Quantum Probability and Infinite Dimensional Analysis: proceedings of the 26th Conference: Levico, Italy, 20-26 February 2005

QP-PQ: Quantum Probability and White Noise Analysis Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy QP-PQ: Quantum Probability and White Noise Analysis VOl. 20:

Quantum Probability and Infinite Dimensional Analysis eds. L. Accardi, W. Freudenberg and M. Schurmann

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Quantum Information and Computing eds. L. Accardi, M. Ohya and N. Watanabe

Vol. 18:

Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds. M. Schurmann and U. Franz

Vol. 17:

Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki

Vol. 16:

Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsuiand A. Hora

Vol. 15:

Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg

Vol. 14:

Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola

Vol. 13:

Foundations of Probability and Physics ed. A. Khrennikov

QP-PQ VOl.

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Quantum Probability and Infinite Dimensional Analysis eds. L. Accardi, W. Frendenberg and M. Schurmann

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Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay

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Quantum Probability and Related Topics ed. L. Accardi

Vol. 8:

Quantum Probability and Related Topics ed. L. Accardi

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Quantum Probability and Related Topics ed. L. Accardi

QP-PQ Quantum Probability and White Noise Analysis Volume XX

Proceedings of the 26th Conference 20 - 26 February 2005

Levico, Italy

Editors

L. Accardi Universith di Roma Tor Vergata, Italy

W. Freudenberg Brandenburgische Technische Univevsitat Cottbus, Germany

M. Schurmann Greifswald University, Germany

v

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QP-PQ: Quantum Probability and White Noise Analysis - Vol. XX QUANTUM PROBABILITY AND INFINITE DIMENSIONAL ANALYSIS Copyright Q 2007 by World Scientific Publishing Co. F'te. Ltd.

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FOREWORD

The present volume contains the proceedings of the 26th Conference on Quantum Probability and Infinite Dimensional Analysis held in Levico, Italy, 20-26 February, 2005. The goal of the conference was to communicate new results in the fields of quantum probability and infinite dimensional analysis. The fact that contributions to this volume range from classical probability, ‘pure’ functional analysis and foundations of quantum mechanics to applications in mathematical physics, quantum information theory and modern mathematical finance shows that research in quantum probability and infinite dimensional anlysis is very active and strongly involved in modern mathematical developements and applications. The conference also served as the mid-term meeting and 4th plenary conference of the Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology of the European Community under contract HPRN- CT-200%00279. As such the conference presented the scientific research done so far within the network, and during special sessions young researchers of the network were given the opportunity to present their research work. This led to interesting discussions and stimulated new collaborations. We gratefully acknowledge the support by the European Community. Special thanks go to Stefanie Zeidler and Augusto Micheletti who took care of the logistics of the mid-term meeting and the conference and to Uwe Jahnert and Augusto Micheletti for help with the editing of these proceedings. Luigi Accardi Wolfgang F’reudenberg Michael Schurmann

V

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CONTENTS

Foreword

V

A Combinatorial Identity and Its Application to Gaussian Measures L. Accardi, H.-H. Kuo and A . I. Stan

1

Feynman Formulas for Evolution Equations with L6vy Laplacians on Manifolds L. Accardi and 0. G. Smolyanov

13

On the Fock Representation of the Renormalized Powers of Quantum White Noise L. Accardi and A . Boukas

26

Powers of the Delta Function L. Accardi and A . Boukas

33

Dispersion Relations in the Stochastic Limit of Quantum Theory L. Accardi and F. G. Cubillo

45

Integral Representation of Positive Operator on Infinite Dimensional Space of Entire Functions W. Ayed and H. Ouerdiane

53

Comparison of Some Methods of Quantum State Estimation Th. Baier, D. Petz, K. M. Hangos and A. Magyar

64

Entropic Bounds and Continual Measurements A . Barchielli and G. Lupieri

79

Generalized q-Fock Spaces and Duality Theorems A . Barhoumi and H. Ouerdiane

90

vii

viii

Covariant Quantum Stochastic Flows and Their Dilation V. Belavlcin and L. Gregory

102

Alicki-Fannes and Hudson-Parthasarathy Evolution Equations A . C. R. Belton

128

Boson Cocycle as the Second Quantization of the Boolean Cocycle A . Ben Ghorbal and F. Fagnola

134

Functional Integrals over Smolyanov Surface Measures for Evolutionary Equations on a Riemannian Manifold Ya. A . Butlco

145

Quantum Probabilistic Model for the Financial Market 0. Choustova

156

A New Proof of a Quantum Central Limit Theorem for Symmetric Measures V. Crismale and Y. G. Lu

163

On the Most Efficient Unitary Transformation for Programming Quantum Channels G. M. D’Ariano and P. Perinotti

173

Stability Analysis of Quantum Mechanical Feedback Control System P. K. Das and B. C. Roy

181

Markov States on Quasi-Local Algebras F. Fidaleo

196

Some Open Problems in Information Geometry P. Gibilisco and T. Isola

205

Introduction to Determinantal Point Processes from a Quantum Probability Viewpoint A . D. Gottlieb

212

ix

Note on the Time Operator T. Hida

224

On the Dynamical Symmetric Algebra of Ageing: Lie Structure, Representations and Appell Systems M. Henkel, R. Schott, S. Stoimenov and J. Unterberger

233

An Analytic Double Product Integral R. L. Hudson

241

On Generalized Quantum Turing Machine and Its Application S. Iriyama and M. Ohya

251

Dynamics with Infinite Number of Derivatives for Level Truncated Noncommutative Interaction L. Joukovskaya

258

A Logarithmic Sobolev Inequality for an Interacting Spin System Under a Geometric Reference Measure A . Joulin and N . Privault

267

To Quantum Mechanics through Gaussian Integration and the Taylor Expansion of Functionals of Classical Fields A . Yu. Khrennikov

2 74

Hyperbolic Quantization A . Khrennikov and G. Segre

282

Discrete Energy Spectrum in Discrete Time Dynamics A . Khrennikov and Ya. Volovich

288

Convolution Associated with the Free cosh-Law A . D. Krystek and L. J. Wojakowski

295

A Theorem on Liftings of Statistical Operators J. Kupsch

302

X

Positive Maps between Mz (C) and Ad,(@). On Decomposability of Positive Maps between Mz(C) and M,(C) W. A . Majewski and M. Marciniak

308

Thermodynamical Formalism for Quasi-Local C*-Systems and Fermion Grading Symmetry H. Moriya

319

Micro-Macro Duality and an Attempt Towards Measurement Scheme of Quantum Fields I. Ojirna

323

The L6vy Laplacian Acting on Some Class of LCvy Functionals K. Sait6 and A . H. Tsoi

. 330

On the General Form of the Integral-sigma Lemma in Symmetric Fock Space K. Schubert

338

Spatial Eo-Semigroups are Restrictions of Inner Automorphism Groups M. Skeide

348

A Characterization of Poisson Noise S i Si

356

On Two Conjectures in Segal-Bargmann Analysis S. B. Sontz

365

Note on Quantum Mutual Type Entropies and Capacity N. Watanabe

373

A COMBINATORIAL IDENTITY AND ITS APPLICATION TO GAUSSIAN MEASURES

LUIGI ACCARDI Centro Vito Volterra Facoltd d i Economia Universitd di Roma “Tor Vergata” 00133 Roma, Italy E-mail: accardiQvolterra.mat.uniroma2.it HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, L A 70803, U.S.A. E-mail: kuoQmath. lsu.edu AUREL I. STAN Department of Mathematics The Ohio State University at Marion 1465 Mount Vernon Avenue Marion, OH 43302, U.S.A. E-mail: stan. [email protected]

2000 Mathematics Subject Classifications: 05335, 60H40. Key words and phrases: commutator, annihilation operator, creation operator, neutral (preservation) operator. Assuming that a probability measure on W d has finite moments of any order, its moments are completely determined by two family of operators. The first family is composed of the neutral (preservation) operators. The second family consists of the commutators between the annihilation and creation operators. As a confirmation of this fact, a characterization of the Gaussian probability measures in terms of these two families of operators is given. The proof of this characterization relies on a simple combinatorial identity.

1

2

1. Introduction

Let d be a positive integer and p a probability measure on the Bore1 subsets of Rd. We assume that p has finite moments of any order, (i.e., SRdIxiIPdx < 00, for all p > 0 and i E {1,2,. . . ,d } , where xi, denotes the i-th coordinate of a generic vector x = (21,x2,.. . ,xd) in Rd). For all non-negative integers n, we consider the space F, of all polynomial functions f(x1,x2,.. .,xd), of d-variables, such that d e g ( f ) 5 n, where d e g ( f ) denotes the total degree o f f . Because p has finite moments of any order, it follows from Holder’s inequality that F, C L 2 ( R d , p ) for , all n 2 0. Moreover, F, is a closed subspace of L 2 ( R dp, ) , since F, is finite dimensional, for all n 2 0. Thus, for all n 2 0, we may define the spaces Gn := F, 8Fn-1, where F, 8F,-l denotes the orthogonal complement of F,-1 into F, with respect to the inner product given by p, and F-1 := { 0 } is the null space. We define the Hilbert space H as the direct sum of the orthogonal subspaces {G,},>o ( i e . , ‘H := en2OGn). Let V be the space of all polynomial functions of d-variables: 2 1 , 2 2 , . . . , x d , and of arbitrary degree, in which any two polynomials, that are equal p-almost surely, are considered to be identical.

For all i E {1,2,. . . ,d } , we denote by Xi the operator of multiplication by the variable xi. This operator is defined on the space V, which is dense in H, as ( X i f ) ( x l , x 2 , . . ,xd) := x i f ( x 1 , ~ 2 ., ..,Q). The following lemma leads us to the notion of creation, neutral (preservation), and annihilation operators. See [2] for details.

Lemma 1.1. For all n 2 0 and i E {1,2,. . . ,d } , XiG,IGk, for all k # n - 1, n, n 1, where I denotes the orthogonality relation with respect to the inner product generated by the probability measure p.

+

From Lemma 1.1,it follows that, for all n 2 0 and i E { 1 , 2 , . . .,d } , XiG, c Gn-1 CBG, @ Gn+l. This means that, for all f E G,, X i f = fi,,-1+ fi,, fi,n+l, for some fi,n-1 E Gn-1, f i , n E Gn, and fi,n+l E Gn+l. We define three family of operators D;(i) : G, -+ Gn+l, DR(i)f := fi,,+1, D:(i) : Gn -+ G,, D:(i)f := fi,,, and D,(i) : G, -+ Gn-I, D;(i)f := fi,n-l, and observe that the restriction of the multiplication operator Xi to the space G,, XilG,, satisfies the relation:

+

XilGn for all n 2 0 and i E {1,2,.

= D:(i)

..,d}.

+ D:(i) + D,(i),

3

We extend now this family of operators, by linearity, to the whole space V of polynomial functions. That means, if cp = Cn,Ofn, where for each n 2 0, f n E G,, and only a finite number of the terms-fn are not zero, then we define, for all n 2 0 and i E { 1 , 2 , . . . ,d } ,

c c 00

a+(i)cp :=

D,+(i)fn,

n=O 00

aO(i)cp:=

D:(i)fn,

(3)

n=O

and

c 60

a-(i)cp :=

D,(i)fn.

(4)

n=O

Equality (1)becomes now the following fundamental relation of this theory:

xi = U + ( i ) + U O ( i ) + a-(i),

(5)

for all i E {1,2,. . . ,d } . a+(i) is called a creation operator, uo(i) a neutral or presemation operator, and a -(i) an annihilation operator, for all i E {1,2,..., d } .

A probability measure p on Rd,having finite moments of any order, is called polynomially factorisable, if for any non-negative integers i l , i2, . . . , id, E[xi,'x> . . . x:] = E[xf']E[x?] . . .E[x:]. A probability measure p that is a product of d probability measures p1, p2, . . . , p d , on R, each of them having finite moments of any order, is clearly polynomially factorisable by Fubini's theorem, but the converse is not true and this was shown by a counterexample in [2]. If, for each i E { 1 , 2 , .. . , d } , we regard X i as the random variable that associates to each outcome x = (x1,22,. . . ,Zd), from our sample space, Rd,its i-th coordinate xi, then the notion of polynomially factorisability can be understood as a weak form of independence of the random variables XI, X 2 , . . . , xd. If one is dissatisfied with the fact that our sample space is the particular one Rd,and would prefer to have a general theory for an arbitrary sample space R and arbitrary random variables Y1,Y2,. . . , Yd defined on 0, having finite moments of any order, then he(she) can take p to be the joint probability distribution of these random variables and in this way the whole theory is moved on Rd.The random variables Y1,Y2,. . . , Yd defined on R, are thus replaced by the coordinate

4 random variables X I , X2, . . . , x d on Rd. Therefore, one can see that by working only on Rd we do not loose anything from the generality of this theory. The following theorem was proved in [ 2 ] : Theorem 1.1. A probability measure p o n Rd, hawing finite moments of any order, is polynomially factorisable, if and only if for any j , k E {1,2,. . . , d } , such that j # k, any operator f r o m the set {a-(j), ao(j), a+(j)} commutes with any operator from the set {a-(k), ao(k), a+(k)}. 2. A Combinatorial Identity

We will prove now an identity that will have remarkable connections with the standard Gaussian probability measure on R. Lemma 2.1. For all natural numbers n, we have:

1 Ij1< j z

C <...

+

( j l - l > + ( j z - 3)+ . . . ( j n - ( 2 n - 1)) = ( 2 n - I)!!,

<j, 52n

where a+ := max(a, 0 ) , for all real numbers a Proof. Let us consider 2 n people called: a1, a2, . . . , azn, such that any two of them are of different ages. We assume that a1 < a2 < ... < a2n, where ai < aj means that person ai is younger than person aj. We would like to split these 2 n people into n disjoint teams (couples) of two people each. Each couple will play a game of tennis with any other couple. We can count the number of such partitions in two different ways. The first way is to use the multinomial coefficient to count the number of ways in which we can form team 1, team 2 , . . . , team n. After fixing team 1, team 2 , . . . , team n, if for example we move the two people from team 1 into team 2, and the two people from team 2 into team 1, leaving all the other teams unchanged, then from the playing tennis point of view the teams are the same, they have only changed their names (team 2 is called now team 1, and team 1 is team 2, but the players within each team have remained the same). Thus by permuting the names (counting) of the teams we over-counted the number of partitions n! times. Hence, the total number of partitions is:

(2,;n,,2),

5

Therefore, we have obtained the number from the right-hand side of the formula from our lemma. The second way is to order first the two people within each team and then to order the teams. Since the team ( a i , a j ) is the same as the team (aj , a i ) , to avoid over-counting, we will always specify a team by listing first the younger member of the team and second the older member of the team. Thus for example ( a l ,a2) is a legitimate team, while (a2,a l ) is not. We also order the teams among themselves, by saying that team 1 is less than team 2 if the older person from team 1 is younger than the older person from team 2. To avoid over-counting we will select the teams in their increasing order (that means the older member of team 1 is younger than the older member of team 2 who in turn is younger than the older member of team 3 and so on). We count now the number of partitions, by selecting first the older member from each team in their increasing order. Let ajl be the older member from team 1, a j , the older member from team 2 , and so on. Since we have ordered the teams, we have 1 _< jl < j 2 < . . . < j , 5 2n. Let us fix for the moment j1, j 2 , . . . , j,. We need to select now the younger member from each team, whom we call a i l , ai, , . . . , aim. Since the younger person from team 1, a i l , is younger than his(her) team-mate ajl , we must have 1 5 i~ < j ~ Therefore, . there are jl - 1 choices for 21, namely: 1, 2 , . .. , jl - 1, if j 1 > 1, and no choice (room) if jl = 1. Thus, we can say that in both cases, the number of ways to select a younger partner for person ajl is (jl - 1)+. Let us assume now that the partner ail of ajl has been chosen and fixed and let us proceed to select the partner ai, of aj, from team 2. Since aiz is younger than aj,, we must have a2 < j 2 . Since there are j 2 - 1 numbers (positions) less than j 2 , namely: 1, 2 , . . . , j 2 - 1, out of which two are already occupied by a1 and j1, we can see that there are j 2 - 3 choices left for 22, if j 2 > 3, and no choices if j 2 5 3. Thus the number of ways in which 22 can be selected is ( j 2 - 3)+. Similarly we can see that after Zl and 22 have been selected and fixed, the number of ways in which we can select a younger partner for the player aj, is ( j 3 - 5)+, + and so on. Finally, there are (jn - ( 2 n - 1)) ways to select a younger partner for the player aj,. Using the generalized principle of counting, we conclude that if the older players from each team: aj,, a j , , . . . , aj, have been fixed and ordered as jl < j 2 < < j,, then the number of ways in which we can pair each of them with a younger partner is: (jl - 1)+(j2 - 3)' . . . ( j n - ( 2 n - 1))'. Thus the total number of partitions is: Cl<jl<jz<...<jn12n(ji - 1)+(j2 - 3)'. . . (jn - ( 2 n - 1))' and so, our identity is proved.

6

3. Standard Gaussian Probability Measure

In this section we characterize the standard Gaussian probability measure on Rd in terms of the neutral (preservation) and commutators between the creation and annihilation operators. Theorem 3.1. The standard Gaussian probability measure on Rd,i.e., the probability measure given by the density function

is the only probability measure, on Rd, having finite moments of all orders., such that f o r i, j., k E { 1 , 2 , . . .,d } , ao(i)= o

(6)

[ a - ( j ) , a + ( k ) ]= d j , k I ,

(7)

and

where [ u - ( j ) , u+(k)]:= a - ( j ) a + ( k ) - a + ( k ) a - ( j ) denotes the commutator of u - ( j ) and a+(k), and 6 j , k = 1, if j = k , and 6 j , k = 0, if j # k, is the Kronecker symbol. Here I denotes the identity operator of the space V of all polynomial functions of d real variables: 2 1 , 2 2 , . . ., X d . I n equahties (6) and (7), the domain of ao(i),a - ( j ) , and a+(k) is considered to be V . Proof. (e) If p is the standard Gaussian probability measure on Rd, then it is well-known that relations ( 6 ) and (7) hold. (+) Let us assume now that p is a probability measure on Itd,having finite moments of all orders, such that for i, j , k E {1,2,. . . ,d } , uo(i)= 0 and [ a - ( j ) , u + ( k ) ]= d j , k I . We will compute all the mixed moments of p. Since for all j # k , any operator from the set { a - ( j ) , u o ( j ) , u + ( j ) } commutes with any operator from the set { u - ( k ) , ao(k),u+(k)}, using Theorem 1.1, we conclude that p is polynomially factorisable. This means that, for we have all monomials z?x: . .

