 0 itnplies $ ( A * A ) = 0 for A € % . [Uhlmann's definition] Let 9 be a complex linear space and p , q be two Moreover, let H ( 9 ) be the set of all positive hermitian forms a on seminorms on 9. 9 satisfying la(x, y)( d p ( x ) q ( y ) for all x, y ~ 9 Then . the quadratical mean QM(p, q) of p and q is defined by QM(p, q)(x) = ~up{a(x,x)'''; a ~ H ( 9 ) } , ~ € 9 ,
(4.4)
384
30
MASANORI OHYA
and there exists a function p,(x) of t E [0, 11 for each x E Y satisfying the following conditions [Sl, 521: (1) For any x ~ 9 p,(x) , is continuous in t, (2) pi12 = Q M b , 41, (3) P ~ / Z= Q M b , P A (4) PO+ I)/' = QM(pt, 4). This seminorm p t is denoted by QI,(p, q) and is called the quadratical interpolation from p to q. It is shown [52] that for any positive hermitian forms a,p, there exists a unique function QF,(a, p) of tE[O, 11 with values in the set H ( 9 ) such that QF,(a, p)(x, x)'/' is the quadratical interpolation from a(x,x)''' to p(x, x)"'. The relative entropy functional S(alP)(x) of a and p is defined as S(alP)(x) = -liminf(l/t){QF,(a, B)(x, x)-a(x,
4)
(4.5)
1-0
for X E ~ Let . 2 be a *-algebra d and cp, $ be positive linear functionals on d defining two hermitian forms cpL, $" such as cpL(A, B ) = cp(A*B) and $R(A, B) = $(BA*). Then the relative entropy of cp and $ is defined by S($lcp) = s($RIcpL)(I).
(4.6) If 9is a von Neumann algebra fn and cp, $ are normal positive linear functionals on fn, then the Uhlmann relative entropy is shown [57] to be equal to the Araki relative entropy. For a C*-algebra d and two positive linear functionals cp, $ on d, Uhlmann's definition can be directly applied. Further, by considering the GNS of the functional cp II/ and the canonical extensions cp", representation TC (= IT,+,) $" of cp, $ to ~ ( d )we " ,have the following [57]:
+
THEOREM 4.1. I n the above notations, the relative entropy S(cplII/) in a C*-system is equal to S(cp"l$") for its canonically extended von Neumann system. Both definitions can be thus used for states in C*-systems. Let us show that the expression for the relative entropy of two density operators Q and a can be derived from the Uhlmann expression. For normal states cp, $ on a von Neumann algebra B ( X ) such that cp(.) = tre. and $(.) = tra with density operators e and a, we get QF,(i+bR,cpL)(I,I ) = tre'-'af,
(4.7)
hence S($lq) = S(J/RJ~L)(I) = -liminf,,,(llt){QFt(J/R, = - lim inf,,,(l/t)tr(e'
cpL)(l, W $ R ( I ,I)}
-'a' - e) = tre (loge - log a).
(4.8) Here we summarize the fundamental properties of the relative entropy. For notational simplicity, we write a theorem in the von Neumann algebraic terminology. Namely, let cp, $ be normal states and {cp,}, {$"} sequences of normal states on a von Neumann algebra %.
385
31
QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
THEORLM 4.2. (1) Positivity: S f cpl$) 3 0. (2) Joinr Convexity: S(Iv$ (1- 1.)$2 licp, (1 - 2)ca,) < ?.S(I/~ Icp for a n y 2r[0, 11. (3) Additivity: ~ ( $ 1 0 $ 2 i c p 1 @ c p J = Icpl)+S(IC121V2). (4) Lower Semicontinuity: I f lirn II tjn- $ Ii = 0 and lim I/ cp,, - cp /I
+
+
+(1-2)S ( $ 2 Icp2)
w,
< lim infS($,Icp,).
n-
= 0,
then S($lcp)
n+m
-”
Moreover, if there exists a positive nunzber 2 satisfying
J !,I~
< kp,,
n-r m
then lim S($nIVn)
=
’($1~).
n-rm
(5) Monotonicity: For a channel A* from 6 to SV*$lA”cp)
e,
< Wid.
(6) Lower Bound: ~ ~ $ - c p ~ ~< 2 /S($lcp). 4
The proofs of (1)-(4) are given in [49, SO], and the proof of ( 5 ) is essentially given in [52], that of (6) in [54]. The relative entropy is related to the concept of sufficiency, and it can be used to classify some equilibrium states and stationary states [54, 57, 60, 62, 631. 95. Compound state and mutual entropy
As discussed in Introduction, when a state cp changes to another state (p under a physical transformation, we ask how much information of q is correctly transmitted to @, and the amount of this information is expressed by the mutual entropy (information) i n CDS. We like to formulate this mutual entropy in QDS for two states cp and @ = A*cp with a channel A*, so that we first set the compound state of the initial state cp and the final state @ expressing the correlation existing between these two states as an extension of the compound measure given by (1.7). The compound state @ on the tensor product C*-algebra d B 8 of two states cp on .dand (p on d should satisfy the following properties: (c. I ) @ ( A @ I )= cp(A) for any A E . ~ ; (c. 2) @(I@B)= @(B)for any BE^; (c. 3) the expression for @ contains the classical expression as a special case; (c. 4) @ indicates the correspondence between each elementary component (pure state) of cp and that of @. There are several states satisfying the above two conditions (c. 1) and (c. 2). For instance, the direct product state Q0 of cp and @ given by @o =
(Po($
(5.1)
is such a state, which corresponds to the direct product measure in CDS. We call a state satisfying the conditions (c. 1) and (c. 2) a quasicompound state. Let us define the “true” compound state having all the above conditions. Such a compound state is given through the decomposition (2.7) of the state 9. For a state cp in a weak”
386
32
MASANORI OHYA
compact convex subset Y of 6 and a channel A*, let p be an extremal decomposition measure of cp. A compound state @, of cp and A*cp with respect to Y and p was introduced in [58, 641
This state obviously satisfies (c. l), (c. 2) and (c. 4) because the measure p is pseudosupported by exY. The condition (c. 3) is indeed satisfied When d and d are Abelian algebras with measurable spaces (52, F)and (0, F),respectively, and cp is a probability measure on 52, the extremal decomposition of cp is unique and given by cp = j6,dcp,
(5.3)
R
where 6, is the Dirac measure concentrated at a point x E 52. Put A ( x , Q ) = A*6,(Q) for any x E 52 and Q €9. Then A is the Markov kernel defining the classical channel, and we have
s
@,,M2 Q ) = 6,(P)A*6,(Q)dcp = y l p ( ~ M xQVcp , = !4x, QWcp R
(5.4)
P
for any P E 9, Q E 9. Thus our compound state defined by (5.2) is the desired one, but the uniqueness condition of the compound state is an open question. This compound state might play a similar role as the joint probability in CDS although the joint probability does not exist in QDS [65]. Now let us formulate the mutual entropy representing the information trans. mutual entropy mitted from an initial state ( P E 6 to the final state A * c p ~ 6 The w.r.t. an initial state ~ E Y the, decomposition measure p and a channel A* are defined by
I:(%
A*) = S(@,YI@J, (5.5) where S(.l.)is the relative entropy for two states in a C*-algebra. The mutual entropy w.r.t. an initial state ~ E and Y a channel A* is now defined by
IY(cp; A*) = limsup(I:(cp; A*); ~ E F , ( Y ;E ) } , (5.6) where E 2 0 and F , ( Y ; E ) is the subset of the set M , ( Y ) such that F , ( Y ; E ) = ( ~ E D , ( Y ) ; SY(cp) < H ( p ) < S Y ( p ) + &< + a > (F,(Y; 0) = { ~ E D , ( Y ) ; SY(cp) = H ( p ) } )or F , ( Y ; E ) = M , ( Y ) when S”(cp) = co.The above sets M , ( Y ) , D , ( Y ) and the functional H ( p ) are those introduced in Section 2. Note that the mutual entropy (5.5) should be used when the decomposition measure is fixed. In the sequel we use the simple notations @, I,(cp; A*) and I(cp; A*) when Y = 6. Before discussing the fundamental properties of the mutual entropy, we introduce another mutual type
387 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
33
entropy for an initial state cp and a final state $. We call it the quasimutual entropy and denote it by Zo(cp, $). Let G,, be the set of all quasicompound states in 6 for cp and t,k, and let Y o be cp@$. Here it is not necessary that $ is connected to cp through a channel. We define the quasimutual entropy for cp and Ic/ by (5.7)
Z0(cp, $) = s u P { S ( ~ l ~ oYE ) ; Gqc,!€J G Yo},
which will .be used to define the &-entropy for QDS. In the remainder of this section, we assume that d and d are von Neumann algebras acting on Hilbert spaces 2 and 2, respectively, and the states denoted by cp, cp, and $ are normal states on a von Neumann algebra d.Furthermore, let X and 3 be positive natural cones for d and d,respectively. Two states cp, and cp2 are said to be orthogonal to each other (denoted by cpl I cp2) if their supports s(cpl) and s(cp2) are orthogonal, where the support s(cp) of cp means the smallest projection E satisfying q(1- E ) = 0. The measure p E M , ( Y ) is said to be orthogonal if ( J w d p ) l ( w d p ) is satisfied for every Bore1 set Q in 9'. Q
Y/Q
A channel is called normal if it sends a normal state to a normal state. The following lemmas are easily proved. LEMMA5.1. For any normal channel A*, cplIq2implies cpl@A*cp,Icp2@A*cp2. LEMMA5.2. For x,z in X and y , w in 5,we have Ax@y,z@w
=~x,z@Ay,w.
THEOREM 5.3. For a normal channel A* and a normal state cp, $ a measure p is in the set F,(G; E ) n D,(G) and is orthogonal, then I,(cp; A*) = JS(A*wlA*cp)dp< S(cp)+~. B
Proof: It sufices to prove the theorem for the case cp = pIql + p 2 c p 2 , cpl I q 2 . Let x, x,, y , y, ( k = 1, 2) be the vectors in positive cones such that cp(A) = (x, Ax), cp,(A) = (xk~Ax,), A*(p(A) ( y , A y ) and A * ( P k ( A ) = ( Y k r Ayk) for any then S($l+$21$) = S($llIc/)+ According to Theorem 3.6 of [SO] (i.e., if $11$2r +S($ll$) for any $) and the above Lemmas 5.1 and 5.2, we obtain I,(cp; A*) = S(@,I@o) = S(P1c p ~ @ ~ * c p 1 l @ 0 ~ + ~ ~ ~ 2 c p 2 @ ~ * c p 2 1 @ 0 ~ = PlS(Cpl@~* cp1 P
o ) +112S(cp2@~*cp21@0)
+
+ P l h P l +P2lOgP2 = P1 ( X , @ Y l , +P2(XZ@Y29
(log~Xl@Yl,x~Y)xl@Yl)+ (log~x2@~2,xoY)x2@Y2) +
388
34
MASANORI OHYA
+P"logP, +P210gP2 = Pl (XI 9
+ P1
( h v L , , x ) X 1)
(10gdx~,x)x2)+P2(J'2, (]08dy2,y)J1z)
+h(x2,
+
+ Pl l o g k + P2lOgP2 = PI S((P,l(P)
+PI
+ PllogPl +PZS((P, 190)+&logP2 +
W * ( P I
+ P2%4*cp2 I A * d
lA*cp)
= S ( P ~ (+ PP ~~
~~~I(P)+P~~S(~*(P~~~~*(P)+ + P 2 S(A*(P2 In*(P) = PlSM* 91In*cp) + P 2 W * v21n*cp). The second inequality is obvious for the case of S ( q ) = have A*) = &4m*cpnln*(P)( =
IJcp:
+ CO. When S(cp) < + a,we
p(n*4A*44&4
n
G
C/lnS((PnIq) n
= S(cP,(P,l(P~+(-cPn~og~n) n
I1
= -1PnlogPn
< S((P)+E,
n
where we used the monotonicity S(A*(p,IA*(p, < S(cp,lq) and the assumption p€EF,(G;E), Q.E.D. When a? is the full algebra B(X'), any normal state cp is described by a density operator e such as ( ~ ( 4=) treA for any A E ~ Then . our entropy S ( ( P )defined by (2.8j is equal to that of von Neumann: S ( q ) (= S ( e ) ) = -trelogc. Every Schatten decomposition e = 1,E n , En = Ix,) (xnl (i.e., An is the eigenvalue of e and x, is its
1 fl
associated eigenvector) provides every orthogonal measure in D,(6) defining the entropy S((P).Since the Schatten decomposition of e is not unique unless every eigenvalue A, is nondegenerate, the compound state @ given by (5.2) is expressed as QE(Q)= troEQ,
Qed632,
with gE= ~
,E,oA*E,,
n
where E represents a Schatten decomposition { E n ) . Then the mutual entropy for ('0 drd the channel A* is given by I(q;A*) = sup{1,((P; A*); E = { E n ) of e ) ,
389 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
35
with
I,(cp; A*) = S(a,loo) = tro,(logo,-logo,), where oo = @ @ A * @ . Since every Schaten decomposition is discrete and orthogonal, for a state cp given by a density operator Q such as cp(A)= tre A , we have the following fundamental inequality. THEOREM 5.4. 0 d Z(cp;A*) d min(S(cp), S(A*cp)}. Sketch of
the proof: According to Theorem
5.3, we have ZE(y; A * )
d min (S(cp), S ( A * q ) } for every Schatten decomposition E. This theorem follows by taking the supremum over E, Q.E.D. This theorem means that the information correctly transmitted from the input system to the output system is less than that carried by the initial state. When we send an information through a channel, we have to consider the efficiency of the communication. This efficiency is measured by the mutual entropy; namely, we ask for which channel is the mutual entropy larger. THEOREM 5.5. For a state cp given b y cp(.) = t r p and a channel A*, we have (1) if A* is deterministic, then Z(cp; A*) = S(cp); (2) if A* is chaotic, then I(cp; A * ) = 0; ( 3 ) ifA* is ergodic and q is stationary for a time evolution c1, = A d U , , and if every eigenvalue of e is nonzero and nondegenerate, then Z(cp; A*) = S(A*cp). Sketch of the proof: (1) For the Schatten decomposition X;1,Ek of
Q,
A*Ek is
k
a pure state because A* is deterministic. This fact and the equality of Theorem 5.3 conclude (1). (2)Since A* is chaotic, the compound state oE is equal to oo. Thus Z(cp; A*) = S(Bolo,) = 0. (3) As e is a-invariant, every pure state E , appearing in the Schatten decomposition becomes a-invariant, hence ergodic! The ergodicity of A* implies that A* E , is ergodic and pure. Therefore the conclusion follows because of S(A*E,) = 0, Q.E.D. Before closing this section, we provide the definition of the quantum mechanical &-entropyas a generalization of the &-entropyintroduced by Kolmogorov [66] for random variables. Let (a,9, p ) be a probability space and M(O) be the set of all random variables. Then for any pair of random variables f and g, there exist the joint probability measure pfs and the direct probability measure pf@p,, by which we define a functional I"(f,g ) like the mutual entropy in CDS by I"f9
9 ) = S(Pf9IPf@Pg).
(5.8)
390
36
MASANORI OHYA
The &-entropy S ( f ; E ) for f e M ( S 2 ) and any
E
> 0 is defined as
where d is a metric on the set of random variables in the probability space and M d ( f ;E ) is the set of all random variables h satisfying the following inequality: { l d ( f , h)2dpL)1/2d E. Analogously, we define the &-entropyS(cp; E ) of a state cp in QDS as follows: Let 61cpp;E ) be the set of all states 1 / / 6 ~ satisfying IIcp-$II < E. Then (5.10) q c p ; 4 = inf{lO(cp,*); * € 6(cp; E ) } , where Io((p, $) is the quasimutual entropy defined by (5.7). This &-entropyand a bit more restricted &-entropy in QDS can be used to define fractal dimensions for quantum systems [67]. 56. Construction of channel for optical communication processes
In this section we present the mathematical construction of a channel describing some optical communication processes [35]. Let d = B(%) be the set of all bounded linear operators on a separable Hilbert space 2, 6= G ( 2 )being the set of all normal states (density operators) on d.In order to discuss communication processes, we need two dynamical systems: an input system ( d l ,6,)and an output system ( d 2G, 2 )acting on the Hilbert spaces 2, and X2,respectively. Here we determine the general form of a channel for an optical communication process by taking account of direct effects of noise and loss existing in the course of the communication process. In addition to the Hilbert spaces 2, and S2, we need two more Hilbert spaces XI and X, describing the effect of noise and loss, respectively. (Noise) G(Xl) c
--
c -
- , - - A ,
Let v E 6(Xl) be a state representing the noise and a, IT, y the following maps: (1) the map a is an amplification from B ( 2 J to B ( X 2 @ X 2 )given by a(A) = A@Z for any A E B ( % ~ ) , (2) the map .n is a completely positive map from B(X2@X2) to B(X1OX,)with z(Z) = I describing the physical mechanism of the channel, (3) the map y is given by y(Q) = tr,,vQ for any Q E B ( % ~ O X ~ ) . Then we define a mapping A from B ( X 2 ) to B ( X i ) such that A = ~OITOU.
391
QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
31
We next consider the dual maps of a, n, y: (1’) the dual map a* of a is a map from G(X,@X,) to G(X,) such that a*(O) = trx2B, (2’) the dual map n*: G(X,@Yl)+G(X,@X,) is given by trn*(B)W = trOn(W) for any OEG(X,@X,) and any WEB(X,@X,), (3’) the dual map y*: G ( X , ) + G ( X l @ X l ) is given by y * ( e ) = e@v. Therefore, once we know the noise v and the mechanism of the transformation n, we can write down a channel explicitly as A*
= a*on*oy*
or equivalently, (6.1) A* (el = trx*n* (e0 v ) for any e E G(X,). (6.1) is a mathematical expression of the channel for an optical communication process introduced [35]. We now construct a more concrete model of the channel for an attenuation process. A quantum system composed of photons is described by the Hamiltonian H = a*a+ 1/2, where a* and a are creation and annihilation operators of a photon, respectively. The Schrodinger equation H x ( q ) = Ex(q) is easily solved. The eigenvalue En is E , = n+ 1/2 (n 2 0) and the eigenvector x,(q) to E, is (l/(nl’zn!)l/z)Hnx x (2’/’q)exp( -q2/2), where H,(q) is the nth Hermite function. The Hilbert space of this system is the closed linear span of the linear combinations of x,(q) (n = 0, 1, 2,. . .). Our model for an optical communication process is considered as follows: When n, photons are transmitted from the input system, m , photons from the noise system add to the signal. Then m2 photons are lost to the loss system through the channel, and n, photons are detected in the output system. In this model, the Hilbert spaces X,, X,, XI, X, and their coordinates are denoted in Table 2. System Input
Hilbert sp.
Xl
-
CONS
Coordinate
xk”(q1)
41
+
According to the conservation of energy (n, + m , = n2 m,), we assume the following linear transformation among the coordinates q l , t , , q,, t , of the input; noise, output and loss systems, respectively:
392
38
MASANORI OHYA
For simplicity, we put rn, mapping n* = V ( . )I/* by
= 0.
By using this linear transformation, we define the
V(xhi’@J#’)(q2, t 2 )
= xi2’OyV’(~q2-pf2, pq2 +at,)
where cy), after performing calculations, is found to be
so that the channel A* is expressed by
with the noise v = Iyb’)) (yh’)l E G X l ) due to the ‘‘zero point fluctuation” of electromagnetic field (yb‘) is a vacuum state vector in XI). Now a2 can be regarded as the transmition efficiency y for the channel. This expression (6.2) provides us a concrete expression for the error probability of some optical communication processes as seen in the next section.
97. Rigorous derivation of error probability The mathematical formulation of quantum control theory was initiated by Helstrom [68] on the basis of the work [69], and many works related to this topics, for instance, quantum estimation and quantum decision, have appeared [7&78]. Here we derive a general formula for the error probability in quantum control communication processes. In particular, the error probabilities for a coherent state or a squeezed state taken as an input state are given. Let us first explain quantum coding. Suppose that we encode an information by some procedure and we represent the information by a sequence of alphabets L’dk), where dk)is an element in the set %? of some proper symbols. Then it is necessary for us to express this symbol or the sequence of symbols by some quantum code representing optical signals. This expression is called the quantum mechanical coding. Let tibe the quantum code corresponding to a symbol ci€%?. In the sequel we take = (0, I},
=
{to, ti}
for simplicity. One expression of the quantum code represents the state of a photon; for instance, tois the vacuum state and 5 , is another state such as coherent or squeezed state. Two states (quantum mechanical codes) 5b’) and (5’) in the input system are transmitted to the output system through a channel A*. We here introduce a general formula for the error probability as an application of the construction of a channel A* given by (6.1) in Section 6. The signal “0” represented by the vaccum state
393 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
39
&’) goes always to “0” because we have only disturbance due to the zero point fluctuation. However the signal “1” represented by some other state ti1)can be read either “0” or “1” at the output system due to the loss of information. Therefore the error probability qe that the signal “1” is recognized as the signal “ 0 is given by where
pPCM e
=
c
N C j d ( l -qelN-j,
(7.2)
j=t,+l
pPPM e
=
(7.3) 9e, where NCj = N ! / { N - j ) ! j ! j . Now, using the concrete channel (6.2) constructed in the previous section, we can calculate the error probability when we take the coherent state or the squeezed state as an input state. When the input state is a coherent state e = l6)(6l with the eigenvector 16) associated to the eigenvalue 6 for the annihilation operator a of photon [38, 791, the error probabilities qe, PzCMand PzPMare given by 4, = exp(-qIQ12), N
pPCM e
=
c
Ncj{exp(-~ Ie12)}j{1 - exp( - v 1012)}N-’,
j=to+l
pPPM = e
4,
according to the general formulae (7.1H7.3). These formulas are exactly the same as those used in usual discussions of optical communication processes [76, 771. When the input state is a squeezed state [S0-82] denoted by 2, p) which is the eigenvector of the annihilation operator b = l a + pa*: bl& 2, p ) = (26+ pe)l8; 2, p), the error probability q, is given as 4, = ~exPC{(1-?)~-1}Iyl2+C1-(I -vl)2zl{~r2/(22)+pY2/(2~}1,
where z = { ~ l ~ 2 - ( l - ~ ) z ~ p Other ~ 2 } ~ error ’. probabilities PECMand PfpMare obtained in terms of 4, from (7.2) and (7.3), respectively.
$8. Dynamical change of entropy As discussed in Introduction, the irreversibility of a physical system is understood in terms of the entropy change, in particular, the entropy increase. However, this entropy change can not be achieved from fundamental equations of quantum
394
40
MASANORI OHYA
mechanics. We consider the dynamical change of the entropy in this section and that of the mutual entropy in the next section. Concerning the change of entropy, we address ourselves to the following questions: (El) For a dynamical change of a state cp to (p under some external effects, under what conditions does the entropy S ( q ) increase: S(cp) Q S ( @ ) ? (E2) When a state depends on time, for which time evolution does the entropy S(cp,) converge to some definite value: the existence of a state t+b and the limit lim S(cp,) = S ( $ ) ? f+m
(E3) Under what conditions are both (El) and (E2) satisfied: S(cp,)tS($) ? We study these problems in this section, but it is not so easy to get complete answers to them, so we also consider the following converse question to the above: (E4) Under what conditions is the entropy invariant in the course of dynamics ? Here we discuss the above problems from a general standing and in the linear response framework. Let us consider the case when the state change is caused by a time independent or time dependent channel; namely, cp A* cp or A,*cp. The following theorem might be one of the most general results concerning the question (El). Let 2 and 9 be the Hilbert spaces corresponding to the input and the output systems, respectively, and A* be a channel from G(A?), the set of all density operators on 2, to G(9). THEOREM 8.1. Ifthe dual map A ofthe above channel A*: G ( 2 ) + G(2’)satisfies the equality trA(g) = tre for any e E G ( 2 ) , then S(A*q) 2 S ( e ) . The proof of this theorem is based on the convexity of the entropy and the convex analysis of operators (see [83-851 for the details). Some special cases of this theorem have been discussed by several authors such as Nakamura and Umegaki [37], Davis [87] and Ingarden and Urbanik [88]. When we measure an operator Q having a discrete spectral decomposition such as Q = x q n P n , ---f
@
+o
n
e
describing the system suffers change after the measurement of Q, viz. = rQ@ = c P n @ P n .The transformation rQis a channel and satisfies all
the state
n
conditions of Theorem 8.1. Moreover, this I‘, is nothing but a conditional expectation from B ( 2 ) to the von Neumann algebra ‘2R generated by the operators Q and I : ‘2R = {Q}”. For this T,, not only the inequality S(T,@) 2 S ( e ) but also the necessary and sufficient condition satisfying the equality S(r,e) = S ( e ) are shown in [37]. That is [S(r,e) = S ( e ) iff e E {Q}’]. We here give a simpler proof of this fact: If e E {Q}’, then r,e = e, hence the equality holds. Conversely, if the equality s(rQ@) = s(@) holds, then 0 = S(rQ@)-s(@) = -trT,(e)logT&)+treloge = - trr,(@logr,(e))+ tr eloge = - trelogr,(e)+
+ tr@log@= s(@lrQ(@))+
395
QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
41
According to the inequality (6) of Theorem 4.2, we have r,(e) = e, so that e E {Q}'. Ingarden and Urbanik called the entropy S(T,g) the Q-entropy and showed [88] that it has similar properties as the von Neumann entropy such as positivity, concavity, additivity, etc. From Theorem 8.1 we can conclude that a time dependent channel A: satisfying all conditions of the theorem yields a monotone increase of the entropy S(A:e). Thus it is interesting to find physical systems having these properties or to find some conditions under which a suitable time dependent channel can be constructed from the first principle of quantum mechanics. Now we discuss the problem (E2) in a more concrete dynamics having irreversibility in itself, namely, in the linear response dynamics. A mathematical rigorous treatment of the linear response theory (Kubo theory) has been discussed by several authors [89-911. Here we introduce the entropy for the linear response theory and study the dynamical change of the entropy in the linear response dynamics. Let a physical system be described by a triple (d, 6,a@)), where a2 = B(A?), 6 is the set of all density operators on 2 and a,(.) = U;U:, U , = exp(itH) with a Hamiltonian H . Take a state cp E 6 such that cpf) = t r p . We suppose that cp is a faithful (i.e., cp(A*A)= 0 - A = 0) KMS state at = 1 w.r.t. a, and the KMS Hamiltonian H is lower bounded. When the physical system described by the above state cp is perturbed by an a-analytic self-adjoint operator LV(AE (0, 1)) in d,the perturbed time evolution a: and the perturbed state cp" are expressed as follows: @:(A)=
c ( W J d
n20
0
tl
...
1,
a,(A)I...l]
(8.1)
for t 2 0 (the case t < 0 is due to exchange of 0 and t in the integral domain) and #(A)
w = 1 (-1). n2O
= cp(W*AW)/cp(W*W),
j d t , . . .j d t , ait,(v)...aitn(v). O
... < t , < 1 / 2
(8.2) (8.3)
Taking the first order approximation w.r.t. il in (8.1) and (8.2), we have
1
q"*'(A) = v ( A ) - A 1dscp(Aais(V)+ h ( A )~ ( v ) ,
(8.5)
0
where we need some computation to obtain (8.5) from (8.3) [91]. In the sequel we call q".' and (a?')* cp a "linear response perturbed state" and a "linear response state", respectively, although they do not exhibit the positivity of a state and the linear response time evolution is not an automorphism of d . $9'
396
42
MASANORI OHYA
It is known [92] that cp" satisfies the KMS condition at p = 1 w.r.t a r , hence ' p y is an equilibrium state for the Hamiltonian H + AV. Therefore it is natural to expect the state change under :lt such that w* - lim Cur)* cp = cp'.
(8.6)
t+ m
That is, when a perturbation AV acts on the system in equilibrium described by cp and H , we should observe that the state cp gradually approaches to new equilibrium state for the Hamiltonian H + AV. Regardless of (8.6), the entropy of the time dependent state is always equal to that of cp (i.e., S((ur)*cp) = S(cp)) because a : is implemented by a unitary operator generated by H f 2 V . This means that the entropy of a state is a dynamical invariant even when the state changes to a different equilibrium state like (8.6), which contradicts the macroscopic behavior of physical systems in our real world. This result is not surprising because quantum mechanics does not contain the irreversibility in itself as discussed before. The linear response theory succeeded to explain some irreversible phenomena, so we hope that the entropy of the linear response state might change in time. From the assumptions imposed on our dynamical system and cp, we obtain by a simple calculation
(tlr)*cp
cp(a:*'(A)) = tre"J(t)A, cp'-l(A)
= tre"9'A
with 1
= (1-2
Q"J
S a,(V)ds + tre V ) p , 0
t
2
0
0
e"' ' ( t ) = ( I - iA S tl - s( V)ds+ iA S tl -
.
+ i ( V)ds)e
Since the set of all trace class operators is an ideal for d,both eV*' and e"" ( t )are trace class operators, so we define normal states O ( t ) and 8 by e(t) = lev'' @)lltrle",'
@)I,
where IA( = (A*A)'''. The entropy S(@"*' (t)) of the linear response state e'" ( t )and the entropy S(ey3') of the linear response perturbed state e'" are defined as s($'S1
(t)) = - trO(t)logO(t),
S(ev.l)= -trOlog8. If @",'(t)2 0, then our entropy is identical to that of yon Neumann because of tr@"V1(t)= 1.
397
QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
Now, our problem is as follows: When the linear response state
e'"
43
( t )approaches
to the linear response perturbed state gv*' in the weak topology (i.e., lim trev,'(t)A t'rn
= treV"A
for any A in d), under what conditions does the entropy S(e",'(t)) dynamically change to S(e"+') ? Here we assume the existence of the limit w - lim Q"" (t) = Q"" , but this existence is realized under some ergodic conditions t+cn
[91]. Then we can prove the following lemma. LEMMA8.2. I f e'.'(t) weakly converges to ev*l as t + co, then we have (1) ~/O(t)-O~~l+O as t-+co, where II.II1 = trJ.1, (2) there exist positive constants a, b such that ae < lev,' (t)l < be for a suficiently small 2. Sketch of the proof: (1) Since tre".' (t) = tre",' = 1, the following implications hold: $'*'(t)+e",'(weak)+ IIe"3'(t)-~v-111 -to* ~ / ~ " ' * ' ( t ) - -+~O"+~ ' ~11 I~~ " ~ ' ( t ) l --le".'I 11 + O and Itrl@""(t)l-ttrle".'l I+O*O(t)-+O(weak)+ ~~€l(t)-Ol~l+O. (2) Since cp is faithful and normal, q(ay.'(A)) can be expressed as [90, 911
cp(ar*'(A))= cp(A)- (TA* x , T ( I - exp( - itH))V x ) , where T = { 1 - exp( - H ) ] / H . This fact with the lower boundness of H implies the conclusion, Q.E.D. Applying (7) of Theorem 2.1, we get THEOREM 8.3. If QV3' (t) weakly converges to e",' as t + co and S(e) isfinite, then S(QV*l (t)) converges to S(e",') as t -+00. This theorem might provide a reason why the linear response theory has been useful for the description of some irreversible phenomena. Finally, we discuss the problem (E5). For two dynamical systems (d, 6, a) and (2,G, a) and a channel from 6 to 6, let 9 'be a weak* compact convex subset of 6 and 9 be that of 6. If we have a certain one-to-one correspondence between 9 'and 9 by the channel A*, then the equality S"(cp) = Sp(A*cp) is expected. We can prove the following intuitively trivial facts about the entropy invariance [22]. THEOREM 8.4. For a stationary channel A*, the followings hold: (1) I f A* is injective and a *-homomorphism, then S K ( q )= SK(A*cp)for any cpEK(4.
(2) I f A * is a bijectionfrom exl(a) to exI(a), then S'(cp) = S'(A*cp)for any c p I(a). ~ (3) If A* is a bijection on 6 , then S(cp) = S(A*cp) for any cp E 6. The entropy change in the linear response dynamics for more general quantum systems can be similarly discussed, and the entropy change for some open systems [39,93] and for some quantum stochastic processes [95] might be worth considering for the study of the irreversibility, which will be discussed elsewhere.
398 44
MASANORI OHYA
$9. Dynamical change of mutual entropy As discussed in Introduction, we study the time development of the mutual entropy defined by (5.6) when the state is changed by a time dependent channel A:. Here we assume that d i s a von Neumann algebra acting on a Hilbert space YP with & = d and A ( R + )= { A t ;t E R + } is a dynamical semigroup on a von Neumann algebra d (i.e., A ( R + ) is a weakly* continuous semigroup and A,* is a normal channel for each t E R + ) and there exists at least one faithful normal stationary state $ w.r.t. A: (i.e., A,*$ = $ for all t e R ' ) . Define two subsets &(A) and d ( C ) of d such as &(A)
=
{ A ~ d At(A) l = A , t€R+}, A,(A*A) = A,(A*)A,(A),t~ R ' } .
d ( C )= { A
Then &(A) is a von Neumann subalgebra of d and there exists a unique conditional expectation € from d to &(A) [l5, 951. THEOREM 9.1. Under the same conditions as those of Theorem 5.3, $&(A) = d ( C ) . holds and d is type I , then Z,(cp; A:) decreases in time and approaches I,(cp; €*) as t-03.
Proof: Since &(A) = d ( C ) , it is known [95] that A:o weakly converges to €* w for any normal state o.The assumption that d is o-finite and type Z implies the
norm convergence of A:o
[96]:
IlA:cp-€*cp(I -+0 where cp
= Cp,,cp,
and
IIA,*cpn-€*cpnII-0,
t+m,
with cp,,Icp, (n # m). As there exists a constant
A,,€R + satisfying
< il,,cp for each n, the inequality A:cp,, d A,,A:cp holds for all t E R + . Therefore (4) of Theorem 4.2 applies and we obtain 'p,,
lim S(A:cp,,lA:p)
= S(€*cp,l€*(p).
t'rn
This equality and the equality of Theorem 5.3 proves the existence of lim I,(cp; A:). t'rn
This limit is decreasing in time because S(A~+,cp,JA~+, cp) ,< S(A:cp,,I@cp) for all S E R + ,Q.E.D. THEOREM 9.2. If & = B ( 2 ) and &(A) = d ( C ) holds, then for any normal state (1) I(@;A:) decreases to I(@;€*) as t -+ + 00, and (2) there exists only one stationary state w.r.t. A: iff I(@;€*) = 0 for all e.
+
e
Proof: The convergence of ZE(e;A:) to I&; B*) as t + 00 is proved in Theorem 9.1. Hence we obtain (1) by taking the supremum over E . The statement (2) is easy to prove, Q.E.D.
399 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
45
THEOREM 9.3. When A ( R ) is unitarily implemented, the mutual entropy I(@; A:) is equal to the entropy S(@). Theorem 9.1 and item (1) of Theorem 9.2 show that the information of the state cp is lost in time when A: is dissipative. Item (2) of Theorem 9.2 tells that the
information of an initial state is completely lost in equilibrium if there exists only one stationary state for A:. These results might be quite natural for dissipative processes from physical points of view. About ten years ago, Toyoda conjectured [97] that the quantum measurement process should be treated similarly as a communication process of information. This program is now progressing [32] by using a communication channel and the mutual entropy.
Acknowledgements The author thanks Professors H. Umegaki and R. S. Ingarden for their hospitality and encouragement. He also thanks Professor Hiai for his critical reading and useful comments. REFERENCES [l] [2] [3] [4] [5] [6] [7]
[S] [9] [lo] [ll] [12] [13] [14] [l5]
[16] [17] [18] [19] [20]
J. von Neumann, Die Mathematischen Grundlagen der Quantenmechanik, Springer, Berlin, 1932. C. R. Shannon, Bell System Tech. J., 27 (1948), 379 and 623, G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems, J. Wiley & Sons, 1977. H. Haken, Synergetics -An Introduction, Springer-Verlag, Berlin, 1978. S. Kullback and R. A. Leibler, Ann. Math. Stat., 22 (1951), 79. A. N. Kolmogorov, I E E E on Information Theory, 2 (1956), 102. I. M. Gelfand and A. M. Yaglom, Amer. Math. SOC. Transl., 12 (1959), 199. H. Umegaki, J . Math. And. Appl., 25 (1969), 41. H. Umegaki and M. Ohya, Entropies in Probabilistic Systems (in Japanese), Kyoritsu Publishing Company, 1983. M. S . Pinsker, Information and Information Stability of Random Variables and Processes, HoldenDay, Inc., 1964. R. Ash, Information Theory, Interscience, 1965. S . Guiasu, Information Theory with Applications, McGraw-Hill, 1977. R. Haag and D. Kastler, J . Math. Phys., 5 (1964), 848. M. Tomita, Quasi-standard von Neumann Algebras, Mimeographed Notes, Kyushu Univ., 1967. M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications, Lecture Notes Math., 128, Springer, 1970. S . Sakai, C*-algebras and W*-algebras, Springer, Berlin, 1971. H. Ezawa, Structure of quantum mechanics (in Japanese), in: Quantum Mechanics I I , ed. by H. Yukawa and T. Toyoda, Iwanami Publishing Company, 1978. 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I , Springer, New York, 1981. 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I I , Springer, New York, 1981. M. Takesaki, Theory of Operator Algebras I , Springer, New York, 1983.
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[21] H. Umegaki, M. Ohya and F. Hiai, Introduction to Operator Algebras (in Japanese), Kyoritsu Publishing Company, 1985. [22] M. Ohya, J . Math. Anal. Appl.. 100 (1984). 222. [23] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, 1970. [24] E. H. Lieb and M. B. Ruskai, J . Math. Phys., 14 (1973), 1938. [25] A. Wehrl, Rep. Math. Phys., 6 (1976), 15. [26] A. Wehrl, Rev. Mod. Phys., 50 (1978), 221. [27] G. Lassner and G. A. Lassner, Rep. Math. Phys.. 15 (1979). 41. [28] W. Thirring, Quantum Mechanics of Large Systems, Springer-Verlag, 1983. [29] H. Umegaki and M. Ohya, Quantum Mechanical Entropies (in Japanese), Kyoritsu Publishing Company, 1984. [30] R. Haag, N. M. Hugenholtz and M. Winnink, Commun. Math. Phys., 5 (1967), 215. [31] G. Choquet, Lecture Analysis I , 11, 111, Benjamin, New York, 1969. [32] M. Ohya, Symmetry breaking and quantum measurement (in preparation). [33] M. Ohya and T. Matsuoka, J . Math. Phys., 27, No. 8 (1986), 2076. [34] M. Ohya, J . Math. Anal., Appl., 84 (1981), 318. 1351 M. Ohya, I E E E on Information Theory, 29 (1983), 770. [36] M. Ohya and N. Watanabe, Japan J . Appl. Math., 3, No. 1 (1986), 197. [37] M. Nakamura and H. Umegaki, Math. Jap., 7 (1962), 151. [38] J. R. Klauder and E. C. G. Sudershan, Fundamentals of Quantum Optics, Benjamin, New York, 1968. [39] K. Kraus, Ann. Phys., 64 (1970), 311. [40] M. Ohya, Kodai. Math. J., 3 (1980), 287. [41] H. Umegaki, Tohoku Math. J., 6 (1954), 177. 1421 J. Tomiyama, Proc. Japun Acad., 33 (1957), 608. [43] L. Accardi and C. Cecchini, J . Func. Anal., 45 (1982), 275. [44] H. Umegaki, Kodai Sern. Rep., 14 (1962), 59. [45] G. Lindblad, Commun. Math. Phys., 33 (1973), 305. [46] G. Lindblad, Commun. Math. Phys., 40 (1975), 147. [47] F. Schlogl, Ann. Phys., 45 (1967), 155. [48] A. Connes abd E. Sterrmer, Acta. Math., 134 (1975), 289. [49] H. Araki, Publ. R I M S Kyoto Uniu., 11 (1976), 809. [SO] H. Araki, Publ. R I M S Kyoto Uniu., 13 (1977), 173. [SI] W. Pusz and S. L. Woronowicz, Rep. Math. Phys., 8 (1975), 159. 1521 A. Uhlmann, Commun. Mat. Phys., 54 (1977), 21. [53] H. Spon, J . Math. Phys., 19 (1978), 1227. [54] F. Hiai, M. Ohya and M. Tsukada, Paclfic J . Math., 96 (1981), 99. [55] R. S. Ingarden, Int. J . Engng. Sci., 19 No. 12 (1981), 1609. [56] R. S. Ingarden, H. Janyszek, A. Kossakowski and T. Kawaguchi, Tensor, N . S., 37 (1982), 105. [57] F. Hiai. M. Ohya and M. Tsukada, Pacific J . Math., 107 (1983), 117. [58] M. Ohya, Lecture Note in Math., 1136 (1985), 397. [59] M. Donald, Commun. Math. Phys., 105 (1986), 13. [60] D. Petz, Commun. Math. Phys., 105 (1986), 123. [61] R. S. Ingarden, Geometry of Thermodynamics, preprint. [62] H. Umegaki, Kodai Math. Sern. Rep., 11 (1959), 51. [63] S. Gudder and J. P. Marchand, J . Math. Phys., 13 (1972), 799. [64] M. Ohya, L. Nuouo Cimento, 38 (1983), 402. [65] K. Urbanik, Stud. Math., 21 (1961), 317. [66] A. N. Kolmogorov and V. M. Tikhomirov, Amer. Math. SOC. Translation, Ser. 2, 17 (1961), 277.
401 QUANTUM INFORMATION THEORY AND THEIR APPLICATIONS
[67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [SO] [Sl] [82] [83] [84] [85] [86] [87] [SS] [89] [90] [9l] [92] [93] [94] [95] [96] [97]
47
M. Ohya, Fractal dimensions for general quantum states, to appear. C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976. C. W. Helstrom, J. W. S. Liu and J . P. Gordon, Proc. IEEE, 58 (1970), 1578. R. S. Ingarden, Fortschritte der Physik, 12 (1964), 567. R. S. Ingarden, Fortschritte der Physik, 13 (1965), 755. A. S. Holevo, Problemy Peredachi Informatsii 8 (1972), 62. A. S. Holevo, Problemy Peredachi Informatsii 9 (1973), 31. R. S. Ingarden, Rep. Math. Phys., 10 (1976), 43. A. S. Holevo, Rep. Math. Phys., 2 (1977), 273. R. J. McEliece, IEEE. Trans. Information Theory, 27 (1981), 393. 0. Hirota, K. Yamazaki, M. Nakagawa and M. Ohya, Trans. IECE Japan, E69 (1986), 917. M. Ohya and N. Watanabe, Eflciency of modulations in optical communication processes, to appear. R. J. Glauber, Phys. Rev., 131 (1963), 2766. D. Stoler, Phys. Rev., D1 (1970), 3217, 4 (1971), 1925. J. N. Hollenhorst, Phys. Rev., D19 (1979), 1669. H. P. Yuen, Phys. Rev., A13 (1976), 2226. J. Bendat and S. Sherman, Trans. Amer. Math. Soc., 79 (1955), 58. M. Nakamura and H. Umegaki, Proc. Japan Acad., 37 (1961), 149. W. Donogue, Monotone Matrix Functions and Analytic Continuation, Springer, 1974. T. Ando, Topics on Operator Inequality, Res. Rep. of Hokkaido Univ. C. Davis, Proc. Japan Acad., 37 (1961), 533. R. S. Ingarden and K. Urbanik, Acta Phys. Polon., 21 (1962), 281. J. Naudts, A. Verbeure and R. Weder, Commun. Math. Phys., 44 (1975), 87. A. Verbeure and R. Weder, Commun. Math. Phys., 44 (1975), 101. M. Ohya, Rep. Math. Phys., 16 (1979), 305. H. Araki, Publ. R I M S Kyoto Univ., 9 (1973), 165. E. B. Davies, Quantum Theory of Open System, Academic Press, New York, 1976. L. Accardi, A. Frigerio and J. T. Lewis, Publ. R I M S Kyoto Univ., 18 (1982), 97. A. Frigerio, Commun. Math. Phys., 63 (1978), 269. G. F. Dell'Antonio, Commun. Pure Appl. Math., 20 (1967), 413. R. Toyoda, Quantum mechanics and physics for information (in Japanese), in: Quantum Mechanics 11, ed. by H. Yukawa and T. Toyoda, Iwanami Publishing Company, 1978.
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0 1983 IEEE Repnnlc4 wth permmioh IEEE TRANSACTIONS ON INFORMATION TFEORY, VOL IT - 29, NO 5. SEPTEMBER 1983
so-called mutual information which plays an essential role in information theory. In order to formulate it rigorously, we have to know proper relations between input states and output states connected with each other by a communication channel. There have been some attempts [7], [lo] to proceed with this program. This short note is one more attempt to formulate the communication process with the following aims: 1) construct a suitable compound state of an input state and its output state generated through a communication channel; 2) formulate the mutual information by using the compound state constructed and study its fundamental properties; 3) examine a simple process of photontransmission in OUT formulation in order to show that our formulation is indeed an extension of the usual one. 11. QUANTUM DYNAMICAL SYSTEM We briefly review the density operator method to describe a quantum dynamical system. Since von Neumann's famous book [25] appeared, a quantum mechanical system has been described by the method of Hilbert space. Let % be a separable complex Hilbert space and B ( % ) be the set of all bounded linear operators on %. A self-adjoint operator of B( %) might be called a physical observable. We consider only bounded operators in this correspondence. The expected value of A E B ( % ) in a state vector Q E % is given by (a, A Q ) , where ( . , ') is the inner product of %. More generally, let T ( X ) + , ,be the set of all positive trace class operators (density operators) on % with trp = 1. That is, 1 for any T ( % ) + , , = (p E B(%)lp > 0, trp = 2"(@", pa" We call a complete orthonormal system (CONS) (an)of density operator a (normal) state on B ( % ) as usual [21]. A dynamical change of a quantum system is described by either the Heisenberg picture or the Schradinger picture. The Heisenberg picture provides the dynamics of observables and the SchriMinger picture provides that of states. These pictures are completely equivalent for Hamiltonian dynamics. In our notation, the time evolution dynamics of a quantum system is given by a one-parameter group a ( R ) of automorphisms of B(%) such that a , ( A ) = u , A u _ , for anyA E B ( % ) and all t E R , the set of real numbers, where u , is a unitary operator on % generated by the Hamiltonian H of the system: u, = exp(itH). Thus our quantum dynamical system is described by a triple ( B W ) ,V'W+,,,a ( R ) ) .
k;
111. QUANTUM MECHANICAL CHANNEL In information theory we have to consider two dynamical On Compound State and Mutual Information in systems: an input system and an output system. An input system is described by a triple ( B ( % , ) , T ( X ) + , , a ( R ) )and an output Quantum Information Theory system is described by another triple ( B ( % , ) , T(%,), MASANORI OHYA T ( R ) ) ,where T ~ ( A=) o,Ao_, for a unitary operator u, on %,. These time evolutions a , , T, characterize the input states and the Absrracf-A quantum mechanical compound state of an input state and output states respectively. They have nothing to do with the its output state generated through a communicationchannel is constructed. dynamics of a channel. As in classical case (cf. [23]), we often The mutual informationof quanlum communication theory is defined by need to prepare a stationary state as an input state in quantum using the compound state, and its fundamental properties are studied. case. Information (entropy) of the input system is transmitted to the I. INTRODUCTION output system. A channel exhibits all dynamical effects for the The communication of information by using a laser is an information transmission. In Shannon's theory [2], [ 181, the inforimportant technique. Laser action is a typical quantum effect, mation of a system is carried by a probability distribution of and therefore we should reconstruct information (communica- events, and a channel induces a change of this probability distrition) theory in terms of quantum mechanical language [3], [6], [7], bution. The concept of state in a quantum dynamical system can [XI, [lo], [13], [14]. This program is not so easy to complete. One be regarded as an extension of that of probability distribution. of the main difficulties is to formulate in a systematic way the Therefore the information of quantum system is carried by a state, and a channel provides a dynamical change of states. It is shown [23] that the dual map of a channel in classical Manuscript received February 3. 1982: revised November 10. 1982. theory transforms any positive bounded function of an input The author is with the Department of Information Sciences, Science Universystem to that of an output system. By analogy with this fact, a sity of Tokyo, Noda City, Chiba 278. Japan. 01983 IEEE
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5. SEPTEMBER 1983
quantum mechanical channel might be formulated as follows [7], [13]: A mapping A* /rom T ( X , ) + , lo , T ( X , ) + , ,is said to be (1 channel if its dual map A from B(ilc,) to B ( X , ) satisfies the following two conditions: a) A is complete!v positive (i.e., /or any positive n X n-matrix (Q,,,) with Q,, E B(Kz), the matrix (AQ ) is positive for all n E N ) , and b) A I , = I , /or the identirv oper&r I, on ‘X, ( k = 1,2) (we denote every identity operator by / i n the sequel when no confusion occurs). The physical significance of the above conditions is the following. 1) We need “complete positivity” instead of “positivity” because the tensor product A: b A; of two channels A: and A; should also be a channel, and this is not the case in quantum systems unless both A , and A ? are completely positive [21, p. 2181. 2) The condition b) is necessary for a channel to transfer a state to another state. These conditions are very weak so that most of physical transformations satisfy them [14], [27]. Therefore our definition of a channel might be general enough in order to Construct a mathematical framework of quantum communication. By Stinespring’s theorem 1191, 126) a completely positive normal map A is expressed as AQ = Z,V;QVk for any Q E B(X,), where V, is a bounded operator from ‘XI to 3c, (i.e., V, E B(‘Xl, ‘ X I ) )with Z,V;V, = I . Since trA*pQ = trpAQ, we have . particular when X I A*p = Z,V,pV; for any p E T ( ‘ X , ) + , ,In = ‘X, and Vk is a projection for each k , the change of p to A*p is often called a quantum measurement [4, p. 151. Finally we note that a channel considered in this note is supposed to be memoryless. In particular, we do not deal with the transmission of sequences of states. IV. COMPOUND STATE a channel s A’ to an output system and the resulting stat-curies all information of the input system in a certain physical situation. We should like to know how much information is correctly transferred to the output system. For this purpose, we have to find some relation between constituents of p and those of A’p. The compound system of the input and output systems is described by the pair (B(‘X), T ( ‘ X ) + , , ) ,where ‘X is the tensor product Hilbert space of ‘XI and ‘X,: ‘X = ‘XI b ‘X,. If an input state p is given and its associated output state A’p is observed, then a compound state a of p and A’p should satisfy the following conditions: a) a E T(‘X)+,,,b) tra(A b I ) = trpA foranyA E B ( X , ) , a n d c ) t r a ( I b B ) = t r A * p B f o r a n y B E B ( ‘ X , ) . There are many such states, for example, a state defined by a. = p b A*p (3.1)
An input state p is transmitted t h o u
is a typical state satisfying the above conditions. We call this state a. a trivial compound slate. However, it does not provide for any correlation between p and A’p. Not every state satisfying a), b), and c) is suitable for expressing the correlation. Thus we have to construct a proper compound state. Once an input state p is given, we can obtain the eigenvalues (A,) of p and have a spectral decomposition of p such that
with A‘
(3.5)
= 1°A)(Okl?
where 0, ( k = 1,2.. . . ) are mutually orthogonal normalized eigenvectors of p. In the decomposition (3.4), the eigenvalue of multiplicity n is repeated precisely n times. Furthermore, this decomposition is unique if and only if no eigenvalue is degenerate. For instance, if the multiplicity of an eigenvalue p of p is two and I,and ‘P2 are its orthonormal eigenvectors, then two vectors p d = 2-’/’(’PI Y,j are defined by a , = Z - 1 / 2 ( * l + 12) also eigenvectors of p associated with p. Hence p has the following two different expressions: ~
+ pP2)(’P2I)
p
=
p(l*,)(IIl
p
=
p(la,)(a,l + lQ,)(Q,l)
+ other terms.
(3.6)
+ other terms.
(3.7)
and
Each E, appearing in the decomposition (3.4) is a pure state (i.e., E: = E,) and is here called an elementary event. Thus the input state p can be considered as an event composed of elementary events E, occurring with probability A,. These elementary events are transmitted to the output system through a communication channel A*. Then we should find a connection among the input constituents ( E x ) and the output constituents (A‘E,) in order to express the correlation existing between p and A*p. We introduce a compound state representing this connection: ot = Z k A k E , b A*E,.
(3.8)
This compound state does depend on the decomposition (3.4) of p. that is, on the choice of E = ( E x ) . It is clear that aE satisfies the conditions a), b), and c) mentioned previously. The compound state introduced here plays the role of the joint probability in classical theory. In the following sections, we shall see that our construction of the compound state is appropriate for discussing quantum communication.
INFORMATION V. MUTUAL In this section, we define the mutual information by applying the compound state constructed in the previous section. The information (entropy) of an input system described by a state p is given by von Neumann [25] as S(P)
=
-trplogp.
(4.1)
This quantity represents the randomness or uncertainty of p. We now have the following question: when we send information carried bv a state p to an output s-vstem through a channel A*, how much information can be transmitted to a receiver observing the ourput state?
In Shannon’s communication theory, the mutual information gives this amount. It is a partial information of the input state gained by observing the output state. We have to define the mutual information in our quantum dynamical systems. In the sequel, we denote the mutual information by I ( p ; A*j for an input state p and a channel A*. where Pk is the projection from %to the eigenspace associated to When the mutual information I(p; A*) is properly defined, it the eigenvalue A,. The projection P, is further decomposed into should have the following properties: if there exists a one-to-one one-dimensional projections [ 191, though this decomposition is correspondence between constituents of p and those of A*p. then not always unique. That is, we can expect that a receiver gets all information of p. namely, I(p; A*) = S(p). On the other hand, if there does not exist any relation among constituents of p and those of A*p, then I(p; A*) where n , is the dimension of P, and E k , , is the projection to the should be zero. Except for these two extreme cases. I(p; A*) one-dimensional subspace of ‘X generated by an eigenvector 0,,,should be larger than 0 and less than S(p). The Kullback-Leihler information [9] is applied when we associated with A,: E,,, = p k , J ) ( Q ,I k in the Dirac notation. formulate the mutual information in classical information theory. By relabeling the indices of ( E,, ,), p can be written as Umegaki first extended the Kullback-Leibler information to P = ZkAkfik (3.4) noncommutative dynamical systems and called it the relative 3
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IEEE TRANSACTIONS ON INFORMATION THEORY,
information (entropy) 1241. The relative entropy has been further generalized and studied by Araki [I], Lindblad [Ill, Uhlmann [22], Hiai. Ohya. and Tsukada [5], and others. Using the relative entropy we formulate the mutual information with respect to an input state p and a communication channel A* as follows: J ( P : A*)
=
SUPE(S(Wo)IE = ( E k ) ) ,
=
tr 01. log 0,
=
5.