.zF,

E[zf'x$ . . .zy] = E[z2;]E[z$ ]*. E[x~], where E denotes the expectation with respect to p. Since the mixed moments of p are the product of their corresponding marginal moments, it is enough to focus on computing the following moments: E[xi],E[x:],E [ z ? ].,. . , where i E {1,2,. . . ,d} is an arbitrary fixed subscript. Let i E { 1 , 2 , . . . ,d } and m E M be fixed. Let 4 := 1, be the constant polynomial 1. We call 4 the vacuum vector. To obtain the

7

monomial xy,from the constant polynomial 1,we must apply the multiplication operator Xi, to the vacuum vector $, repeatedly m-times. That is, xy = X i X i . . . X i $ = X y $ . Sincexi =a-(i)+ao(i)+a+(i) andao(i) = 0 , we have Xi = a-(i) a+(i). Because the index i has been fixed, we will denote a - ( i ) and a + ( i ) shortly by a- and a+, respectively. We denote by (-, -), the inner product with respect to p. Then, we have:

+

where el E { -, +}, for all j E { 1 , 2 , . . . ,m}. Actually, it will be better to count the epsilons from right to left since otherwise ern is applied first to and so on. Therefore, we have: the vacuum vector 4, then

E [ X Y= ~

C...,

'l,',,

(aBm...aL2aB1$,~).

(8)

Em

0, such that Let j E {1,2,. . . ,m). If el = -, then there exists kJ : Gk, -+ Gk3-1, and for this reason we call a'3 a backward step. Here, if kl = 0, then Gk,-1 = G-1 := { 0 } is the null space. If eJ = then there exists kj 2 0, such that a'3 : Gk, + Gk3+1,and for this reason we call a'3 a forward step. If m = 2n 1 is odd, where n is some non-negative integer, then in every term (a'm ..-a'Za'l$,+), of the sum from formula (8), the number of forward steps is different from the number of backward steps. Since we start from Go (because $ E Go) and we do a number of forward steps and a different number of backward steps, it is clear that either aem. . . a'za'l $ = 0, if the number of backward steps is greater than the number of forward steps, or a'- . . .a'zacl 4 E Gk, for some k L 1 (k is actually odd, but this is not important for us), if the number of forward steps is greater than the number of backward steps. Because Gk i s orthogonal to Go, for all k 2 1, and 4 E Go (here we refer to the $ that appears after the comma in (a'- . . . a'2u'1$, +)), we conclude that in both cases (a'm . . . aazaa14,4) = 0, for all choices of € 1 , €2, . . . , cm. Thus the sum from formula (8), is a sum of zeros, and so E[xrn]= 0, for all odd positive integers m. If m = 2n is even, where n is a positive integer, then the only non-zero terms from the sum c , , , , , (...,CZn (aczn. .aaza'l $ , 4 ) ,are eventually those in

a']

+,

+

-

8

which the number of forward steps is equal to the number of backward steps. For these terms, since aEm . . a'2a'14 E Go, we can see that a E z n . . . aE2ael4 is a constant polynomial, therefore a number, and so (aQn . . . aazael474) = aczn..-aE2a"q5. Since exactly n of €1, €2, ..., EZ,, are equal to - (the other n being equal to +), let us call the positions of these minus-epsilons < j,. That means ~j = -, if by j i , j 2 , . . . , j,, where jl < j 2 < j E { j i , j 2 , . . . ,h}, and Ej = +, if j E { 1 , 2 , . . .,2n} \ { j l , j 2 , . . .,j,}. Formula (8) becomes now:

We observe first that if

then j 1 > 1, j 2 > 3, . . . , j , > 2n- 1. Indeed if there exists k E { 1 , 2 , . . . , n}, such that j k 5 2k - 1, then there are at most 2k - 2 epsilons before E j k = -, out of which k - 1 are negative, namely ejl, ej2, . . . , ejk-l, and therefore, there are at most k - 1 positive epsilons left before e j k . Thus a'ik . . . a E 2 a E= 1 40, since we start out from the vacuum space Go and do more steps backward (k)than we do forward (at most k - 1). This is the reason why the factors (jl - l)+,( j 2 - 3)+, . . . , ( j , - (272 - 1))' and not j l - 1, j z - 3, . . . , j , - (2n - 1) will appear later in this proof.

Claim 1: For all k E M, we have:

Indeed, this can be checked by induction on k. Since [a-, a+] = I , formula (10) holds for k = 1. Let us assume that it holds for k and prove that it

9

also holds for k

+ 1. This is true since:

a- ( a + ) k + l = ( a - ( c ~ + ) a+ ~)

+ [a-, (a+)k]) a+ = + k(a+)"') a+ = (a-a') + k(a+)k = (a+)k (a'a- + [.-,a']) + k(a+)k = ( a + ) k(a'a- + I) + k(a+)k = (a+)"+'a- + ( a + ) k+ k(a+)k = ( ~ + ) ~ + ' a+- (k + l)(a+)k. Thus we can see that a-(a+)"+' = ( u + ) ~ + ~ + u -(k + l ) ( ~ + and ) ~ sub= ((a+)%-

tracting (a+)"'afrom both sides of this equality we obtain that the commutator of a- and is [a-, = (k l ) ( ~ + ) Thus ~ . according to the principle of mathematical induction, formula (10) holds for all positive integers k .

+

10

We now interchange a- and ( u + ) ' ~ - ~if, j~ > 5, and so on until all the aoperators disappear. Each time we interchange a- with a power of a+, the power of a+ decreases by 1, as we can see from formula (10). Because the total power of a+ was initially n ( n forward steps), we can see that after n interchanges all the a+ operators disappear, too. Since 4 = 1, we obtain in the end:

if j i > 1, j 2 > 3, . . . , and jn > 2n - 1. If at least one jk is less than or equal to 2k - 1, for 1 5 k 5 n, then (~+)2n-~nu-(u+)~n-l-~*-lu-.

.. ~

u + ~ ~ z - ~ - j ~ u - ~ u= + ~0,j ~ - ~ ~

as was explained before. Thus we can say that, for all 1 _< j l 272, we have:

< j 2 < .. . <

jn 5

(a+)2~-jn

a-

a-

(u+)jn-l-jn-l

= (jl - l ) + ( j Z - 3)+ . . * ( j ,

.. .(u+)jz-l-jl

a-

- (2n - l))+.

(u+)jl

-14 (11)

Applying now formulas (9) and (11) we obtain

E [ x ~=~ ]

c

(jl

- l ) + ( j 2 - 3 ) + . .. ( j ,

- (2n - l))+.

1531<jz<...<3,52n

Using now Lemma 2.1, we conclude that

E [ P ]= (2n - l)!!, for all positive integers n. This formula holds also for n = 0, if we define (-l)!!:= 1. Therefore we can say that, for all non-negative integers il, i 2 , . . . , id, ~ [ x f ' x ?. . .x:] = (il - 1)!!(i2 - I)!!. (id - I)!!, if il, iz, . . . , id are all even numbers, and E[xFx? . .]z: = 0, otherwise. Thus p has the same mixed moments as the standard Gaussian probability measure on Rd. Since there is only one probability measure on Rd having the same moments as the standard Gaussian probability measure (namely the standard Gaussian measure itself), we conclude that p must be the standard Gaussian measure. 0

11 4. Final Comments

1) The above characterization of the standard Gaussian probability measure is just a particular case of a theorem that states that if p and Y are two probability measures on Rd, having finite moments of all orders, such that they have the same neutral (preservation) operators: aE(i) = aE(i), and the same commutators between the annihilation and creation operators: [ u ; ( j ) , a i ( k ) ]= [ u ; ( j ) , a $ ( k ) ] for , all i, j , k E { 1 , 2 , . . . , d } , then p and Y have the same moments of all orders: E p[xyxt . ..xy ] = E, [x? : x . . .xy 1, for all non-negative integers i l , i 2 , . . . , i d . We believe that this theorem is a fundamental theorem of the theory of Interacting Fock Spaces and we will publish it in a forthcoming paper. 2) We have proven Lemma 2.1 by counting, the number of partitions of 2n

different people into n teams, in two different ways. From the proof of Theorem 3.1, we can see that the second way of counting the teams is strongly connected with the fact that, for the standard Gaussian probability measure on R, u0 = 0 and [a-, u+] = I . How about the first way of counting the number of teams? The first way uses the multinomial coefficient This way is also strongly connected to the Gaussian random variables if one recalls the Wick formula for calculating the expectation of a product of the form JI . J 2 . . . where J1 J2, . . . , Jm is a finite sequence of jointly normally distributed random variables with mean zero. This expectation is computed in terms of the complete Feynman diagrams. Namely, a complete Feynman diagram of order m is a graph composed of m vertices and some edges connecting these vertices, such that each vertex is connected to exactly one other vertex (no vertex is connected to itself, no vertex is connected to two different vertices, and no vertex is left unconnected to some other vertex). We place J1 on one vertex, (2 on another vertex, and so on Jm on the last vertex. If we choose a complete Feynman diagram y, in which J1 is connected to J i l , J2 to &, , and so on ( n to ti,,, then we define:

(2,:r,,2).

cm,

~ ( 7:= ) E[Jlti11E[bEzzl* . . E[EnJz,],

(12)

where E denotes the expectation. If we denote by r(m) the set of all complete Feynman diagrams of order m that can be obtained using 6 , &, . . . , Jm as vertices, then the following formula (called Wick formula) holds:

..

~ [ t 1 ~ *<m] 2 =

C

v(Y).

-rEr(m)

(13)

12 See [6] for a proof. In particular, if = & = .. = & , = X , where X is a standard Gaussian random variable, then we can see t h a t there are no complete Feynman diagrams of order m, if m is odd, while if m = 2n is even, there are exactly ( z ,...,2 ) / n != (2n- l)!! complete Feynman diagrams y made from &, (2, . . . , Gn, and for each of them v(y) = 1. Therefore, if m is odd, E [ X m ]= 0, while if m is even, the Wick formula yields: +

2

c

(m-l)!!

E [ X y=

1 = (m-l)!!.

-(=l

In this way we can see the strong connection between the standard Gaussian probability measure and each of the sides of the combinatorial identity from Lemma 2.1. References 1. Accardi, L. and Boiejko, M.: Interracting Fock space and Gaussianization of probability measures, Infinite Dimensional Analysis, Quantum Probability

and Related Topics 1, (1998), 663-670. 2. Accardi, L., Kuo, H.-H., and Stan, A.: Characterization of probability measures through the cannonically associated Interacting Fock Spaces, Infinite Dimensional Analysis, Quantum Probability and Related Topics 7, (2004), 485-505. 3. Accardi, L., Lu, Y.G., and Volovich, I.: The QED Hilbert module and interacting Fock spaces; I I A S Reports No. 1997-008 (1997) International Institute for Advanced Studies, Kyoto. 4. Accardi, L. and Nahni, M.: Interacting Fock spaces and orthogonal polynomials in several variables; in “Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads”, N. Obata et al. (eds.), World Scientific (2002), 192-205. 5. Chihara, T.S.: A n Introduction to Orthogonal Polynomials. Gordon and Breach, 1978. 6. Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Math. 129, Cambridge University Press, 1997. 7. Kuo, H.-H.: White Noise Distribution Theory. CRC Press, Boca Raton, FL, 1996. 8. Obata, N.: White Noise Calculus and Fock Space. Lecture Notes in Mathematics 1571, Springer-Verlag, New York/Berlin, 1994. 9. Parthasarathy, K.R.: A n Introduction to Quantum Stochastic Calculus. Birkhauser, 1992. 10. Szego, M.: Orthogonal Polynomials, Coll. Publ. 23,Amer. Math. SOC.,1975. 11. Wiener, N.: The homogeneous chaos, American J. Math. 60 (1938), 897-936.

FEYNMAN FORMULAS FOR EVOLUTION EQUATIONS WITH LEVY LAPLACIANS ON MANIFOLDS

L. ACCARDI Centro Vzto Volterra Facolta’ d i Economia Universita’ da Roma ”TOTVergata” 00133 Roma, Italy E-mail: accardiQvolterra.mat.uniroma2.it

O.G. SMOLYANOV M. V. Lomonosov Moscow State University Faculty of Mechanics and Mathematics Moscow, 119992 , Russia E-mail: [email protected] A Feynman formula is a representation of a solution of Cauchy problem for an evolution partial differential (or pseudodifferential) equation by a limit of some Gaussian or complex Gaussian finite dimensional integrals when their multiplicity tends to infinity. Some Feynman formulas for solutions of heat and Schrodinger type equations with Levy Laplacians on the infinite dimensional manifold which is the set of mappings of a real segment into a Riemannian manifold are obtained.

1. Introduction

One calls a Feynman formula a representation of a solution of Cauchy problem for an evolution partial differential (or pseudodifferential) equation by a limit of finite dimensional integrals whose integrands contain some Gaussian, or complex Gaussian, exponents, when the multiplicity of integrals tends to infinitya[5]. In this note we obtain Feynman formulas for aFor the heat equation these multiple integrals coincide with some finite dimensional approximations of Wiener integrals; for the Schrodinger equations such multiple integrals coincide with integrals which are used in a (coming back to Feynman himself) definition of sequential Feynman path integral. Hence in the first case the limits of multiple integrals coincide with the Wiener integral and in the second case they coincide with Feynman path integrals; so in both cases the Feynman-Kac type formulas are corollaries t o the Feynman formulas.

13

14

solutions of heat and Schrodinger type equations with Levy Laplacians on the infinite dimensional manifold which is the set of mappings of a real segment into a Riemannian manifold. The Levy Laplacian on functions on such manifolds is defined using a combination of methods which were exploited in [3], to define the Levy Laplacian on functions on vector spaces, and in [7] to define the Volterra Laplacian on functions on the same infinite dimensional manifolds. The formulated definition of the Levy Laplacian on such manifolds is equivalent to the definition from [2] but the present definition is better adapted to the derivation of Feynman type formulas. The idea of the proofs of the main results is based on reducing the derivation of the Feynman formulas for solutions of equations on manifolds to the derivation of the similar formulas for solutions of analogous equations on vector spaces, and use the results from [3]. A similar idea was applied to equations with the Volterra Laplacian in [6] where only equations on finite dimensional manifolds were considered. Such approach is motivated by consideration of the situation when the Riemannian manifolds are assumed to be embedded into some Euclidian spaces (it is possible due to the famous Nash theorem) and a construction of the surface measure is used (see [4] and [7]). In particular we use a Chernoff theorem (formula), which is as related, to representations for solutions of Scrodinger type equations on manifolds and also to representations for solutions of similar equations on vector spaces by integrals over trajectories in the phase space 5 , as the famous Trotter formula is related to representation for solutions of some simple Schrodinger equations by integrals over trajectories in the configuration space. To indicate both difference and analogy of properties of Levy Laplacians and Volterra Laplacians we formulate also some results, essentially from [7], related to Volterra Laplacians. Due remark in the footnotel one can also consider the obtained results (for the case of the heat equation) as a construction of a Levy Brownian motion on the space of trajectories in a Riemannian manifold (see Remark 3 below and [9], [8] and [3]). Let us finally notice that the increasing interest to the Levy Laplacian is related to the fact that the Yang-Mills gauge fields are Harmonic functions for the Levy Laplacian (see [l]and references therein). The presentation in the paper has partly a formal character; in particular we omit some analytical assumptions about classes of functions for which the given formulas are valid.

15

2. Preliminaries and notations

For any metric space E and each a > 0 the symbol C([O,a ] ,E ) denotes the metric space, of all continuous maps of [0,a] into E , equipped with the E ) denotes the subuniform metric; if x E E then the symbol Cz([O,a], space of C([0,a ] ,E ) consisting of those functions that take value z at zero. If E is a (finite dimensional) Riemannian manifold and n E M then the symbols Cn([0,a],E ) and Cg([0,a ] ,E ) denote the infinite dimensional manifolds of n times continuously differentiable functions which are elements of C([O,a], E ) (respectively of Cz([O,a ] ,E ) ) ;we assume that Cn([O,a ] ,E ) and Cg([0,u ] ,E ) are equipped with the topology of uniform convergence of functions and of theirs derivatives up to order n (this topology can be defined by a metric). The symbol W,',,,([0,a ] ,E ) (x E E ) is defined as follows. Let E be isometrically embedded into Rk for the proper k, let W i ([0,a ] ,Rk)be the usual Hilbert-Sobolev space (of all absolutely continuous functions on [0,a] taking values in E , vanishing at 0 and having square integrable derivative, equipped with the Hilbert norm defined by (1g1I2= I1g'(t)I&dt) and let, for any z E E , gz E C([O,a],E) be defined by g z ( t ) = z for any t E [O,a];then W,',,([O,a],E) = ( W i ( [ O , a ] , R k+gz) ) n C([O,a],E); this definition does not depend on the embedding of E into Rn;one can also assume that W&.([O,a ] ,E ) (3Cg([O,a ] ,E ) if n 2 1) is equipped with the natural structure of an (infinite dimensional) Hilbert manifold generated by the Hilbert space structure of W;( [0,a ] ,Rk). Let E be a Banach space and H be its Hilbert subspace (this means that H is a vector subspace of E and that the canonical imbedding of H into E is continuous). If B is a continuous positive selfadjoint operator in H then the Gaussian cylindrical measure on E with the (H-)correlation operator B is the image, with respect to the canonical embedding of H into E , of the cylindrical Gaussian measure in H with the correlation operator B (of course such cylindrical measure on E may not be a-additive). Under the same assumptions the (cylindrical) Wiener process on [0, a] taking values in E , or (cylindrical) Brownian motion in E (defined on [0,a ] ) , with correlation operator B , starting at x E E , is the starting at x E E (cylindrical) Markov process on [O,a] taking values in E and defined by the transition probability P B ( t ,z, -) = P E ( t ,.) z where t > 0, x E E and PE(t.)is the Gaussian cylindrical measure on E with the H-correlation operator t B and the mean 0. The (cylindrical) probability on C([O,a ] ,E ) generated by this process is called the (cylindrical) Wiener measure with the (H-)correlation operator B ; it is concentrated (in a natural sense) on

Jt

+

16

Cz([0,a ] ,E ) . Then P(.,.,.) is the integral kernel of the resolving operator for the Cauchy problem for the heat equation in E with the Laplace operator A B defined by the identity (ABg)(z) = tr(Bg”(z)),where g”(z) is the second defivative, with respect to H , of a function g and the symbol tr means the trace of the corresponding operator in H ; the operator A B is called the (Volterra) Laplacian corresponding to the operator B (see the next section for some details). 3. Laplace operators

In this section we define Volterra Laplace and Levy Laplace operators acting on functions defined on some spaces of continuous mappings of [O,a] into a Riemannian manifold. Our main goal is to investigate the Levy Laplacians; but it is instructive to consider both the Levy Laplacian and more traditional Volterra Laplacian in a parallel way. Let G be a compact Riemannian manifold of dimension d, which is a submanifold of Iw”. The Volterra Laplacian and the Levy Laplacian acting on functions defined on Wi,,,([0,a ] ,G) (x E G) are defined as follows. Let F beafunctiononWi,,([O,a],G) a n d l e t + E C;([O,a],G). Foreacht E [O,a] let { z ; , z $ ,..., be an orthonormal basis in the tangent to G at +(t) space, which is obtained by the parallel transport, along +([O, t ] ) ,of a fixed (not depending on t) orthonormal basis in the tangent to G at x space. For each r = 1 , 2 , ...,d let the symbol cf denote the geodesic which passes through +(t) in the direction of the vector 2.: Let also {e}! be an orthonormal basisinWi([O,a],Iwl)andlet,foreachk= 1 , 2 , ..., d , p ~ W a n d f o r h a v i n g sufficiently small module real number a , the function E C i ([0,a ] ,G) be defined as follows: +k”(t) is an element of the geodesic c i , the distance of which to +(t) , along ci, is equal to Iae:(t)I and the direction of +(t) to that element coincides with the direction of zk, if a e r ( t ) > 0 and is opposite otherwise. For the same k , p let a real valued function Fk,p of a be defined by Fk,,(a) = F($Lyp).

zi}

+kJ’

Definition 3.1. If there exists a continuous function A v F : Wi,,([O,a ] ,G ) ---f Iw whose value, for any E C;([O,a ] ,E ) , is defined by A v F ( + ) = ~ k , p ( F k , p ) ” (then 0 ) we say that F belongs to the domain of the Volterra Laplacian A and call the function A v F the value, which that operator takes on F .