SEPTEMBER
1983
Z,,. m&,,.
dQ,, 8 t,, a,*,,.
8
t.)
.log($,. 0 T n , , . a E Q n8 q n , ) =
n,Zt,.. m , x k h k ( Q n
xjj,
3
Eh@n,)(*m, A*Ek*,.)
.log P,X,(Q,.. E,Qn)(qm,,A*E,'Pm)
(4.2)
tr aE(Iog at: - log q,),
IT - 29, NO.
and the orthonormality of Q,,, we have
A*En*",,) log h 9 , ( t . 3A*&*",)
where a, and a. are compound states defined in Sec!ion IV and S(a,~a,) is the relative entropv of aE from a. given by S(a,(a,)
VOL.
= Z ~ , , m Z n , ~ ~ n ( * n l ~ = =
(4.3)
Z,,.n,(*n,9hA*E,,log L A * E , Y m )
Z,,,.,LIogh,,(*,,,,, A*&*",) + Z,, , A S
In order to discuss the properties of the mutual information I ( p ; A*), we need a few definitions concerning the channel A*.
=
1,) A*€,, 1% A*E,,*", ) Z,,h,,logh,trA'E,, + P,,h,,trA*E,,logA*E,,
I ) A channel A* is said to he deterministic if A* sends a pure state in T ( X l ) + , lto a pure state in T('JC2),,1and t h s
= Z,,h,,logh, + X J , , ( - S ( A * E , , ) ) . In the course of the above calculation, we used the fact that A*€, CorresDondence is one-to-one and orthoeonal 1i.e.. E. I E ~ . " is a, state. hence tr A'E,, = I . In the same manner, we obtain A*'E, i A*E,)., 2) A channel A' is s a d to be chaotic if A*E - A*€ for any -tra,loga,= -Z,,h,logh,, + S ( A * p ) (This pair E , , E . amearinn in the decomvos!tiin (3.4). . . means ;ha( A i c a n n G distinguish elementary events of p . ) Thus, 3) A channel A * is said to he stationilly if A 7, = a, A holds S(aEI%O) = S ( A * P ) - Z , , h , , S ( A * E , , ) (i.e.. A ( T , ( A ) )= a , ( A ( A ) )for anyA E B(3c,)) for all r E = Z,,~.,S(A*E,IA*P) R.
-
Y
0
0
The ergodicity of a channel is an important concept in classical information theory [23]. In [ 131 we introduced ergodic channels in quantum systems and studied their dynamical properties. 4) A channel A * is said lo be ergodic if A* is stationary and A* sends any extremal a-invariant state in T ( X l ) + , lto an extremal 7-invariant state in T ( % ) +, ,.
Recall that a state p is a-invariant if trpa,(A) = trpA holds for any A E B ( ? C , ) and all r E R. An a-invariant state is a state describing a thermodynamical equilibrium of a physical system with a time evolution a , . Furthermore. an a-invariant state is extremal if it cannot be expressed by a convex combination of other a-invariant states. An extremal a-invariant state is often called an ergodic state a , , and it describes a pure phase in thermodynamical equilibrium. For more detailed physical meaning of such states, consult [ 15. ch. 61. We have the following results. Theorem 1: For an input state p and a channel A'. the inequalities 0 < I(p: A*) < min(S(p), S(A*p)) hold. Theorem 2: For an input state p and a channel A*, we have I ) if A* is deterministic, then I ( p : A*) = S ( p ) ; 2) if A* is chaotic, then I ( p : A*) = 0; 3) if p is an a-invariant state whose eigenvalues are nonzero and nondegenerate. and if A' is ergodic, then I ( p ; A*) = S( A'p).
Let us prove these theorems. First of all. we prove the following lemma. Lemmo 3: For a decomposition (3.4) of an input state p and the compound state a,$ defined by ( 3 . Q we have
Q.E.D. Proof of Theorem I : The positivity of S(a,lao) implies that of I ( p : A*). As shown in the proof of Lemma 3, S(a,la,,) d S ( A * p ) , from which the inequality I ( p ; A') < S(A*p) follows. Let us show another inequality I ( p ; A*) < S ( p ) . Since A is a completely positive and identity preserving map, it is known [ 1 I], [221 that S ( A * p , ( A * p 2 ) 4 S(pIlp,) for any pI and p z in T ( 3 c l ) + , lHence . from Lemma 3, we have S(%I,J")
c ~,AJ(EnIP) =
Z,;h,,(trE,,logE,, - trE,logp)
=
- 11 Z,,X,, E,,log p
=
- trplog p = S ( p ) .
Taking the supremum over E . we get I ( p ; A') d S(p).
Q.E.D.
Proof of Theorem 2: I ) Since A* is deterministic, for each elementary event E, of p. A*E, is a pure state and A'p = Z , h , A*€, is an extremal decomposition of A*p into pure states (i.e., a decomposition of type (3.4)). Hence, S(A*p) = S ( p ) = -Z,h,logh,. From Lemma 3, we have S(a,ia,,)= S ( A * p ) because A*€, is pure (so S(A*E,) = 0). Therefore we get I(P;A*) = S ( p ) . 2) Since A* is chaotic, the compound state a, is equivalent to a,. Indeed at
=
X,h,E, 8 A*€,
=
Z,XhEk 8 A'p
=
p 8 A*p
= a,,.
Thus we get I ( p : A*)
=
suprS(a,la,,)
=
S(a,,,la,,)= 0
3) When p is a-invariant, every pure state Ex appearing in the S ( a,Ja,,) = Z,X,S(A*E,IA*p). (4.4) decomposition (3.4) is also a-invariant. Let us prove this fact. Since p is a-invariant, [ u , , p ] = 0 holds for all r E R. From the Proof: Let (Q,,) be a complete orthonormal system of XI assumptions for the eigenvalues of the state p , the equality containing all eigenvectors of p and (t,) be a CONS of ?c2 [ u , , E,] = 0 holds for all r and k . Thus Ex is a a-invariant state, containing all eigenvectors of A*p. Then (Q,, 8 Tn,)becomes a hence ergodic. Since it is assumed that A* is ergodic, A'E, is an CONS of the tensor product Hilhert space 3c = X I 8 3c2. The ergodic state for 7,. We now show that this ergodic state A*E, is hy. also pure. Suppose that A*€, is not pure; then we have an relative entropy .. of the compound states ar and an - is given s(a,la,) = tr(a,Iog oh - 0,log a o ) . extremal decomposition of A*€, into pure states such as A*€, = Put E,, = lQ,,)(Q,,l for every n. Then pE,, = h,,E,, if Q,, is an Z,,p:,O,:. The 7-invariance of A*& reduces that of 0: as discussed eigenvector of p and p E,, = 0 otherwise. According to this fact above. This contradicts the fact that A*€, is an ergodic state.
405 IEEE TRANSACTIONS ON INFORMAnON THEORY, VOL. 11-29, NO.
TABLE I SIMPLE MODEL
Thus A*E, is a pure state. From this result we have I(p;
A’)
because of S ( A * & )
=
=
supES(o,luo)
=
773
5 , SEPTEMBER 1983
S(A*p)
Q.E.D.
0.
Annihilation Hilbert Soace Creation OD. Hamihonian Eieenvalue
CONS
Part 3) of Theorem 2 tells that for an ergodic channel the mutual information is equal to the information carried by the output state A*p; hence an ergodic channel is the most effective channel for a-invariant states having nondegenerate eigenvalues. It is seen from Theorems 1 and 2 that the mutual information defined by (4.2) satisfies most of the properties desired of an and any W E B ( X , 0 X I ) . The information gain. The mutual information i s given by a com- for any u E T( X t o T ( % , @ X I ) + , is given by pound state; therefore our construction of a compound state will dual map r*: T(’Xt)+,l t r r * ( p ) Q = trpT(Q) for any p E T ( X l ) + , land any Q E he on a right track for quantum communication theory. B ( % , 0 XI). It is easily seen that I?* i s expressed as r * ( p ) = p Before closing this section, we make a few more remarks. Remark I : Let us briefly discuss the correspondence between @ 6. Therefore, once we know the noise E and the mechanism of the our formulation and that of Shannon. Suppose for simplicity that both input and output classical systems have n events. In classical transformation n, we can write down a channel explicitly such theory an input state p i s a probability distribution (q,);= I of that A* = n* r* events, and a channel A* is given by a n x n transition matrix (5.2) [ p , , ] ; : , =,. Thus we have the following correspondences: or equivalently, a) output state probability distribution A * p = A’p = try, IT*(, B E ) (5.3) (Z;=tp,,qI)Y-, ( = ( T i ) ) ; b) compound state joint probability distribution for any P E T(%i)+,i. (P(;,;));:,=I withp(i.1) =p,,q,; If neither noise nor loss exists in the course of information c) mutual lnformation I = Z,. , p ( i , ;)(log p ( i . j ) log r,ql). transmission, then A* = n* because X I = C and %, = C. the set of all complex numbers. In this case the information of the Remark I : The capacity of the channel is an important qnantity for analyzing the transmission of information. In quantum input system should he completely transmitted: I ( p ; A*) = S( p ) . communication theory, it might be given by taking the supremum Therefore Il* should be deterministic. We apply our formulation to a very simple model of informa. we need of I ( p ; A’) over p in a certain subset S of T ( X , ) + , ,As more mathematical preparation to discuss the capacity and its tion transmission. This model has been discussed in [ZO],and we properties and we are still working on this subject, we do not briefly sketch it. A transmitter sends photons with frequency in the laser range, and a receiver detects them. The system comtreat it here. posed of photons is described by a Hamiltonian H = 0 ’ 0 + 1/2, VI. A CONSTRUCTION where a* and a are creation and a n d l a t i o n operators of phoOF CHANNEL AND AN ton, respectively. Here we take hv = 1 for simplicity. The APPLICATION We have two aims in this section. One of them i s to determine Schrdinger equation H O ( x ) = r 4 ( x ) is easily solved [16, p. the form of a channel A* by t&ng account of direct effects of 1821: the eigenvalue c n is given by c,, = n + 1/2 ( n t N , the set noise and loss existing in the channel. and another is to examine of all natural numbers) and the associated eigenvector is given by the validity of our formulation by considering a simple process of G,r(x) = ( I / ~ ) H , ( J Z x ) e x p ( ~ x ’ / 2 )for each n , where H , ( x ) is the nth Herrmte function. The Hilhert space X of this photon transmission. In addition to the Hilbert spaces X I and X2, we here use two system is the closed linear span of linear combinations of 4,, more Hilbert spaces X I and M, in order explicitly to describe ( n = I , 2,3. . . . ). In this model the Hilbert spaces X I , X 2 . MI, considered above and their related quantities are denoted some external effects to the input and output states. For instance and a state in ’JC, induces a noise into the channel and a state in X I in a Table I. When n t photons are sent through a channel having n 2 phoindicates a loss of information at the output system. Let 5 be a state in XI (i.e., t E T(3C1),,,)describing the noise tons as its noise, let rn, photons be detected and m 2 photons be lost. Then a relation a) n , + n 2 = m , + m 2 should hold because in the channel. We consider the following maps: of conservation of energy (photon number). Since a = I / & ( x + a / a x ) and a* = l / f i ( x - a / a x ) , the relation a) is satisB(x2) B(X, B B(x,B f,B ( % , ) fied under, for example. a linear transformation of coordinates (5.1) xI. x , , y , , y 2 such that b) x i = a.vl Py2 and x 2 = Pv, + ay, with a 2 + p’ = I . We now suppose for simplicity that the noise These maps are defined as follows. The map a is an amplification is due to the “zero point fluctuation” of electromagnetic field, from B ( X I ) to B ( X , 0 X,) given by a ( A ) = A 8 I for any that is, n z = 0. Hence the noise source i s described by a state A E B ( X , ) . The map II is a completely positive map from 5 lQo~x2))(Qo(x2)l. For an input state E,, = 14,,(~l))(~,,~.~t)l. B ( 3 c , @ % , ) t o B ( X l 0 3 C l ) w i t h n ( l ) = I . T h e m a p ~ i s g i v e n the change of the state €,, 0 5 under the transformation h) by r ( Q ) try EQ for any Q E B ( K , B XI), where try i s a follows from the change of the vector 4,,(xt)B 4,(x2) such that partial trace wiih respect to the Hilbert space X I [4, p. I50\. It is easy to show [ 121 that these maps are completely positive. ~ 4 , ~ ( xeoo(x2)(= ,) c,(.~~,J,~)) @,,(XI) B 4 0 ( x 2 ) hence A = r ,. II a is also completely positive. We next consider the dual maps of a. n. and r. The dual map = Z;=”c,Y,(.v,) BQ,,-,(y,). a* of a is a map from T ( X Z0 X,), to T ( X 2 ) + ,such , that tra’(0)A = t r f l a ( A ) f o r a n y f l E T ( 3 t 2 B M 2 ) + , a n d a n y A E where c’; = J n ! / ; ! ( n - j ) ! ( - P ) ” - ’ a ’ . By (5.3) we have B ( X , ) . Thus, a’(@) = try fl. The dual map II*: T (Xl @ M t ) + , t - T ( % 2 B X 2 ) + , l given by t r I I * ( o ) W = t r o I I ( W ) A’€,, = %,V(P @ E)V* = tr.%p’,, )(*,rl = Z;=,,lc;\’14,)( 4,l
-
,
-
Q
~
x,) 2
a,)
~
-
-
0
,
il
406 IEEE TRANSACTIONSON INFOEMATION THEORY, VOL. IT -
114
Therefore for an input state p = Z:$I,,E,~ (0 d N d m ) with P.,h., = 1 and A, ii X ( k = j), the compound state and the trivial compound state hroduced in Section IV are given by at = Z,,X,E, 0
A*En = Z~~oZ;_oX,)c;'12E, @ S,,
a. = XnAnEnB
Z,hhA*Eh = ~ , , , ~ P ~ = o X , X ~ l C@~8,, I*E,
where 8, is defined by S, = I@,)(@,/ E T ( 3 c 2 ) + , ,Thus . we can calculate the mutual information as follows: I ( p ; A*)
=
-L,A,logX,
- Z,N_,(Z~-,X,~C;~~)
.log (z:=,A"lC,Y)+ X"X;=o( A,lc;l2) log (A,ic;l*) =
B,N_,P~=,A"IC;I2Iog (I#/(
X:=,AnlC;12)).
This result is also obtained from the Shannon's formula with an initial probability (A") and the conditional probability (lcJ2) given above. As the above model is very simple, this conditional probability can be calculated [20] without introducing a channel A' explicitly. Once the conditional probability is known (e.g., when every eigenvalue of p is nondegenerate and A' is given explicitly, it can be obtained by taking proper CONS'S at the input and output systems), Shannon's expression for the mutual information is equivalent to ours. Without formulating a channel and a compound state precisely, neither have we a general rule to compute the mutual information (the joint probability) nor can we study quantum communication processes in general or the properties of a channel and the mutual information in particular. ACKNOWLEDGMENT
The author wishes to express his gratitude to Professor H. Umegaki for his valuable comments and constant encouragement. He also thanks the referees for their valuable suggestions and advice, which enabled him to revise his correspondence in the present form. REFERENCES
H. Araki. "Relative entropy of
states of von Neumann algebras," Puhl. R I M S . Kyoto Univ.. "01. I I . pp. 809-833, 1976.
R. Ash. Informnoon Theory. New York: Interscience. 1965. E. B. Davies. "Informadon and quantum measumment." f E E E Trans. Inform. Theory. pp. 530-534. 1977. , Quantum Theory of Open System. New York: Academic, 1976. F Hiai. M. Ohya, and M. Tsukada. "Sufficiency. KMS condition and relative entropy in von Ncumann algebras." PonfrcJ. Moth.. vol. 96, pp. 99-109. 1981. C. W. Helstrom, J. W. S. Liu, and J. P. Gordon, "Quantum mechanical communication theory," in Proc. IEEE. vol. 58, pp. 1578-1598, 1970. A. S. Holevo, "Problems in the mathematical theory of quantum communication channels." Rep. Moth. Phys., vol. 12. pp. 273-278, 1977. ,"Information theoretical aspects of quantum measurement." Probl. Peredub InJorm.. vol. 9. no. 2, pp. 31-42, 1973. S. Kullback and R. A. Leibler, "On information and sufficiency," Ann. Morh. S f o l i x . vol. 22, pp. 79-86, 1951. R. S. Ingarden. "Quantum information theory," Rep. Moth. Phys., vol. 10. pp. 43-73, 1976. G. Lindblad, "Completely positivc maps and entropy inequalities." Commun. Moth. Ph.v>.,vol.4O,pp. 147-151. 1975. , "On the generators of quantum dynamical semigroup$." Commun. Math. Phys.,vol. 48, pp. 119-130, 1975. M. Ohya. "Quantum ergodic channels in operator algebras," 3. Moth. Anal. Appl.. vol. 84,pp. 318-328. 1981. -, "Entropy transmission in C*-dynamical systems," preprint. D. Ruelle. SfaItsimdMechnnrcs. New York: Benjamin. 1965. L. I. Schiff. Q W I I U ~ Mechonro. 2nd ed. New York: Wiley, 1968. R Schatten. Norm I d d r of Completely Continuour Operororr. New York: Sp"ngcr-Verlag. 1970. C. E. Shannon. A Mrrrhemnrtcol Theory of Commun8co1,orr. Urbana. IL: UNV. Illinois. 1949. N. F. Slincspnng. "Positive functions on C*-algebras." in Proc. Amer. Math. So'.. "01. 6, pp. 211-216, 1955. H. Takahasi. Informor,on Theory OJ Qurrntum Mechanrcol Channels, AdU Y ~ C Ptn ~ Commun,curron Svrremr. Vol. I . New York: Academic. 1966, pp. 227-310.
-
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M.Takesaki. T h e q o/Operulor Algehm I. New York: Springer-Verlag. 1981. A. Uhlmann, "Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an Interpolation theory," Commun. Mach. Ph.vr.. "01 54. pp. 21-32. 1977. H. Umegaki. "Representation and entremal properties o f averagjngoperators and their applications to information channels." J . Morh. A n d Appl.. "01. 25. pp. 41-73. 1969. "Conditional expectation in an operator algebras IV," Kodoi Mvrh. Sem. Rep.. vol. 14. pp. 59-85, 1962. I. von Neumann. Die Mothemotirchen Gnmdlagen der Quonrenmechonrk Berlin: Springer. 1932. K. Kraus. "General slate changes in quantum theory," A n n Phyr.. vol. 64. pp. 311-335. 1970. V. Gotini. A. Ftigerio. M. Verti, A. Kossakowrki. and E. L. G. Sudarshan. "Properties of quantum Markovian master equations," Rep. M o d . Ph.vr.. vol. 13. DO. 149-173. 1982.
-.
407 LETTERE AL NUOVO CIMENTO
VOL.
38,
N.
11
12 Novembre 1983
0 1983 Societa Italiana di Fisica
Note on Quantum Probability. M. OHYA Department of Information Sciences, Science University of Tokyo Noda Oity, Ohiba 278, J a p a n (ricevuto il 3 Agosto 1983) PACS. 03.65.
- Quantum theory; quantum mechanics.
Summary. - When a state of a physical system dynamically changes to another state, it is important to know the correlation existing between the initial state and the final state. This correlation is described by a compound state (measure) in classical systems. In this note, we show a way how t o construct such a compound state in quantum systems which is an extension of the classical compound state.
It is rather important in many physical sciences to study the dynamical change of states of a system. One of the most general description of this state change for classical systems is suggested in the communication theory of Shannon. A state of a classical dynamical system is expressed by a probability measure on that system and its dynamical change is generally considered as follows (1) : let X , H be compact Hausdorff spaces and Fx, Fybe their Bore1 fields, respectively. We denote the set of all regular probability measures (states) on ( X , Fx) by P ( X ) and on (P,Fp) by P(P). A mapping 1 : X x F y + Rf satisfying the following two conditions is called a channel: i) l ( ~ , E P ( P) for each fixed m E X and ii) l (-,Q ) is a continuous measurable function on X for each fixed Q E gP.This mapping is often called a transition (or Markov) kernel and is useful to study, for instance, information transmission and stochastic processes. A channel so defined provides a mechanism of state change. Namely, B state q E P ( X ) is transferred to a state y E P(P) under a channel I such a8 m )
Moreover, in order to study the process of state change and the property of a channel itself, we need a compound state (joint probability) indicating the correlation existing
(l)
402
H. UMEGAEI: J . Math. Anal. AppZ., 25. 4 1 (1969).
403
NOTE ON QUANTUM PROBABILITY
between the initial state q and the final state y. The compound state @ is given by
61
for any Q1E F x and Q2E gr. In quantum dynamics, we take two 0*-systems ( d ,G(d))and (g,G ( A ) ) ,one of which describes an initial (input) system corresponding to ( X , .Fx.P ( X ) ) above and P(Y ) ) . Here d another describes a final (output) system corresponding to ( Y ,gF, (respectively, a)is a O*-algebrawith unity I d (respectively, 1%) and G ( d ) (respectively, G ( 9 ) )is the set of all states (i.e. normalized positive linear functionals) on d (respectively, 9)(2). Then let us consider a mapping A* from G ( d )to G ( a )such that its dual map A : :g+d is completely positive ( 2 ) with AI9 = Id. This mapping A* is called a channel between two quantum-dynamical systems (3). I n particular, when& = 0 ( X ) ,the set of all = C ( P ) , the formula (1) defines a channel A* from continuous functions on X, and P ( X ) to P(P) (Le. y = A * q ) because every probability measure q on ( X , .Fx)can be regarded as a state on C(X) by the Riesz-Markov-Kakutani theorem. We meet several channels in several fields of physics. For example, time evolution automorphism group, dynamical semi-group and conditional expectation on a certain algebra are typical channels. Now it is well known (4) that the joint probability measure does not generally exist in quantum systems. Hence it has been difficult to define a compound state describing the correlation existing between an initial state q E G ( d ) and its final state A*g, E G ( 9 ) . The aim of this note is to construct such a compound state and to show that our compound state is an extension of the classical one given by (2). For an initial state g, and the final state A*q, a compound state @ on d 09 of q and A * q should satisfy the following two conditions: i) @ ( A@ 1%) = g,(A) for any A E d and ii) q(Id @ B ) = A* q ( B )for any B E 9. There exist many states satisfying these conditions, for instance, @, = q @ A * q is such a state. But this state does not carry any correlation between g, and A*q. For any weak *-compact convex subset Y of G ( d ) ,there exists a maximal measure p such that g, is the barycentre of p and p is pseudosupported by the set e XY of all extreme points of 9’in the sense that p(Q) = 1 for every Baire subset Q of 9’ with Q 3 e x.5p (6). In this case, we write g,=
(3)
s
odp.
(0x9)
Note that the above maximal measure p is not always unique, and we denote the set of all such measures by M J Y ) . We now construct a true )) compound state of g, and A*p. For each p E M , ( 9 ) , define (4)
It is easy to see that this state @P satisfies the conditions i) and ii) mentioned above. Let us now show that the compound state defined by (4) is indeed an extension of the (9 M. TAKESAKI:Theory (*)
(‘)
of operator algebra I (Berlin, 1981). M. OBYA: J . iWath. Anal. A p p l . , 84, No. 2, 318 (1981). K. URBANIK:Studia Math., 21, 113 (1961).
409 404
M. OHYA
classical one. When 9' = P ( X ) , the extremal decomposition of a state p E P ( X ) is unique and given by
(5) where 6, is the Dirac measure concentrated a t a point $ E X . Since (A*8,)(Qz)= = 1(x,Qz) for any Qz E Fy, we have
for any Q1 E and Qz E g y . We finally consider the case of al = C(Zl) = C(Z1) G I , where C ( 2 ) is the set of all compact operators on a separable Hilbert space 2. Then G ( . d ) contains the set T(Pl)+,lof all positive trace class operators on Xl with unit trace and so does G ( a ) . Moreover, a channel A* is a trace-preserving completely positive map from T(21)+,l t o T(Pz)+,l.In this case, if an extremal decomposition of a state e E T ( 2 1 ) + ,is 1 given by e = C Anen, then our compound state is
+
n
U=CAnen@A*@n. n
Among these compound states, the following is the most important: U,
=
C &,En @ A*En . n
The symbols appearing in (6) mean the following: a ) 1, is the eigenvalue of e, and the eigenvalue of multiplicity rn is repeated precisely m times. b ) En = [sn)
*** The author wishes to express his gratitude to Prof. H. UNEGAKI for his interest in and valuable comments t o the present work.
R. QCHATTEN: Norm ideals of completely continuous operators (Berlin, 1970). M. OHYA: O n compound slate and mutual informalion i n quantum informalion theory, to appear in I E E E Information Theory. (') M. O n Y a : Remarks on dynamical change of states, preprint. (K)
41 0 Vol. 84, N o . 2, December 1981 Reprinted from JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS All Rights Reserved by Academic Press, N e w York and London
Quantum Ergodic Channels in Operator Algebras MASANOFU OHYA Department of Information Sciences, Science University of Tokyo, 278 Noda Ciry, Chiba Japan Submitted by G.-C. Rota
The quantum ergodic channel is studied by operator algebraic methods. The ergodic and KMS channels are introduced and their dynamical properties are discussed.
INTRODUCTION After Shannon's work [19], information theory deals not only with entropy (information) itself but also with its transmission. This mechanism of transmission has been revealed by the so-called channel between input sources and output receivers. Laser physics has been much developed and its usefulness t o information theory has recently been realized. It is therefore necessary to formulate communication theory in terms of quantum mechanical language. Many authors have tried to find a mathematical basis of information theory [2, 9-12, 231. The present paper is one trial along this line. In Section 1, we introduce several channels and formulate the problems to be investigated in this paper. In Section 2, we discuss the existence of ergodic channels introduced in Section 1. In Section 3, we characterize the ergodic channel and prove a theorem of the Radon-Nikodym type for a stationary channel. In Section 4, we study the KMS and weakly mixing channels and consider some relations between such channels. 1. PRELIMINARIES
The classical communication channel has been described in the following way: Let , 'A Y be compact Hausdorff spaces and S?x, Bybe their o-fields, respectively. In information theory we call (X, gX) an input (message) space and ( Y , . 9 y ) an output (message) space. We denote the sets of probability measures by P ( X ) on ( X , 9x)and by P ( Y ) on ( Y, g Y ) A . mapping 318 Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
41 1 QUANTUM ERGODIC CHANNELS IN OPERATOR ALGEBRAS
319
v : X @ 9, +R is said to be a channel from the input space to the output space when the following conditions are met: +
v(x, -) E P ( X ) for each fixed x E X ,
(1) (2)
v ( . , Q) is measurable on
(X, 3*) for each fixed Q in 9,.