+

Definition 3.2. A Wiener process (Brownian motion) in C([O,a ] ,G ) starting at x E G is the starting at gz E C([O,a],G)homogeneous Markov

17 process in this space whose transition probability is the Green function of the heat equation with the Volterra Laplace operator A v from the preceding definition; the corresponding Wiener measure on C([0,a], C([0,a],G)) is denoted by W,.

Remark 3.1. One can show that the measure W , can be generated, as a surface measure, by the Wiener measure on C([0,a], C([0,a],an))whose W,l([O, a],R”)-correlation operator is the identity in W i([0,a],Rn). Below we identify C([O,a],C([O,a],G)) and C([O,a] x [O,a],G) and we denote the probability on C([O,a] x [O,a],G), which is the image of W,, with respect to the mapping of C([O,a],C([O,a],G)) on C([O,a] x [O,a],G) defined by this identification, by the same symbol W,. Let {e,”} beanorthonormalbasisinLz([O,a],R1), { e k } C C2([0,a],R1), cp E C:([O,a],G) and let the function ( P ~ PE C;([O,a],G) (where k = 1,2, ...,d , p E N and a is a real number having sufficiently small module) be defined as follows: c p ? P ( t ) is an element of the geodesic c i , the distance of which to p ( t ) , along c i , is equal to lae,”(t)I and the direction from cp(t) to that element coincides with the direction of z:, if ae,”(t) > 0 and is opposite otherwise.

Definition 3.3. If there exists a continuous function A L F : W,’,,( [0,a ] ,G) +. R whose value, for any cp E C:( [0, a ] ,G), is defined by 1 k=d,p=r a ~ F ( c p= ) limr+w F ~ k = l , p = l ( F k , p ) ” (then 0 ) we say that F belongs to the domain of the Levy Laplacian A L and call the function A L F the value, which that operator takes on F . Remark 3.2. When G = Rk then the definitions of the Levy Laplacian and of the Volterra Laplacian coincide with the corresponding classical definitions (see e.g. [3], [8], [9] and references therein). 4. Feynman formulas

In this section we give some formulas, which can be called Feynman formulas, for solutions of Cauchy problems both for Schrodinger and heat equations (with potentials), on spaces of continuous mappings of [0, a] into a Riemannian manifold, generated by the Volterra Laplacian and by the Levy Laplacian. Like in the preceding section our main goal is the Levy Laplacian but it seems again quite instructive to consider those two classes of equations (generated respectively by the Levy Laplacian and by the

18

Volterra Laplacian) together. For the case of Volterra Laplacian we use some results from and also some corollaries to those results. Let p be the metric in the Riemannian manifold G generated by the Riemannian structure and let scal(q) and m(q)be the scalar and the (vector valued) mean curvature of G at q E G; we assume that G is a Riemannian submanifold of an Euclidian space and that 11 . 11 is the norm in that space. If q j E G , j = 1 , 2 , 3 , 4 , then the number p(ql,qz;q3,q4) is defined as follows. For any z , z E G, the symbol y(z,z) denotes a shortest geodesic between x,z , and the symbol 70- - the geodesic which passes through q1 in the direction of the vector a’, which is obtained by the parallel transport, along the geodesic y(q3,41), of the vector a that is tangent at 43 to the geodesic y(q3, q4) and is directed from 43 to 44. Let qi E 70 be such that the distance of qi to q1, along the geodesic yo (in the direction of a’), is equal to the distance between 43 and 44 along y(q3,44). Then p l ( q l , q 2 ; 43, q4) is the length of y(q2, q i ) and pz(q1, q2; q3,q4) = 1142 - qi11; if the distances between q j are sufficiently small then pj (q1, q2; 43, q4) depend on qj continuously ( j = 1,2). on [0, m) x G x G x G x G The nonnegative functions pg, Qg,pg, are defined by: .7 P g ( t , 41,427 q37 44)) e- (Pl(Sl~Sz;S3~¶4))2 2t

QL

Qg(t’ql’ q27q3’q4) =

.

PV(t,ql,4zr43,44) SGp~(t,Q1,4ZrQ3,44)dQ4’

.

P 3 , Q1, Q2,43,44)) = e-

(P2(¶1+7z;S3,¶4))2

9%

PV(t,41,Q2r43~44)

2t

9

Q1, q2743, q4) = ~ G P ~ ( t , Q 1 , * 2 , q 3 , 4 4 ) d Q 4 ~ The nonnegative functions p i , Q f i , p g , Qg on [0,m) x G x G x G x G are defined by: (P(91,92)2+(P(93,94))2 .7 P i ( t , Q l , q z r q 3 , 4 4 ) ) = e2t PL (t,4l ,QZ

943 $44)

Qi(t7q1’q27 q 3 7 q 4 ) = SG ~ ~ P ~ ( t ~ 4 1 , 4 2 ~ 4 3 r Q 4 ) d P Z d 4 4 ~ 1142-¶1112+1194-93112 . P k ( t , 41,427 43744)) = e2t l PL (t,ql&’Zt43~44)

Qg(t’ “’ q27q3’ q4) = S, SG P~(t,41,42,43144)dQZdQ4’ A function X€J on a subspace of C([0,a ] ,G) is called cylindrical if there exist such different t l , t2, ..., t k E [0,a] and such a function f on the product of k copies of G that Q(g) = f ( g ( t l ) , g ( t z ) ,...,g( tn)) for allg E C([O,a],G). In this case we say that the function \k is defined by the set { t j } and by the function f . A cylindrical function on a subspace of C([0,a] x [0, a],G) is defined analogously. A partition of the square [0,a] x [0, u] is a family T = { T ( i ): i = 1,2}

19

where T ( i )= { t i j E [O,u] : 0 = t i 0 < t i 1 < t i 2 < ... < tiqi) = a } ; then the sets T ( i )are called partitions of [0,a ] , the numbers d i u m T ( i ) = muz{ltij t + l I , j = 1,2, ...,k ( i ) } are called the diameters of those partitions and the number d i u m T = m a z { d i u m T ( i ) , i = 1,2} is called the diameter of T. The symbol S ( T ) is defined by S ( T ) = {(tl,., t 2 j ) E [0,u] x [0,u] : r = 1 , 2 , ..., lc(l),j= 1,2, ..., lc(2)). Let F be a cylindrical function on C([O,u] x [0,a] ,G) defined by a set { s j : j = 1,2, ..., r } and by a function f and let { T k } be a sequence of partitions of the square [O,u] x [O,a] whose diameters tend to zero and for any k , S ( T k ) 3 { s j } and curd(Tk(i)) = r(lc(i)). Below z E G and qn,o = q0,+ = 2 for all n. The following three propositions were established in 7. Proposition 4.1. Iff is continuous and bounded then

x

QK((t2,n

- t 2 , n - i ) ( t i , k - t ~ , k - l ) ,q n - i , k - i , q n , k - i , q n - l , k ,

qn,k)dqn,k,

The statement will be still valid if QR is substituted by QE. Proposition 4.2. There exists such a probability Si on C([0,u] x [0,a ] ,G ) that for any bounded continuous function f the following identity holds

20

The measure S: is equivalent to the Wiener measure W x and the corresponding Radon-Nykodim density is defined by

Proposition 4.3. There exists such a probability S: o n C([O,u] x [0,a ],G ) that f o r any bounded continuous function f the following identity holds

where

The measure S: is equivalent to the Wiener measure W , and the corresponding Radon-Nykodim density is defined by

&

dS: dWX

-(q)

=

c2e-: J," s," scal(~(tl,tz))dtldtz+B J," J," I l m ( q ( t 1 , t z ) ) l l Z d t 1 d t 2

where c;' =

Proofs can be obtained using a theorem for measures, on a semiring of rectangles of the Euclidian plane, taking values in a commutative group of operators, which is similar to the famous Chernoff theorem lo (see also and estimates from 4). Some similar results could be formulated using, instead of functions p:, Q L , p g , QE, the functions p k , Q k , p k , Qk ; but in this case one need to use a cylindrical "Levy-Wiener measure". In particular Proposition 1 can

25

Remark 5.1. The latter proposition can also be viewed as a definition of a Levy Wiener measure on spaces of mappings of [0, u] into W i ([0, u ] ,G). References 1. L. Accardi, P. Gibilisco, I.V. Volovich. Yang-Mills gauge fields as Harmonic functions for the Levy Laplacian. Russian J. Math.Physics, 2, 235-250, 1994. 2. L. Leandre and I.V. Volovich. The stochastic Levy Laplacian and Yang-Mills equation on Manifolds. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4, 2:151-172, 2002. 3. L. Accardi, O.G. Smolyanov. Representations of Levy Laplacians and Related Semigroups and Harmonic Functions. Doklady Mathematics, 65, 3: 356-362, 2002. 4. O.G. Smolyanov, H.v. Weizsiicker, 0. Wittich. Brownian motion on a manifold as a limit of stepwise conditioned flat Brownian motions Proc. of the Conference dedicated to 60-th birthday of S.Albeverio, Canadian Mathematical Society, Conference proceedings, v. 29, 589-602, 2000. 5. O.G. Smolyanov, A.G. Tokarev, A. Truman. Hamiltonian Feynman path integrals via the Chernoff formula. J . of Math. Phys., 43, 10, 5161-5171, 2002. 6. O.G. Smolyanov, A. Truman. Feynman integrals over trajectories in Riemannian manifolds. Doklady Mathematics, 68, 2, 194-198, 2003. 7. O.G. Smolyanov, H.von Weizsiicker, 0.Wittich. Constructing diffusions on the set of mappings of a segment into a compact Riemannian manifold. Dolclady Mathematics, 71, 3, 391-395, 2005. 8. L. Accardi, O.G. Smolyanov. Extensions of spaces with cylindrical measures and supports of measures generated by the Levy Laplacian. Mathematical Notes, 64, 4, 483-492, 1998. 9. L. Accardi, O.G. Smolyanov. Brownian motion generated by the Levy Laplacian. Mathematical Notes, 54, 5, 1174-1177, 1993. 10. R.P. Chernoff.// Mem.American Math.SocJ974, v.140.

ON THE FOCK REPRESENTATION OF THE RENORMALIZED POWERS OF QUANTUM WHITE NOISE

LUIGI ACCARDI Centro Vito Volten-a, Universith d i Roma Tor Vergata via Columbia, 2- 00133 Roma, Italy E-mail: [email protected]. it ANDREAS BOUKAS Department of Mathematics and Natural Sciences, American College of Greece Aghia Paraskevi, Athens 15342, Greece E-mail: [email protected] We describe the "no-go" theorems recently obtained by Accardi-Boukas-Franz in [l]for the Boson case, and by Accardi-Boukas in [2] for the q-deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN).

1. Introduction

Classical (i.e It6 [5]) and quantum (i.e Hudson-Parthasarathy [6, 31) stochastic calculi were unified by Accardi, Lu, and Volovich in [4]in the framework of Hida's white noise theory by expressing the fundamental noise processes in terms of the Hida white noise functionals at and a: defined as follows. Let L&(Rn) denote the space of square integrable functions on R" symmetric under permutation of their arguments, and let F := @ ~ = o L ~ y , ( R where ") if $ := {$(")}p=o E F , then $(O) E C, $("I E L&,(R") and

{$(n)}F=o

The subspace of vectors $ = E F with @("I = 0 for almost all n will be denoted by DO.Denote by S c L2(R")the Schwartz space of smooth functions decreasing at infinity faster than any polynomial and let 26

27

D := {$ E Fl$(,) E S, C,"==, nl$(,)I2 linear operator at : D --+ F by (Ut$)(,)(Sl,.

. . ,s,)

< m}. For each t

:= & T i $ ( , + l ) ( t , s 1 , .

E R define the

. . , s,)

and the operator valued distribution (cf. [4]for details) a t by

where denotes omission of the corresponding variable. The Hida white noise functionals satisfy the Boson commutation relations

[at,ad] = S ( t - s) [ a i ,ad] = [at,a,] = 0. In order to consider higher powers of the Hida white noise functionals we will use the renormalization

S ( t ) l = cl-l q t ) , c > 0, 1 = 2,3, ....

A complete analysis of the choice of such a renormalization, as well as a discussion of other possible renormalizations can be found in [4]. 2. The Boson-Fock Case

In the Boson case the basic commutation relations, the properties of the Fock vacuum vector @, and the duality relations are

[at,.!]

= S ( t - s)

[at,a s ] a t @= 0

[ a [ ,a!] =

=0

(a,)* = a! (@,@) = 1. Let 7f be a test function space and for f E 'Ft and n, k E {0,1,2, ...} define the sesquilinear form on DO

28

with involution

(B,"(f))*= B m . More precisely, for

4, .JI in DOand k,m 2 0,

@(gf)

= JRd

g(t) f(t) dt =< 97 f >

*

In the following we will use the notation

B," := B,"(X[O,t]). It was proved in [l]that for all t , s E R+ and n, k,N , K 2 0

Multiplying both sides of (2.1) by test functions f ( t ) g ( s )and formally integrating the resulting identity (i.e. taking . . . dsdt), we obtain the commutation relations for the Renormalized Powers of White Noise (RPWN)

ss

where n , k , N , K E {0,1,2, ...},

where bnn,kis Kronecker's delta and

29

where the factorial powers &)

are defined by

d y ) := z(z - 1)

* *

f

(z - y

+ 1)

with do)= 1. In what follows we will use the notation

BZ := B;(xI) where I C R with p ( I ) < $00 is fixed. Moreover, to simplify the notations, we will use the same symbol for the generators of the RPWN Lie algebra and for their images in a given representation. Theorem 2.1. (No-Go Theorem for Boson RPWN). Let C be a Lie sub-algebra of the RPWN Lie algebra with the following properties: (a)

*-

C contains Bg, and B P where the noise operators are defined on

the same interval I and B ; ( X I )= p ( I ) . (ii) the BE satisfy the commutation relations (2.2) . Then C does not have a Fock representation i f the interval I is such that

Proof. If a common Fock representation of the BE existed, one should be able to define inner products of the form

< ( ~ B ? ( x I+) b XI))^)@, ( ~ B ? ( x I4 ) XI XI))^)@ > where a , b E R, the noise operators are defined on the same interval I and @(I) = p ( I ) . Using the notation < 2 >=< @,a:@ > this amounts to the positive semi-definiteness of the quadratic form

A=

[ << B;,(xI)Bi"(XI) >> << %,(XI)

(B,n(XI))2

( q ( X I ) ) 2

( p ( x I ) ) 2 (pa(xI))2

1

> . >

30

Using the commutation relations (2.2) we find that

A=

[

( 2 n )!c2n-zp ( I )

(2n)!c2"-'p(I)

( 2 n ) ! c 2 n - 2 p ( ~ 2(n!)2C2n-2p(1)2 )

I.

+ ((q! - 2(n!)2)C2n-3p(1)

A is a symmetric matrix, so it is positive semi-definite if and only if its minors are non-negative. The minor determinants of A are

and

1 d2 = 2~~("-~)p(1>~(n!)~(2n)!(cp(1) - 1) 2 0 ($ p ( I ) 2 .; Thus the interval I cannot be arbitrarily small.

0

3. The q-Deformed Fock Case

In the q-deformed case, where q E (-1, l ) , q # 0, we start with the q-white noise commutation relations at at - q af;at = q t

- s)

and letting, as in the Boson case,

B z ( f ) := JRd f ( t ) a L n @ d t we obtain the q-RPWN commutation relations

where

31

=

{

;(n-A)(k-A)

[k[klq! A],!

(&,A

+ (1- 6 n , A ) (y),)

if X 5 n and X 5 k if X > n or X > k

and for k = 0 and/or K = 0 the corresponding sums on the right hand side of (3.1) are interpreted as zero. The following theorem was proved in [2].

Theorem 3.1. (No-Go Theorem for q-RPWN). Let q E (-1, 1),q # 0 and c W and n, k 2 0 let B," := B,"(xI)with BOO = p ( I ) .1, the measure of I . Let also the "vacuum vector'' @ be such that B,"@ = 0 whenever k # 0 and let ( x ) := ( @ , x @denote ) the "vacuum expectation" of an operator x. W e assume that (a,@) = 1. Define

for a f i e d interval I

< Bgn B:" > < Bgn (B;)2 > A ( n , q ;I ) :=

< Bin (B;)2 > < (BE)2(B;)2 > For any choice of n and q the matrix A(n,q; I ) cannot be positive semidefinite for all I c W. Proof. Using commutation relations (3.1) we find

32

A ( n , q ; I )is a symmetric matrix, so it is positive semi-definite if and only if its minors are non-negative. T h e minor determinants of A(n,q; I ) are

dl = p ( I ) c

~ [2&! ~ -

~

which is non-negative for all I and

d2 = ~ ( 1c ) ~~[2nIq! ~ .-

~

which, as in the Boson case, is bigger or equal to zero if and only if

which cannot be true for arbitrarily small I.