A probability measure p is transferred to a probability measure p' through a channel v in the manner of
for any Q E 9,. Let B ( X ) and B ( Y ) be C*-algebras of all bounded Baire measurable functions on X and Y, respectively. Let K ( B ( Y ) ,B ( X ) ) be the set of all bounded linear transformations from B(Y) to B ( X ) satisfying for any K E K( B ( Y ) , B ( X ) ) , (1) K 1 = 1, (2) Kf 0 for any f > O in B(Y), and (3) Kf, 0 for any f, 0 in B(Y). Umegaki has shown [23] that a one to one affne correspondence exists between the set of channels and the set K(B(Y),B ( X ) ) above such that
1
1
>
for any f E B(Y). A quantum mechanical dynamical system is described by a triple (.d, G ( d ) , a@)), where d is a von Neumann algebra with unity 1 acting on a Hilbert space R,G ( d ) is the set of all normal states on d, and a ( R ) is a strongly continuous one-parameter group of automorphism of d. We need two such dynamical systems in the information theory, namely, an input system and an output system. An input system is described by a triple G ( d ) , a ( R ) ) above and an output system is described by another von Neumann algebraic triple (3, G ( 9 ) , z(R)) acting on another Hilbert space .?. A message of the input system is sent to the output system through a channel. By analogy with Umegaki's result discussed above, it is natural for us to take a channel for the quantum case as a mapping from G ( d ) to G(.%'). More precisely, a mapping, denoted by A*, from G ( d ) to G ( 9 ) is said to be a channel if it is a dual map of A from 9to d such that (1) A is completely positive [ 1, 4, 14, 15, 20, 211 (i.e., it sends every positive n X n matrix (B,) with entries B, in 3 to a positive n x n matrix (AB,) for a1 n E N ) , (2) A is ultraweakly continuous, and (3) A(Z) = Z (cf. Ref. [ 101). ,'@L(
412
3 20
MASANON OHYA
The set of all invariant states in G(d)under a , (t E R ) is denoted by Z(a) and the extreme points of Z(a) are denoted by exZ(a). We similarly define Z ( t ) and exI(r). A channel A* is said to be stationary (or covariant) if A 0 r, = at 0 A holds for all t E R . The set of all stationary channels is denoted by SC(.pP,.5?) or simply by SC. A channel A* is said to be ergodic if A* E SC and (o E exZ(a) implies A*(o E exZ(r). The set of all ergodic channels is denoted by E C ( d , ,9) or EC. These channels play important roles in usual (classical) information theory (cf. Ref. [23]). In quantum statistical mechanics, the equilibrium state of a dynamical system is an important tool in studying the property of such a system [S, 171. The most general formulation of an equilibrium state is due to the KMS condition [6, 8, 221. A state (o E G ( . d ) satisfies this condition with respect to a , if for any pair A , B E d,there exists a bounded function F A , B ( z of ) z E C analytic in,and continuous on, the strip 0 Im z ,< 1 with boundary values: FA,B(t)= (o(a,(A)B ) and FA,B(t i) = (o(Ba,(A)). Let us denote the set of all KMS states w.r.t. a, by K ( a ) and the set of all extreme points of K ( a ) by exK(a). We also define K ( z ) and exK(t) for r,. A channel A * is said t o be KMS if A* E SC and (o E K ( a ) implies A*(o E K(r). We denote the set of all KMS channels by K C ( d ' , 9) or KC. A channel A * is said to be extremal in SC if A * = AA 7 + (1 - A) A for s o m e A , * , A ? E S C a n d A E ( O , 1) i m p l i e s A F = A F = A * . We introduce one more concept in this section: This concept is somewhat related to that of ergodic channel. We call a state (o weakly mixing (or v-clustering) for a if (o(v,(A) B ) = (o(v,(A)) (o(B)holds for any A , B in -19, where v, is defined as
+
<
::
1 T q,(A) = lim - d t a , ( A ) T-rm T
j,,
for any A E .d. We denote the set of all weakly mixing states for a by WM(a). For t, we define W M ( s ) in the same manner. A channel A* is said t o be weakly mixing if (o E W M ( a ) implies A*(o E WM(r). The set of all weakly mixing channels is denoted by WMC(.d,.9) or WMC. The problems to which we address ourselves are as follows: Do the channels defined above exist? ( 2 ) What kinds of characterizations do we have for such ergodic channels? (1)
Remark.
Some authors define a weakly mixing state Q by limT+m(1/T)
41 3 QUANTUM ERGODIC CHANNELS IN OPERATOR ALGEBRAS
321
dt 1 #(a,@) B ) - #(a,@))#(B)l = 0. This definition is slightly general, but we take the above definition here.
:1
2. EXISTENCE OF ERGODIC CHANNELS
Let 3 be the von Neumann algebraic tensor product of d’ and .9. Let Aut(9l) be the set of all automorphisms of 9l=d’@O.We denote an embedding from 28 into 3 by j ; that is, j ( B ) = 1 0B for any B E 9. Suppose that exI(r) # 0 (resp. K ( r ) # 0 or W M ( r )# 0).We take a state I+? E W M ( r ) )and define A = (1 0I,?) o j ; then it is readily seen that the dual map A* of A is an ergodic (resp. KMS or weakly mixing) channel. More sophisticated channels can be constructed. For example, let us take u E Aut(3) satisfying the intertwining property u o yt = yt o u ( y t = a , @ r t , t E R ) and define A = (1 @ I,?) 0 u for some I+?€ G(9). When tjiE W M ( t ) ,we can show that A* is a weakly mixing channel. Let us sketch the proof For any rp E WM(a),we have A*cp(VT(Bl)B2) = cp 0I,?(4VT(Bl))@ 2 ) ) = cp 0I,?(V,(~(BI)) a@,))
for any B , , B , E 9. It is easy to see that (0
0 I , ? ( V y ( 4 0Bl) A , 04) = V(VJ-41))
c p ( 4 I,?(V?(BI))I , ? P 2 ) 5
for any A , , A 2 E d and B , , B , E g . Since 9 1 = { x i A i @ B i : A i E d ,
B iE 9}-“, we obtain A*V(VT(Bl)
= (0 0 I,?(Vy(‘(B1)))
90
=A*q(VT(Bl))A*p(B2)*
Hence A*cp E WMC. When I+? E K(r), A* becomes a KMS channel: For any cp E K ( a ) , the state q~ 0I,? on 3 satisfies the KMS condition w.r.t. yf. There then exists a bounded function FQ,R(z)of z E C for any pair Q, R E 9l analytic in and continuous on the strip 0 Im z 1 with boundary values: F Q , R ( f= ) cp 0 I,?(y,(Q) R) and FQ,R(f i) = rp 0 W(Ry,(Q)). In particular, put Q = u ( B , ) and R = u(B2)for any B , , B , E 9. Then
< < +
FBI,Bl(f)
FB,.B2(f
Thus, A* is a KMS channel.
=A*p(rf(Bl)
B2),
+ i> =A*~(Bzrt(Bi))*
41 4
322
MASANORI OHYA
By these simple arguments, we can answer question (1) in Section 1 affirmatively under rather plausible conditions (e.g., exZ(7) # 0). We remark here that the condition 0 o yt = y t o u is indeed satisfied for several physical models.
3. ERGODICCHANNEL We give some characterizations of an ergodic channel. Before stating and proving the results, we note one point of terminology; that is, for a subset 9 of S(d),when u,(Q)= u,(R) is satisfied for every a, E .iu, we say that Q is equal t o R mod 9‘. THEOREM 1. If a channel A* is ergodic, then it is extremal in SC mod I ( a ) (i.e., AT = A ; iff u,(A,(B))= a,(A2(B))for any u, E I ( a ) and any B E .%).
Proof. Suppose that there exist AT, A : E SC and (1 -A)A;. We then have
A* =AAt
+
A *p(B) = AA :@)
+ (1
-
A E (0,1)
such that
A) A&@)
for any u, E exZ(a) and any B E 9. Now, it is easy to see that A;u,(rt(B))= A ; p ( B ) ( k = 1, 2 ) for any B E 9, so Atu, E Z(r). Since A * E EC, the state A *p is ergodic, which implies A f a , = A $9 = A *a,. The set I ( a ) is the closure of the convex hull of exZ(a) in the weak*-topology, hence the equality ATu,=ATu,=A*u, holds for every u , E I ( a ) . Therefore, A T = A f = A * mod Z(a). Q.E.D. THEOREM 2. When 28 is simple and . ” P = B ( X ) , a channel A* is ergodic if and only i f A * is extremal in SC mod Z(a).
Proof. The “only if’ part has been proved in Theorem 1. W e show the “if’ part here. Suppose that a channel A* is not ergodic; then there exists an ergodic state a, in Z(a) such that A *a, is not in exZ(t). Put ty = A *a,. The state w can be decomposed into two other invariant states tyl, ty2 such that w=hy, (1 - A ) wz for some 1 E (0, 1). Thus the states wk (k = 1, 2 ) are ’ U@(R)’15, 13, dominated by w. Hence there exist Q$ ( k = 1, 2 ) in r @ ( 9 )n 17, 181 such that Q$ 0,
+
>
and
41 5
323
QUANTUM ERGODIC CHANNELS IN OPERATOR ALGEBRAS
in which no, { Uf : t E R},and Y are, respectively, the representation, oneparameter unitary group, and cyclic vector obtained by the GNS construction theorem [6, 17, 181. Since the von Neumann algebra .58 is assumed t o be simple, the representation no is faithful. Define Zo= A o 77; I . This map E@can be extended t o a complete positive map from B(d;t",)to d' because .d= B(,P). We use the same notation for this extended map. We therefore have
for any B E . d . It is easily shown that n@,Qt are completely positive, so the map A t = &@o Q: o n@is completely positive from 9to d'. Consider a map @ v p e G ( , d ) A t (remember y~ = A*a,), where A ; = A when y~ 6! Z(z). This map is completely positive from 2?to %dewith dv= d for all a, E G ( d ) . We define a projection E from @ Lde t o d such as w ( E ( @ A t ( B ) ) = ) w (A;(B)) for every state w E G ( . d ) . Let us define a map
A,
=E
@At.
0
Then is a Completely positive stationary map and A,(Z) = I. According t o the decomposition of W, we have A* = LA? + (1 - A) AT, which contradicts the extremality of A*. Q.E.D.
Remark. In the proof of this theorem, if the state A*a, is ergodic (i.e., A * q E exZ(z)), then Q t (k = 1, 2) are equal to the unity Z. A completely positive map A : 2 i + d can be represented by the Stinespring theorem [3, 201 as follows: There exist a representation p from 9 into the set of bounded linear operators B(9) on a Hilbert space P and a bounded linear transformation V on R into 9, by which we obtain A ( B ) = V p ( B ) V for any B E 9. As in the GNS theorem for an invariant state, when a channel is a stationary, we expect that there exists a strongly continuous unitary operator T , ( t E R ) such as p ( r , ( B ) )= T,p(B)T P t . We can indeed construct this unitary operator. According to the Stinespring theorem, the Hilbert space 9 above is the completion of .9 @R/./IT w.r.t. an inner product defined below. Here, .2?@ 2 is the algebraic tensor product of .Band 3,the inner product (., .>is given by (a.
a )
(a,0)= Z(4ji,A(BTC,) Y,) for any
a=
B, @
a,, 0 =
I
C , @ YK in .9 @ Z ,and ,H is the
416
324
MASANORJ OHYA
set I52 E .9 O X I (52, 52) = 0 ) . Let us denote the equivalent class of Q by [ Q ] .Then we have (Q,
0 ) = ( [ a ][@I>, ,
p ( B ) [ EB , O
=
[I
BBk
O @k]9
V @= [I 0@ I .
For simplicity of discussion, we assume that a , is spatial (i.e., there exists a strongly continuous unitary operator U , such as a,(A) = U , A U - , for any A E ,r9 and t E R ) . By analogy with the GNS theorem for an invariant state, define an operator S, by
It is then easy t o see that (S,Q, S , 0 ) = (Q, 0 )
for any Q, 0 E .9 P',and (S, 0,
s,0 ) = 0
for any O E . 4 . . S , can be extended to an unitary operator TI on Y'= 8 0 F;!. 4 .. This operator T I is given by
Tl[Ql = [ S f Q I . Now, for any B , Bk E ,9, @, €,P', we obtain
Hence, p(r,(B))= T,p(B) T - , holds for any B E 9. Furthermore, TI V @ = T,[I@ @ ] = [ I @ U , @ ] = VU,@ is satisfied for any 4~E.2". If @ is an invariant vector under U , , then T , V @= V@. We therefore obtain THEOREM 3. If a channel A* is stationary and a, is spatial such that a , ( A )= U , A U - , f o r any A E Lc9, then there exists a strongly continuous oneparameter unitary group { T , I t € R } such that p(s,(B))= T,p(B) T - , for any B E ,z??and T , V @ = V @for any invariant vector @ of P under U , .
41 7 Q U A N T U M ERGODIC CHANNELS IN OPERATOR A L G E B R A S
325
By using this theorem, we have the following Radon-Nikodym theorem for a stationary channel. THEOREM 4. Let A* be a stationary channel. Then there exists a completely positive stationary map B from ,9to dominated by A i f and only i f p ( , ~ ? ) ’n T(R)’ # CZ.
Proof: Suppose that such a completely positive map .? exists. According to Stormer [21], there exists an positive operator Q in p ( 9 ) ’ such that Q 6? CZ and Z ( B ) = V*ep(B) V for any B E .d. It has been shown that T I V @ = [ I @ U , @ = V U l @ for any @ E 2‘ and all t E R. Hence T , V = V U , holds for all E R. Since E is stationary, we obtain r/-*QT,p(B)T - , V = V*Qp(r,(B))V = B(z,(B))
for any B E 9.Thus, QT, = T,Q for all t E R , which means that Q is an operator in p( 9)’ n T(R)’. This concludes ~(9)’ n T(R)’ # CI. Conversely, if we take an operator Q from p ( 9 ) ’ n T(R)’ such as Q 6? CI and 0 Q < I . Let us define a map E as B ( B ) = V*Qp(B) V for any B E 9. Then it is readily seen that Z is a completely positive stationary map from Q.E.D. d to .dand dominated by A .
<
4. WEAKLYMIXINGAND KMS CHANNELS
We study the property of weakly mixing and KMS channels and some relations to ergodic channel. When a channel A* is stationary and weakly mixing, we have A*4(%(Bl)
B2)
=A*4(vT(B*))
for any B , ,B , E ,3and 4 E WM(a). Since A * is stationary, the right hand side of the above equality becomes
4(v&wJ)
4(A(B2)) = 4
(vm(W1))~(~J)
=@(AqT(Bl)
Hence, A(vT(Bl)B 2 )= A ( v , ( B , ) )
mod W W a ) .
418
326
MASANORI OHYA
Conversely, for any A*4(qT(B1)
4E B2)
WM(a), we obtain
=4(A(qT(B1)
B2))
=4(A(qT(B1))A(B2))
= 4(Va(~(Bl))AQ32))= 4(Va(@l)))
4(4B,))
for any B , , B , E 9. Thus A*# E WM(a). Let E , be a projection associated with a state 4 E G ( d ) from the GNS Hilbert space 3,to the set of invariant vectors under the unitary group { U f @ 1 t E R } of 4 (i.e., n,(a,(A)) = q n , ( A ) Urn, and Uf@ = @ for the GNS representation n, and the cyclic vector @). We call a dynamical system (d, a ) G-abelian on an invariant state 4 if for any A , B E d, &[qa(A), B ] )= 0 is satisfied, where is any vector state of E,3,. In particular, when (d, a ) is G-abelian on every invariant state for a, we merely say that (d, a ) is G-abelian. Suppose that the input system (d, a ) is G-abelian and a channel A* is stationary and weakly mixing. Then exI(a) = W M ( a )nI(a). Since A* E W M C n SC, A*# E WM(z), which implies A * @E exI(7). Hence A* is ergodic. For the G-abelian input system (d, a ) and a stationary channel A * , if A is a *-homomorphism, we can readily show that EC = WMC nSC. Let .$be the set of z-analytic elements of 9. Then it is an easy exercise to show that A* E KC if and only if A ( z t ( B l ) B 2 )=A(B,7,+,(BI>)mod K ( a ) for any B , € 3and B , E 9 . Moreover, if the input system ( & , a ) is qabelian (i.e., # ( D * [ q , ( A ) , B ]D )= 0 for every invariant state 4 and any A , B , D E sf) and A is a *-holomorphism, then we can easily see that A * is K M S if and ony only if A* is ergodic. We summarize these facts as
THEOREM 5. For a stationary channel A*, we have: (1) A * € WMC i f and only mod W M ( a )for any B , , B , E .9.
if A(qT(B1)B2)=A(qT(B1))A(B2)
( 2 ) I f the input system (-4, a ) is G-abelian and A* E WMC, then A* is ergodic. ( 3 ) If (,4, a ) is G-abelian and A is a *-homomorphism, then EC = W M C n SC. (4) A* E KC i f and only i f A ( z , ( B l ) B 2 =A(B27,+,(B1)) ) mod K ( a ) for any B , E ,8 and B , E .9. ( 5 ) I f (.d, a ) is q-abelian and A is a *-homomorphism, then KC = EC.
41 9
QUANTUM ERGODIC CHANNELS IN OPERATOR ALGEBRAS
321
CONCLUSION We introduced and studied several ergodic channels (ergodic, KMS, weakly mixing) in this paper. We hope that these concepts are useful to analyze the information theory for quantum systems. One of the applications is the transmission of entropy (information), which is discussed in Ref. [ 161.
ACKNOWLEDGMENTS The author thanks Professor H. Umegaki for valuable discussion and useful comments.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
M. D. CHOI,Ill. J . Math. 18 (1974), 565. E. B. DAVIES,IEEE Trans. Inform. Theory (1977), 530. E. B. DAVIES,“Quantum Theory of Open System,” Academic Press, New York, 1976. C. DAVIS,Proc. Amer. Math. SOC.8 (1957), 42. s. DOPLICHER,D. KASTLER.A N D E. ST0RMER. J . Funct. Anal. 5 (1969), 419. G. G. EMCH,“Algebraic Methods in Statistical Mechanics and Quantum Field Theory,” Wiley. New York, 1971. R. HAAGA N D D. KASTLER,J . Math. Phys. 5 (1964), 848. R. HAAG,N. M. HUGENHOLTZ. A N D W. WINNINK, Commun. Math. Phys. 5 (1967), 215. c. W. HELSTROM,J. W. s. LIU, AND J. P. GORDON,Proc. IEEE 58 (1970), 1578. A. S. HOLEVO.Rep. Math. Phys. 12 (1977), 273. A. S . HOLEVO,Problemy PeredaEi Informacii 9, No. 2 (1973), 31. R. S. INGARDEN, Rep. Math. Phys. 10 (1976), 43. I. KOVACSA N D J. SzuCS, Acta Sci. Math. (Szeged) 27 (1966), 233, K. KRAUS,Ann. Phys. 64 (1971), 311. G. LINDBLAD, Cornmun. Math. Phys. 48 (1976), 119. M. OHYA.preprint. D. RUELLE.“Statistical Mechanics,” Benjamin, New York, 1969. S. SAKAI,“C*-Algebras and W-Algebras,” Springer-Verlag, New York/Berlin, 197 1. C. E. SHANNON, Bell System Tech. J . 21 (1948), 379. N. F. STINESPRING, Proc. Amer. Math. Soc. 6 (1965), 211. E. S T 0 R M E R , Acta M a f h . 110 (1971), 233. M. TAKESAKI, “Tomita’s Theory of Modular Hilbert Algebras and its Applications,” Lecture Notes in Mathematics. No. 128, Springer-Verlag, Berlin, 1970. H. UMEGAKI, J . Math. Anal. Appl. 25, No. 1 (1969), 41.
420 PACIFIC JOURNAL OF MATHEMATICS Vol. 96, No. 1, 1981
SUFFICIENCY, KMS CONDITION AND RELATIVE ENTROPY IN VON NEUMANN ALGEBRAS FUMIOHIAI, MASANORIOHYA AND MAKOTOTSUKADA The sufficiency in von Neumann algebras is discussed with some applications to classification of normal states. It is shown that the concept of sufficiency characterizes the KMS-states and the invariant states with respect t o a modular automorphism group. The relations between the Sufficiency and the relative entropy are established.
Since the investigation of sufficient statistics in abstract measure theoretic terms was initiated by Halmos and Savage [lo], the concept of sufficiency has been developed by many mathematical statisticians in terms of various relations given by comparison of experiments, risk functions within the framework of statistical decision problems and so on. A characterization of sufficiency was given in [12] through the measure of Kullback-Leibler information. The concept of sufficiency was first generalized by Umegaki [22, 231 to the noncommutative case of semi-finite von Neumann algebras with some extension of the Kullback-Leibler information (usually called the relative entropy). Later the related discussions especially concerning the relative entropy for quantum systems have been made by several authors, e.g., Araki [2, 31, Gudder and Marchand 171, and Lindblad [13]. As defined precisely and explained in 55 1 and 4 of this paper, the concept of sufficiency is more or less considered through the informativity of a certain subalgebra with respect t o a given algebra for a dynamical system of interest. Namely, in the case t h a t such a subalgebra is sufficient, the relative entropy on the subalgebra is equal to t h a t on the given algebra. This fact may or may not be a reason why the concept of sufficiency has not been entered into analysis of physical systems, in which the change of entropy is thought of more relevant. The Kubo-Martin-Schwinger (KMS) condition was introduced by these three authors [11,14] as a boundary condition of t h e thermal Green function. Haag, Hugenholtz and Winnink [8] showed t h a t in the operator algebraic framework this condition is a fundamental one describing thermal equiIibrium of quantum systems. The KMS condition through the Tomita-Takesaki theory now becomes a core of studying von Neumann algebras. Under the above historical basis, our main motivation of this Introduction.
99
42 1 100
FUMIO HIAI, MASANORI OHYA AND MAKOTO TSUKADA
work is as follows: How useful for quantum systems is the concept of sufficiency? How much of the related topics of sufficiency, mostly done for the commutative case, can be generalized t o the noncommutative case? Having these questions in our mind, we discuss the sufficiency with some applications t o classification of normal states on the basis of recent development of von Neumann algebras. In § 1 of this paper, we establish definitions and notations used throughout and also give some simple facts. In 82, i t is shown t h a t the concept of sufficiency characterizes the invariant states and the KMS-states with respect to a modular automorphism group. In 8 3 , we prove some formulas on the relative entropy using Araki’s definition of relative entropy. In 84, combining several theorems obtained in the previous sections, we establish some results which indicate the relations between the sufficiency and the relative entropy. As a whole, we like to claim t h a t the concept of sufficiency might be very useful for analysing von Neumann algebras and hence some quantum systems. 1. Cefinition and preliminaries. Throughout this paper, let be a von Neumann algebra with unity I acting on a Hilbert space .P, and 8 be the set of all normal states of %. A dynamical system of physically interest is described by a triple (’2,8,a ) , where a,, t 6 R , is a strongly continuous one-parameter automorphism group of %. A state 9 6 8 is said to satisfy the K u b o - M a r t i n - S c h w i n g w (KMS) b o u n d a y y c o n d i t i o n a t a certain constant p > 0 with respect to a, if for every pair A, B E % there exists a bounded function Fa,a(x)continuous on and holomorphic in the strip OSIm xsp with boundary values: (32
F A , d t )
= F(at(A)B) and
Fa,n(t
+ i P ) = Y(Bat(A))
If 9 satisfies the KMS condition with respect to at, then q~ is proved t o be a,-invariant, i.e., p a t = 9. Considering a,,, we may take p = 1 in the sequel discussions. Takesaki showed in [17] using Tomita’s theory t h a t t o every faithful state ~ J E @there exists a unique one-parameter automorphism group (i.e. , the so-called m o d u lar a u t o r n o r p h i s m g y o u p ) ol‘ with respect to which e, satisfies the KMS condition a t p = 1. I n this paper, a subalgebra always means a von Neumann subalgebra of (32 with I. For a subalgebra !R and a state 9 E@, let E,( - Im) denote the conditional expectation with respect t o !R and 9 (if i t exists), which is characterized as a norm one normal projec-
422 SUFFICIENCY IN VON NEUMANN ALGEBRAS
101
tion from 92 onto 2Jl satisfying 9(A)= q(E,(A[2Jl))for all AEY?(cf. [19, 211). It was shown by Takesaki [18] t h a t for a faithful state ~ € the8 conditional expectation E,(.[2Jl) exists if and only if Dl is invariant under the modular automorphism group 0;. According to [S], for any two faithful states 9, E 8 there exists a strongly continuous function t M U ~of R into the unitary group of (3% which is a pcocycle, i.e.,
+
u,+t = u&(ut)
S,
tER
and which satisfies
a f ( A )= u,~l"(A)u; , tER , AE9. This pcocycle ut is denoted by ut = (D+:0 ~and ) is~called the Connes Radon-Nikodym dwivative of with respect t o q. Some discussions are found in [4, 91 concerning Connes Radon-Nikodym derivatives and conditional expectations. Let S be a subset of 8. A subalgebra YJl is said to be suficient for S if E,(.I%R)exists for each s p ~ Sand for every A € % there exists an AoeYJ!such t h a t
+
A0 = -%(AI 'w a.e. 191
Y
?€
s,
where A = B a.e. [y] means q(IA - BI) = 0. This definition of sufficient subalgebras is somewhat weaker than t h a t in [22]. Also we call 23 to be minimal suficient for S if 2Jl is sufficient for S and any subalgebra being sufficient for S includes 2Jl. For sp, ,$- E 8, it is said t h a t # is absolutely continuous with respect to 9 (we write ,$- < 9)if f o r each A €%, ?(A*A)= 0 implies #(A*A)= 0; t h a t is, ,$- << 9 if and only if s(+) 5 s ( 9 ) where s ( y ) is the support projection of 9. We give here the elementary facts of sufficiency which are readily seen from the definition. (1") Let 9,,$- E 8 with #
+
2. Sufficiency and characterization of states. The following lemma is a restatement of [4, Lemma 1.61 in our terminology. We
423 FUMIO HIAI, MASANORI OHYA AND MAKOTO TSUKADA
102
give the proof for completeness, LEMMA2.1. F o r each subalgebya YJl a n d t w o f a i t h f u l states q,$ E 8, t h e f o l l o w i n g c o n d i t i o n s a r e equivalent: ( i ) YJl i s s u f i c i e n t f o r {q,+}; ( i i ) E,(-[YJt) e x i s t s a n d (D$:D q ) t E YJt f o r e v e r y t 6 R.
+r
Proof. Let $3 = q 1 'Dl and 4 = YJt. Assume that YJl is sufficient for {q,+}. Then the conditional expectation E,(.IYJl)exists and +(A) = $(E;,(A/YJl))for A € % . By [5, Lemma 1.4.41, we have
(D+:Dq),= ( D ( & E ) : D(+oE)), = (Dq:OF.>,6 YJ? for every t E R, where E ( . ) = E,(.IVt). Conversely assume t h a t E,(. IYJl) exists and (D+:D q ) t G YJt for all t 6 R. Since a? = or r YJl, i t follows t h a t u,= (D+: Dq),is a @-cocycle. By [5, Theorem 1.2.41, there exists a unique faithful normal semi-finite weight 3 on YJl such t h a t (Dq: D+),= ut. Define a faithful normal semi-finite weight on 2 by +'(A) = $(E,(A[ n))for A 6 %. Then it follows that
+'
(Dq': D q ) t = (I)$:
=
(D+:Dq),, t € R .
+' +,
Hence we have = so t h a t +(A) = +(E,(A]Vt))for every A em. This shows t h a t 'Dl is sufficient for {q,+}. In this section, let q be a fixed faithful normal state of % and a," its modular automorphism group. Let Z, be the subalgebra consisting of all A € % such t h a t q ( A B ) = q ( B A ) for every B G % . The subalgebra Z, is called the c e n t r a l i z e r of q and is exactly the fixed point algebra of o? (cf. [17,Lemma 15.8]), i.e., 2, = { A € % :&'(A) = A, t € R }.
Let 8 be the center of 92, i.e., 8 = %n %'. Clearly 8 cZ,. Let I ( q ) be the set of all &-invariant states in 8, and K ( q ) be the set of all states in 8 satisfying the KMS condition with respect to G? a t p = 1. Then we have: THEOREM 2.2. ( 1 ) FOYeach
+E@,
+ € I ( q )if a n d o n l y i f Z,
i s s u f i c i e n t f o r {q,+}.
( 2 ) T h e c e n t r a l i z e r Z, i s m i n i m a l s u f i c i e n t f o y I ( q ) .
+
Proof. ( 1) Let 6 (33 and take = (+ + q ) / 2 . Then we easily see t h a t + € I ( q ) is equivalent to 7;p1~I((q)), and the sufficiency of 2, for {q,+} is equivalent to t h a t for {q, Therefore we can assume is faithful. Since Z, is elementwise invariant under a?, that
+
424 SUFFICIENCY I N VON NEUMANN ALGEBRAS
103
there exists the conditional expectation E,(-\Z,) from 8 onto 2,. Hence, in view of Lemma 2.1, it suffices t o show t h a t +sI(cp) if and only if (D+: Dp),6 2, for every t E R. If + 6 I@), then by [5, Lemma 1.2.31 there exists a positive self-adjoint operator h affiliated with 2, such t h a t (D+: DF),= hzt s 2, f o r all t E R. Conversely suppose t h a t (09: Dp),sZ, for every t € R. Since
d ( A ) = (D+:Dp)d(A)(D+:0 ~ ) :
(2.1) we have
p(ol"(A)) =
= q ( A ), A
€
92 .
+
Hence i t follows that q is or-invariant, and thus is oY-invariant (cf. [17,Theorem 15.21). ( 2 ) It follows from (1) t h a t 2, is sufficient f o r every pair {q,+} with +€I(p). Hence 2, is sufficient for I(p). To show the minimality of Z,, let %! be any subalgebra which is sufficient for I ( q ) . We now prove t h a t Z,c%!. Take any positive invertible operator h € Z , with q ( h ) = 1, and define a faithful state + s @ by +(A) = q ( h A ) for A s 2. Then we have G I(?) and (D+: Dp), = hzt. Since (0.1.: Dp), E -9'2 for every t 6 R by Lemma 2.1, i t follows t h a t h ~ m .Thus Z,c!!Jl.
+
THEOREM 2.3. ( 1 ) F o r each +E@, + s K ( q ) if and o n l y if 8 is s u f i c i e n t f o r {p, +}. ( 2 ) T h e c e n t e r 8 is minimal s u f i c i e n t f o r K(p).
+
P?*oof. As in the proof of Theorem 2.2, we can assume t h a t is faithful. If + E K ( q ) , then by [15, Theorem 5.41 there exists a positive self-adjoint operator h affiliated with 8 such t h a t +(A) = p(hA) for A G 52, so that (D+:Dq),= hit E 8 for every t € R. Conversely if (0s:D q ) t €8 for every t e R, then by (2.1) we have 0 2 ~ = oP and hence +€K(p). Thus (1) is proved. The proof of (2) is analogous to that of Theorem 2.2. 3. Relative entropy. When 2 is finite dimensional, for each q and in G3 the relative entropy S(pl+) is defined by
+
S ( 9I+)
=
WP+ log p+ - p+ log p,)
9
+.
where p, and p+ are density matrices for p and Araki [2, 31 extended the relative entropy to the case for normal positive linear functionals of general von Neumann algebras, and studied its several properties such as joint convexity, lower semiconitinuity and monotonicity .
425 FUMIO HIAI, MASANORI OHYA AND MAKOTO TSUKADA
104
In this section, we assume as in [3] t h a t % has a cyclic and separating vector. Let V be a natural positive cone (cf. [I]) for '31 and let 9 and be states in 8. By [I, Theorem 61, there exist unique vector representatives @ and T of y and + in V such that 9 ( A ) = (@, A @ ) and +(A) = (F, AY) f o r all A s % . The operator So,vwith the domain
+
D(S,,,) = %.F+ ( I - .P'(!r))
Sm,y(A!F + 9) = s'(F)A*@ , A E '3t , s"(!F)Q = 0 , where sn(F)(€92) denotes the %-support of the vector Y. Then is a closable conjugate-linear operator (cf. [3]) and the relative modular operator is defined by
_A4,v = (So,v)*ss,,
.
The relative entropy S(ql+) is now given by
Then S(913)2 0, and S(yl+) = 0 if and only if 9 = +. For each I +) denote the relative entropy of the restricsubalgebra %R, let SLm(9 t o 9Jt. By the monotonicity of relative entropy tions of 9 and generally proved in [20], i t holds t h a t
+
S&I+)
(3.1)
5 S(Fl4f-P)
for every subalgebra 9X (also see [2, 3, 13, 231).
THEORZM3.1. For each 9,+E@, lie)
- +ll
5 {2%9I+)YZ.
Proof. By [16, p.311, we can take two normal positive linear functionals (Iyzj(and ~
( 9
and qz such t h a t 9 Let e = s(yl). )is(yz). ~ )
+ = y1-
1 1 9 - $11
Then it follows t h a t
= j/ylI/
+
/I9- .1.ll = (9- +)(e) - (9- +)(I - e ) ZZ
2(9(e) - +(e))
*
Let %R be the subalgebra generated by e and I - e. monotonicity, we have
S(914-1 2 Sd9, I +I
By using the
426 SUFFICIENCY IN VON NEUMANN ALGEBRAS
It was shown in [6] that 2 la - pi 5
{ 2 ( p log; 6 + (1- @)log 1 - a ~
-
105
r2
for 0 5 a, p S 1. Taking a = y(e) and p = +(e), we deduce the desired inequality. 0 THEOREM 3.2. Let y,+ E @ be f a i t h f u l and 9 2 be a subaEgeb9.a such that Y.R c 2,. DeJine E @ by +'(A) = +(&(A 1 Y.R)) f o r A E 2. I f Sdyl+) < 00, then
+'
+
S(+' I+)
=
S(9I +)
-
+'
Proof. First note t h a t is well 9 rY.R is a faithful normal trace, there operator h affiliated with 93 such t h a t Take the spectral decomposition h = Since h,,E!IJl, we have for every A s %
Sm(9I+)
*
defined from 9 2 ~ 2 , . Since exists a positive self-adjoint +(A) = ?(hA) for all As%R. xde(x) and h, = \ ^ x de(x). 0
+'(A) = +(J%(AI an>) = lim 9(hL,E,(A I 93)) ,-== =
lim y(E,(h,A n-m
I a))= lim y(h,A) = y ( h A ) . L,-m
Hence i t follows (cf. [5, Lemma 1.2.31) t h a t (D+': OF),= hzt for all t E R. By the relations
(D+':D+)t = (D+': D y ) , ( D Y : D+)t = htt(D?: D+),, (0.1. D+)t ': = (A,l,,)ttdlp2t,
(DY; O+)t
= (&,,)"A,at
,
where drP= A ~ , we ~ , deduce t h a t ( A ~ , , ~= ) ' h2'(dm,,)'' ~ for all t E R. Moreover since hzt E %R c 2, and
oT(A) = (Ao,y)"A(~'m,yr-"~ , AE%, i t follows that hzt and (Am,,)'t commute. Now let 6 and d be vector representatives of y Y.R and 1 !IJl in a natural positive cone for Y.R. Since 9 5YJl is a trace, it follows t h a t A t , $ = h. By log A$,&= J(1og dg,g)J where J is the modular conjugation operator associated with P (cf. [3, Remark 3.4]), we have
+
r
r
S&I+)=
(3.2)
- (@,
=
(log 4 , P ) @ )
@, (log h)@)= (Y,
which is finite from the assumption.
S(+' I+)
=-
P, (log &,,Y)
(log h ) Y )
,
Therefore we obtain
427 106
FUMIO HIAI, MASANORI OHYA AND MAKOTO TSUKADA
THEOREM3.3. Let y s 8 be f a i t h f u l and 2Jt be a subalgebra such that YJlcZ,. Let + s 8 a n d deJine 6 8 by #(A) = $(E,(A[m)) i s f a i t h f u l o r (b) 5 Xg) f o r some f o r A s % . Suppose either (a) x > 0. If S,(yl+) < 00, t h e n
+
+
I1 +' - + I1 5 Cw(9I+)
+'
+
- S m ( 9 I +))}"'
*
Proof. For the case (a), the desired inequality is immediate from Theorems 3.1 and 3.2. Now suppose that 5 X y for some x > 0. For each E > 0, let qPE = (1 + E)-'(+ c y ) € 8 and define .,bi B 8 by +:(A) = +,(E,(AI 2Jt)). By the convexity of relative entropy (cf. [3,Theorem 3.8]), we have
+
+
Hence (3.3)
Since
+. 5 hy for
each
E
> 0,
by [3, Theorem 3.71 we have
lim S(yI+J = S(FI+>9
€++O
lim ~rn(9l?bPE) = Sm(9116.)
*
€++O
+
Since +: = (1 + &)-'(+' ~y),we obtain the desired inequality by letting & + +O in (3.3). Before closing this section, we have to note that Professor Araki gave us very important comments to some results of our first version of this paper, which make us enable to write them in the above form. 4. Sufficiency and relative entropy. faithful state y 6 8 be fixed as in S 2.
In this section, let a
THEOREM4.1. For each subalgebra 2JtcZ, and each $s@, the following statements hold: ( 1 ) Suppose the condition (a) o r (b) in Theorem 3.3. If 2Jt i s suficient f o r {y,+}, t h e n S,(yI+) = S(yl+), a n d conversely if S 9 t ( ~ l #= - ) S(yl+) < 00, t h e n 2R '. i s suficient f o r {y,+}. ( 2 ) If 2Jt i s suficient f o r {y,+}, t h e n S,(pl(+ y)/2) = S(yI(++y)/2), and conversely if S,(yI(~+y3)/2)=S(yl(++~)/2) < + 00, then 2I'l i s suficient f o r {y,+}.
+
+
428 SUFFICIENCY IN VON NEUMANN ALGEBRAS
107
Proof. ( 1 ) We use the notations in the proof of Theorem 3.2. Let ’D be suffcient for {q,+}, and suppose the condition (a). There exists a positive self-adjoint operator h affiliated with YJ2 such t h a t +(A)= q ( h A ) for all AEYJ~.Then we have +(A) = +(EdA I ‘D)) = d h A ) 7
A %9
as in the proof of Theorem 3.2. Hence i t follows t h a t (D+:D ~ J=) ~ htt, and we have (AQ,yr)’t= ( D y :D+)&
Since hit and ( & , l p ) i t We thus have
(4.1)
=
h-itd”$.
commute, i t follows t h a t h-it and A$ commute.
S ( q 1 +)
= -
=
(v, (-log h + log Ap)Y)
P, (1ogh)F’)
7
by AqF = F. From (3.2) and (4.1), we obtain &(?I+) = S ( q [+). The case (b) is proved from the case (a) by taking limits as in the proof of Theorem 3.3. Assume conversely t h a t S,(yl+) = S ( q I+) < 00. Then it fol= which implies t h a t YJl is sufficilows from Theorem 3.3 t h a t ent for {q,+}. ( 2 ) is immediate from (l), since 9R. is sufficient for {y,.j-} if and only if 91 is sufficient for {q,(+ + q)/2}.
+
+’ +,
The above fact (1)extends the result [23,Theorem 51 which was proved under some strong assumptions. Combining Theorem 4.1 with Theorems 2.2 and 2.3, we have the following:
COROLLARY4.3. ( 1) Suppose the condition (a) or (b) in Theorem 3.3. If + E K ( q ) , then S,(q [ lip) = S ( q[ #), and conversely if S8(qI +) = S(q~,i+) < +a,then + s K ( q ) . ( 2 ) If lip 6 K@), then. S&J I (+ + 91/21 = S(q I (+ + d / 2 ) , and ~ 1 1 2< ) + , then 6 K@). conversely if S,(q 1 (+ 91/21 = S ( qI (+
+
+
00
+
The monotonicity (3.1) says t h a t the restriction of measurement to a subalgebra YJl usually makes i t more difficult to discriminate between two states. From Theorem 4.1, the physical meaning of suf-
429 FUMIO HIAI. MASANORI OHYA AND MAKOTO TSUKADA
108
ficiency might be explained as follows: If a subalgebra 2Jl is sufficient for {y,+}, then we obtain from the measurement of 2Jl as much information a s from t h a t of '32 t o discriminate between 9 and In particular, to distinguish $ 6 I ( 9 ) (resp. E K ( 9 ) )from 9, the measurement of 2, (resp. 8) gives as much information as %.
+
+.
ACKNOWLEDGMENT. The authors wish t o express their gratitude t o Professor H. Umegaki for his valuable advice and constant encouragement. They thank Professor H. Araki for his critical reading and essential comments for this work. Professor J. Tomiyama and the referee kindly indicated the useful paper [4]to the authors. REFERENCES 1. H. Araki, Some properties of modular conjugation operator of v o n N e u m a n n algebras and a non-commutative Radon-Nikodym theorem w i t h a chain rule, Pacific J. Math., 50 (1974), 309-354. 2. ___ , Relative entropy of states of lion N e u m a n n algebras, Publ. RIMS, Kyoto Univ., 11 (1976), 809-833. 3. , Relative entropy f o r states of v o n N e u m a n n algebras 11, Publ. RIMS, Kyoto Univ., 13 (1977), 173-192. 4. F. Combes and C. Delaroche, Groupe modulaire d'une e s p h a n c e conditionnelle duns u n e algibre de won N e u m a n n , Bull. SOC. Math. France, 103 (1975), 385-426. 5. A. Connes, Une classij5cation des facteurs de t y p e III, Ann. Sci. Bcole Norm. Sup. SCr. 4, 6 (1973), 133-252. 6. I. Csiszir, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318. 7. S. Gudder and J.-P. Marchand, Noncommutative probability o n v o n N e u m a n n algebras, J. Math. Phys., 13 (1972), 799-806. 8. R. Haag, N. M. Hugenholtz and M. Winnink, O n the equilibrium states in q u a n t u m statistical systems, Commun. Math. Phys., 5 (1967), 215-236. 9. U. Haagerup, Operator valued weights in v o n N e u m a n n algebras, I , J. Functional Analysis, 32 (1979). 175-206. 10. P. R. Halmos and L. J. Savage, Application of the Radon-Nikodym theorem to the theory of suficient statistics, Ann. Math. Statistics, 20 (1949), 225-241. 11. R. Kubo, Statistical mechanical theory of irreversible processes I , J. Phys. SOC. Japan, 12 (1957). 570-586. 12. S. Kullback and R. A. Leibler, O n i n f o r m a t i o n alzd suficiency, Ann. Math. Statistics, 22 (1951), 79-86. 13. G. Lindblad, Expectations and entropy inequalities f o r f i n i t e q u a n t u m systems, Commun. Math. Phys., 39 (1974), 111-119. 14. P. C. Martin and J. Schwinger, Theory of many-particle systems I , Phys. Rev., 115 (1959), 1342-1373. 15. G. K. Pedersen and M. Takesaki, The Radon-Nikodym theorems f o r v o n N e u m a n n algebras, Acta Math., 130 (1973), 53-87. 16. S. Sakai, C*-Algebras and W*-Algebras, Springer, Berlin. 1971. 17. M. Takesaki, Tomita's Theory of Modular Hilbert Algebras and its Applications, Springer, Lecture notes in math. Vol. 128, 1970. , Conditional expectations in v o n N e u m a n n algebras, J. Functional Ana18. lysis, 9 (1972), 306-321. ~
~
430 SUFFICIENCY IN VON NEUMANN ALGEBRAS
109
19. J. Tomiyama, On the projection o f norm one i n W*-algebras, Proc. Japan Acad., 33 (1957), 608-612. 20. A. Uhlmann, Relative entropy and the Wigner- Yanase-Dyson-Lieb concavity i n an interpolation theory, Commun. Math. Phys., 54 (1977), 21-32. 21. H. Umegaki, Conditional expectation i n an operator algebra, TBhoku Math. J., 6 (1954), 177-181. 22. , Conditional expectation i n an operator algebra, III, K6dai Math. Sem. Rep., 11 (1959), 51-64. 23. , Conditional expectation i n an operator algebra, IV (entropy and information), K6dai Math. Sem. Rep., 14 (19621, 59-85.
~
Received March 13, 1979 and in revised form January 4, 1980. SCIENCEUNIVERSITY OF Toxyo NODACITY,CHIBA278, JAPAN
431 M. OHYA
KODAI MATH. J . 3 (19801, 237-294
ON OPEN SYSTEM DYNAMICS -AN OPERATOR ALGEBRAIC STUDYBY MASANORIOHYA Abstract T h e open system dynamics is rigorously studied within t h e C*-algebraic framework in terms of t h e approach to equilibrium. I t is pointed out t h at every combined s ta te of every s t a t e of a finite system and an equilibrium s t at e describing a n infinite reservoir relaxes t o equilibrium through a n interaction between both systems when th e total Hamiltonian of t h e combined system satisfies some spectral properties.
See. I : Introduction. T h e approach to equilibrium of a dynamical system is one of the most important problems to be solved in quantum statistical mechanics [l, 2, 3, 41. T h e principle aim of this paper is t o study the problem of this type for a finite system interacting with an infinite reservoir in equilibrium. T h e motivation of this work is as follows: Some physicists think that the system to be measured or in which some experiments a r e performed should be finite even if i t is large compared with the radius of an atom. However, if a system is finite and isolated, any time dependent state of the system will not relax t o equilibrium because the basic Schrodinger equation of motion is reversible under time reflection. From evidence accumulated by experiments, most of physical systems relax t o some equilibrium after long time. T h i s fact tells that w e had better treat such finite system as open system, namely, interactions between the finite system and the outside of the system (the so-called reservoir) should be taken into account. We then expect that physically interesting combined states of the system and the reservoir will relax to equilibrium through an interaction between them. In this paper, w e obtain conditions under which such relaxation occurs. We here take the Kubo-Martin-Schwinger (K. M. S.) condition [5, 6, 71 as that of equilibrium of our systems considered since any Gibbs state satisfies this condition and the K. M. S. condition seems most appropriate [8, 91 t o discuss thermal equilibrium in quantum statistical mechanics. Received April 18, 1979
287
432
288
MASANORI OHYA
Sec. 11: Formulation of the Problems. Let As be a Hilbert space of a system S and H S denote a self-adjoint lower bounded Hamiltonian of S. We often call the system S finite when the volume of S is finite or the degrees of freedom of S is finite. Our system S is assumed to have finite volume, so the spectrum of H S becomes discrete. Let our finite system S be described by a triple (,As, Gs, a?), where J s is the C*-algebra B ( A s ) of all bounded linear operators on the Hilbert space As, Ss is the set of all normal states on J s(i.e., the set of all linear functionals q5' on J s such that @(A*A)>=Ofor any A € J S ,@'(ZS)=1 for unity I s of As and @'(Aa)T #(A) for A, A, filtering upwards, in As), and a; ( t E R )is the time evolution automorphism of J s generated by the Hamiltonian H S . On the other hand, an infinite reservoir R is described by another triple (AR, SR, af), where J Ris a C*-algebra with unity IR, S R is the set of all states on J Rand a: ( ~ E Ris) a strongly continuous one-parameter automorphism of dR' We assume that the infinite reservoir is initially in equilibrium described by a faithful K.M.S. state pR a t the inverse temperature p with respect to a;. It is said that the state p R satisfies the K. M. S. condition a t /3 w. r. t. the automorphism a: if for any pair .4, B in d R ,there exists a bounded function FA,&) of the complex number z holomorphic in and continuous on the strip - p S I m z 5 0 such that FA,B(t)=pR(af(A)B) and Fa,B(t-i)=pR(Baf(A))for any t ER. By the Gelfand-Naimark-Segal (G. N. S.) construction theorem, to the state pDxthere correspond a Hilbert space X R , a cyclic vector QR, a representation rrR being a *-homomorphism from the C*-algebra d Rto the set B ( X R )of all bounded linear operators on the Hilbert space X R and a strongly continuous one-parameter unitary group U f such that nR(af(A))=UftnR(A)Uf*for any A = J R and U f B R =B E .
Let us take any faithful state @ESS of S and consider its time development. When the system S is isolated and the state $s is not a?-invariant (i.e. @(af(A))#$S(A)for some A E A ' ) , the expectation value @ ( a f ( A ) )is periodic or at least almost periodic function in t because the system S is finite. Thus ~ $ ~ ( a f ( does A ) ) not relax to equilibrium for all A = J S when time t tends to infinite. That is, the infinite time limit of #Sea? in the weak*-topology does not exist. We hence need to take account of the effect of an infinite reservoir on the finite system S in order to explain such relaxation behavior. T h e initial (non-interacting) combined system of the system S and the reservoir R is described by the following: (1-1) Algebra : d = A S @ J R , (1-2) State : + = @ @ p R ~ ~ = S S @ S R , (1-3) Time evolution : a;=af@aF, (1-4) Hilbert space : A=AS@SR, (1-5) Representation: n=iS@rrR (is is the identity map onto As). Let us introduce an interaction between the systems S and R. T h e interaction will be a bounded self-adjoint element V = V * E J . By Stone's theorem,
433 289
ON OPEN SYSTEM DYNAMICS
there exists a self-adjoint operator HR which generates the unitary one parameter group U f , i. e., Uf=exp(iH"t). We may call H=HS+HR+n(V) the total Hamiltonian of the combined system, which generates the so-called perturbed time evolution automorphism at of J [lo]: (1-6)
at(A)=
n20
in{dt1...Jdt,[c$,(V ) , ...[a!,( V ) , a;(A)]...] OSt16-.6t,51
for t 2 0 (the case t s O is due to exchange of 0 and t in the above integral domain). Considering the combined system and any faithful state on J denoted by #=$'@p", it is natural for us to ask the following question:
'( Under what conditions o n the dynamical system does the limit w*- lim $.cut I t I--
exist and is it identical to a proper eqilibrium (K.M.S.) state at the inverse temperature p with respect to the automorphism at ?" This problem concerns the relaxation process of the combined system. In conventional discussions of physics, one does not worry about such question but takes it for granted. By answering this question, we expect that the restriction of the limiting state to the algebra J sof the system S might be close, in some sense, to the K. M. S. state of S with the same temperature of the reservoir R, and we can also explain some physical phenomena of the so-called relaxation processes; for example, if the temperature of the system is initially different from that of the reservoir, our experience tells that if the system is in contact with the reservoir, then the temperature of the system goes to that of the reservoir. We finally note that our investigation here is concerned with the time development of the combined system but not of the system itself. It is really important to study directly the time development of a state of the system under the effect of some interaction with an infinite reservoir. For this purpose, the technique of conditional expectation invented by H. Umegaki [11] will be essential. This aspect will be discussed elsewhere [12]. Sec. I11 : Relaxation Process. In this section, we study the problem presented in the previous section. As mentioned before, the initial temperature of the system S might be different from that of the reservoir R, or the initial state $s of S might not be a;-invariant. In any case, there exists a trace class operator pS=exp(-pHS)I Tr exp(-pHS) so that the state ps defined by pS(A)=TrpSA for any A € d S satisfies the K.M.S. condition a t the inverse temperature ,B of the reservoir with respect to the automorphism af of the system. Thus the state pS@pRon d satisfies this condition a t p w. r. t. a!=a;@aF. Let us denote this state by p in the sequel discussions. Moreover, for a faithful normal state $' on A s which may not be &-invariant, we denote the combined state with the equilib-
434
290
MASANORI OHYA
rium state y R of the reservoir by $=qiS@(pR as in (1-2). Let u s start by proving several lemmas. LEMMA1. L e t w be a state on J and X be another state on J dominated by i. e., X S h for some A>O. Then f o r any E>O, there exists a n element W i n J such that I X ( A ) - w ( A W ) I <&ll~4lI f o r any A E J . w,
Proof. It is well-known that for any state X dominated by w, there exists a positive operator B in n,,(J)' such as X(A)=(SZ,Bn,(A)BW, where J2 is the G.N.S. cyclic vector induced by the state w. The cyclicity of D implies that for any E>O, there exists an element W in J such that \lBzSZ-~~,,(W)Dl\ < E holds. We hence have IX(A)--(AW)) <EJIAJJ for any A E J . (9. e. d.) LEMMA.2. For the state #=#'@p" introduced aboz>e,there exists an element K i n J such that for any E>O, I#(A)-p(AK)I <EllAll
f o r any A E J . P1,oof. Let us consider two following subsets CV' and LD' C V S = { $ ' ~ S S :$'(A)=(!P',
of L?:
A!Ps) for any A € J S and K'EX~}
and @={$'ES':
y5'5ApS for some R E R + } .
As is known C13, 141,the set CY' is dense in 6 ' and the set LB' is dense in C V ~ because ps is a K.M.S. state. Hence for the state 4' and any E>O, there exists a state o in CY' such that I@(A)-w(A)I <(~/3)llAll holds for any A E J ~ . Moreover, for the above state w, there exists a state X in 9 ' such that Io(A)-X(A)I <(~/3)11All for any A E J S . As X is dominated by ps, according to Lemma 1, there exists an element W in As such as IX(A)-p'(AW)I <(e/3)llAll. We therefore obtain l#(A)-p(AK)l<~llAll for any A E J , where K is taken as K= W@I". (q. e. d.) LEMMA3. For the state #=#'@p", there exist a state $ on J and an element R in d such that ( i ) the state y5 satisfies the K. A[. S. condition at /3 w. r. t. a t , and (ii) f o r any E>O, I#(A)-$(AR)~<E~IAI~ holds f o r any A E J .
Proof. Let us introduce a vector as
4f=Dv@/llDv@JI,
435 291
ON OPEN SYSTEM DYNAMICS
where @ is the cyclic vector associated with p such that @ = P @ O R and ~ p ~ ( i l ) = ( @ ~ , AOS) for any A € J S , and Dv is given by
Dv= x ( - l ) " l d t , . . . l d t
(2-1)
n 50
,irg,,(s(V))...irgtn(n(V)) ,
ort1s.-st,sg/z
where ir0 is the canonical extension of a' to the von Neumann algebra n ( J ) " . T h e above vector Y is always in the Hilbert space &, although Dv is not in the C*-algebra n ( J ) except when V is a aO-analytic element of J. Define a state by +(A)=(% n ( A ) Y )for any A E J . This state satisfies the K.M.S. condition a t p w. r. t. at [lo]. T h e state and the time evolution automorphism at are constructed by 'p and a! through (2-1) and (1-6) respectively. Conversely, it is easily seen that 'p and a: can be constructed back from q5 and at with the interaction-V. Namely, putting @'=QvY/I'///Q'YIj with
+
+
+
QV=
x Jdtl...Jdt,uitl(n(v))...irit,(n(V)) .
n a0
ost~s...rt,af9,2
We can then readily show that @'=@. Moreover, by the simple but rather tedious computations using the boundary properties of the K. M. S. state, we obtain
~ ( ~ ) ~ v Y ) / l l Q v Y l ldzA=) (S~, V, Y ) , where S,"is given by Sv/I (Y, SvY)I and 'p(4=(QV%
SV=
n a0
Jdtl~..~di,iritl(n(V))...irit,(n(l/)) .
ortlr.-rt,sj3
If the interaction V is a a'-analytic (hence a-analytic) element of J ,the above
Sg is given by n(gg), where s"g=s"'/I+(.!?")I
and
Since the set of all a-analytic element of J is dense in J in the norm topology, for any V= V*EJ and any E > O , there exists a a-analytic element V,= V / , * E J such that 11 V- Val[ < E . According to Theorem 3.1 of [lo], we easily obtain
Hence for any E > O , IIQVY/'/lQVY1l-Q"o~/IIQ"o~lili<& holds. It is thus a easy 1 exercise to show that for any E > O , the inequality I'p(A)-+(As"~o)I
Now, by Lemma 2, there exists an element K in 1 such that I$(A)--(o(AK)I
R is an element of J and the inequality I#(A)-+(AR)I < ~ l l A l lholds for any A € J. (q. e. d.)