0

References 1. Accardi L., Boukas A., Fkanz U. Renormaliaed powers of quantum white noise, to appear in Infinite Dimensional Analysis, Quantum Probability, and Related Topics (2005). , Higher Powers of q-deformed White Noise , to appear in Methods of 2. Functional and Topology (2005). 3. L. Accardi, A mathematical theory of quantum noise, Proceedings of the first world congress of the Bernoulli Society, Ed. Prohorov and Sazonov, vol.1 (1987) 4. L. Accardi, Y . G . Lu, I.V. Volovich, White noise approach t o classical and quantum stochastic calculi, Lecture Notes of the Volterra International School of the same title, Trento, Italy, 1999, Volterra Center preprint 375. 5. Ito K., O n stochastic differential equations, Memoirs Amer. Math. SOC.4 (1951). 6. K . R. Parthasaxathy, A n introduction to quantum stochastic calculus, Birkhauser Boston Inc., 1992. ~

POWERS OF THE DELTA FUNCTION

LUIGI ACCARDI Centro Vito Voltem, Universitci di Roma Tor Vergata via Columbia, 2- 00133 Roma, Italy E-mail: accardiQ Volterra.mat.uniroma2.it ANDREAS BOUKAS Department of Mathematics and Natural Sciences, American College of Greece Aghia Paraskevi, Athens 15342, Greece E-mail: [email protected] Our attempts t o establish a Fock representation for the renormalized higher powers of white noise, involve the assignement of a meaning to the powers of the Dirac delta Ck dk) where function. In this paper we give meaning to the expression 6" = n 2 2, dk)is the k-th derivative of the Dirac delta function and co, ...,cn-l E C are arbitrary.

c;:,'

1. Introduction: The Square of the Delta Function

An ill-defined object such as the square of the Dirac delta function was given a meaning by L. Accardi, I. Volovich and Y. G. Lu in [3], motivated by the study of the square of white noise, as follows: Let S = S(R) be the Schwartz space on the real line, let

S o = { ( b E S : (b(0)=0)={x7+b(x) : $ E S }

(1.1)

and, for n E {1,2, ...} define

Notice that, for each n, fn(x) is a discontinuous function and so { f n ) ~ ~ ~ is O not o a sequence of "very good functions" in the usual sense of the theory of generalized functions or distributions (cf. [4]). For all 4 E S 33

34

where we have used the substitution x =

y, and so

lim fn(x) = 6(x) n++m in the sense of generalized functions. To give a meaning to a2(x) we notice that for E SO

+

=o thus, as a distribution on SO,

We remark at this point that if we try to extend this construction to higher powers of the delta function by working in SOthen the above limit becomes infinity. As we will see in the subsequent sections, a space of smoother functions is needed and that will cause the derivatives of the delta function to appear. Returning to (1.6), let F be the extension of limn++m ft(x) to all of S. For any $ E S we have

+(x) = d.1

- 4(0)

+ 4(0) N z )

(1.7)

where $J E S is arbitrary with $(O) = 1. Since +(z) - 4(0) $(x) E So and F is zero as a distribution on SO,applying F to both sides of (1.7) we obtain

35

F(4J)= +(O) F(dJ) and so, by the arbitrariness of $, we may define

F = c ~ i.e

b2 = c b

(1.10)

where c E C is arbitrary. This particular renormalization of the square of the delta function turned out to be very fruitful in relation to the study of the squares of the Hida white noise functionals (cf. [l]and the references within). The obvious generalization bn=c,6

(1.11)

where n 2 2 and c, E C is arbitrary, has not been very easy to use in order to prove the existence of a Fock space representation for the Lie algebra associated with the higher powers of the white noise functionals (cf. [2] ). 2. The Square of the Delta Function Revisited In this section we describe a method for defining the square of the Dirac delta function that allows for a generalization to higher powers. Definition 2.1. For k E {0,1, ...} we define

b ( k ) ( 4 J )=

(-1)Q‘”‘O)

Theorem 2.1. On S,

b2 = c16

+ c2 6‘

36

where c1, c2 E C are arbitrary. Proof. As in Section 1, for n E {1,2, ...} define

Then, for

4 E SI

=o Thus, the generalized function F defined by { f ~ } ~ ' ~(the " natural candidate for h2) is equal to zero on Si.Now let 4 E S. Then x ~ ( z E) So and

x

= a1 ( X I

+ $642 + ( X I

(2.6)

where I) E S is arbitrary with + ( O ) = 1 and a l ( x ) is defined by

it follows that a1 (0) = a: (0) = 0 and so a1 ( x ) E S1. Applying F to both sides of (2.6) we find

where c E C is arbitrary, and so

F = cb’ on SO.To extend the definition to S let = QO(Z)

(2.10)

4 E S and write

+ 4 ( 0 ) $‘(z)

(2.11)

where $’ E S is arbitrary with +(O) = 1 and a!o(x)E SOis defined by QO(2)

= 4 k ) - 4 ( 0 ) Nz)

(2.12)

Applying F to both sides of (2.11) we find

F = CI 6 + ~2 6‘

(2.15)

J2 = c16 + c2 6’

(2.16)

i.e.

on S, where c1, c2 E C are arbitrary. We remark that such an expression for d2 was also obtained in [3] by using a different regularizing sequence. 0

38

3. The Cube of the Delta Function

Theorem 3.1. On S,

Is3

where c1

c2 c3

E

= c16

+ c2 6' + c3 6l'

C are arbitrary.

Proof. As in Section 2, for n E {1,2, ...} define

f n ( z )=

{

n

if -1

< 2 5 -1

-

n

0 otherwise

Then, for q5 E SZ

nlym

f:(x) +(x) dx

x 3 $(x) dx

= lim = lim n-+m

-L J' 8n

(3.3)

y3$(;y)dy 1

-l

=o Thus, the generalized function F defined by {f~}~I~O0 (the natural candidate for S 3 ) is equal to zero on S2. Now let q5 E S. Then x2 $(x) E & and

x2 4(x) = a2(.)

+ 4(0) x2 $(x)

where $ E S is arbitrary with $ ( O ) = 1 and az(x)is defined by

(3.4)

39

where c E C is arbitrary, and so

F = cb“

(3.9)

on Sl. To extend the definition to SO,let x 4 E SOand write

2

4(x) = a1>.(

+ 4(0) 2 $(.)

(3.10)

where $ E S is arbitrary with $ ( O ) = 1 and a1(x)E S1 is defined by

Ql(Z)

=x

($(.I

- 4(0)

Applying F to both sides of (3.10) we find

where c1, ...,c5 E C are arbitrary, and so

(3.11)

40

F = ~ 1 6+ ’ ~2 6”

(3.14)

on SO,where c1,c2 E C are arbitrary. To extend the definition to S let d E S and write

4(.)

=

+ 4(0) +(.I

(3.15)

where $J E S is arbitrary with + ( O ) = 1 and ao(z)E SO is defined by

sob> = 4(.) - 4(0) +(.I

(3.16)

Applying F to both sides of (3.15) we find

where cl, ..., cs E

C are arbitrary, and so (3.19)

s3 = c16 + c2 6’ + c3 6/’ on S, where c1, c2, c3 E C are arbitrary.

4. The General Case

Theorem 4.1. If k 2 2 then, on S,

where c ~..., , Ck-1 E c are arbitrary.

(3.20) 0

41

Proof. As in Sections 2 and 3, for n E {1,2, ...} define

lilim

x k $(x) dx

f k ( x ) $ ( x ) d x = lim l 2kn

/

1

1 yk$(;y)dy

= lim

-

= lim

1 1 ~ $ ( 0 ) ykdy 2 n -1

n++m

n++m

=o

-l

/

Thus, the generalized function F defined by { f ~ } ~ 5 (the ~ "natural candidate for h k ) is equal to zero on Sk-1. Now let 4 E s. Then xk-' 4(x) E S k - 2 and Zk-'

4(Z) = ak-l(x)

+ 4(0) x k - l $(z)

where $ E S is arbitrary with $(O) = 1 and

where

(Yk-l(x)

is defined by

(4.4)

42

+

F(&' 4) = F ( a k - 1 ) $(O) F(z"' = 0 4(0) c1

+

$)

(4.8)

C1

(k - l)! -- c1 (k - l)! (-1) - ,(p-q2k--1

x=o

(k-l)(&l+) 8

k-1

4)

where c E C is arbitrary, and so

F =c on

sk-2.

p - l )

(4-9)

To extend the definition to s k - 3 let x k - 2 zk-2 4(5)= a k - 2 ( 2 )

(+(.I

sk-3

+ +(O) x k - 2 $(z)

where $ E S is arbitrary with $(O) = 1 and

Qk-2(2) = xk-2

+E

ak-2(z)

- +(O)

$(.))

E

sk-2

and write

(4.10) is defined by

(4.11)

Applying F to both sides of (4.10) we find

F = c1 6 ( k - 2 ) + c2 J(k-1)

(4.14)

on s k - 3 , where c1,c2 E C are arbitrary. Continuing in this way we find that

43 k- 1

F =

C cm drn)

(4.15)

m=l

on 81,where c1, ..,Ck-1 E C are arbitrary. To extend the definition to S let E S and write

4 b ) = sob) + 4(0) N z )

(4.16)

where $ E S is arbitrary with $ ( O ) = 1 and oo(z) E SOis defined by

Applying F t o both sides of (4.16) we find (4.18)

1=1

k- 1

k- 1

=

C

el

(-1)l S(l)(f$)

1 =o k-1

=

c

21 S("(q5)

1=0

where

6, cl,...,&-I

E

C are arbitrary, and so k-1

F

=

C m=O

i.e.

~ , 6 ( ~ )

(4.19)

44 k-1

(4.20) m=O

on S, where cl, ...,ck-1 E C are arbitrary.

0

References 1. L. Accardi, A. Boukas, The unitarity conditions for the square of white noise, Infinite Dimensional Anal. Quantum Probab. Related Topics , Vol. 6, No. 2 (2003) 1-26. 2. L. Accardi, A. Boukas, U. F’ranz, Renormalized powers of quantum white noise, to appear in Infinite Dimensional Anal. Quantum Probab. Related Topics (2005) . 3. L. Accardi, Y.G Lu, 1.V Volovich, White noise approach to classical and quantum stochastic calculi, Lecture Notes of the Volterra International School of the same title, Trento, Italy, 1999, Volterra Center preprint 375. 4. M. J. Lighthill, An introduction to Fourier analysis and generalised functions, Cambridge University Press (1958).

DISPERSION RELATIONS IN THE STOCHASTIC LIMIT OF QUANTUM THEORY *

L. ACCARDI Centro Vito V o l t e m . Universitci degli Studi d i Roma “Tor Vergata”. 00133, Rome, Italy. e-mail: [email protected] .uniroma2. it

F.G. CUBILLO Departamento de Ancilisis Matemcitico. Universidad de Valladolid. Valladolid, Spain. e-mail: fgcubil [email protected]. es.

4 7005,

We apply new techniques based on the distributional theory of Fourier transforms t o study, in the stochastic limit of quantum theory, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schrodinger equation obtained in such limit. This approach permits us to dispense with the “analytical condition” and other restrictions usually considered and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field.

1. Introduction

The stochastic golden rules [l, 21, which arise in the stochastic limit of quantum theory as natural generalizations of the Fermi golden rule, provide a natural tool to associate a stochastic flow, driven by a white noise equation, to any discrete system interacting with a quantum field. The stochastic limit captures the dominating contributions to the dynamics arising from the cumulative effects, on a large time scale, of small interactions; the physical idea is that, looked from the slow time scale of the system, the field looks like a very chaotic object: a quantum white noise, i.e. a &correlated

* 26th Conference on Quantum Probability and Infinite Dimensional Analysis. Levico Terme, 20-26 February, 2005. 45

46

(in time) quantum field also called master field. The new evolution is an approximation of the original one which preserves much nontrivial information on the original complex system related to its decay and shift properties. In this work we study, from an analytical point of view, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schrodinger equation obtained in such limit, which contains the quantum mechanical fluctuation-dissipation relations. This approach permits us to dispense with the analytical condition and other restrictions usually considered - see Section 2 - and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field. To be precise, we shall see that, for the region rl where the dispersion function is regular and not constant, every Bohr frequency of the system in its range gives rise to an independent master field, which is a quantum white noise concentrated over the corresponding resonant surface, whereas both the rest of Bohr frequencies and the open regions r a j where , the dispersion function is constant, give rise to zero master fields, except for the resonant case, see Theorem 4.1. In a similar way we will show that the regions rajdo not contribute to the drift term whenever the resonant case is not present, whereas for the region rl we obtain the usual expression, see Theorem 5.1. The contribution of the singular regions of dispersion has not been completely determined yet. 2. Preliminaries

In what follows we shall consider quantum systems describing the interaction of a discrete spectrum system S with free Hamiltonian r

and Bohr frequencies w = E~ - E r t , ( E ~E, ~ EJ Spec H s ) , and a bosonic quantum field as reservoir R with free Hamiltonian (on Fock space)

HR :=

J

dkw(k)~+(k)~(k),

where w ( k ) is the dispersion function, a k ( k ) are the creation and annihilation densities, and the reference vector is mean zero Gaussian and gauge invariant, with covariance of the form

47

We will assume that the total Hamiltonian has the form

H(’) := Ho

+ XHI = H s + H R + XHI,

where 1 is a real coupling parameter and the interaction Hamiltonian H I is of dipole type, i.e.a

HI =

C (058 A ( g j ) + Dj 8 A * ( g j ) ) , j

where Dj are system operators and

A * ( g j ) := / d k g j ( k ) a + ( k ) , A ( g j ) := / d k g * ( k ) a ( k ) , being the functions g j the cutoff or form factors. Often we will simplify the notations by omitting the symbol 8. In the stochastic limit approach we consider the time rescaling t t t / X 2 in the solution U,’” = eitHoe-itH‘x’ of the Schrodinger equation in interaction picture:

a

-U(’) at

= -iXHI(t)

U,“),

H I ( t )= eitHoHIe-itHo 7 and study the limits, in a topology to be specified, of the rescaled interaction Hamiltonian and of the rescaled propagator: 1 lim - H I

A-0

x

lim

A-0

(+)

=: ht,

~ $ =:1 ut. ~

In canonical form this reduces to find the limit of the rescaled creation and annihilation densities 1 eTi+ ( 4 k I - w ) af,w(t, k ) := a (k), (2)

x

*

obtaining the white noise Schrodinger equation &Ut = -ihtUt, whose normally ordered form is the quantum stochastic differential equation

dUt = (-idH(t) - Gdt)Ut,

(3)

aThe asterisk * denotes the Hermitian conjugate for operators and the complex conjugate for scalars. For distributional densities we use the symbol + instead * .

48

where

is called the martingale term and 1 Gdt := lim A-0

A2

1 l1 (!$) t+dt

dtl

dt2 (HI

HI

(s))

(4)

is known as the drift term. Among the usual assumptions to achieve this program we have the following: 0 0

0

the cut-off functions g j are Schwartz functions; the dispersion function w ( k ) and the cut-off functions g j are related by the following analytical condition:

the (d-1)-dimensional Lebesgue measure of the surface {k : w ( k ) = 0 ) is equal to zero (this implies, in particular 6 ( w ( k ) )= 0).

In this work we apply new techniques, based on the distributional theory of Fourier transforms [ 3 , 4 , 5 , 6 ] ,which permit us to dispense with the above conditions and to establish the dependence of the stochastic golden rules on certain properties of the dispersion function w ( k ) . 3. The Dispersion Function

In what follows we shall assume that the dispersion function Rd 3 k w ( k ) E R is such that w ( k ) 2 0 for all k E Rd and we can write =

H

rl u r2u r3,

where: (i)

rl is an open set of Rdin which w ( k ) is a C'-function and V w ( k ) # 0 for every k E rl. We shall denote by I'i the range of the restiction of w ( k ) to rl,i.e. I?:

:= Rang(wlr,),

and assume that the boundary zero.

8ri of I'i

has Lebesgue measure

49

(ii)

raj

r2 = UFaj, being an open subset of Rd where the dispersion function w ( k ) is constant and equal to a j , i.e.

w ( k ) = aj, (iii)

Q k E raj.

R d \ ( r l U rz),that is I’3 contains the boundaries of rl and I’2 and other possible regions of singular points of the dispersion function w ( k ) . r3 =

4. Convergence of the Rescaled Densities Let us study the convergence, in the sense of correlators, of the rescaled creation and annihilation densities given in Eq.(2). To simplify the notation we restrict our attention to the vacuum reference vector, so that N ( k ) = 0 (see Eq.(l)). The extension of the results to the general case is immediate. Moreover, because the mean zero Gaussianity, we have only to prove the convergence, in the sense of Schwartz distributions [5], of the covariance

i.e. we must calculate, for any Schwartz test functions lim

A-0

/

4, cp, f

dtdt’dkdk’$(t)cp(t’) f (k)g(k’)(ax,,(t, k)u:,,,(t’,

and g,

k’)).

The following theorem shows that, on rl, every Bohr frequency w in the open range of the dispersion function gives rise to an independent master field, which is a quantum white noise concentrated over the resonant surface w ( k )-w = 0, and the rest of Bohr frequencies’giverise to zero master fields, while, on the open regions raj where the dispersion function is constant, the limit does not exist in the resonant case aj = w = w’ and again gives rise to zero master fields otherwise.

Theorem 4.1. Under the conditions for w ( k ) given above, in the sense of Schwartz distributions, i.e. in S’(R2d+2): ( a ) Over rl, i f w doesn’t belong to the boundary d A-0 lim (ax,&

k)a:,,t (t’,k’))

= bu,,,2~b(t

-

t’)b(k

(b) Over each raj,

-

jrl

k’)b(w(k) - w)Xr:(w).

of I’i,

50

The proof of this result cast some light on the resonant case aj = w = w‘ of item (b): Over each rajthe final expression in our calculations isb 2T A-0 lim x2

4’

(y ) v A(%$)

= lim A-0

x

lQj

dk f ( k ) g ( k )

4‘(0) ~ “ ( 0 ) d k f ( k ) g ( k ) , raj

which is equal to zero when q5”(0) = 0 or ~ “ ( 0=) 0, or f m otherwise. Thus the limit also exists in this case, and is equal to zero, if we restrict our attention to test functions with zero mean in time. What happens over functions of the form

I’3

or when w E

Xi? For example, for dispersion

w ( k ) = klP7 CL

> 0,

we have rl = Wd\{O},r2 = 8, r3= { 0 } , r: = ( 0 , ~and ) ar: = {0}, that the frequency of interest is w = 0. We obtain in this case

-

if d

0,

-

p

SO

> 0,

b(t - t’)6(k - k’)b(k), if d - p = 0. When d - p

< 0, our techniques do not give an answer.

5. The Drift

As Eq.(4) shows, the drifi term G d t in the stochastic Schrodinger equation given in Eq.(3) is the limit of the expectation value in the reservoir state of bWe use the following conventions: The Fourier transform transform f V of a test function f E S ( W d ) are given by

fA

and the inverse Fourier

so that f A v = f v A = f . The Fourier transform F A and the inverse Fourier transform FV of a distribution F E S’(Rd) are defined by the relations (FA, f A )= (F,f),

(FV, f V )= (F>f),

being dual pair (., .) antilinear on the left and linear on the right.

51

the second term in the iterated series solution for the rescaled Shrodinger equation in interaction picture. In the following theorem we show that the open region r2 does not contribute to the drift term whenever the resonant case ak = w is not present, whereas for the region we obtain the usual expression for the drift. The contribution of the singular region r3 to the drift has not been determined yet.

Theorem 5.1. Under the conditions f o r w ( k ) given above we have: (a) If rz is not empty and n o Bohr frequency w of the system coincides with one of the values (Yk, then the contribution of the region r2 t o the drift t e r m is zero, whereas i f any of the Borh frequencies w of the system coincides with one of the values ak, then G does not exist. (ii) Otherwise

G=

((giIgj);E; ( D i )E w ij

(Oj)

+ (gilgj);*Ew (Di)E; ( D j )

w

+ The part corresponding t o the singular region r31,

where, for each Bohr frequency w , the E w ( D j )are system operators defined by Ew(Dj) :=

1PE,-wDjPEr, Er

Fw := {

E E ~

E Fu

Spec H s : E~ - w

E

Spec H s } ,

and the explicit f o r m of the constants (gilgj); is r

The constants (gi 1gj)s are called generalized susceptivities and have an important physical interpretation. In some sense they contain all the physical information on the original Hamiltonian system and can be considered as the prototype of quantum mechanical fluctuation-dissipation relations, cf. [ l ] .

52

Acknowledgements

F.G. Cubillo is grateful to L. Accardi and Centro Vito Volterra for support and kind hospitality. References 1. L. Accardi, Y.G. Lu, I. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, Berlin, 2002. 2. L. Accardi, S.V. Kozyrev, Quantum Interacting Particle Systems. In Quantum Interacting Particle Systems, World Scientific, Singapore, 2002, pp. 1193. 3. I.M. Gelfand, G.E. Shilov, Les Distributions, Dunod, Paris, 1962. 4. V.G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. 5. L. Schwartz, Mkthodes Mathkmatiques pour les Sciences Physiques, Hermann, Paris, 1966. 6. E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, Oxford, 1948.

INTEGRAL REPRESENTATION OF POSITIVE OPERATOR ON INFINITE DIMENSIONAL SPACE OF ENTIRE FUNCTIONS

W. AYED Institut PrLparatoire a m Etudes d ’Inghieurs de Nabeul, 8000 Nabeul, Tunisia E-mail: wided. [email protected] H. OUERDIANE Department of Mathematics Faculty of Sciences of Tunis University of Tunis El-Manar 1060 Tunis, Tunisia. E-mail: habib. [email protected] In this paper, we give a new criterion for positivity of generalized functions and positive operators on test functions space of entire function on the dual of a nuclear space N’ and with 0 order growth condition, denoted by F’o(N’). This definitions are used to prove that every positive operator has integral representation given by positive Radon measure and these measures are characterized by integrability conditions. This new criterion of positivity can be easily applied t o several examples.

1. Introduction The main purpose of this paper is to introduce a new and useful criterion of positivity of generalized functions and operators. This enable us to prove an integral representation of such distributions and operators. In the first section, we summarize some results needed in this paper. In the second section, we reformulate the usual definition of positive generalized functions in two infinite dimensional variables. Then, we prove an integral representation of such generalized functions by means of positive Radon measure. In the third section, we define positive operators in L(.Fo(N’),.Fp(M’)*) among the positivity of their kernel. Then we prove, (see theorem 3.1) that every positive operator has an integral representation. The Radon mea53

54

sures associated to a positive generalized function or positive operator are characterized by integrability conditions of Fernique type. In the next, we assemble a general framework which is necessary for our paper, see [3, 2, 1, 12, 91. Let N and M be two complex nuclear F’rkchet space whose topology is defined by a family of increasing Hilbertian norms (1.;, p E N} (resp. { \.Iq; q E N}). For p , q E N,we denote by N p (resp Mp) the completion of N (resp. M ) with respect to the norm 1., (resp. I.Ip). Then:

N = projlimp,JVp,

M = projlimq,ooMq

Denote by N-, (resp. the topological dual of the space N p (resp. M q ) . Then by general duality theory, the strong dual space N’ (resp. M’) can be obtained as:

N’ = indlimp+ooN-p,

M’ = indlim,+,M-,

Due to the nuclearity of N (resp. M ) , the strong and the inductive limit topology of N’ (resp. M’) coincide. Let M p @ N p be the Hilbert space direct sum, then the direct sum M @ N is by definition:

M

@ N = projlim,,,M,

@ Np

Similarly,

( M @ N)’ = M’ @ N’ = indlim,,,M-,

€3

N-,

We fix a pair of Young functions (0, cp), and we define the following space of entire functions of two variables:

Eql(N-,

@ M-p, (&Cp),

(m1, m2))

55

becomes a projective system of Banach spaces and we put:

F(e,vp)(N’ @ M’) = projlimp,,,,,,,,loEx~(N-p

@ M-p,

(0, cp), (mi,m2))

which called the space of entire functions on M’@N’ with (0, cp)-exponential growth of minimal type. Similarly,

( E X P ( N p @ Mp, (0, cp),

(m1,m2)LP

E

N,ml > 0,7732 > 01

becomes an inductive system of Banach space and the space of entire functions on N @ M with (0, cp)-exponential growth of finite type is defined as:

G(e,p)(N@ M ) = indlimp+~;ml,mz-+m EZP(Np @ Mp, (0, cp),

(m1, m2)).

Denote by F(e,+,)(N’@ MI)* the strong dual of the test function space F(e,v)(N’@ M’). For any ( t , q ) E N x M , we consider the exponential function e(t,,,) : N’ @ M’ + C defined by:

e ( ~ , , , ) ( z 1 ,:= 4 et@,(zl @ a= ) exp((z~,E)+ ( 2 2 , ~ ) ) = et 8 e&l, ~ 2 ) . It is easy to see that e(E,,,) E F(e,vp)(N’@ M’) there is a unique topological isomorphism

F(e,vp)((N@ MI’)

. We recall, from [ll],that

Fe(N’)GFv(M’)

(1)

which extends the correspondence eC@,,tf et 8 e,,. Denote by X (resp. Y ) the real Frdchet nuclear space whose complexified is N (i.e. N = X iX) (resp. M = Y i Y ) and X’ (resp. Y’) is the strong dual of X (resp. Y ) . We recall the following theorem from [9]:

+

+

Theorem 1.1. Let M and N be complex nuclear Re‘chet spaces and let 0 and cp be Young functions. Then the Laplace transform is a topological isomorphism:

L,: F(e,v)(N’@ M’)* + G(e*,v*)(N @M) where 0* and cp* are the conjugate functions respectively of 0 and cp and they are given by

e*(x) = sup(tx -qt)), tao

cp*(x)= sup(tx - cp(t)), 2 0. tao

56

If N = M and cp = 8, we write simply Fe(N’ @ N’) = F(e,e)(N’@ N’). Denote by L(Fe(N’),.Fe(N’)*)the space of all linear continuous operators from .Fe(N’) to Fe(N’)*. From the nuclearity of the space Fe(N’), we have by the Schwartz- Grothendieck kernel theorem

C(.Fo (N‘),Fp(M‘)*)

.Fe

(,I)*

6~~(M’)*

(2)

F[e,+,)(N’ CBM‘)*. Since the kernel ZK of an operator 2 E C(F~(N’),.Fp(M’)*) is an element of F(e,p)(N’@M’)*l the symbol of Z is by definition the Laplace transform of EK, so we obtain the following relation: A

Z ( J C B ~=) ((E~, ecBe,)) = ((E~, ece,)) = L ( E ~ ) ( ~ C B ~ J, ) ,E N , r~ E M . (3) Using the theorem 1.1, we get the following analytic characterization of continuous operators from .Fo(N‘) into Fo(N’)*:

Theorem 1.2. A function 0 : N @ N + CC is the symbol of some Z E C(.Fe(N’), Fe(N’)*)if and only if 0 E G‘p ( N @ N ) . In the next, we suppose that the two Young functions 8 and cp satisfy the additional conditions:

then, we obtain the following Gelfand triples; (see [3, 111)

where 7il i E (1, 2) are respectively the Gaussian measure on the strong dual of X and Y (see [ 5 ] ) given via Bochner Minlos theorem by the characteristic functions:

Remark 1.1. In the case where M = {0}, this implies that Fo+,(N’ CB M’) = Fe(N’), and all further results proved for the two infinite dimensional variable test functions space Fe,p(N’@ M’) are also valid in .Fe(N’).

57

2. Positive generalized function in two infinite dimensional variables In this section, using the involution defined in the equation (7), we define positive generalized function in F(o,,,p)(N’@ M’)* which generalize the classical one. Then we give an integral representation of such generalized functions. For this, we shall recall (see [ll])that .F(o,+,)(N’@ M’)* is a nuclear algebra with the involution * defined for any f E F(o,,+,)(N’ @ M’)* bY

f*(z,w):= f ( Z ,

a),

z E N’,

w E M’.

In the following, using the isomorphism (l),we remark that for any Fe(N’) and f2 E F,(M’), we have: f = f1 8 f2 E F(e,+,)(N’ @ M’).

(7) f1

E

Definition 2.1. A generalized function CP E F(o,,,p,(N’@ M’)* is positive if for any f = f l 8 f2, where fl E Fo(N’) and f2 E F,(M’), we have ((@,ff*)) 0.

z

We recall (see [ll,12, 13, 17]),that a generalized functions CP E F(o,+,)(N’@ M’)* is positive in the classical sense if for any f E Fe,+,(N’@ M’), such that f(a:+iO@y+iO)bO

V(z,y) E X ’ X Y ’

we have ((CP7.f))

b 0.

In the following, we denote by F(O,+,)(N’@M’);~ the set of classical positive generalized functions and by F(o,,,) (N’@M’); the set of positive generalized functions given in the definition 2.1.

Lemma 2.1. Any classical positive generalized function CP E F(o,+,)(N’@ M’)* is also element of F(o,+,p,(N‘ @ M’):. Proof Let CP E F(o,+,)(N’@M’);c and consider f = f l B f 2 , where f l E Fo(N’) and f2 E F,(M’), then we have ((CP,ff*))O 2 0 because for any a: E X’, y E Y‘: (ff*)(a:

+ io @ y + 20) =. ( f I

+ io @ y + i0)l2 2 0.

Theorem 2.1. For any CP E F(o,+,)(N’@M’);,there exist a unique positive Radon measure pa on X’ @ Y’ such that for all f E F(o,+,)(N’@ M’) one

58

has: P

So, the function Ca is a characteristic function. Then using the BochnerMinlos theorem, see [7, 5, 81, there exist a unique positive Radon measure pa such that for all ( E , q ) E X x Y : ((Q,ez(E@s))) = CP.€.(E,d =

1

X’CBY’

“XPi((GE) + (Y,v))dPa(z@?4). (9)

Then, it is sufficient to extend the equality (9) to each f E F(e,+,)(N’@M’). It is clear that (9) is verified on the algebra & spanned by the exponential functions {ei(E,a);(E, q) E X x Y } which is dense in F((e,+,)(N’ @ M’). Let

59

f E F(e,,+,)(N’@M’) and ( f n ) n E ~a sequence in E which converges to f,for the topology of F(e,,+,p)(N’ @ M’). Then

-

((a,(fn - f d f n - fm>*))

(10)

L,$y,

(11)

- fm>(z@ Y)I

I(fn

2dP a@@Y).

Since ( f n ) n E ~converges to f,it is Cauchy sequence in E and using the continuity of @ and also the continuity of the product on F(e,,+,)(N’@M’), then taking the limit in the equation (lo), we obtain that the sequence ( f n ) n ~ is Cauchy type in L2(X‘@Y’,p+),so it converges in this space. Denote by 1 the limit of ( f n ) n E ~with respect the norm L2(X’ @ Y’,p a ) . Because the topology of F(e,,+,p)(N’ @ M’) is finer than the topology of L2(X’ @ Y’,p a ) , so each neighborhood of 1 in L2(X’ @ Y‘,p a ) is a neighborhood of 1 in F(B,,,)(N’@ M’), then it will contain f too, then 1 = f p a . a . e . Now using the theorem of dominated convergence, and the fact that the measure pa defined by Bochner theorem is supported by X - , @ Y-, for some p > 0, we conclude that:

=

J

@ @ y)dCLa(z @ Y)

/

f b @ Y)dPa(.

X’$Y‘

= XJ$Y‘

@ Y).

Corollary 2.1. A n y classical positive generalized function @ has an integral representation given by the relation (8). Moreover:

Fe,,+,(N’CBM’);, = Fe,,+,(N’CBM’);.

(12)

Remark 2.1. In fact corollary 12 gives us a new criterion of positive generalized functions that we will use in the next to obtain our aims. Moreover contrary to the usual definition of positive generalized functions, this criterion dos not require a definition of a class of positive test functions. In this way, this criterion becomes more pratique. Using the relation (4),we obtain the following triple:

Fe,,+,(N’@ M’)

c L ~ ( x ’x Y’,71

72)

c F,,,+,(N’ @ M’)*

~

60

which implies that every @ E Fo,,+,((N' @ MI)* can be interpreted as a gaussian distribution. Then by theorem 2.1, we get the following corollary:

Corollary 2.2. let cf, E Fg,,+,(N'@ MI): and pa the associated measure given by the equation (8). T h e n cf, can be interpreted as a generalized Radon Nikodym derivative of the measure pa with respect the standard gaussian measure 71 @I 7 2 :

In the following, we will give a characterization of the Radon measure defined in theorem 2.1.

Theorem 2.2. Let p a finite measure o n X' @ Y' equipped with the Bore1 u-algebra of B(X' @ Y ' ) . The measure p represent a positive generalized function @ E Fo,,+,(N'@ MI): if and only if it verifies the two following properties: (i) There exist q > 0 such that the measure p is supported by X - q @ Y - q . (ii) There exist m l , m2 > 0 such that

Jx-,

ee(mlIrI-,)+,+,(n2I1/I-,)dp(a:

@

y ) < 00.

@Y-,

(13)

To prove this theorem, we will use the two following lemma with prove similar to those given in [12] for one infinite dimensional variable:

Lemma 2.2. Let p be a measure which represents a positive generalized function cf,. T h e n there exist m'l > 0, m'2 > 0 and p , q E N satisfying q > p such that for any (E, 77) E X q x Yq and for any n, 1 E N , we have:

Jx-,

$Y-,

(xmn,P " ) ~ ( P , d p ( x @ 9)

I II L(@>Ile*,,+,*,--p,--p,n'l,m'z

121

( 2 W W e m2

*

~2,Cp;l

El7l:7l.:l

(14)

Lemma 2.3. Let p be a measure which represents a positive generalized function CP. T h e n there exist m'l > 0, m'2 > 0 and p , q E N satisfying q > p such that f o r any n, 1 E N,we have:

61

3. Positive operator in L(Fe(N‘),F,(M’)*) In the following section, using the results of the previous section, we are able to define positive operators in L(Fe(N’),Fq(M’)*). Then we give an integral representation for such operators. The case N = M and ‘p = B correspond to the White noise operators studied by [9]. For every E E L(Fe(N’),Fq(M’)*) the associated kernel denoted by E K E (FO(N’)GF+,(M’))*satisfies the following relation:

fag)), f E Fe(N’),9 E F ~ ( M ’ ) and the symbol of E E L(Fe(N’),Fq(M’)*) is defined by: -4 6 , rl) = ( F Kec, 8 e,)) = ((+, e(&3,))) E E N , rl E M . ( ( ~ f , g )=)

(16)

Definition 3.1. An operator E E L(F~(N‘),Fq(M’)*) is positive if its kernel EK is an element of Fo,,+,(N’@ M’):. Theorem 3.1. For any positive operator E E L(Fe(N’),F,(M’)*) there exists a unique positive Radon measure p~ o n XI @ Y‘ such that for all f E Fe,,(N’ @ M I ) , one has: ( ( E K ,f)) =

1

X‘$Y’

+

f ( x 20 a3 y

+ iO)dp=(x@ y ) .

Moreover, the measure p~ is characterized by the following integrability conditions: ( i ) There exist q > 0 such that the measure p~gis supported by X - , Y-, . (ii) There exist ml,m2 > 0 such that

@

Proof Since E K E Fe,q(N’ @ MI):, then by theorem 2.1, there exists a unique positive Radon measure pa on X‘ @ Y‘ such that:

JX’$Y’f ( x + io CBy + iO)dps(X CB y ) ,

<< EKlf >>=

~f

E

Fo,,+,(N’CBM I ) .

62

T h e characterization of pz is a consequence of theorem 2.2. We deduce from theorem 3.1, t h e equation (16) and t h e corollary 2.2, the following corollary that can be needed to characterize unitary solution of quantum stochastic differential equations. I n a future paper, we generalize this new criterion and we apply this results to several concrete examples in order to give regularity property for solution of some quantum differential equations.