436
292
MASANORI OHY A
Let us find conditions under which every state 4=$"@pR ( @ E S ~ ) relaxes, under the time evolution at, to a n equilibrium (K. M. S.)state a t ,6 w. r. t. a t , as physically expected. T h e spectrum of a Hamiltonian is one of the most important quantities of physics, and most of physicists a r e interested in the property of i t ; for example, how dense i t is in R. In our case, the system considered is finite, so the spectrum of t h e Hamiltonian HS is discrete, which forbids the approach to equilibrium. Therefore w e needed some interaction with suitable reservoir. T h e Hamiltonian t o be studied was the so-called total one H = H S + H R + n ( V ) . What w e ask is the following: Which conditions do w e have t o impose on the total Hamiltonian H ? In other words, which interaction V do w e have to choose ?
THEOREM 4. I f the rank of the projection E to the null space of H is one and the spectrum of H consists of {0} and absolutely continuous one, then the time evolution automorphism at admits unique K . M . S . state $J in the representation space 4E and the limit w*-lim #oat is equal to $. Proof. In Lemma 3, w e constructed the state satisfying the K. M. S. condition w. r. t. at. For any A, B E J , let us consider $(Aat(B)), which is equal to ( y, n(A)exp(+itH)n(B)?F) because the K. M. S. state $ is a,-invariant. According to the spectral decomposition of H,w e have
When t tends to infinite, the above expression becomes (n(A)* y, E n ( B ) Y ) because of the spectrum properties of H. Since the rank of E is one, E d B ) p =(p,n ( B ) y ) Yfor any B E J . We thus obtain lim $(Aa,(B))=$(A)$(B)
ltl-==
for any A, B E J . Namely the state $ is clustering for at. This fact tells [13] that $ is the unique K.M.S. state of A. Let us now consider
I $MA))-$M) I
?
which is less than
I $(at(A))-$(at(A)R)I + I $(at(A)R)-$(A)I r where R is a n element in J obtained in Lemma 3. T h e first term of the above expression is again less than E ~ I A Ibecause ~ of Lemma 3. We now estimate the second term : I$(at(A)R)-$(A) I *
m=
As shown that the state $ is clustering for at and R is in A,
437 ON OPEN SYSTEM DYNAMICS
293
lim I(t)=O Itl-m
because of $(R)=l.
We hence have w*-lim
q50crt=$.
1t1-m
(4. e. d.)
We finally see what happens to the state $ when the strength of the interaction V becomes very weak but the interaction still has the properties of Theorem 4.
THEOREM 5. Under the conditions of Theorem 4, we hazle w*-lim limq50crt=ps on d S . IlViI-m
1ti-m
Proof. From Theorem 4, we have only to show w*-lim +=ps on dS. llvll-o
As discussed in the proof of Lemma 3, the unique K. M. S. state $ is given for any A E A , where T V is given by through $(A)=(Tv@,?r(A)TV@)
TV=DV/IIDV@lI with Dv defined in the proof of Lemma 3. Therefore the following inequality holds :
I +(A)-p(A)I S2IlU- Tv)@llII All . As mentioned in Lemma 3, for any V = V * E A and any E>O, there exists a such that //TT@-TVo@ll <E. We hence have aO-analytic element V,= V , * E ~ I $(A)-p(A)/~ ~ l l ~ ~ - ~ v a //All ~ @ l .l T h e above inequality together with the facts that TV0=O=Zimplies that w*-lim $ IIVU+O
= y on A. As the restriction of the state p to J s is y s , we have the conclusion. (q. e. d.) This theorem shows that if we can choose the interaction so that its strength is sufficiently weak but the total Hamiltonian H = H S + H R f n ( V ) still satisfies the condition of Theorem 4,then the limiting state of 4 under I/ V\l-+O is enough close to the equilibrium state y , that is, any state q5' on As approaches to the equilibrium state y s in the above sense. T h e theorem 4 will be somewhat related to the derivation of equilibrium state [15, 161, about which we will discuss elsewhere. T h e conditions of Theorems 4 and 5 might be realized in some physical models.
A c kn ow I e dge n a en t T h e author thanks Professors H. Umegaki, D. Kastler, H. Ezawa, K. Nakamura and F. Hiai for fruitful discussions and useful comments to the related topics of this work. He also thanks Professor K. Kunisawa for his kindness.
438 MASANORI OHYA
REFERENCES F. HAAKE, “Statistical T r e a t m e n t of Open System by Generalized Master Equation”, Springer-Verlag, 1972. E. B. DAVIES, “Quantum T h e o r y of Open Systems”, Academic Press, 1976. K. HEPP A N D E. H. LIEB, Helv. Phys. Acta. 46, 573, 1975. G.G. EMCII AND C. RADIN, J. Math. Phys. 12, 2043, 1971. R. KUBO, J. Phys. Soc. Japan, 21, 570, 1957. P.C. MARTINA N D J. SCHWINGER, Phys. Rev. 115, 1342, 1959. R. HAAG, N. HUCENHOLTZA N D M. WINNIK, Commun. Math. Phys. 5, 215, 1967. G. L.SEWELL, “ T h e Description of Thermodynamical Phases in Statistical Mechanics”, 1974. G.G. EMCH AND J . F . KNOPS, J. Math. Phys. 12, 2043, 1971. H. A R A K I , R. I.M.S. Kyoto, 9, 165, 1973. H. UMEGAKI, Tohoku J. Math. 6, 177, 1954 and 8, 86, 1956., Kodai Math. Sem. Rep. 1 1 , 51, 1959 and 14, 59, 1962. M. OHYA, in preparation. M. TAKESAKI, “Tomita’s T h e o r y of Modular Hilbert Algebras and i t s Applications”, Springer-Verlag, 1970. J. DIXMIER, “Les algsbres d’ophrateurs d a n s l’espace Hilbertien”, GauthierVillars, Paris, 1969. R. HAAG, D. KASTLER A N D E. TRYCI-I-POHLMEYER, Commun. Math. Phys. 38, 173, 1974. D. KASTLER, Private Communication. SCIENCES, DEPARTMENTOF INFORMATION THESCIENCE UNIVERSITY OF TOKYO, 278, NODA CITY, CHIBA,JAPAN.
439 Vol. 16 (1979)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
MASANORI OHYA Department of Information Sciences, The Science University of Tokyo, Noda City, Chiba, Japan* and Institute of Fundamental Studies, Department of Physics and Astronomy, University of Rochester, Rochester, New York, U.S.A. (Received June I S . 1977-Revised
February 28. 1978)
The problems of a dynamical process in the linear response theory are studied in the C*-algebraic framework. We outline the domain of validity of the linear response approximation by considering the similarity and difference between the linear response dynamics and the exact dynamics. It is shown that the linear response method is identical to the exact method when we consider the return to equilibrium of all locally perturbed states. As far as the stability of a dynamical system is concerned, the linear response theory is shown to differ from the exact theory. It is pointed out that the phase transition in a dynamical process does not occur in the linear response theory.
In Section 11, we define the perturbed dynamical systems under a perturbation V Y* E %: (3,q“, a“) in the “exact” sense (E.S. for brevity) and (3, q”.’, a“-’) in the “linear response” sense (L.R.S. for brevity). In that section, we formulate the problems to be investigated in this paper. In Section 111, it is shown that the linear response method is identical to the exact method when we consider the return to equilibrium of all perturbed states. In Section IV, we study the stability of the dynamical system (%, q , a), and it is pointed out that the linear response theory differs from the exact theory. In Section V, the phase transition in a dynamical process is discussed and it is shown that we do not have such transition in the linear response dynamics. In Section VI, we illustrate general results obtained in the previous sections by the X - Y model. =
I. Introduction Whereas the linear response theory [ l , 21 has succeeded to explain many physical phenomena, it is nevertheless true that one does not know how to control this theory rigorously. Recently some work [3, 41 has been done to formulate the linear response theory in the von Neumann algebraic framework. Our work is an attempt to study further
*
Mailing address. 13051
440 306
M. OHYA
the linear response theory in the C*-algebraic framework with the purpose of determining the domain of validity of the linear response approximation. An infinite dynamical system is described by a triple (%, p, a), where % is a C*algebra with unity I, rp is a state on 'u and a is a continuous homomorphism from the set of real numbers R to Aut('U), the set of automorphisms of 21. We call a t , t E R, a time evolution (automorphism) of 'u. I t is assumed that at satisfies the following continuity: limI\a,(A)-Al( = 0 for any A E 'u. We further assume that the above state cp satisfies 1-0
the K.M.S. condition [5] with respect to at a t an inverse temperature B (we take B = 1 for simplicity). So our dynamical system in the sequel is (3,p , a) with the K.M.S. condition. Since the K.M.S. state p is time invariant under at, there exists a strongly continuous one parameter group of unitary operators U, implementing a t : 17,(at(A)) = UIITp(A)U-t, defined through U,@ = C?, where ITpis the G.N.S. representation and C? is the cyclic vector induced by 9. Moreover, by Stone's theorem [7], there exists a self-adjoint operator H such that U, = exp(itH). I n the following sections, we denote by @ and 2, the canonical extensions of p and a, from the C*-algebra to the von Neumann algebraIT,(%)", the weak closure ofIT&8) [8].
II. Formulation of
the problems
I n this section, we consider the effect of a perturbation on the dynamical system a). We then formulate the problems t o be investigated in the following sections. If our dynamical system (%, cp, a) is disturbed by a perturbation AV,where Y is a self. adjoint element of 21 and A is a positive real coupling constant, then the exact perturbed system is denoted by a new triple (3,p", a"), which is precisely defined below [9, lo]: The exact perturbed state p" is given by, for any A E 'u,
(21, rp,
(2.1)
where @"
5
~ p ( W ~ ~ ) C ? / l l I T p ( ~ ~=* ITp(WV)@, )C?ll
Wi';2=
(-;I)" l d t l n>O
...
1
(2.2)
dtnai,,(V) ... ait,(V).
(2.3)
O
This perturbed vector @' is also cyclic and separating for 17,(2l)" 191. The above W& (hence W") is actually a n element of 21 [9, 101, provided that the perturbation V is a n a-analytic element of 3. However, Up(W&)C? is an element of the G.N.S. Hilbert space .y;", for every V = Y* E 'u. The exact perturbed time evolution automorphism ay of 'u is given by the uniformly convergent Dyson series: For any A E 21,
arw
=
c(W
naO
dt,
041,s ...sf.sf
df,[at,(J?,[... ,[.t,(v>,44)1
...I1
for t 1 0 (the case t < 0 results by exchange of 0 and t in the above integral).
(2.4)
441 DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
307
We here note that the above perturbed state p l y satisfies the K.M.S. condition with respect to the perturbed time evolution a; [9, 101. We now consider the linear response dynamical system denoted by a triple @, y, a), which is obtained by linearizing the exact perturbed dynamical system w.r.t. 1.The linear response functional ply*' is given by 1/2
(ply";
A)
=
~sLx-~,(V)A/\
(p;A)-A\pl; / 0
s
112
-?.(p;
dsAais(Vj)+d(p; A>
(pl;
J9
(2.5)
0
for any A E 'u. The linear response time evolution a [ - ' is given by I
a,V*'(A) = a,(A)+ilSds[a,(Y), a,(A)I
(2.6)
0
for any A E 'u and t > 0 (the case t < 0 results by exchange of 0 and t in the above integral). We study, in the exact dynamics (see, e.g., [lo, 11, 12]), the existence of the limits w*- lim qV at = q ~ y ,lo*- lim p c([ = pl* and discuss the properties of the limits 9: 0
t-*W
0
r-*m
and y7+, provided that they exist. In the linear response dynamics, the following questions are pertinent to our investigations: (1) Under which conditions on V do the limits rp1-lE w*- lim pl".' 0 a, exist? r-im
(2) Under which conditions on V are rp';.' equal to pl? (3) Under which conditions on V do the limits pl*, = lo*- lim rp
0
a[* ' exist? (If they
1-fm
exist, the system is said to be stable in L.R.S. under the perturbation V [4].) (4) Under which conditions on V are pl*, equal to pv* ' ? Comparing the results for the above questions with those for the exact dynamics, we ask (5) Can we outline the domain of validity of the linear response approximation? By analogies to the exact dynamics [lo, 1 1 , 12, 13, 15, 211, the question (1) may be regarded as the problem of the approach to equilibrium, the question (2) is of the return to equilibrium and the questions (3), (4) are of the stability of the dynamical system, all in L.R.S. under a perturbation V. The question (3) bas been studied by Verbeure and Weder [4], whose C*-algebraic version is as follows:
THEOREM 11.1. For the dyriarnical system ('u, pl, a) arid a perturbation V = V" E a, the dytianiical systein is .stable in L.R.S. under V if arid ordy if exp(itH)Uv(V)@ weakly coilverges in the G.N.S. Hilbert space X q as t --f kco. Moreover, in this case, we have
442 M. OHYA
308
In the above theorem, T is a self-adjoint positive operator defined by T = ( A -l)1~2/(lnA) and A = exp(-H), and Eq is the projection on the null space of H. This theorem is the C*-algebraic version of the Theorem 3 of [4], and it can be readily proved. Before closing this section, we remark that the linear response functional qV*' may not be a state on 91 in general unless certain conditions are satisfied. The linear response time evolution ay* introduced by Verbeure and Weder [4] in the algebraic framework is not an automorphism of 'u.
m. Return to equilibrium In this section, we consider the time development of q V * after removing the perturbation V dynamically. Namely, we study the questions (1) and (2) given in the previous section. The question (1) of the approach to equilibrium can be solved as follows.
THEOREM 111.1. For the dynamical system (3, q , a) and a perturbation V = V* E 'u, the limits p;'. = w*- lim vv*' at exist if atid only ifexp( - itH)&(V) @ weakly converges 0
t-*m
in the Hilbert space .Tqas t (q1.l; A ) =
--f
00.
-@,
Moreover, iti this case, we have
1 7 , ( ' 4 E p q ( V ) @ ) + J ( q ;A )
for any A ~ 2 l . We need a lemma to prove this theorem. LEMMA111.1. For the dynamical system (21, 9, a) arid a pertrrrbation V = V* E 91, we have
) =
1
~ S S < P ;Aais(V)>+lb
A)
0
for any A E%.
Proofi
According to the definition (2.5) of qv", we have only to show that
112
1
ds(y; Aai,(V)) for any A E 'u. Let us first prove the above
ds(v; a - i S ( V ) A ) = 0
1/2
equality for any a-analytic element f = f*E 91. Since v is a K.M.S. state w.r.t. a t , the equality (v; U - ~ ~ ( ? ) A=) (v; A E - ~ ~ + ~ (holds $ ) ) for any A E '%. This immediately yields the desired equality, We now consider the case of a general perturbation V = V* E 'u. Then it suffices for us to show that for any V = Y* E 'u and any E > 0, there exists an a-analytic element f = f*E 3 such that 1
1
I$ds(v; Aais(fl>-Sds(p,; Aa,,(V)I < 0
0
E.
DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
It is easily seen that for any A
309
€3,we have
Therefore,
I (TY7+,(A)*@,IIq(7-V)@)I= I(TD+,(A)*@, Tn,(fG I I T17,(4*@llIIT(fl+,(~ -flJV))@I I. =
V)@)/
(3.1)
Let '& be a subset of the von Neumann algebra % E D+,(%)'' defined by !fl = { df iil(Q) x xJ(t): Q E %, f E where is the Fourier transform of functions in the space 9 of infinitely differentiable functions with compact supports. Any element of % is &analytic. We introduce a sesquilinear form on the von Neumann algebra % in such a manner that (R,Q)# = (TR@,TQ@) for any R, Q in 8.We then obtain a Hilbert space 8, by closing % w.r.t. the above sesquilinear form. It is known [3] that !fl is dense in 92#. It is an easy exercise to show that for any Q E and any E > 0, there exists an a-analytic element 7= f* such that IIQ-I&(F)Il# < E . Hence, for any V = V* E 2I and any E > 0, there exists an a-analytic element = f*E such that (3.1) is less than E. (Q.E.D.)
s},
5
Proof of Theorem 111.1: According to the above lemma and the time translational invariance of the state under ar, we have
v
1
(pYs1;a t ( 4 ) = (p;n>-~Sds(p;a,(A)a,,(V))+I(p; A )
Since U,= exp(itH) and U r @ = 0,( y ; a,(A)als(V))is equal to (U+,(A*)@,exp(-itH-sH)D+,(V)@).Hence, if exp( -itH)n,(V>@ weakly converges in the Hilbert space &'q as r 3 rt: 00, then the limits lim (pY*'; a,(A)) exist for any A E a. Conversely, if the t-tfa,
above limits exist for any A E %, then exp(-itH)Dv(V)@ weakly converges in X q because the set {D+,(A*)@: A E %} is dense in &'+,.Finally, if the weak limits w- lim exp(-itH)IT+,(V)@ exist in A?+,, then they are equal to E&l+,(V)@ 161. We t-jm
thus conclude
(d*'; A ) = . (Q.E.D.) A remarkable feature is that the above theorem can be proved under the same condition of the Theorem 11.1. Thus we have an interesting consequence: For the dynamical system (a, p, a), the approach to equilibrium occurs in L.R.S. under a perturbation V if and only if the dynamical system is stable in L.R.S. under V. We might not have this situation in the exact dynamics. Now let us answer the question (2). Before stating the result, we introduce one concept.
444
3 10
M. OHYA
DEFINITION 111.1. The projection E, is said to be one-dimensional for A
E: 91
if
E&,(A)@ = ( @ , f l , ( A ) @ ) @ . We can answer the question (2) as
THEOREM111.2. For the ajmamical system (3, p , a ) and a perturbation V = V* E 91, the retlcrn t o equilibrium in L.R.S. occurs wider V fi and only gcp:.' exist and E, is onedii?iensionalfor V. Reinark. The existence of pI*' is always in the weak sense in our forthcoming discussions, so we merely say that p:~' exist as in the above theorem. Pro08 The "if" part can be proved as follows: When pZS1exist, they are given by
for any A E 'u. E, is assumed to be one-dimensional for V, we thus obtain the equality (@,U,(A)EJ7p(V)@)= (y; A ) (p; V ) , which immediately implies (pi,'; A ) = (p; A ) for any A E 'u, i.e. return to equilibrium. Conversely, if the return to equilibrium in L.R.S. occurs under a perturbation V = V* E 91, then for any A E 3,(pis1; A ) = (9;A), which implies (@
9
17,(A) E,17,(V) @)
=
(0f l , ( A ) @) t
(@
3
n,(V@).
Hence we obtain (&(A*)@, ( ~ , G ( V ) -(@9fl,(VP))
@) =
0.
As Up(%)@ is dense in Ifq, the above equality shows that (E,U,(V)-
(@,fl,(V)@))@= 0 ,
which means that Ev is one-dimensional for V. (Q.E.D.) Using this result, we have a remarkable property which tells us the similarity between the linear response dynamics and the exact dynamics.
THEOREM 111. 3. For the dynarnical system (X, p, a), the following two conditions are equivalent: (1) The return to equilibrium in E.S. occurs under all perturbations V = V* E 3. (2) The return to equilibrium in L.R.S. occurs under all perturbations V = V* E '8. Proofi Since q~ is a K.M.S. state with respect to cr,, the return to equilibrium in E.S. occurs under all perturbations V = V* E '8 if and only if the dynamical system ('u, p , a) is clustering (i.e., lim (p; a , ( A ) B ) = (p; A ) (9; B ) for any A , B E 3) [lo]. I-+m
Moreover, the dynamical system is clustering if and only if it is stable in L.R.S. under all V = V* E 3 and r-clustering (i.e., q ( y ; a,(A)B) =
445 DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
31 I
clustering if and only. if the projection Ev is one-dimensional [6]. Any element of 'u is expressible as a sum of two self-adjoint elements of 'u. We therefore arrive at the assertion of this theorem by virtue of Theorem 111.2. (Q.E.D.) This theorem can be realized [12] by some dynamical systems; for example, the X- Y model [15] and the free Fermi gas [16]. We will discuss the X - Y model in Section VL.
IV. Stability of a dynamical system In this section, we investigate the question (4) and discuss the difference between the linear response dynamics and the exact dynamics. We will show the conditions on the perturbation V under which the linear response functional $"' is achieved by a purely dynamical process from the initial equilibrium state y. The answer of this question is IV. 1. The linear response functional THEOREM linear response time evolution ar*' as t -+ co (i.e,,
vV*' is yV*l =
dynamically achieved by the w*-lim pl 0 if and only i f I- m
ccr.')
the dynamical syslem (a, y , a) is stable in L.R.S. under the perturbation V = V* E 2 and the projection Eq is one-dimensional for V.
Proofi The "if" part can be shown as follows: According to Lemma 111.1, yV*' is given by 1
=
$
< y ; A ) - 2 M y ; Aa,,(V))
+ A
0
for any A
€41. It is easily seen that
s
1
M y ; Aai,(V>> = ( @ , q ~ ) T 2 ~ , ( V ) @ ) .
0
Since the dynamical system (a, y , a) is stable in L.R.S. under V, we have from Theorem 11.1 lim
which turns out to be equal to
The projection E, is assumed to be one-dimensional for V, so we obtain
Hence we conclude
446 312
M. OHYA
Conversely, the "only if" part is shown as follows: If the equality (4.2) is satisfied, then the dynamical system is stable in L.R.S. under the perturbation V and the equality (4.1) holds. Hence, as proved in Theorem 111.2, the projection E, is one-dimensional for V. (Q.E.D.) According to the above theorem and Theorem 111.2, we can readily prove COROLLARY IV. 1. I f the djmarnical system (%, 9,u) is stable in L.R.S. under a perturbation V = V* E % and the projection Ep is one-dimensionalfor V, then w*- lim limp,
0
s-fm 0
t-m
hv*'o a, = P
When the dynamical system (2, q , u) is clustering, it is stable in L.R.S. under all perturbations and 7-clustering [4].Applying the above theorem and corollary, we obtain
THEOREM IV.2. I f the dynarnical system (3,cp, u) is clustering, then (1) qV*' = w*-lim q '.0',u for all perturbations V = V* E 3, and I-m
(2) p , = w * - I i m l i m q o a r * ' o a , f o r a U V = V * E % . s-fm
t-m
This result shows a sharp contrast between the exact theory and the linear response theory. Namely, we can not assert the same fact for the exact dynamics. This aspect will be illustrated in Section VI by an exactly solvable model: X - Y model.
V. Phase transition in dynamical process When there exist several different thermodynamical phases at an inverse temperature /? (= I), it is interesting to consider the change in a dynamical process from one state to another. For instance, we have metastable states and stable states for some physical systems [17, 18, 191, we then ask when a metastable state goes over to a stable state by the effect of a perturbation. In our terminology, if we have two different states 9 and of an infinite dynamical system, which satisfy the K.M.S. condition with respect to the time evolution a, of the system, the question of interest is whether the transition from 9 to y can occur dynamically through a perturbation V = V* E %; that is, w = w*lim pV 0 at in E.S. and y = w*-lim p,".' 0 a, in L.R.S. We assume that the state cp is an t-m
t-a,
extremal K.M.S. state and y is another K.M.S. state, both w.r.t. u t . In exact dynamics, such transition possibly occurs [20] under some perturbation. However, in the linear response dynamics, we have a negative result as discussed below.
LEMMAV.l. For the dynamical system (3,91, a), i f q is a K.M.S. state-w.r.t. a, and
TI*' exist as states on % under a perturbation V = V* E %,
then y1.l are vectw states of q.
447
DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
I7J.Z)". Let {Py>be an increasing net of projections in I&(%)" jection on the closure of
U P,,Jf'+,).
313
converging to P (a pro-
Then
Y
I(?;*';
Py-P>l
< I(+;
Py-P>l'
Q+-I l ( + ; ~ v ( V ) ) l ) + ~ *l l ( ~ Y - - ~ ) @ I l
ll-EqL7q(V)@ll,
which goes to zero as y -+ co because $ is a normal state on ITq(%)". This means that @I*' are normal states on I7,(%)". Since p is a K.M.S. state w.r.t. a,, 9';~'are vector states of p (see Theorem 4, p. 233 of Ref. [24]). (Q.E.D.) Let p and y be two K.M.S. states as assumed above. We now suppose that the desired transition occurs in L.R.S. under V = V* E 8 (i.e., y = w*-limp"*l 0 a, = p!.'). Since the canonical extension express p!*' as [14]
@? of &*'is a
1-m
faithful normal state on I7&!l)", we can thus
(p;-l; A ) = (+y;U&4))= (@, R*17,(A)R@),
where R is a self-adjoint positive operator affiliated with the center Z = lIV(fl)"nflq(w'. Since p is assumed to be extremal, the von Neumann algebra ITv(%)'' is a factor: Z = CI. Thus there exists a constant ,u E C such that R = ,uZ.As
THEOREM V.l. For the dynanzical system (a, p, a), i f p is extremal atnong the K.M.S. states w.r.t. a,, then the phase transition in the dynamical process does not occur in L.R.S. under any perturbation V = V* E %. This theorem might tell us that we need at least some non-linear terms to discuss the phase transition effected by perturbations.
VI. X - Y model We illustrate some of general results obtained in the previous sections by the X - Y model. The X- Y model has been rigorously studied by several authors [15, 21, 221. We mainly follow the paper of Emch and Radin [21] here. Let us consider a one dimensional lattice Z. Every site of Z is occupied by a particle with spin 1/2. Thus to every site k E Z , an algebra is associated, and to every finite is associated. These algebras are given by region A c Z, an algebra
aA
'uk
c
= {21Ik+~2~+-IJ~~ky'1-2,IE4 ~ ~ : =
%A =
0
1, 2 , 3, 4)},
2lk,
kcA
where 6 . Y . ' are the Pauli matrices and 1, is the identity matrix at the site k E Z. We then define an algebra of local elements: 'u, = U a A .The quasi-local C*-algebra of our ACZ
lattice system is the inductive limit of the above algebra:
=
-5. ACZ
448
3 14
M. OHYA
The local Hamiltonian H , (ti is a positive iiiteger associated with a finite region 11. Z)of the X- Y niodel is
= [ - t z , n] c
where 6 is some real constant in (0, 1). The time evolution generated by H, is $(A) = exp(itH,)A exp(-itH,) for any A E 2 ' 1 and t E R. The canonical equilibrium state cp,, on 2 with an inverse temperature = 1 is given by P)" = exp(-H,,)/Trexp(-El,,). I t is then shown [22] that there exists a one parameter group of automorphisms CI, of % and a state v on 91 such that lim IIcc:(A)-ct,(A)[I = 0 and lim I(qn; A)-(cp; A ) [ = 0. Then n-m
n-m
y becomes an extremal K.M.S. state w.r.t. CI,. L e t d and w, v, be the even C*-subalgebra of 2 and the restrictions of cp and c ( ~t o this subalgebra respectively (cf. [21]). The state w turns out t o be a n extremal K.M.S. state w.r.t. v,. The dynamical system of the X- Y model is now set by the triple (d, w ,v ) ~ It can be shown [21, 231 that (d, v) is asymptotic Abelian in norm (i.e., lim II[A, cct(B)]II r-+m
0 for any A , B ~ d ) hence , the dynamical system (d, co, Y ) is clustering. As discussed in the proof of Theorem 111.3, the return to equilibrium in E.S. occurs under all perturbations V = V* ~d (i.e., w*- lim coy 0 v, = ro). On the other hand, the perturbed dynamical =
r-rfm
system in E.S. (d, w v ,vv) is not always asymptotic Abelian. For example, consider a perP- I
turbation V = V*
{ ( I + ~ ) ~ ~ ~ ~ + ' + ( l - ~ ) (up :~integer u ~ + , 2} I), then the local
= k=
-D
is left pointwise invariant under the perturbed time evolution vIy ; algebra d I - , + , , namely, for any A p - l , , there exists B e d such that II[B,YI~(A)]II = II[B, All1 # 0. We therefore know that for such perturbation V = V$' eat', the equality w v = wiklim w 0 YY is not satisfied. According to the Theorems 111.3 and IV.1, we however have I-tm
the following situation in the linear response dynamics.