References 1. M. Ben Chrouda, M. El Oued and H. Ouerdiane: Convolution Calculus and application to stochastic differential equations, Soochow Journal of Mathematics, Vol. 28, No. 4 (2001), 375-388. 2. M. Ben Chrouda and H. Ouerdiane: Algebra of operators o n holomorphic functions and Applications, Journal of Mathematical Physics, Analysis and Geometry, Vol. 5, (2002), 65-76. 3. R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgui: U n the'ordme de dualite' entre espaces de fonctions holomorphes ic croissance exponentielles, Journal of Functional Analysis, Vol. 171, (2000), 1-14. 4. I. M. Gelfand and N. Ya. Vilenkin: Generalized functions, Vol. 4, Academic Press, New York and London , (1964). 5. T. Hida: Brownian Motion, Springer-Verlag, New York, (1980). 6. T. Hida, H. H. Kuo, J. Potthoff and L. Streit: White Noise, An Infinite Dimensional Calculus, Kluwer Academic Publishers, Dordrecht, (1993). 7. H. H. Kuo: White Noise Distribution Theory, CRC Press, Boca Raton, (1996). 8. N. Obata White noise calculus and Fock space, LNM, No. 1577, (1994). 9. U. C. Ji, N. Obata and H. Ouerdiane: Analytic characterisation of generalized Fock space opeartors as two-variables entire functions with growth condition. Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol: 5, NO. 3, 395-407, (2002). 10. N. Obata: Coherent state representations in white noise calculus, Can. Math. SOC.Conference Proceedings, Vol. 29 ,517-531, (2000). 11. H. Ouerdiane: Infinite dimensional entire functions and Application to stochastic differential equations, Notices of t h South African Math. Society, 35 (2004), NO, 1, 23-45. 12. H. Ouerdiane and A. Rezgui: Representation integrale de fonctionnelles analytiques positives, Canadian Mathematical Society Conference Proceedings Vol. 28, 283-290, (2000). 13. H. Ouerdiane and A. Rezgui: U n theorem de Bochner-Minlos avec une condition d'integrabilite' Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 3, No. 2, 297-302 (2000). 14. H. Ouerdiane: Noyaux et symboles d'oprateurs sur les fonctionnelles analytiques gaussiennes, Japanese Journal of Math. vol 21, N1, (1995). 15. H. Ouerdiane and N. Privault: Asymptotic estimates for white noise distrib-

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utions, Probability Theory/ Complex Anatysis, C. R. Acad. Sci. Paris, Ser. 1338, p. 799-804, (2004). 16. F. Trhves: Topological Vector Space, Distribution and Kernels, Academic Press, New York and London, 1967. 17. Y. Yokoi: Positive generalized White noise functionals, Hirichima Math, Vol 20, 137-157, (1990).

COMPARISON OF SOME METHODS OF QUANTUM STATE ESTIMATION

TH. BAIER* AND D. PETZ+ Department f o r Mathematical Analysis, Budapest University of Technology and Economics H-1111 Budapest, Muegyetem rkp. 8-8, Hungary E-mail: tbaierQmath. bme.hu, petzamath. bme.hu

K. M. HANGOS~AND A. MAGYAR Process Control Research Group, Computer and Automation Research Institute H-1518 Budapest, P O B o x 53, Hungary E-mail: hangosQsc1. sztaki. hu, amagyarQsc1. srtaki. hu

In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used t o quantify the error. It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.

1. Introduction

The aim of quantum state estimation is to decide the actual state of a quantum system by measurements. Since the outcome of a measurement is stochastic, several measurements are to be done and statistical arguments lead to the reconstruction of the state. Due to some similarities with X-ray tomography, the state reconstruction is often called quantum tomography [2]. More precisely, in physics-related books, journals and papers, tomography refers to both the state and parameter estimation of quantum dynamical *Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology. tSupported by the Hungarian grant OTKA T032662. *Supported by the Hungarian grant OTKA T042710.

64

65

systems where the term state tomography is used for the first, and process tomography is applied for the second case [14, 6, 51. The engineering literature contains also papers related to state and parameter estimation of quantum systems but they term it identification for the case of parameter estimation [13, 11 and state filtering for the case of state estimation [12]. In this paper the estimation of the state of a qubit is discussed. This is the simplest possible case of quantum state estimation where no dynamics is assumed and the measurements are performed on identical copies of the qubit. Therefore, the state estimation problem reduces to a static parameter estimation problem, where the parameters to be estimated are the parameters of the density matrix of the qubit. The methods of classical statistical estimation are used to develop state estimation of quantum systems in the first group of papers [8, 6, 171. This approach suffers from the fact that the state estimation is usually based on a few types of measurement (observables) that are incompatible, thus there is no joint probability density function of the measurement results in the classical sense [9]. The most common way of statistical state estimation is the maximumlikelihood (ML) method that leads to a convex optimization problem in the qubit case (see below). The other way of computing a point estimate of the state of a quantum system is to use convex optimization methods such as in [12, 131. Here one can respect the constraints imposed on the components of the state but there is no information on the probability distribution of the estimate. The efficiency of the ML estimate, its asymptotic properties and the Cram&-Rao bound can be used to derive consequences on the asymptotic distribution of an estimate and on its variance. This approach has been used for optimal experiment design in [12]. A lower bound on the estimation error for qubit state estimation is derived in [7]. It is natural to require that any state estimation scheme should be unbiased and should converge in some stochastic sense to the true value if the number of samples (measurements done) tends to infinity. The basis of the comparison is then a suitably chosen measure o f f i t (for example averaged fidelities with respect to the true density matrix, or the variance of the estimate). The fidelity and the Bures-metric defined therefrom was used to derive optimal estimators of qubit state in [3]. Fidelity has been used to evaluate the performance of an estimation scheme [4] for the so called "purity" of a qubit (i.e. the length of its Bloch vector) in the context of Bayesian state estimation.

66

Large deviations can also be used to analyze the performance of state estimation schemes [ll],when the qubit is in a mixed state. An optimal estimation scheme is also proposed based on covariant observables. The aim of this paper is to investigate the properties of two state estimation methods, the Bayesian state estimation as a statistical method and the least squares (LS) method as an optimization-based method by using simulation experiments. The simplest possible quantum system, a single qubit, a quantum two level system, is applied, where we could compute some of the estimates analytically. 2. Preliminaries about two level systems

The general state of a two level quantum system is described by a density operator p, which is a positive operator on the Hilbert space C2,normalized to Tr p = 1. On the one hand, p is represented in the form of a 2 x 2 matrix, and on the other hand by the so-called Bloch vector s = [sl,s2, s3IT. With use of the Pauli matrices 1 0 01 =

[; I

,

c2=

[p -;]

,

c3=

[o-l]

9

the correspondence between the density operator p and the Bloch vector s is given by the expansion 1 p = -(I s1c1 s2c2 s3e3), 2 where the constraint

+

+

+

is satisfied. The correspondence between p and s is affine. Thus the state space of a spin system is represented by the three dimensional unit ball, called the Bloch ball. Observables, i.e. physical quantities to be measured, are represented by self-adjoint operators acting on the underlying Hilbert space [16]. A self-adjoint operator A has a spectral decomposition A = &Pi. The different eigenvalues X i of the operator A correspond to the possible outcomes of the measurement of the associated observable and the ith outcome occurs with probability Prob (Xi) = Tr pPi, where P i is the projection onto the subspace of the corresponding eigenvectors. Consequently, the expectation value of the measurement is

C:.,

(A),., := xXiProb(Xi) = TrpA. i

67

3. Measurements on qubits

For the state estimation, we will consider 3n identical copies of qubits in the state p. On each copy in this passel, we perform a measurement of one of the Pauli spin matrices { m l , m2, m 3 } , each of them n times. The possible outcomes for each of this single measurements, i.e. the eigenvalues of the mi, are f l and the corresponding spectral projections are given by

For the sake of definiteness, we assume that first 01 is measured n times, then u2 and then 03. The data set of the outcomes of this measurement scheme consists of three strings of length n with entries 5 1 :

Dl = { D l ( j ): j

(i = 1 , 2 , 3 ) .

= 1,e.a , n }

(3)

The predicted probabilities of the outcomes depend on the true state p of the system and they are given by 1 1 Prob(Dl(j) = 1) = Tr (pP:) = 2 ( 1 + ( m i ) p ) = ~ ( 1si). (4)

+

4. Quality of the estimates

As a measure of distance between two states of a system, i.e. between two density operators p and w , the fidelity F ( p , W ) = Tr

J1

pTwpT

(5)

can be considered [15, 141. It fulfills the properties F ( P , W ) = WJ, P),

F(p,w)=l

* p=w,

0 I F(P,W) I 1

F(p,w)=O

*w l p .

For spin 1/2 systems the fidelity can be calculated from the eigenvalues A1 and A2 of the operator A = p a w p i as F(P,W)

=

A +A.

These eigenvalues can be computed from Tr A and Det ( A ) as

If we express T r A and Det A in terms of the Bloch vectors s (resp. r ) of p (resp. w ) , the fidelity can be written as

68

The quality of the estimation scheme for a true state p can be quantified by the average fidelity between the true state and the estimates wi (1 5 i 5 m):

if m estimates are available. Alternatively, the Halbert-Schmidt distance d(p,w) := J T q j T p

(7) can be used as a measure. In terms of the Bloch vectors, this reduces to &(si - ~ i ) The ~ . average Hilbert-Schmidt distance is given by .

m

Remember that for an efficient estimation scheme x ( p , m ) must be small, while @(p,m)should be close to 1. 5. Bayesian state estimation

First we give a brief summary of the Bayesian state estimation. In the Bayesian parameter estimation, the parameters 8 to be estimated are considered as random variables. The probability P(8 1 0") of a specific value of the parameters conditioned on the measured data Dn is evaluated. Afterwards, the mean value of this distribution is used as the estimate. If the measured data is a sequence of outcomes, as in our case, it can be split into the latest outcome D"(n) of D" and Dn-l, the preceding. Then the conditional distribution of the parameter becomes

P ( e I D"(n),D"-') and the Bayes formula

can be applied resulting in the following recursive formula for P(8 I 0")

69

In our state estimation, we have three data sets Or,i = 1,2,3, corresponding to the three directions, see ( 3 ) . The estimation is performed for the three directions independently (and afterwards a conditioning has to be made). The probabilities P ( D l ( n )1 Din-’, 0) have the form

If we denote by l ( i ) the number of +l’s in the data string DT,then (8) becomes

where Pf(v)is an assumed prior distribution, from which the recursive estimation is started. For the sake of simplicity we assume that P ~ ( v has ) similar form with parameters K and X in place of n and t , respectively. (These parameters might depend on i, but we neglect this possibility.) After a parameter transformation we have a beta distribution,

where C is the normalization constant and u E [0,1].It is well-known that the mean value of this distribution is

mi =

+ +

l(i) 1 X n+lc+2

and the variance is

+ + X)(n - q i ) + 1+ K - A)

(l(2) 1 (n

+ + 2)2(n+ + 3 ) K.

K.

(12)

The above statistics (11) can be used to construct an unbiased estimate for si in the form bi = 2

qi) +1+x -1

n+K+2 after the re-transformation of the variables. Since the components of the Bloch vector are estimated independently, the constraint ( 1 ) is not taken into account yet. Thus, a further step of conditioning is necessary. We simply condition (b1, & , & ) to ( 1 ) : rJi. -

JJJ U i f ( u l ) f ( u 2 ) f ( U 3 ) dul dU2 du3 JJ.f f ( ’ l L l ) f ( U 2 ) f ( u 3 )du1 du2 du3 ’

(14)

70

where both integrals are over the domain and

( ( ~ 1 u, 2 , u g ) : u!

+ ui + uz I 1)

f(u2):= P ( s i p ; ) ( u z ) .

Then the conditioned estimate of si will be 2(fiZ

- 1).

The justification of the proposed conditioning procedure is the subject of another publication. 6. Least squares state estimation

We have the data set (3) to start with. If f l in the string Da,then the difference 7ra

:= T i ( + )

~i(f is )the

relative frequency of

-Ti(-)

is an estimate of the ith spin component si (i = l , 2 , 3 ) . As a measure of unfit (estimation error) we use the Hilbert-Schmidt norm of the difference between the empirical and the predicted data according to the least squares (LS) principle. (Note that in this case the Hilbert-Schmidt norm is simply the Euclidean distance in the 3-space.) Then the following loss function is defined: 3

L ( w ) = d 2 ( r , T )=

j = llr1I2 ) ~

(rj - ~

+ 1 1 ~ 1 -1 ~2 r .

T

(15)

j=1

where r is the Bloch vector of the density operator w . An estimate of the unknown parameters s = [sl,s2, s3IT is obtained by solving the constraint quadratic optimization problem:

L(w) llrll 5 1

Minimize subject to

The above loss function is rather simple and we can solve the constrained minimization problem explicitly. In the unconstrained minimization, two cases are possible. First, 1 1 ~ 1 15 1, and in this case the constrained minimum is taken at r = T . When the unconstrained minimum is at T with 1 1 ~ 1 1> 1, then it is clear from the 3-dimensional geometry that the constrained minimum is taken at

r=-.

I1

11n-11

71

7. Simulation experiments The aim of the experiments is to compare the properties of the above described least squares and Bayesian qubit state estimation methods. The base data of the estimation is obtained by measuring spin components 01, C T ~ ,and 03 of several qubits being in the same state i.e. having just the same Bloch vector s. The number of the measurements of each direction is denoted by n in what follows. The same measurement data had been used for the two methods. The Bayesian method was applied with conditioning and also without it to analyze its effect. The measurements were performed on a quantum simulator for two level systems implemented in MATLAB [lo]. An experiment setup consisted of a Bloch vector s to be estimated and a number of spin measurements performed on the quantum system. The internal random number generator of MATLAB was used to generate "measured values" according to the probability distribution of the measured outcomes. In this way a realization of the random measured data set is obtained each time we run the simulator. Each experiment setup was used five times and the performance indicator quantities, the fidelity, the Hilbert-Smith norm of the estimation error and the empirical variance of the estimate were averaged. 8. Results of the experiments

The fidelity (5) of the real Bloch vector and the estimated one, variance of the estimations (12), and the Hilbert-Schmidt norm (7) of the estimation error were the quantities which have been used to indicate the performance of the methods. 8.1. Number of measurements

The first set of experiments were to investigate the dependence between the performance indicator quantities and the number of measurements n. Fidelity. It was expected that the fidelity goes to 1 when n goes to infinity. Fig. 1 shows the experimental results for estimating a pure state spUPe = [0.5774, 0.5774, 0.5774]*. The result of the Bayesian estimation (dotted line) shows the weakest performance because of the conditioning feature of the method: the conditioned joint probability density function gives worse estimation, than the original one (dashed line). On the other hand, the original Bayesian without conditioning tends to give defective Bloch vector estimates with length greater than one. The price of the

72

.i

.................... i

I ’.. rI::::.]

.: .........

0.98

.........(.I.

0.9,

.............. $ ...... ...........I... ....

w

#?

~

:

Bayesian WihD”1 regld.

..a

o,gs&.; ............ i... ..........................

p.g.k

1: ..........

0.94

,,,,.; ..................

:

dan

.& ................ .............. .J

1...

.................

..............

..........

.................i................... i ..................................

...................

i

0.93 :...............;................. > .................. :............... j ................ 0.92

0.9,

: ............ ....................

~

................. ........... ..;.............

................ ...................: ........... ~

ZOO

~

..>................. <.................

400 600 8W number o f m ~ ~ l ~ r e m e n t S

1000

0

200

400 600 number of measurements

800

IWO

Figure 1. Fidelity as a function of n for a pure state ( s p u r = ) and a mixed state (srnized)

validity of the Bayesian method with conditioning is the precision for (near) pure states. It is apparent that the least squares estimation does not have the above problem. The situation is a little bit different for estimating mixed states (.smized = [0.3, -0.4,0.3IT). It can be seen that the two kinds of Bayesian estimation differ only for small n’s. When n is greater than 25, the conditioning has no traceable effect, i.e. the Bayesian estimation with and without conditioning gives the same result. Least squares method also works a little bit better for mixed states than for pure states, at least for larger n’s. It can be seen that pure states are challenge for both methods but least squares handles this difficulty a bit better. In order to investigate more deeply the behavior of the estimates with low number of measurements we show the variation of the fidelities as a function of the number of measurements in the interval n = [5,150] for both the pure and mixed states above (see Fig. 2). It was expected that the Bayesian estimates outperform the LS one for low number of experiments, but it is only true in the case of mixed states. For pure states the overly conservative conditioning of the Bayes method causes a bias. In addition, one can notice, that the effects related to the low number of measurements can be seen only when n < 25. Hilbert-Schmidt norm. For Hilbert-Schmidt norm, it was expected to decrease to zero in the limit. The experiments seem to come up to expectations (Fig. 3). In the case of pure states the same phenomena is noticeable as for fidelity. If one zooms on the low number of measurement region in Fig. 3 then the picture in Fig. 4 results. Here we can see the same effects as for the fidelity, but in a less exposed way. Thus fidelity seems

73

..............................

o,g

t

.........

0.55

....~.,,.................. ..............

...................

.......... ...........

8 0.-

50

150

100

0.82

50 100 number 01 measurnem

numbst01 masunmntr

Figure 2.

150

Fidelity as a function of low n for a pure state ( s p U r e and ) a mixed state

(hized)

Hllbe~-SchmlUmrm as a hlncUOnOf the rwmberol measurements -8-

................

1. .............,.;............ -e

............... i...

LS UUmauo" Bayeslsn WiUlDul regul.

.*. Bsyulan

..........I... .............:..

I 200

400

600

number 01 measirments

5W

1040

200

Figure 3. The Hilbert-Schmidt norm as a function of mixed state ( s , i Z e d )

TZ

400 500 number 01 measurements

800

1000

for a pure state (spure)and a

to be a more sensitive indicator of performance than the Hilbert-Schmidt norm. Variance. The variance of the estimates were computed for the Bayesian estimation before conditioning. As it was expected, there is no apparent difference between the variance for the three spin components S ~ , S Z and , sg and the variance decreases with n. The fact that the state to be estimated is a pure or a mixed state also does not have any effect on the result (Fig. 5). The same effect can be seen if one focuses on the low number of measurement region, as seen in Fig. 6.

74 Hliben-ScHmidl o m as a IuncUon 01 he number of measuremenu

HilbertSchmiU norm a$ a hlndion of he number al measurements

1w

50

number of mea$urements

1w

50

150

number of measummm

Figure 4. The Hilbert-Schmidt norm as a function of low n for a pure state (spUre)and a mixed state ( s , i s e d )

I

0.018

...........

t'

-.

-c Spin component x

-* Spinmmpnsnty ,..............:.. .............:.. : . spinmmponmz

.;

........... ,..................,............. ..................

.............. i................... > ..............< . .............

260

4w

-

sw

n u m k 01 mearummenu

-8w

1WO

i

j

.................

............. ~ ....................................................... . OO

2W

400 BW n u d w of maaslurannk

8W

1WO

Figure 5. Variance as a function of n for a pure state (spure)and a mixed state (s,ized)

8.2. The length of the Bloch vector

During the second set of experiments the length of the Bloch vector was varying. Its direction was s = [0.5774, 0.5774, 0.5774IT. The expectation to fidelity was to be relatively independent of the Bloch vector length IIsII. The experiment results can be seen in Fig. 7. The first picture shows the case n = 100, where, in spite of the big variance, the conditioned Bayesian shows an increase near the pure state (Ilsll = 1). At n = 900 it is more apparent that LS and conditioned Bayesian methods (both have certain conditioning feature to avoid faulty estimates near llsll = 1, see (18), (14)) have worse performance near pure states. Fig. 8 shows fidelity between llsll = 0.9 and llsll = 1 for n = 900, where the above mentioned phenomena can be seen more clearly.

75 Val-

as P t v d n 01 h.number of meawmmmr

....................... ......................................................... .......

..............................

.......................

Figure 6. Variance as a function of low n for a pure state ( s p U r e ) and a mixed state (Smised)

1eogm 01the Blwh wcmr

Figure 7. Fidelity as a function of llsll for n = 100 and n = 900

As it was expected, the Hilbert-Schmidt norm seems to be constant for varying Bloch vector lengths, Fig. 9 shows the simulation results. For relatively small n the variance is rather big but increasing the number of measurements it can be seen that the Hilbert-Schmidt norm is almost constant. Near llsll = 1 there is a small increasing for the conditioned Bayesian method. The expectation for variance was to be independent of Bloch vector length. Fig. 10 shows the results with the same variance-scale as in Fig. 5. The first graph is the results for 100 measurements, the other one is for n = 900. The result are in accordance with Fig. 5. As it was expected, the two graphs can be regarded as constants.

76 Fldsllty as a Imdm of h e the BlDh vecm length

..............

0.994 ....

f

0.993'

0.9

j..............

y$z2& 0.92

,.................:...............................

,og"l, .........................

0.94 0.96 length 01 h e BloEh v&or

Figure 8. Fidelity as a function of HllbertSchmidtmrm as B function01 Ihe Ihe Bloch vecUlr lenglh

'.'\1

i... .............. 0.98

I 1

llsll for n = 900

Hllben-Schmidl norm as a lurrlion of the Ihe BloCh veclor 1-h

+ LS Bhtimsllon

--

B*~lmwilho"l~1.

. . . . . . . . . . . . ....................................................................

lnglh of the BloCh veclor

Figure 9. Hilbert-Schmidt norm as a function of JJsJJ for n = 100 and n = 900

9. Conclusion

The performance of two state estimation methods, the Bayesian state estimation as a statistical method and the least squares (LS) method as an optimization-based method is investigated in this paper by using simulation experiments. The fidelity and the Hilbert-Smith norm of the estimation error as well as the empirical variance of the estimate are used as performance indicator quantities. The variation of these quantities as functions of the number of measurements and the length of the Bloch vector are computed. It is found that fidelity is the best indicator for the quality of an estimate from the investigated three performance indicator quantities from both qualitative and quantitative point of view. For state estimation of a single qubit the region of the 'low measurement number' being n < 25 and the 'large measurement number' n > 200 has been determined experimen-

0,014

~

.............

..................

j

............... ..............

1 ; ..................

j

. '

0.016

-c Spinmmponenlx Splncomponsnty nenll

-*

........... . ~ .,., ~

i

77

j

................ ................ ................

0,014

.........

1.

............................

..::..: ~

4

0.01 2

1

........... ;..............................

;

:

*

Spin-mponentx spinmmponenty Spin component z

.(I..

. . . . . . . . . . *..

..........

~

.................i..................;................. ;... .............

.................. ~

....................................

....................................

0.00

0

0.2

0.4 0.6 length of the Blmh W S ~ M

0.8

1

lengthof the BloCh vector

Figure 10. Variance as a function of llsll for n = 100 and n = 900

tally. As for the comparison of the different state estimation methods we have found that the Bayesian method could outperform the LS estimation only in the case of mixed states for low number of measurements (below n = 25). The investigated methods were found to be quite sensitive to the length of the Bloch vector, i.e. to the fact if a pure or mixed state was the one to be estimated. The methods that are not informed about the purity of the state can perform quite bad if they are used to estimate the state of a pure or "nearly pure" state. It is also found that the way of conditioning is critical for the methods capable of estimating both pure and mixed states. The simple length constraint of the least squares method (in (18)) seems to work quite effectively, thus a version of the Bayesian estimation method with LS-type constraining is a good candidate of an improved stochastic state estimation method. To handle somehow the difficulties related to estimating nearly pure states one should avoid to use a flat geometry on the state space but one should use a suitably defined special Riemannian geometry instead. References F. Albertini and D. D'Alessandro. Model identification for spin networks. Linear Algebra and its Applications, 394:237-256, 2005. L.M. Artiles, R. Gill, and M.I. Guta. An invitation to quantum tomography. Journal of the Royal Statistical Society (B), 67:109-134, 2005. E. Bagan, M. Baig, R. Munoz-Tapia, and A. Rodriguez. Collective vs local measurements in qubit mixed state estimation. Phys. Rev. A , 61:061307, 2003. E. Bagan, M. A. Ballester, R. Munoz-Tapia, and 0. Romero-Isart. Measuring

78

5. 6. 7.

8. 9. 10. 11. 12.

13.

14. 15. 16. 17.

the purity of a qubit state: entanglement estimation with fully separable measurements. arXiv, quant-ph/0505083:vl, 2005. G.M. D’Ariano, L. Maccone, and M.G.A. Paris. Orthogonality relations in quantum tomography. Physics Letters A , 276:25-30, 2000. G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi. Quantum tomography. Quantum Tomographic Methods, Lect. Notes Phys., 649~7-58, 2004. M. Hayashi and K. Matsumoto. Asymptotic performance of optimal state estimation in quantum two level system. arXiv, quant-ph/0411073:vl, 2004. C. W. Helstrom. Quantum decision and estimation theory. Academic Press, New York, 1976. Z. Hradil, J . Summhammer, and H. Rauch. Quantum tomography as normalization of incompatible observations. Phys. Lett. A , 199:2&24, 1999. Mathworks Inc. MATLAB software system, 2001. http://~.mathvorks.com/. M. Keyl. Quantum state estimation and large deviations. arXzv, quantph/0412053:v17 2004. R. L. Kosut, I. Walmsley, and H. Rabitz. Optimal experiment design for quantum state and process tomography and Hamiltonian parameter estimation. arXiv, quant-ph/0411093:vl, 2004. R. L. Kosut, I. Walmsley, and H. Rabitz. Identification of quantum systems: Maximum linkelihood and optimal experiment design for state tomography. IFAC World Congress, 1.l:Prague (Czech Republic), 2005. M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. A. Uhlmann. The ”transition probability” in the state space of a *-algebra. Rep. Mathematical Phys, 9:273-279, 1976. J. von Neumann. Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, 1957. J. RehaEek, B . 4 . Englert, and D. Kraszlikowski. Minimal qubit tomography. Physical Review A , 70:052321, 2004.

ENTROPIC BOUNDS AND CONTINUAL MEASUREMENTS

ALBERT0 BARCHIELLI Politecnico d i Milano, Dipartimento di Matematica, Piazza Leonard0 da Vinci 3$ 1-20133 Milano, Italy. E-mail: [email protected] GIANCARLO LUPIERI Uniuersitci degli Studi d i Milano, Dipartimento d i Fisica, Via Celoria 16, 1-20133 Milano, Italy. E-mail: Giancarlo. [email protected] Some bounds on the entropic informational quantities related t o a quantum continual measurement are obtained and the time dependencies of these quantities are studied.

1. Introduction

In the problem of information transmission through quantum systems, various entropic quantities appear which characterize the performances of the encoding and decoding apparatuses. Due to the peculiar character of a quantum measurement, many bounds on the informational quantities involved have been proved to hold [l, 2, 3, 4, 5 , 6, 7, 81. In the case of measurements continual in time, these bounds acquire new aspects (family of measurements are now involved) and new problems arise. A typical question is about which of the various entropic measures of information is monotonically increasing or decreasing in time. We already started the study of this subject in Refs. [9, lo]; here we apply to the case of continual measurements the new techniques developed [6, 7, 81 for the time independent case. 1.1. Notations and preliminaries

We denote by C(d;B) the space of bounded linear operators from d to B, where A,B are Banach spaces; moreover we set C(d) := L(d;A). Let 31 be a separable complex Hilbert space; a normal state on C(X) 79

80

is identified with a statistical operator, T(7-l)and S(7-l) c T(7-l)are the trace-class and the space of the statistical operators on 3-1, respectively, and llPll1 := Tr ( P , 4 := Trx{Pa), P E 777-4, a E C(7-l). More generally, if a belongs to a W*-algebra and p to its dual M * or predual M , , the functional p applied to a is denoted by ( p , a ) .

m,

1.1.1. A quantum/classical algebra Let (R, 3,Q) be a measure space, where Q is a a-finite measure. By Theorem 1.22.13 of [ll],the W*-algebra L"(R, 3,Q) 18 C(7-l) (W*-tensor product) is naturally isomorphic to the W*-algebra L" ( R , 3 , Q; C(7-l)) of all the C('H)-valued Q-essentially bounded weakly* measurable functions on R. Moreover ([ll],Proposition 1.22.12), the predual of this W*-algebra is L' ( R , 3 , Q; 7(.FI)), the Banach space of all the 7('H)-valued Bochner Q-integrable functions on R, and this predual is naturally isomorphic to L'(R, 3,Q)@T('H)(tensor product with respect to the greatest cross norm - [ll],pp. 45, 58, 59, 67, 68). Let us note that a normal state a on Loo(R, 3,Q; C(7-l))is a measurable function w H ~ ( wE)7(7-l), ~ ( w2)0, such that Trx{a(w)} is a probability density with respect to Q.

1.2. Quantum c h a n n e l s a n d e n t r o p i e s 1.2.1. Relative and mutual entropies The general definition of the relative entropy S(ElII) for two states C and n is given in [12]; here we give only some particular cases of the general definition. Let us consider two quantum states a, r E S(7-l) and two classical states q k on L m ( R , 3 , Q ) (two probability densities with respect to Q). The quantum relative entropy and the classical one are

S,(al.r) = Tr.F1{a(loga - log7)},

(14

We shall need also the von Neumann entropy of a state r E S(7-l):

S,(T) := -Tl'{TlOgT}. Let us consider now two normal states ak on L" (a, 3,Q; L(7-l))and set qk(w):= Tr{ak(w)},~k(w) := ak(w)/qk(w) (these definitions hold where the denominators do not vanish and are completed arbitrarily where the

81

denominators vanish). Then, the relative entropy is

We are using a subscript “c” for classical entropies, a subscript “q” for purely quantum ones and no subscript for general entropies, eventually of a mixed character. Classically a mutual entropy is the relative entropy of a joint probability with respect to the product of its marginals and this key notion can be generalized immediately to states on von Neumann algebras, every times we have a state on a tensor product of algebras [6, 7, 81. 1.2.2. Channels Definition 1.1. ([12] p. 137) Let M i and M z be two W*-algebras. A linear map A* from M z to M i is said to be a channel if it is completely positive, unital (i.e. identity preserving) and normal (or, equivalently, weakly* continuous). Due to the equivalence [13]of w*-continuity and existence of a preadjoint

A, a channel is equivalently defined by: A is a completely positive linear map from the predual M I , to the predual M2*, normalized in the sense that (A[p],1 2 ) 2 = (p, l l ~ ) ~ Vp , E M I , . Let us note also that A maps normal states on M I into normal states on M2. A key result which follows from the convexity properties of the relative entropy is Uhlmann monotonicity theorem ([12], Theor. 1.5 p. 21), which implies that channels decrease the relative entropy.

Theorem 1.1. If C and II are two normal states o n M I and A* is a channel from M2 -+ M I , then S(ElII) 2 S(A[E]lA[II]). 1.3. Continual measurements

Let us axiomatize the properties of a probability space where an independent-increment process lives and that ones of the a-algebras generated by its increments. The probability measure Q1 we are introducing will play the role of a reference measure. Assumption 1.1. Let dard Borel. Moreover:

(X, X , Ql) be a probability space with ( X ,X ) stan-

82

1. s 1. t } is a two-times filtration of sub-a-algebras: X,S X$ c X for0 r I s 5t I T; (2) V t 2 0, X t is trivial; (3) X; = X+ for o 5 s 5 t; (1) { X ; , 0

c

A

T:T>t

(4) X /

=

v v

X ; for o I s
T:S
(5)

x=

x,";

t:T>O

(6) f o r 0 I rI s 5 t 5 T , X l and X$ are Q1-independent.

Continual measurements are a quantum analog of classical processes with independent increments [14, lo]. As any kind of quantum measurement, a continual measurement is represented by instruments [15, 16, 171, but, as shown in [7],instruments are equivalent to particular types of channels. Here we introduce continual measurements directly as a family of channels satisfying a set of axioms (cf. also [18, lo]). Assumption 1.2. Let 'Ft be a separable complex Hilbert space. For all s, t , 0I s 5 t , we have a channel

% : L1( X ,X,", Q i ;7(31)) L1(X, X,", Q i ;I(7-l)) such that

By points (3), (4) of Assumption 1.2 and (6) of Assumption 1.1, one gets: V C TE~ L1 ( X ,X,", Q 1 ; 7(7l)), 0I s 5 t, IEQ1

[At"[aslIxt$] = A t " [ E Q 1 b s l ] .

(3)

Here IEQ, and IEQ, [.lX;] are the classical expectation and conditional expectation extended to operator-valued random variable. Let us also define the evolution

u(t,s)[7]:= EQ, [Aq[7]],

7

E 7('Ft),0

5 s 1.t ;

(4)

u(t,s)is a channel from 7('Ft) into 'T(7l). By points (2), (3), (4) of Assumption 1.2, for 0 1. r I s 5 t , CT,E L 1 ( X ,X , " , Q 1 ; 7 ( 7 l ) ) ,we get

u(t,~ o u)( s , r ) = ~ ( t , r ) ,

EQ,

[~t"[~~,]Ix,0] =u(t,s)[as].

(5)

83

The quantum continual measurements is represented by the operators in the sense that they give probabilities and state changes. If ~0 E S(3-1) is the initial state at time 0 and B E A’: is any event involving the output in the interval ( O , t ) , then Tr{A:[v~](z)}Ql(dz) is the probability of the

A:,

s,

A2p1(z)

is the state at time t , conditional on the result event B and %{At [sol(z)) z (the a posteriori state). Instead, tr(t,O)[vo] represents the state of the system at time t , when the results of the measurement are not taken into account (the a priori state). 2. The initial state and the measurement 2.1. Ensembles

In quantum information theory, not only single states are used, but also families of quantum states with a probability law on them, called ensembles. An ensemble { p , p } is a probability measure p(dy) on some measurable space (Y, y ) together with a random variable p : Y -+S(3-1). Alternatively, an ensemble can be seen as a quantum/classical state of the type described in Section 1.1.1. Given an ensemble, one can introduce an average state iJ E S(3-1) -

P := W P I =

s,

4dY) P(Yh

(6)

the integrals involving trace class operators are always understood as Bochner integrals. Finally, the average relative entropy of the states p(y) with respect to ;cj is called the “X-quantity” of the ensemble: X{P,P} :=

/Y

P(dY)Sq(P(Y)I;cj) = E, [Sq(PlP)] .

(7)

This new quantity plays an important role in the whole quantum information theory [3, 20and can be thought as a measure of some kind of quantum information stored in the ensemble. 2.2. The letter states

Let us consider the typical setup of quantum communication theory. A message is transmitted by encoding the letters in some quantum states, which are possibly corrupted by a quantum noisy channel; at the end of the channel the receiver attempts to decode the message by performing measurements on the quantum system. So, one has an alphabet A and the letters Q E A are transmitted with some a priori probabilities 8.Each

84

letter cr is encoded in a quantum state and we denote by p i ( a ) the state associated to the letter Q as it arrives to the receiver, after the passage through the transmission channel. While it is usual to consider a finite alphabet, also general continuous parameter spaces are acquiring importance [19, 201.

Assumption 2.1. Let ( A ,A,Qo) be a probability space with ( A ,d) standard Bore1 and let oi be a normal state o n L" ( A ,d,Qo;L(7-l)). Let us set

qi is a probability density and { f l , p i } is the initial ensemble. The average state and the X-quantity of the initial ensemble are

2.3. Probabilities and states derived f r o m 70

For 0 5 r 5 s 5 t we define:

Then, qt and &(z) are states on L(7-l),4; is a state on L"(X, X f , Q1) and 5; a state on L" ( X ,X f , Q1; L(7-l)).We have also

Moreover, there exists a unique probability PI on ( X , X ) such that P1(dz)IxP = g ( z ) Q l ( d z ) for all t 2 0. Also P1(dz)Ixt = 4;(x)Qi(dz) holds.

85 2.4. The general setup

It is useful to unify the initial distribution and the distribution of the measurement results in a unique filtered probability space. Let us set:

R :=A x X , UO

:= fi o T O ,

w := ( a , % ) ,

7r0(w) : = a ,

7rl(w)

qo := qi o TO = I l ~ ~ o l,( 1 PO := pi

0

=

:= z ,

(13a)

go , (13b)

II0 0 II 1

3 := A 8 X , Q := Qo 8 & I , 3; := { A x Y : Y E X,"}, 30:= { B x X : B E A } , = o { B x Y : B E A, Y E X,"}, . 3 t := 3 0 V

(13c) (134 (13e)

By defining A: := 11 8 A:, we extend to L1(R,FS,Q;T('H)) z L'(A, A, Qo) 8 L 1 ( X ,X:, Q1; I ( 7 - l ) ) . Similarly, we extend U(t, s) to L1(R,FS,Q;I(7-l))N L1(R,FS,Q)BT(7-l). Let us also set: ut := A:[(To],

CT:

:= 5: 0 7 ~ 1= A:[v~],

qt := (IotJJl,

(14a)

In the computations of the followingsections we shall need various properties of the quantities we have just introduced; here we summarize such properties. Let r, s, t be three ordered times: 0 5 r 5 s 5 t . Then, ot and uf are states on Loo(R,3t,Q; C(7-l)) and EQ[qt(3;] = EQ[qCIF:] = 4: , EQ[4:13~] = 4; 7 EQ[fltIF;] = EQ[g;IF:] = > EQ[fltl3s]= U ( t , s ) [ u ~, ] EQ[OClFs] =U(t,s)[U;],

=q s EQ[qtl3~]

7

EQ[gt"IFsl= V t

7

rlt = EQ[gt]I

nt = A,"[gs],

(154 (15b) (15~) (154

We have that {qt,t 2 0) is a non-negative, mean one, Q-martingale. Then, there exists a unique probability P on (R, 3 ) such that tlt 2 0 P(dw)IFt = qt(w)Q(dw).