THEOREM VI.1. For the dynamical system ( d ,w ,v) of the X-Y niodef, (1) the return to equilibrium in L.R.S. (i.e., w*- lim coy, ' 0 v, = co) occurs wider a f l l-fm
perturbations V = V* ~ dand , (2) the linear response functional my* ' is dynati~icalfyachieved (!.e., w*-lim (o o vr-' I-m
= w v *') for all perturbations V = V* E 2.
Acknowledgment The author thanks Professor G . G. Emch for valuable discussions and critical comments. H e also thanks Professor H. Umegaki for valuable suggestions and kindness a t Tokyo Institute of Technology. He finally thanks the referee for critical comments.
449 DYNAMICAL PROCESS IN LINEAR RESPONSE THEORY
315
REFERENCES
[I] R. Kubo: J. Phys. Soc. Japart 2 1 (1957), 570. [2] H. Mori: Prosr. Tlieor. Phys. 33 (1965), 423. [3] J. Naudts, A. Verbeure and R. Weder: Cornnrrot. Mdth. Plrys. 44 (1975), 87. [4] A. Verbeure and R. Weder: Coninrun. Math. Phys. 44 (1975), 101. [5] R. Hang, N. Hugenholtz and M. Winnik: Comnrrrrt. Math. Pliys. 5 (1967), 215. [GI G. G. Emch: Algebraic methods in statistical rnechrrics and qriantrm field theory, Wiley, 1972. d~ Acad. Press, 1972. [7] See, for example, M. Reed and B. Simon: ~ U I ~ C t i O nanalysis, [8] See, for example, page 200 of [GI. [9] H. Araki: R.I.M.S., Kyoto 9 (1973), 165. [lo] D. W. Robinson: C*-algebras artd qrrantirm statistical nrechanics, Lccture in Varenna School, 1973. [Ill G. G. Emch: J. Math. Pliys. 7 (196G), 1198. [121 M. Ohya: Thesis at University of Rochester, 1976, and references quoted therein. 1131 See, for example, 0. E. Lanford 111 and D. W. Robinson: Conrnrnrf. Math. Plrys. 24 (1972), 193. [I41 M. Takesaki: Toniita’s theory of modular Hilbert algebras and its applications, Springer-Verlag, 1970. [151 D. B. Abraham, E. Barouch, G. Gallavotti and A. Martin-Lof: Phys. Rev. Letters 25 I1 (1970), 1449. [lG] See page 305 of [a. [17] G. G. Emch and J. F. Knops: J. Mdth. Phys. 11 (1970), 3008. [IS] 0. Penrose and J. L. Lebowitz: J. Stat. Mecli. 3 (1971), 211. 1191 G. L. Sewell: Letterr a1 Nuovo Cinrento 10 (1974), 430. [201 See Chapters 111 and V of [12]. [211 G. G. Emch and C. Radin: J. Math. Phys. 12 (1971), 2043. I221 H . Araki: Cornniun. Math. Plrys. 14 (1969), 120. [23I H. Narnhofcr: Acta Phys. Arrstrica 31 (I970), 349. [241 J. D i x m k : Les algdbres d’opirateirres dam l’espace Ifilbertieit, Paris 1957.
Stability of Weiss king model Masanori Ohya
’
The Science University of Tokyo. Department of Information Science, 278, Noda Chi& Japan and Institute of Fundamental Studies, Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627 (Received 21 June 1977)
The problems of stability and approach to equilibrium of the Weiss king model are studied. Our investigations are performed in the exact and linear response senses in order to compare both theories. The change of a metastable state of the Weiss Ising model is discussed under local perturbations.
1. INTRODUCTION The problem of stability of a dynamical system with infinitely many degrees of freedom encountered in quantum statistical mechanics has been recently paid much Verbeure and Weder3 attention by several authors. have studied this problem in the linear response theory, Ohya‘ has considered similar problems in the linear response and exact theories. In this paper, we apply some of the general results obtained in Refs. 3 , 4 to the Weiss k i n g model (WIM for brevity in the following) formulated in the operator algebraic framework. In Sec. 11, we review the WIM briefly. In Sec. 111, we show that the WIM provides us a nontrivial example of the work by Verbeure and Weder3 concerning the stability of a dynamical system in the linear response sense (LRS f o r brevity). In Sec. IV, we study the dynamical behavior of locally perturbed states of the WIM by bounded self-adjoint operators belonging to the quasilocal C*-algebra of this model. Namely, we look at the time development of the perturbed states after removing the perturbations dynamically. The discussion of that section i s of the approach to equilibrium in both linear response and exact dynamics. We also discuss the dynamical change of metastable states in the WIM in Sec. IV. The change of a metastable state to a stable state under some local perturbation i s extremely interesting to b e investigated in quantum statistical mechanics and quantum measuring processes. Although we first have to show the existence of metastable states in a dynamical system considered, we fortunately know’ that there exist metastable states in the WIM, provided that the temperature i s below a critical point. We a r e thus a t the stage in the WIM to consider the dynamical change of the metastable states. We want to know whether there exists a local perturbation under which a metastable state goes to a stable state or not.
We take the inverse temperature of the WIM 0 = 1 throughout this paper.
particle from other particles surrounding i t as the field determined by an averaging procedure over the system of interest. Thus the representative particle can be statistically treated in the mean field. Conversely, we may determine the average field which the representative particle exerts on i t s neighbors. If all particles a r e identical, then every mean field should b e coincident (requirement of self-consistency of mean field). This self-consistency requirement determines the mean field of the system precisely. We can study statistical prope r t i e s of the system by this mean field. In this section, we briefly review one of the mean fields: the Weiss Ising model, in the algebraic framework. Following Emch and Knops, we consider a one-dimensional lattice Z (this restriction can b e lifted, but we take i t f o r simplicity). The quasilocal C*-algebra of this lattice system 2 is given by
PI = g PI O = ,=
hF,%h
@
PI , (A is a finite region of Z )
k t h
PI,,{x,l,+x,~+x,u.,+x,u;:x,Ec (j=1,2,3,4)}, where o:”“ a r e the Pauli matrices and I, is the identity matrix a t the s i t e k of the lattice 2. The local C*-algebra91 A enjoys two properties, named “isotony” and “locality. ’” The local Hamiltonian of a ferromagnetic system considered here is given by
where B is an external magnetic field on the system along the z axis and J , , , ( A ) a r e r e a l coupling constants on A with Jk.,(A)=J,,,(A)=J,,.,~(A), Jk,,(A)=O. In order to study more about our system, let u s introduce the following notations: H(A)=
c H,,
kE A
H a = - [B +B,(A)]u;,
I I . WEISS ISING MODEL When we treat a physical system composed of many interacting particles, we have a powerful approximation, the so-called mean field method, which will b e considered as follows: Observe a particle in the system, then we can regard the effect to this representative =’Mailing address. 967
We can assume B=O i n the sequel discussions without loss of generality. The idea of the mean field approximation of the ferromagnetic system comes f r o m that we replace the above volume dependent B,(A) with some averaged volume in-
ReprintedWith permission from M Ohya. J. Matn. Phys 19(5), 967 (May 1978).‘€ 1978. I Amencan Instituteof Physlcs.
967
45 1 dependent one. Since the above local Hamiltonian H(A) is bounded, it defines a time evolution automorphism of % : T:(A)=exp[itH(A)]A exp[-ilH(A)] f o r anyA E%. We then ask whether and in which sense the time evoluexists. Emch and tion T t after taking the limit A Knops' have studied this question under the following conditions: (i) Stability condition: Zit,, IJk,#(A)I .$ck f o r any finite A C Z containing k , where c k is a A-independent constant. (ii) Van d e r Waals condition: lim,,- J,,*(A) = O f o r all k , i E A. (iii) Periodic condition: J , k - t ~ ( A ) = J , k ~ # ~ .w~h( eAr)e,g is a period of the lattice. ~
It i s then shown that the volume infinite limit of r f ( A ) f o r any A t PI exists for each representation n such that i t s domain i s restricted to the quasilocal C*-algebra% , but i t s range cannot be in general the representation space n(%)but the weak closure n(%)". After taking the limit A - m , the representation dependent time evolucan b e extended to the automorphism ?{, the tion it canonical extension of rt (see the r e m a r k a t the end of this section), on the von Neumann algebra n(nl)". Here the time evolutions T t and 7, should b e written a s 7: and 7: if we want to show their representation dependence explicitly, but we take the former expressions for simplicitly. We now notice that if the von Neumann algebra n(%)" is a factor, then the operator II(Bk), in the appropriate limit5 of Il(B,(A)) a s A -, becomes a multiple of identity because of n ( B k ) sII(P[)"n n(%)'. We next have to construct an extremal KMS state 'p in the following
-
(a) n,(er)"n (b)
n,(%)'=cr,
1d 1 ( ~ ; 7 , ( n , ( A ) ) n , ( B ) ) f ( t )
=J"d t ( ? ;
n.(B)"7t(nq(A)))f(t-i)
for any finite regions A , A'of Z , a n y A c % , , BE%,., and a n y f t B , +.he s e t of the Fourier-transforms of functions in the s e t D of infinitely differentiable functions with compact supports, The above 5; in (b) is the canonical extension of 'p to It,(%)" (see the r e m a r k below) and n, is the GNS representation associated with $9. This construction can be carried out by the following facts: (1) 'p is a locally normal state; that is, for any finite region A of Z 'p[%
., = exp[- H(A)l/Tr[-
exfl(A)l,
c B~,,u;.
'p
('p;AB)=(V;A)b;B)
968
(2.6)
(2.7)
CZ.
Let u s finally notice that the canonical extension ? of 'p from PL to n,(%)" is a faithful normal extremal KMS state with respect to T i . I t R e m a r k : (1) The canonical extension 7, of rt to n,(%)" is defined by ?,(Q)=U,QU., f o r any Q t I l , ( d " , where U, is a strongly continuous one-parameter unitary group implementing Y t on n,(%)". (2) The canonical extension 5; of 'p to Il,(?l)" is defined by ('p;Q)=(&,Q*) f o r any Q E n,(%)", w h e r e & i s the GNS cyclic vector induced by 'p.
111. STABILITY OF WEISS KING MODEL Verbeure and Weder3 have studied the stability of dynamical systems in the linear response theory. Let a dynamical system be described by M,6, a ) , where A is a C*-algebra with an identity operator I , @ i s a KMS state with respect to the time evolution a t ( t E R ) , a strongly continuous one-parameter group of automorphisms of A. Verbeure and Weder introduced the linear f response time evolution a s follows: For any A ~ f and aperturbation X V = X V * d with x t [ O , l ] ,
oly.'(A)=a,(A)ciXSo'ds[a,(v),a,(A)I
(3.1)
f o r t > 0, and a y " c 4 ) = a , ( A ) 4 i X ~ o d s [ ~ , ( at(A)l V),
(3.2)
for t < 0, These a r e obtained by linearizations w. r. t. h of the exact time evolution defined by
ayY(A)=C (iXY 1dt, ...J dtn[atl(V), "-0
O*t,S...
',,St
[. ..[at,(V),at@)]...11
(3.3)
for t >- 0, and
=c( i X ) " J dt, ...J dt,
respectively with A n A t = * .
(2.3)
(v ;03 = tanmk, *,
(2.4)
Bk,w=2Sk[('p;u:)l,
(2.5)
J. Math. Phys., Vol. 19, No. 5, May 1978
.
(2.2)
4)= ('p ;0 : ) = 0,
l*tl'.**
X[att(v),[. .[a,"(v), at(A)I.-. I1
(3) We have the self-consistency equations such that ('p;
n,(%).
(5) it i s an automorphism of It,(%,,) f o r any finite A
"-0
is a product state; that is,
for any A , B in %, and %,
(4) 7, i s an automorphism of
ar(A)
k t A
(2)
k tA
rk
sion f r o m Jk to i s always possible by the HahnBanach theorem. We can thus construct an extremal KMS locally normal state 'p. For this state 'p, the following important properties hold:
(2.1)
where, from n,(Bk)= B k , Ik(Bk, t C ) , H(R) is redefined as H(A)=-
where f his the extension of a positive bounded linear functional Jk on a C*-algebra C ( Z ) of all bounded functions on Z. Here Jk is defined on a subspace C,(Z), f o r ) The extenwhich J k [ f ] = l i m A . , ~ i t n J k , l ( A ) f ( iexists.
(3.4)
for t < 0. The C*-algebraic versions of the definitions of stability in LRS of the dynamical system @, a ) due to Verbeure and Weder a r e as followings.
u,
Definition III. 1 : A dynamical system M,$ , a ) i s said to be stable in LRS under a perturbation V = V" EA if the weak* limits @ + , l = w * - l i m t - , , @ ~ a ~exist i under V. Maranori Ohya
968
452 Definition 111,2: A dynamical s y s t e m Iil, @ , a )is s a i d to be s t a b l e in LRS i f i t is s t a b l e i n LRS under all perturbations V= V* EA.
Moreover, U,&(V)+ = exp(itfl)n,(V)b
= exp[itH(A)]&(V)* =it (&(V))b. A s we have s e e n in the previous section, the WIM As q is a product s t a t e , the Hilbert s p a c e fl, can b e an should b e originally t r e a t e d by the t r i p l e (PI, q ,r ) , but incomplete d i r e c t produce s p a c e denoted by % s z f l a , s i n c e r : d o e s not e x i s t as a n automorphism of 91 f o r a w h e r e f l , is i s o m o r p h i c to C2 generated by four-dimeng e n e r a l s t a t e q on P I , the WIM h a s to b e considered through the von Neumann a l g e b r a i c t r i p l e (n,(PI)", ?,?I.s i o n a l v e c t o r s a t the s i t e k . Hence b can b e w r i t t e n as BkEzbaby cyclic v e c t o r s ak f o r BW,). Choosing However, f o r a n e x t r e m a l KMS s t a t e q , the t i m e evolu= ~ , ~ ~ t ~ , ~ * ~ ( ~ \ ~ - { h ~ ~ :fko ri any f h )X,) infl, tion 7, c a n h e a n automorphism not only of n,(ql)" but = B k t n f l k , (x, U,n,(V)*) is equal to (x,,exp[itH(A)] (since a l l r e p r e s e n t a t i o n s a l s o of n,(PI), and hence of X I&(V)b,), w h e r e O , = @ a t n Q , . Since A i s a finite region are faithful). We thus define the l i n e a r r e s p o n s e time of Z and X, is a r b i t r u y e l e m e n t in ff,, the function t by replacing a , , A , XV with ?,, n,(A), evolution ?*'"'*I cR exp[itH(A)] l&(V)@,) is a l m o s t periodic i n t . Xn,(V) i n (3.1) and (3.2). I t is readily shown by m e r e l y T h i s f a c t implies12 that if the weak l i m i t w-lim,-*JI, extending the von Neumann algebraic r e s u l t s obtained Xn,(V)b e x i s t s , then Utn,(V)* is identical to n,(V)*. by V e r b e u r e and W e d e r 3 to the C*-algebraic dynamical Since U,II,(V)b=7~(n,(V))* and is s e p a r a t i n g f o r s y s t e m Iil, @ , a )that the dynamical s y s t e m (A, @ , a )is the von Neumann a l g e b r a n,(PI)", we have ?:(II,(V)) c l u s t e r i n g [i. e., l i m * - * ~ ( @ ~ a , ( A ) B ) = ( @ ; A ) ( ~f;oAr ) = n,(V). As t h e quasilocal C * - a l g e b r a z i s s i m p l e , any A , B E A ] if and only if it is qt c l u s t e r i n g [i. e. , the r e p r e s e n t a t i o n n, is faithful; hence [exp[itH(A)], V] ?I,(@; a t L 4 ) B ) = ( @ ; A ) ( @ B ;) f o r a n y A , B t - A , w h e r e qt ac= 0. Thus we immediately conclude that V is i n ?I is a mean o v e r t t R ] and s t a b l e in LRS. T h i s f a c t imcording to the f o r m s of H ( A ) and o u r local C*-subalgehra p l i e s that if a dynamical s y s t e m is c l u s t e r i n g f o r i t s ql;~ Conversely, if t h e local perturbation V=V*EPI, KMS s t a t e (e. g., X - Y model, f r e e F e r m i g a s ) , Deis in%:, then i t is obvious that U,n,(V)+ weakly confinitions 111.1 and El. 2 become equivalent, T h e r e f o r e , v e r g e s in fl, as t i m e tends to infinite. f o r s u c h s y s t e m s , the problem of stability in LRS is not s o attractive. However, the WIM is even not qt c l u s t e r Next we have to c o n s i d e r f o r a g e n e r a l perturbation ing, so that i t m a y be interesting to study the problem V = V* t B Namely, w e have to show that the weak l i m i t of stability f o r this model. W e a s k a following question: w-limt-*-UtnV(V)* e x i s t s i n f l , if and only if V=V* Under which conditions on V = V * f A ' is the WIM s t a b l e =PI'. T h e "if" p a r t is shown as follows: F o r any V in LRS? In o t h e r words, can we d e t e r m i n e the s e t of t PI' and any e > 0, t h e r e e x i s t s V, = Vt E PI; s u c h that p e r t u r b a t i o n s u n d e r which the WIM is s t a b l e i n LRS? IIV- V,Il ( e . Hence we obtain
x
- (x,,
+
~
B e f o r e answering this question, l e t u s introduce a C*-subalgehraPI' of PI s u c h that
IIu,n,(v)b-
+l
PI'=j$
9(V)*
II -c l l u * ~ ~ ( v ) + - ~ ~ ~ ~ ( v , ) * l l
l u ~ ~ ~ v , ~ ~ -+nl ~l n~ d~v, ,~~ *~ l- l~ ~ ~ ~ ~ * I l
= z Iln,(v)-n,(v,)I/=z
= b l l k + p2";: P I 3 f i 2 E c}. We then have Theoren? 111.1: The WIM is s t a b l e i n LRS under a perturbation V = V* E PI i f and only if VEPI'. Proof: According to V e r b e u r e and Weder, the WIM is s t a b l e in LRS under a perturbation V= V* E PI i f and only if U,n,(V)b =exp(itfl)n,(V)b weakly converges i n the GNS Hilbert s p a c e fl, a s t i m e tends to infinite, w h e r e H is the infinitesimal g e n e r a t o r of the unitary one-parameter group U,implementing 7,. L e t u s f i r s t c o n s i d e r the c a s e when the perturbation V= V* t PI is local; that is, t h e r e e x i s t s a finite region A of Z such that V = V* E 91,. A s w e have d i s c u s s e d , the t i m e evolution 7, is a n automorphism of nV(%,),hence of PI,. T h e r e s t r i c t i o n 7: of 7, to PI, is generated by the local H a m i l t o n i m H ( A )= ZacABk, T h e e x t r e m a l KMS product s t a t e @ is identical to exp[-H(A)VTr exp[-H(A)] on the local a l g e b r a PI,. T h e r e f o r e , the r e s t r i c t i o n of to PI, is a KMS s t a t e with r e s p e c t to the t i m e evolution 7.; If the WIM is s t a b l e in LRS under the local perturbation V, U,n,(V)*) converges to a definite v a l u e for any x ~ f l ,as t i m e t e n d s to infinite.
-
&.
(x,
969
J. Math. Phys., Vol. 19, No. 5. May 1978
II~-~~ll
Conversely, l e t u s prove the "only if" part. If t h e r e e x i s t s V = V * E PI s u c h that w - l i m t + J t n ~ ( V ) * e x i s t s in flv, then as the l o c a l algebraPI,=U,czql, is n o r m d e n s e i n s , f o r any V= V* 6 PI and any L > 0, t h e r e e x i s t s V, = V t t P I , such that Ilk'- V,II < € ; hence i t is e a s i l y s e e n that
llu,n,(v)*- u,n,(v,)*ll
<E.
T h e r e f o r e , if w-lim,.,,UJl,(V)Q e x i s t s in fl,, then w-lim,.,,U,n,(V,)* e x i s t s i n & , too. Since Vo is a local e l e m e n t of PI, t h i s V, should b e in the a l g e b r a s ; as s e e n b e f o r e . T h i s f a c t i m p l i e s that the perturbation V=V* E PI is i n the a l g e b r a 9l' b e c a u s e '3; is n o r m dense in 9.E. D.
z3.
F r o m this t h e o r e m , we can divide the p e r t u r b a t i o n s into two c l a s s e s . T h e first class is the set of perturbations under which the WIM is s t a b l e i n LRS. Under the p e r t u r b a t i o n s of the second c l a s s , the WIM is not s t a b l e in LRS. The following theorem t e l l s u s that when V= V* f gZ, the l i n e a r r e s p o n s e approximation is e x a c t as far as the stability of the WIM is concerned. T h e o r e m 111.2: F o r any perturbation V = V * t%', w e have Maranari Ohya
969
453 (1) w*-lim $ a ~ ( " ) * ' = q on
,
t-*-
(2) w*-lim 'p.Y("'=q onx. t-*-
Remark: (a) The above exact perturhed time evolution T""'is given by (3.3) and (3.4) under the replacements of at and XV with ?, and Xn,(V). (b) This theorem can be read that i f the WIM is stable in LRS under a perturbation V = V* ~ 9 1 ,then it i s stable in exact sense under the same V. Proof: According to Verbeure and Weder, we have lim t-t-
(a;+")*l(nv(v)))
study the time development of such perturbed states in the WIM. A s discussed i n the previous section, the WIM is described by a triple (&(PI), 'p,?) f o r an extremal KMS state q. For a perturbation XV=XV*E %with X E [0,1], wy denote the exact perturbed WIM by a triple (tlw(g), and the linear response WIM by (n,(91), p' ' * I ) . They a r e given a s f ~ l l o w s ' " ~ : The exact perturbed and linear response time evolutions ~ w ' v ' , ~ ~ ' vrespectively ' * l a r e given in Sec. III. Here, let u s introduce the exact perturbed state 5nv'v' and the linear response functional For anyA
$""',:::Y""'"'j ,+'
~*'")*'.
EPI,
(7V'");n,(A))=
=($;&(A))- Xlim (TII,(A*)Q,T[l- exp(itH)]n,(V)*), *-*where T = [ ( A - 1)/lnA]"* and A=exp(-H). seen in the proof of the theorem 111.1
(*",n,(~)*"),
*"=n,(wy,,)*ilI~.(wrlz)*
(4.1)
11,
(4.2)
We have
x:,,,m,cv)),
exp(itMn,(V)@= n,(v)*, ($"n,'"'*l;
for any V=V* E 3'. We hence have
&(A))= ($;IIq(A))
- x(?;n,w
(Tn,(A*)*, T [ 1 - exp(itH)]n,(V)*)=O, which implies limt-*-($; ~"'"'"(n,(A)))= (+; n,(A)) = ( q ; A ) for any A E % . Let u s prove (2). Simple computation tells u s
+A($;
Jo'ds
:*,,(n,(v)))
nmx?; n m ) ,
where $""""' i s obtained by linearizing and some computations.
'
+"'
(4.4)
w. r. t. X
The motivation of this section is the following: (i) Under which conditions on V = V * e % , do the limits (APproach to equilibrium in LRS under V. )
Hence
Zv'")-'= w * - l i m , ~ , , ' p " ~ ' v ' ~ l ~exist i , on II,(B)?
($; T ( " ' ( n w ( A N = (a;G?*("'(n@(AH) =(q;A)+iX~fds(a;[n,(V),~(")(n,(A))I).
Therefore, we have only to show that the limits lim+*- Iids($; [n,(V), Tw'"'(n+,(A))]) exist f o r any A i n P I O = U A ~ , P I , . Since ?,(n,(v))=n,(v) f o r any V=V* EX', the perturbed time evolution is given by
T'v'
~'"'(n,(~))=exp[itn,(~)l T , ( ~ . ( A ) ) exp[- itn,(v)l. Let us take any V, = V$ from%: , where A, i s a finite region of 2 . For a local elemeRtA of PI, there exists a finite region A of Z such that A E PIA. According to the locally normality of the state rp, we obtain
(+;
[n,(VJ, T '"'w,
(A ))I)
= T r exp[-H(A U Ao)][V,, exp(itV,) exp[itH(A)]A X
exp[- i t H ( A ) ]exp(- itVo)l/Tr exp[-H(A U A,)], (3.5)
-
where H ( Q ) = C E n B&,.o; with Q = A o r A U A,. above (3.5) i s equal to zero due to the forms of Vo. A s PI; i s norm dense in PI", we conclude (2) continuity.
The H ( Q ) and
by Q, E , D.
IV. APPROACH TO EQUILIBRIUM One way of analyzing the structure of statistical mechanics is to study the time development of states which have been perturbed from equilibrium. If a perturbed state relaxes to another state as time tends to infinite, we may say that the approach to equilibrium occurs under that perturbation. In this section, we will 970
J. Math. Phys.. Vol. 19. No. 5 , May 1978
(ii) Under which conditions on V=V*E 91, do the limi t s $?("I = w*-lirn,.,,$"a'"'o?, exist on n,(%)? [ ~ p proach to equilibrium in ES (exact sense) under V.] (iii) Is the linear response approximation useful to study some dynamical processes? We study these questions through the WIM. The answer concerning the first two question is Theorem IV. 1: For the WIM initially in equilibrium described by an extremal KMS state rp, the approach to equilibrium i n LRS occurs under a perturbation V = V* e 91 i f and only if the approach to equilibrium in ES occurs under the same V, Proof: A s we have shown' that for a dynamical system satisfying the KMS condition, the approach to equilibrium in LRS occurs under a perturbation V=V* belonging to the C*-algebra of the dynamical system if and only if the dynamical system is stable in LRS under the same V, Therefore, f o r the WIM, when the approach to equilibrium in LRS occurs under a perturbation V = V* E PI, the WIM is stable in LRS under this perturbation V. According to Theorem 111.1, this perturbation V is in the C*-algebra ? II . A s seen in the proof of Theorem III.1, the equality ?t(nv(V))=&(V) holds for all ~ E R . It is easily checked that &(Wr,,)*=eexp[- +X&(V)]@i s satisfied for V=V* E I". Thus we obtain
(P"); Yt(",(a))) = (exp[- + ~ n , ( ~ ) l l pFt(nVIA)) , exp[- +~n,(v)l*)/
ll e x d - W ~ v ) I * / l Z , Masanori Ohya
970
454 which is equal to (?%("I; IlJ.4)) because of [n,(V)p tn,(%") for any positive integer n and [LI,,R]=O f o r any R t n,(%'). Hence the approach to equilibrium in ES occ u r s under V = V * t P I ' . Conversely, if the approach to equilibrium in ES occurs under a perturbation V = V L d, then the limits limt-*m(*n*c"'.,~,(n~(a))) exist uniformly in X E [0,1] f o r a n y A E % . According to the defi(*n""';?t(n&4)))is expressed by a nition of $nq("', power s e r i e s of X. Moreover, i t is readily seen that the function X E [ O , l ] - {@~("';?~(n.@))) is differentiable at all X for any A E PL and each t e r m of the power s e r i e s of X is uniformly bounded in t. Hence
under a perturbatiofl hV=XV* E %isgiven by (4,1). We then switch off the perturbation a t t = O , that is, we look unat the time developnlent of the perturbed state *nw( d e r "7. It i s interesting to know when the limit ?jf,(") =w*-lim,-,~%("' o?t exists and whether $ and $f'"' ape disjoint. The f i r s t question has been answered by Theorem IV. 1. We a r e interested in the second question a r e disjoint, then here. If these states $ and there is possibility of having the desired transition from the metastable state to a stable state. Due to Theorem IV.1, the limit ~ ~ ~ l " ' = w * - l i m t - ~ + nexists ~ ( " ' ~if ~and t only if the perturbation V i s in the C*-algebra%#. As shown in the proof of that theorem, i s identical to 5%'"'.I t is easily seen that the state Gn*("' is faithful normal on the von Neumann algebra It@(%)". Thus the states 5 and Gnvc"' cannot be disjoint. This fact implies that there does not exist a perturbation V = V* t % under which a metastable state of the WIM goes to a stable state.
exists f o r anyA EPL, This concludes that the approach to equilibrium in LRS occurs under V . Q. E, D.
ACKNOWLEDGMENTS
The above theorem tells u s that the linear response method i s a good approximation of the exact method a s far as the approach to equilibrium i s concerned for the WIM. Let u s consider the dynamical change of metastable states in the WIM. In physical systems, we often encounter metastable states which satisfy the same equilibrium condition of stable states. A stable state gives an absolute minimum to the f r e e energy density of the system. On the contrary, a metastable state gives a relative minimum to the f r e e energy density. A metastable state however easily changes to a stable state (or mixture of stable states) by a local external disturbance, The rigorous interpretation of the existence of metastable states and the transition f r o m a metastable state to a stable state has not much done yet. In the WIM, we, however, know5 that the metastable states exist below some critical temperature. These metastable states a r e extremal KMS states a s the stable state. L e t u s discuss the above transition of a metastable state appeared in the WIM under the effect of local perturbations. We ask whether there exists a perturbation which causes such transition. Let b e a metastable state satisfying the KMS condition with respect to 71'. The locally perturbed state +nv(v' of $
*
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J. Math. Phys.. Vol. 19. No. 5, May 1978
"'
+:+""'
@'"'
The author thanks Professor G . G. Emch for valuable discussions and suggestions of this work, He also thanks Professor H. Umegrtki f o r his interest to the author's work and his kindness at Tokyo Institute of Technology.
IR. Haag,D. Kastler, and E. Trych-Pohlymeyer, Commun. Math. Phys. 38, 173 (1974).. 2J.L. Lebowitz, M.Aizemann, and S. Goldstein, J, Math. Phys. 6, 1284 (1975). 3A. Verbeure and R. Weder, Commun. Math. Phys. 44, 101 (1975). 4M. Ohya,
"Dpamical Process in Linear Response Theory," prepr int Emch and J.F. Knops, J. Math. Phys. 12, 2043 (1971). %.G. Emch, J. Math, Phys. 7, 1198 (1966). ?G.L. sewell, ~ e t t ~ . u o v oCimento 10, 430 (1974). 8A. Daneri, A. Lainger, and G.M. Prosperi, Nucl. Phys. %.G.
.
33, 297 (1962). k2.G. Emch, Algebraic Methods in Statistical Mechanics and Quantvm Field Theorv huiley, New York, 1972). 'OR. Haag, N. Hugenhoitz, and M. Winnik,Commun. Math. Phys. 5, 215 (1967). "M. Takesaki, Tomita's Theory of Modular Hilbert A l z e b m and Its Applications (Springer-Verlag, Berlin, 1970). "M. Winnik,in Fundaniantal Pobleins in Statistical Mechanics, edited by E.D.G. Cohen North-Holland, Amsterdam, 1975). %.W. Robinson, lecture at Varenna School, 1973.
Maranori Ohya
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List of Publications 1. Papers 1 M.Ohya (1974) Dynamical process in quantum statistical mechanics, U.R Tech. Rep., 25, No.2, 12-25. 2 M.Ohya (1975) Remarks on noncommutative ergodic properties, U.R Tech. Rep., 26, No.3, 30-38. 3 M.Ohya (1978) Stability of Weiss Ising model, J . Math. Phys., 19, No.5, 967-971. 4 M.Ohya (1978) On linear response dynamics, L. Nuovo Cimento, 21, No.16, 573-576. 5 M.Ohya (1979) Dynamical process in linear response theory, Rep. Math. Phys, 16, No.3, 305-315. 6 M.Ohya (1980) On open system dynamics -an operator algebraic study-, Kodai Math. J., 3, 287-294. 7 F.Hiai, M.Ohya and M.Tsukada (1981) Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J . Math., 96, No.1, 99-109. 8 F.Hiai, M.Ohya and M.Tsukada (1981) Relative entropy in quantum dynamical systems, Res. Rep. Inform. Sci. of TIT, A-77. 9 M.Ohya (1981) Quantum ergodic channels in operator algebras, J. Math. Anal. Appl., 84, No.2, 318-327. 10 M.Ohya (1983) Note on quantum probability, L. Nuovo Cimento, 38, No.11, 402-404. 11 F.Hiai, M.Ohya and M.Tsukada (1983) Sufficiency and relative entropy in *algebras with applications in quantum systems, Pacific J . Math., 107, N O . ~ , 117-140. 12 M.Ohya (1983) On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, No.5, 770-774. 13 A&%HIJ, @& W (1984) B ~ ~ ~ ~ ~4? a>%%, ZW%E$%2i%,J87-A, No.6, 548-552. 14 M.Ohya (1984) Entropy Transmission in C*-dynamicaI systems, J . Math. Anal. Appl., 100, No.1, 222-235. 15 A&%WII (1984) DNA El12 b D t'-fi@T, Semi. Appl. Func. Anal., 6, 45-56. 16 M.Ohya (1985) State change and entropies in quantum dynamical systems, Springer Lecture Notes in Math., 1136, 397-408. 17 M.Ohya (1985) Construction and analysis of a mathematical model in quantum communication processes, Scripta Thechnica, Inc. , Elect. Commun. Japan, 68, No.2, 29-34. 18 M.Ohya (1985) A Mathematical Analysis of DNA Sequences, Symp. Appl. Funct. Anal., 8, 36-47. 19 M.Ohya and Y.Fujii (1986) Entropy change in linear response dynamics, IL
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456
Nuovo Cimento, 91B, No.1, 25-30. 20 M.Ohya and T.Matsuoka (1986) Continuity of entropy and mutual entropy in C*-dynamical systems, J. Math. Phys., 27(8), 2076-2079. 21 M.Ohya and N.Watanabe (1986) A new treatment of communication processes with Gaussian Channels, Japan J . Appl. Math., 3, No.1, 197-206. 22 O.Hirota, K.Yamazaki, M.Nakagawa and M.Ohya (1986) Properties of error correcting code using photon pulse, Trans. IEICE, E69, No.9, 917 - 919. 23 M.Ohya, M.Tsukada and H.Umegaki (1987) A formulation of noncommutative McMillan theorem, Proc. Japan Academy, 63, Ser.A, No& 50-53. 24 K.Yamazaki, O.Hirota, M.Nakagawa T.Umeyama and M.Ohya (1987) Effect of error correcting code in photon communications with energy loss, Trans. IEICE, E70, No.8, 689-692. 25 M.0hya (1987) Information theoretical analysis of phylogenetic trees for molecular evolution, Symp. Appl. Func. Anal., 10, 87-101. 26 M.Ohya (1988) Fractal dimensions for general quantum states, Symp. Appl. Func. Anal., 11, 45-57. 9@$ 27 A%%WII , 7!?E%%,KEEI 465 (1988) f3%N@'sp4?3%?~Kk;135% E)%@fX%&, %3@$$$%4g9$?%*%, J71-B, No.4, 533-539. 28 A%%WIJ (1988) ~ ' s p 4 ~ D & 9 E t 9 & ? $ - # ~ @ ~ D F. u~ 2k'-$k jltB4s33 g-, % 78 & 2 %, 'sp% 500 %-,B124B140. 29 k%%W (1988) IJ 7&4Z3-E)6796t@%i~fi~~, @+/%%E!4Z??2%, 71, No.3, 295-297. 30 M.Ohya (1989) Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys., 27, 19-47. 31 A%%WI], $$*fix(1989) Breiman E)$ZB-D$fi,@, @?1@4g%%% (A), J72-A, 2057-2060. 32 M.Ohya (1989) Information theoretical treatments of genes, Trans. IEICE, E72, No.5, 556-560. 33 A%%WlJ, WJ3 @, A%@* (1989) 7iI @fi@JE)BPJ4L&, Viva Origino, 17, No.3, 141-151. 34 A%%HlJ (1990) $RED E z2b Y- k 7 7 3 .P/b&jC, %d52T%@f%, 80, D138-D149. 35 A%%WIJ, %@GLH (1990) &3YPJ4@%!Z33SiZk;1?6% @@D%%4L, 8z +,/gj@'sp4S*A3A m = ~ = a w f t % , J73-B-I, NO.3, 200-207.
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B3$ (1990) , M 9 6 t ~ ~ H ~ \ f ~&!# ~!I 4 N0.2, 30-37. 37 M.Ohya (1991) Fractal dimensions of states, Quantum Probability and Related Topics, 6, 359-369. 38 M.Ohya (1991) Information dynamics and its applications to optical communication processes, Springer Lecture Notes in Physics, 378, 81-92. 39 M.Ohya and HSuyari (1991) Rigorous derivation of error probability in coherent optical communication, Springer Lecture Note in Physics. 378,
36 A%%WlJ, l&~%A.*
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203-212. 40 M.Ohya (1991) On fuzzy relative entropy, Symp. Appl. Func. Anal., 13, 105-115. ?&%&!J, & k ~ (1991) %?%4& 23&3, %3'1$3%%4$%2%$*%, J74A, 41 No.7, 1075-1084. 42 N.Muraki, M.Ohya and D.Petz (1992) Entropies of general quantum states, Open Systems and Information Dynamics, 1, No.1, 43-56. 43 N.Muraki and M.Ohya (1992) Note on continuity of information rate, Illinois J . of Math., 36, 529-550. 44 k%%HlJ,;Ik 33%(1992) ~ ~ 4 ~ ~ ~ ~ ~ C - F k~ +Oh5 ~ ~ f 83, %?f$SB%4iZ$%?%jX%, J75-A, No.12, 1859-1864. 45 M.Ohya and Y.Uesaka (1992) Amino acid sequences and D P matching: a new method of alignment, Information Sciences, 63, 139-151. 46 M.Ohya, S.Miyazaki and K.Ogata (1992) On multiple alignment of genome sequences, IEICE Trans. Commun. E75-B, No.6, 453-457. 47 M.Ohya and S.Naritsuka (1993) On fuzzy relative entropy, Open Systems and Information Dynamics, 1, No.3, 397-408. 48 M.Ohya (1993) Quantum entropies and thier maximizations, Maximum Entropy and Bayesian Methods, 12, 189-194. 49 M.Ohya and N.Watanabe (1993) Information dynamics and its application to Gaussian communication processes, Maximum Entropy and Bayesian Methods, 12, 195-203. 50 L.Accardi, M.Ohya and H.Suyari (1994) Computation of mutual entropy in quantum markov chains, Open Systems and Information Dynamics, 2, No.3, 337-354. 51 F.Hiai, M.Ohya and D.Petz (1995) McMillan type convergence for quantum Gibbs states, Archives of Mathematics, 64, 154-158. 52 T.Matsuoka and M.Ohya (1995) Fractal dimensions of states and their applications t o Ising model, Rep. Math. Phys., 36, 365-379. 53 M.Ohya and T.Matsuoka (1995) Simulated annealing and its application to Cobb-Douglas economic model, Open Systems and Information Dynamics, 3, No.3, 357-368. L.Accardi, M.Ohya and H.Suyari (1995) An application of lifting theory to 54 optical communication processes, Rep. Math. Phys., 36, 403-420. 55 M.Ohya (1995) State change, complexity and fractal in quantum systems, Quantum Communications and Measurement, Plenum, 2, 309-320. 56 L.Accardi, M.Ohya and H.Suyari (1995) Mutual entropy in quantum Markov chains, Quantum Communications and Measurement, Plenum, 2, 351-358. 57 M.Ohya and N.Watanabe (1995) A mathematical study of information transmission in quantum communication processes, Quantum Communications and Measurement, Plenum, 2, 371-378. b--P-EU?8 Z~ 58 A%#HIJ, $!H8I@Z(1996) K & O 7 7 3 P M ~ C ? ? R L \ ~
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458
I1 ID@%$ D&$fi, %,3%7@S!4$%!2%3%, J79-A, No.9, 1590-1599. 59 N.Muraki and M.Ohya (1996) Entropy functionals of Kolmogorov-Sinai type and their limit theorems, Letter in Mathematical Physics, 36, 327-335. 60 L.Accardi, M.Ohya and N.Watanabe (1996) Note on quantum dynamical entropies, Rep. Math. Phys., 38, No.3, 457-469. 61 M.Ohya and D.Petz (1996), Notes on quantum entropy, Studia Scientiarum Mathematicarum Hungarica, 31, 423-430. 62 M.Ohya and N.Watanabe (1996) Note on irreversible dynamics and quantum information, Contributions in Probability, Undine, Forum, 205-220. 63 S.Miyazaki, HSugawara and M.Ohya (1996) The efficiency of entropy evolution rate for construction of phylogenetic trees, Genes Genet. Syst., 71, 323-327. 64 M.Ohya (1997) Complexity, fractal dimension for quantum states, Open Systems and Information Dynamics, 4, 141-157. 65 %%%HIJ, 4\E % (1997) 81%%hYK k 6 3 A-x%&D%B, Z3%?@S4S %2%*%, J80-A, No.12, 2138-2144. 66 L.Accardi, M.Ohya and N.Watanabe (1997) Dynamical entropy through quantum Markov chain, Open Systems and Information Dynamics, 4, 71-87. 67 M.Ohya, D.Petz and N.Watanabe (1997) On capacities of quantum channels, Probability and Mathematical Statistics, 17, 179-196. 68 %%%HI, 3R@$L#, &* 3 (1997) X 94 X F$3?2%%iZ 16% 9 @$D &g,%3818$6t%!$%!%%i*8, J80-A, No.5, 809-817. 69 k%%WlJ9B@$LW, , @n E S (1997) H % = 2 9 9 blo-k'XCzL6I%&E @Elj%Dfi@fi, %%dk%%.f$4$%2%3C%l%, J80-A, No.6, 1022-1029. ?@%$$ (1997) !?I @WE#i@?&?2JS%L ~ F ' A ~ - ? I L 77 4 % 2 b 70 k%%HIJ, 2T D & & , 5ig;'F,M@34s".^a +zzmB&, J80-A, No.4, 684-691.
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(1997) J %{ZT-D@%%B%$%~?U - 4 , Viva Origino, 25, 223-246. 71 k%%WI 72 S.Furuichi, M.Ohya and H.Suyari (1997) Computation of mutual entropy in quantum amplifier processes, Quantum Communication, Computing, and Measurement, 3, 147-155. 73 M.Ohya and N.Watanabe (1997) Quantum capacity of noisy quantum channel, Quantum Communication, Computing, and Measurement, Plenum, 3, 213-220. 74 M.Ohya and H.Ogata (1997) Multiple alignment for amino acid sequences by dynamics programing, Electronics and Communications in Japan, Part 3, 81, No.4, 684-691. 75 K.Sato and S.Miyazaki and M.0hya (1998) Analysis of HIV by entropy evolution rate, Amino Acids, 14, 343-352. 76 M.Ohya (1998) Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics, 37, No.1, 495-505. 77 #k @, A%WWiJ, iM; ?f3(1998) $$ED773 P)k&Z%Hb\f~i&f3%%
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8, No.2, 187-197.
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78 M.Ohya, D.Petz and N.Watanabe (1998) Numerical computation of quantum capacity, International Journal of Theoretical Physics, 38, No.1, 507510. 79 M.Ohya and N.Watanabe (1998) On Mathematical treatment of FredkinToffoli-Milburn gate, Physica D, 120, 206-213. 80 K.Inoue, M.Ohya and H.Suyari (1998) Characterization of quantum teleportation by nonlinear quantum channel and quantum mutual entropy, Physica D, 120, 117-124. 81 A%%H (1998) B ~ - F - ~ + I ~ O ~ Z S @E3-f8$&5@4ssA*A** B ~ Z ~ a+ZTartR ~ ~ - C ,UJC\, J81-A, N0.12, 1638-1643. 82 %$i!B& I@ %, -,A%%H (1998) g Q 4 - r + 1 ~ O ~ ~ ~ ~ 4 ~ ~ ~ ~ %3-J1%%B!43!?52%jXi%, J81-A, No.12, 1707-1714. 83 H.Ogata, M.Ohya and H.Umeyama (1998) Amino Acid Similarity Matrix for Homology Modeling Derived from Structural Alignment And Optimized by the Monte Carlo Method, J . Mol. Graphics and Mod., 16, 178-189. 84 M.Ohya (1998) Foundation of entropy, complexity and fractal in quantum systems, Probability Towards the Year 2000, 263-286. 85 A.Kossakowski, M.Ohya and N.Watanabe (1999) Quantum dynamical entropy for completely positive map, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2, No.2, 267-282. 86 L.Accardi and M.Ohya (1999) Compound channels, transition expectations, and liftings, Appl. Math. Optim., 39, 33-59. 87 M.Ohya (1999) Fundamentals of quantum mutual entropy and capacity, Open System and Information Dynamics, 6, No.1, 69-78. 88 S.Furuichi and M.Ohya (1999) Quantum mutual entropy for JaynesCummings model, Rep. Math. Phys., 44, 81-86. 89 S.Furuichi and M.Ohya (1999) Entanglement degree in the time development of the Jaynes-Cummings model, Letter in Math. Phys., 49, 279-285. 90 H. Hirano, K. Sato, T . Yamaki and M. Ohya (1999) Study of the relation between the variation of HIV and the disease, progression by entropy evolution rate, Viva Origino, 27, 91-106. 91 K.Inoue and T.Matsuoka and M.Ohya (2000) €-entropy and fractal dimension of a state for a Gaussian measure, Open Systems and Information Dynamics, 7, No.1, 41-53. 92 #k @, kJIIR-, A%%HIJ(2000) &+=9 I- Y--?ZA%XRB2Hb\ I z X ‘c’Y 1/2 %OA&XOlfig$fi, El$JZH&E%%%ijXi%, 10, No.1, 51-69. 93 K.Inoue, M.Ohya and K.Sato (2000) Application of chaos degree to some dynamical systems, Chaos, Soliton & Fractals, 11, 1377-1385. 94 M.Ohya (2000) Complexity in dynamics and computation, Acta Applicandae Mathematicae, 63, 293-306. 95 M.Ohya and N.Masuda (2000) NP problem in quantum algorithm, Open Systems and Information Dynamics, 7, No.1, 33-39.
460
96 M.Ohya and K.Sato (2000) Use of information theory to study genome sequences, Rep. Math. Phys., 46, No.3, 419-428. 97 V.P.Belavkin and M.Ohya (2001) Quantum entropy and information in discrete entangled states, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4, No.2, 137-160. 98 K-H.Fichtner and M.Ohya (2001) Quantum teleportation with entangled states given by beam splittings, Commun. Math. Phys., 222, 229-247. 99 K.Sato and M.Ohya (2001) Analysis of the disease course of HIV-1 by entropic chaos degree, Amino Acids, 20, 155-162. 100 K S a t o and M.Ohya (2001) Can information measure be one of markers to estimate disease progression in HIV-l-infected patients?, Open Systems and Information Dynamics, 8, 125-136. M.Ohya, 1.V.Volovichand N.Watanabe (2001) Quantum Logical Gate Based 101 on Electron Spin Resonance, Quantum Information, 3 , 143-156. 102 M.Ohya,T.Nishioka and K.Sato (2001) Phylogenetic Relation of HIV-1 in the V3 Region by Information Measure, Quantum Information, 3, 127-142. 103 K-H.Fichtner and M.Ohya (2002) Quantum teleportation and beam splitting, Commun. Math. Phys., 225, 67-89. 104 K.Inoue, M.Ohya and 1.V.Volovich (2002) Semiclassical properties and chaos degree for the quantum baker’s map, J. Math. Phys., 43-2, 734755. 105 V.P.Belavkin and M.Ohya (2002) Entanglement, quantum entropy and mutual information, Proc. R. SOC.Lond. Ser. A Math. Phys. Eng. Sci. 458, N0.2017, 209-231 106 K.Inoue, 1.V.Volovichand M.Ohya (2002) A Treatment of Quantum Baker’s Map by Chaos Degree, Quantum Information, 4, 87-102. 107 M.Ohya (2002) An Information Theoretic Approach to the Study of Genome Sequences: An Application to the Evolution of HIV, Springer Lecture Notes in Computer Science, 2509, 50-57. 108 M.Ohya and 1.V.Volovich (2003) On Quantum Capacity and Its Bound, Infinite Dimensional Analysis and Quantum Probability, 6, No.2, 301-310. 109 A.Kossakowski, M.0hya and Y.Togawa (2003) How can we observe and describe chaos?, Open System and Information Dynamics, 10, No.3, 221233. 110 M.Ohya and 1.V.Volovich (2003) New quantum algorithm for studying NPcomplete problems, Rep. Math. Phys., 52, No.1, 25-33. 111 M.Ohya and 1.V.Volovich (2003) Quantum computing and chaotic amplifier, J. Opt. B, 5, No.6, 639-642. 112 &ZA, 4&@33-, GHWPJ?, %%%HI1 (2003) % E z Y b n Y-?2ML\k 7f q‘ %. Y b Vl&B, @%%I%!?%$%$#$&, 13, No.3, 49-57. 113 M.Ohya (2003) Information Dynamics and its Application to Recognition Process, A Garden of Quanta : Essays in Honor of Hiroshi Ezawa, World Scientific Pub. Co. Inc., 445-462.
461
114 K-H.Fichtner, W.Freudenberg, M.Ohya (2003) Recognition and teleportation, Quantum probability and infinite dimensional analysis, QP-PQ: Quantum Prob. White Noise Anal., Quantum Probability and InfiniteDimensional Analysis, World Sci. Publishing, 15, 85-105. 115 K.Inoue, M.Ohya, 1.V.Volovich (2003) On quantum-classical correspondence and chaos degree for baker's map. Fundamental aspects of quantum physics (Tokyo, 2001), QP-PQ: Quantum Prob. White Noise Anal., World Sci. Publishing, 17, 177-187. 116 L.Accardi and M.Ohya (2004) A Stochastic Limit Approach to the SAT Problem, Open Systems and Information dynamics, 11-3, 219-233. 117 M.Ohya (2004) Foundation of Chaos Through Observation, Quantum Information and Complexity edited by T.Hida, K.Saito and Si Si, 391-410. 118 T.Mastuoka and M.Ohya (2004) A New Measurement of Time Serial Correlations in Stock Price Movements and Its Application, Quantum Information and Complexity edited by T.Hida, KSaito and Si Si, 341-361. 119 M.Ohya (2004), On quantum information and algorithm, Mathematical Modelling in Physics, Engineering and Cognitive Sciences, 10, 451-468. 120 M.Ohya (2005) Quantum Algorithm for SAT Problem and Quantum Mutual Entropy, Reports on Mathematical Physics, 55, No.1, 109-125. 121 T.Matsuoka and M.Ohya (2005) Quantum Entangled State and Its Characterization, Foundations of Probability and Physics-3, American Institute of Physics, 750, 298-306. 122 K-H Fichtner, W.Freudenberg and M.Ohya (2005) Teleportation Schemes In Infinite Dimensional Hilbert Spaces, J. Math. Phys. 46, 102103-14. 123 T.Miyadera and M.Ohya (2005) Quantum Dynamical Entropy of Spin Systems, Reports on Mathematical Physics, 56, No.1, 1-10, 124 W. Freudenberg, M. Ohya and N. Watanabe (2005), On quantum logical gates on a general Fock space, QP-PQ: Quantum Probability and White Noise Analysis, World Sci. Publishing, 18, 252-268. 125 K-H Fichtner, W.Freudenberg and M.Ohya (2006) Recognition and Teleportation, Quantum Information, 5, 1-17. 126 T.Miyadera and M.Ohya (2006) On Halting Process of Quantum Turing Machine, Open System and Information Dynamics, 12, No.3, 261-264. 127 F.-H.Fichtner, T.Miyadera, M.Ohya (2006) Fidelity of Qunatum Teleportation Model Using Beam Splittings, QP-PQ: Quantum Probability and White Noise Analysis, Quantum Information and Computing, World Sci. Publishing, 19, 113-130. W.Freudenberg, M.Ohya, N.Turchina, N.Watanabe (2006) Quantum Logical 128 Gates Realized by Beam Splittings, QP-PQ: Quantum Prob. White Noise Anal., Quantum Information and Computing, World Sci. Publishing, 19, 131-148. 129 S.Iriyama, M.Ohya and 1.V.Volovich (2006) Generalized Quantum Turing
462
Machine and its Application t o the SAT Chaos Algorithm, QP-PQ: Quantum Prob. White Noise Anal., Quantum Information and Computing, World Sci. Publishing, 19, 204-225. 130 A.Kossakowski and M.Ohya (2006) Can Non-Maximal Entangled State Achieve a Complete Quantum Teleportation?, Reconsideration of Foundation-3, American Institute of Physics, 810, 211-216. 131 L.Accardi, T.Matsuoka and M.Ohya (2006) Entangled Markov Chains are Indeed Entangled, Infinite Dimensional Analysis Quantum Probability and Related Topics, 9, No.3, 379-390. 132 A.Kossakowski and M.Ohya (2007) New Scheme of Quantum Teleportation, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 10, 3, 411-420. 133 M.Ohya (2008) Adaptive Dynamics and Its Applications to Chaos and NPC Problem, Quantum Bio-Informatics: From Quantum Information to BioInformatics, World Sci. Publishing, 1, 181-216. 134 K.-H. Fichtner, L. Fichtner, W. Freudenberg and M.Ohya, Quantum Models of the Recognition Process - On a Convergence Theorem, submitted t o Journal of Mathematical Physics. 2. Other publications
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lI8p6r 2%B 1-20, BBH?, No.305, 78-83 ; No.307, 46-53;N0.309, 7177; N0.311, 78-83 ; N0.314, 76-83iN0.316, 75-83 ; N0.318, 76-83 ; N0.322, 72-76;N0.323, 78-83iN0.326, 80-83iN0.330, 7478iN0.337, 79-83iN0.342, 79-83;N0.346, 68-73iN0.351, 70-75;N0.35 1, 70-75;N0.353, 57-63;N0.358, 76-83;N0.371, 61-67iN0.373, 59-64iN0.385, 77-83.
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