(16)

Moreover, P(da x X ) = E(da), P(dw)lF;

= &(w)Q(dw)

,

P(A x dz) = Pl(ds), rlt = E P [ P ~=U(t,s)[qs]. ]

(17) (18)

86

3. Mutual entropies and informational bounds

Here and in the following we shall have always 0 5 u 5 T 5 s 5 t.

3.1. The state qt and the classical information Let us consider the state qt and its marginals E~[qtlF,.]= q,., IEQ[Q~~F,T] = q:. Then, we can introduce the classical mutual entropy:

Note that I c ( t ,t ) = 0. For mation gain:

T

= 0 we have the input/output classical infor-

By applying the monotonicity theorem and the channel EQ[O(F,]to the couple of states qt and qT&, we get SC(qtI%qtr)

2 SC(EQ[qtlFs]IEQ[qrqtrIFs]) = Sc(qs1qrq;)r

(20)

which becomes IC(T,

The function t

H

t>2 I C ( T , ).

*

Ic(s,t ) is non decreasing.

3.2. The state us and the main bound

A useful quantity, with the meaning of a measure of the “quantum information” left in the a posteriori states, is the mean X-quantity z ( s , t ) :=

s,

P(dw) sq(Pt(w)Ie%4) = E P [Sq(Ptld)] .

(22)

The interpretation as a mean X-quantity is due to the fact that x ( s , t ) = E P [EP [Sq(ptI&) IF;]]. But by Eq. (7) and Ep[ptlF;] = et”, Ep[efIF;I = &, we have that Ep [Sq(ptle:) IF;]is a random X-quantity. Note that -

X(t,t) =

s,

P(dw) S,(Pt(w)lr]t) =: X{P,PtI.

Let us consider the state

(T,

(23)

and its marginals EQ[n{(Ts}\F,]= q,.,

E Q [ ( T ~=)(T~:. ] Then, we have the mutual entropy s(aSlqT‘;)

= I C ( T , s)

+ z(T7s ) *

(24)

87

For

T

= s and for

T

= s = 0 this equation reduces to

S(gslqsvs) = x{P,ps},

S(~OIqov0)= x{P,P O } = x{pi, Pi).

(25)

By applying the monotonicity theorem and the channel A: to the couple of states us and qrul, we get S(gsIqrgQT)2 s(A,”[cs]IA:[qrgQT])= S ( g t I q r 4 ,

(26)

which becomes -

X(T, s) - X ( T , t ) 2

Therefore, the function t For T = s we get

H~

&(T,

t)-U

T , s)

2 0.

(27)

( st ), is non increasing.

S(gslqsvs)

2 S(gtlqs4)7

(28)

which gives the upper bound for I,:

0 L I,(% t ) I x{P, ps) - T ( S , t ) . For s

= T = 0,

(29)

it reduces to

The bound (30) is the translation in terms of continual measurements of the bound of Section 3.3.4 of [7], which in turn is a generalization of a bound by Schumacher, Westmoreland and Wootters [5] Equation (30) is a strengthening of the Holevo bound [3] Ic(O,t) 5 x { e , pi}.

3.3. Quantum information gain Let us consider now the quantum information gain defined by the quantum entropy of the pre-measurement state minus the mean entropy of the a posteriori states [l,2,4]. It is a measure of the gain in purity (or loss, if negative) in passing from the pre-measurement state to the post-measurement a posteriori states. In the continual case, we can consider the quantum information gain in the time interval ( s ,t ) when the system is prepared in the ensemble { e , p i } at time 0 or when it is prepared in the state vT at time T :

88

By this definition we have immediately Iq(rl t ) = Iq(r,s)

+

t),

Iq(u;r, t ) = Iq(u;r, s)

+ Iq(u;s, t ) .

(32)

It has been proved [4]that the quantum information gain is positive for all initial states if and only if the measurement sends pure initial states into pure a posteriori states. As in the single time case [6, 7, 81, inequality (27) can be easily transformed into an inequality involving I,:

Let us take an initial ensemble made up of pure states: ~ i ( a=)p ~ i(a), V a E A. Let us assume that the continual measurement preserve pure states: the states p t ( a , x) are pure for all choices of t ,a , x. Then, the von Neumann entropy of pt(w) vanishes and we have Iq(slt ) = 0 for all choices of s and t. From the second of Eqs. (32) and Eq. (33) we get

I,(u; r, t ) - Iq(u;r, s)

= Iq(u;s, t ) 2

Ic(u,t ) - Ic(u,s) 2 0,

(34)

i.e. the function t H Iq(u;r, t ) is non decreasing for “pure” continual measurements. In particular, by taking u = r = 0 we have

L l ( O ; O , t ) = Sq(r10) -

L

Wd~)Sq(Q%)).

(35)

For a continual measurement sending every pure initial state into pure a posteriori states, V ~ E O S(3-I) the quantum information gain 1,(0; 0, t ) is non negative, non decreasing in time and with Iq(O; 0,O) = 0.

Acknowledgments Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00279, QP-Applications, and by Istituto Nazionale d i Fisica Nucleare.

References 1. H. J. Groenewold, A problem of information gain by quanta1 measurements, Int. J. Theor. Phys. 4 (1971) 327-338. 2. G. Lindblad, An entropy inequality for quantum measurements, Commun. Math. Phys. 28 (1972) 245-249. 3. A. S. Holevo, Some estimates for the amount of information transmittable by a quantum communication channel, Probl. Inform. Transm. 9 no. 3 (1973) 177-183 (Engl. transl.: 1975).

89 4. M. Ozawa, On information gain by quantum measurements of continuous observables, J. Math. Phys. 27 (1986) 759-763. 5. B. Schurnacher, M. Westmoreland, and W. K. Wootters, Limitation on the amount of accessible information in a quantum channel, Phys. Rev. Lett. 76 (1996) 3452-3455. 6. A. Barchielli and G. Lupieri, Instruments and channels in quantum information theory, Optics and Spectroscopy 99 (2005) 425-432; quant-ph/0409019. 7. A. Barchielli and G. Lupieri, Instruments and mutual entropies in quantum information theory, t o appear in Banach Center Publications; quantph/0412116. 8. A. Barchielli and G. Lupieri, Quantum measurements and entropic bounds on information transmission, t o appear in Quantum Inform. Compu.; quantph/0505090. 9. A. Barchielli, Entropy and information gain in quantum continual measurements, in P. Tombesi and 0. Hirota (eds.), Quantum Communication, Computing, and Measurement 3 (Kluwer, New York, 2001) pp. 49-57; quantph/0012115. 10. A. Barchielli and G. Lupieri, Instrumental processes, entropies, information in quantum continual measurements, Quantum Inform. Compu. 4 (2004) 437-449; quant-ph/O401114. 11. S. Sakai, C*-Algebras and W*-Algebras (Springer, Berlin, 1971). 12. M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993). 13. J. Dixmier, Les Algbbres d’oplrateurs duns 1’Espace Hilbertien (GauthierVillars, Paris, 1957). 14. A. S. Holevo, Statistical Structure of Quantum Theory, Lect. Notes Phys. m67 (Springer, Berlin, 2001). 15. E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). 16. M. Ozawa, Quantum measuringprocesses of continuous observables, J. Math. Phys. 25 (1984) 79-87. 17. M. Ozawa, Conditional probability and a posteriori states in quantum mechanics, Publ. R.I.M.S. Kyoto Univ. 21 (1985) 279-295. 18. A. Barchielli, Stochastic processes and continual measurements in quantum mechanics, in S. Albeverio et al. (eds.), Stochastic Processes in Classical and Quantum Systems. LNP 262 (Springer, Berlin, 1986), pp. 14-23. 19. H. P. Yuen and M. Ozawa, Ultimate information carrying limit of quantum systems, Phys. Rev. Lett. 70 (1993) 363-366. 20. A. S. Holevo, M. E. Shirokov, Continuous ensembles and the X-capacity of infinite dimensional channels, quant-ph/0408176.

GENERALIZED q-FOCK SPACES AND DUALITY THEOREMS

ABDESSATAR BARHOUMI Department of Mathematics Faculty of Sciences of Tunis University of Tunis-El Manar 1060 Tunis, Tunisia E-Mail: abdessatar. [email protected]. tn HABIB OUERDIANE Department of Mathematics Faculty of Sciences of Tunis University of Tunis-El Manar 1060 Tunis, Tunisia E-Mail: habib. ouerdiane@ipein. M U . tn In this paper a new concept of q-symmetric tensor product is defined, where q is a parameter in the interval (-l,l]. Duality theorems are established for new spaces called generalized q-Fock spaces and some of their features are indicated.

1. Introduction

Recent development in quantum probability has led to the q-deformations of many physical constructions. Intensive mathematical investigations of quantum mechanical models for various q-deformed concepts have been pursued during the last decade, their importance spread up to the quantum field theory and the quantum statistical mechanics. These include various attempts to replace algebras, describing space-time structure or hidden symmetries, by their q-analogues called quantum algebras (or quantum groups). In [4, 51, Bozejko and Speicher studied the q-analogues of Brownian motions and investigated q-Gaussian processes with Kiimmerer in [7], which are governed by usual independence for q = 1 and free independence for q = 0. Their constructions were based on the so-called q-Fock space over a Hilbert space H and oriented to the interpolation between the bosonic, canonical commutation relation (CCR), at q = 1 and the fermionic, canon90

91

ical anti-commutation relation (CAR), at q = -1. A certain method for the construction of deformed Fock spaces was developed in [6],which is based on self-adjoint contraction on tensor product space H @I H , and the braids relations play an important role. Their construction is based on a deformation of the usual symmetrization operators by a parameter q E (-1, l),and includes the q-Fock space as the special case which will suggest us to make it better adapted to the analytical aspects. In the other hand, an explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed by Leeuwen and Maassen in [20]. Having these topics, we are interested in their infinite dimensional analytical q-deformed analogues where we keep in mind the framework of nuclear algebras of entire functions, tracing back to P.Krbe [14], developed by Ouerdiane and his collaborators. See e.g. [9] and references cited therein. This paper has the following structure. In section 2, using BozejkoSpeicher's self-adjoint strictly positif operator [4], we introduce a new tensor product denoted @I, and we give some of its properties. In particular, we show that our q-tensor product fulfills the inclusions

Ha"

H'q"

5

H B n ; q E (-l,l],

here H a n stands for the usual symmetric n-fold tensor powers. Section 3 is devoted to the presentation of the called generalized q-Fock spaces according to our q-symmetric tensor product with an auxiliary positive sequences On,, provided by a given Young function 0. The special choices of the Young function will yields many well-known, as well as new, interesting cases. In Section 4 we establish a duality theorem for generalized q-Fock spaces. 2. q-Symmetric Tensor Product

Throughout our paper we will often use the language of q-calculus. So, let us recall and adapt some results for the reader convenience. Let q E (-l,l] and n E N be given. The natural number n has the following q-deformation

+ + ... + qn-'

[n], = 1 q

with [O], = 0.

Then the q-factorial is defined as

[n],! = [I], [2], ... [n], with [O],! = 1.

92

Let S, denote the symmetric group of all permutations of the set (1, ...,n} ) the number of inversions of the permutation (T E S, defined and I ( ( T denote bY

{

I ( ( T )= # ( i , j ) E (1, ..,n}2 ; i < j and ( ~ ( i )> ( ~ ( j ). } Later on we need the next result. For different proofs see e.g. [4] and [3].

Lemma 2.1. For any n E N, q 6 (-1,1], we have the following identity

In q-theory, the most elementary q-hypergeometric series are the qexponential series. Here we introduce the following q-exponential function.

Definition 2.1. We define the q-exponential function Eq,for z E C,by

It is easily verified that, for q E (-1,1], the series in the right hand side of (2) have a radius of convergence R = 00. Moreover, for q = 1 the q-exponential function coincides with the usual exponential function, i.e., El(z) = ez , z E

C.

For two locally convex spaces XI,X2 let X1gX2 denote the algebraic tensor product of X1 and &. The completion of X1gX2 with respect to the n-topology is called the n-tensor product and is denoted by XI 8 X2. If both X1 = H , X2 = K are Hilbert spaces, H @ K stands for the complete Hilbert space tensor product. In particular H @ , denote the n-fold Hilbert tensor power of n identical copies of H , n E N. Let q E [-1,1] be fixed in the following. For each (T E S, let fiz be defined on the product vectors of H @ , by fi~<1@ . . ' 8
93

and

Remark that our normalization of PR' is motivated by the identity (1). Moreover, we recall here that in [4]its proved that has the norm 1. For the particular cases q E {-1, 0, l}, Pp' and are orthogonal projections and we have the two closed subspaces

Ppl

P2,

{

HQn = t

HA^

=

Ppl

1 P,["]t= t }

E H@ln

{t E H

B I ~P

~ =Sr } ,

which are respectively, the well-known n-fold symmetric (Bosonic) and anti-symmetric (Fermionic) tensor product of H . In order to have an interpolation between these bosonic and fermionic concepts, we make the following q-deformation.

<

Definition 2.2. An element E H@lnis said to be q-symmetric if P p ' t = t . The closed subspace of such q-symmetric elements, with respect to the inner product of the Hilbert space H@ln,is called the n-fold q-symmetric tensor product of H. We denote this complete Hilbert space by H@qn. Unlike the three particular cases q = -1, q = 0 and q = 1, for q E (-l,l)\{O}, is not a projection. For the proof, see e.g. [4]. Let : H*n --+ H@qn denotes the orthogonal projection on H@qn. We shall use the following notation

Pkl

~1 ~q

..

Bq

tn := ~F't1 B

* * *

B

tn

7

ti E H .

Then, it holds that the closed subspace generated by {[I Bq . . . gq& ; ti E H } coincides with H@qn. By Lemma 2.1, we easily see that for every 5 E H , Pkl<@n = which implies the following inclusions <@'"

Han

c H@qnc Han

;

q E (-l,l].

(4)

Moreover, for q E (-1, l), the inclusions in (4)are strict as shown in the following example. Example. Let H be a two-dimensional Hilbert space and let B = (el, e2) be an orthonormal basis of H . It is known that

BB.,:= {ei, 8 ei2 @ ei3 ; 21, 22, 23 E

{1,2}}

94

where

I.=(:!)

l + q q + q 2 q2+q3 1+q3 q + q 2 q2+q3q+q2 l + q

(

; A,=

q+q2

Therefore, the characteristic polynomial of M, is

x q ( X ) = ([3],! - XI4 (aq - XI2 ( P q - XI2 where cyq

= (1 - q ) ( 1

+ qI2

;

Pq

= (1

+ Q)(1- d 2 .

Pq # [3],! and

Remark that, for any q E ( - l , l ) \ { O } ,

1

aq= [3],! H q = -2

hence, if q = -3,1 [3],! is an eigenvalue of [3],! PF1 and the associated eigenspace has dimension 6 . Assume that q = and put

-a

6=

C il,i

z , it

(221

+

22)

ei, €3 ei, €3 ei3,

95

3. Generalized q-Fock Spaces Let N be a complex nuclear l?r&het space whose topology is defined by a family of increasing hilbertian norms {I. I p ; p E N}. For p E N we denote by N p the completion of N with respect to the norm l.lp. Then

N = projlim N p . P--t+oO

Denote by N-, the topological dual of the space N p . Then by general duality theory, the strong dual space N' can be obtained as

N' = indlim N-, P-++oO where the strong dual topology and the inductive limit topology of N' coincide due to the nuclearity. In a similar way, one has

and this equality is a topological isomorphism when N@gnis equipped with the r-topology and N:9n is equipped with the projective limit topology. Furthermore, we have the topological isomorphism

nEP=,

We denote by (., .) the canonical bilinear form on N' x N

96

Lemma 3.1. Let p E N and p' > p such that the natural embedding i,),, : Npj c--) N p is of Hilbert-Schmidt type. Then'$:i : N:qn c--) N:qn have a Halbert-Schmidt operator satisfying the inequality

Moreover, for every

fn E

N:qn, we have lfnlp

I IfnIp'

lliP~,Pll;s

(7)

*

Proof. Let { + k , n } k E p l be an orthonormal basis of N:',. Then, by completing {+k,n}kENl we obtain an orthonormal basis {$i,n}iEN of N:". By the general duality theory, there exist a unique isometric anti-linear isomorphism cp -+ cp* from N:" onto N?; such that (cp*,$)

= (cp,$),

;

cp, $ E N:"

where the right hand side is the hermitian inner product of the Hilbert space N:". Hence, the Fourier expansion of cp E N:" is expressed in the form

i=l

i=l

$z,n}iEW

Moreover, as is easily verified, { becomes a complete orthonormal basis of N?;. The Fourier expansion of @ E N?; is expressed in the form M

00

i=l

i=l

Consequently, taking nuclearity into account, we compute

~

follows

as desired. Now, let us compute the norm estimate of an arbitrary f n E N;qn

k=O

k=O

~

HS

as ~

~

97

where (7) is taken into account.

0

In all the remainder of this paper, we take 0 < q 5 1, although most of the formulas remain correct when -1 < q < 0. The limiting case q + 0 is of some mathematical interest in progress in the frame of free probability. This can not be discussed here for lack of space and will appear elsewhere. Let 8 : R+ -+ IR+ be a continuous, convex, increasing function satisfying e ( x ) = +m lim -

x-++cy)

x

8(0) = 0,

and

(8)

such a function is called a Young function. For a Young function 8 we define a positive sequence {8n,q}nENby := inf Eq P(7-11 , n = 0 , 1 , 2 ,... r>O

rn

where Eq is the q-exponential function defined in (2). We preserve the notation 8;,q for the sequence associated to the q-polar function 8; of 8, defined by

We remark that the 1-polar function 8; coincides with the well-known polar function 8*, associated to 8, given by

For p E

N and y > 0 we define the Hilbert space

where f n E N:qn and

This is called the generalized q-Fock space with parameters p and y. By a standard argument we verify that F;,,(N,) ; p 2 0, y > 0} is a projective system of Hilbert spaces. Thus, define

{

F; ( N ) := projlim F;,,(N,) ,-+Oo;

,LO

98

which is called the space of formal q-power series of N .

Theorem 3.1. $ ( N ) is a Fre‘chet reflexive nuclear space. Proof. F: ( N ) is the projective limit of reflexive Frkchet spaces, then it is a reflexive Frkchet space. Let p 2 0 and y > 0 be fixed . Our aim is to find convenient y’< y and p’ > p such that the natural embedding 1;:

-

: F,q,@P4

is of Hilbert-Schmidt type. For any n E

F&(NP)

N,let {$j,n}jENbe an orthonormal

+

basis of

& = {8n,py’?&,n}j EN. of Fi,,,(N,t) and we can

and put

an orthonormal basis

Then,

{ &},

becomes

estimate the norm of the

I,,’P,’ 9P in the following way

To conclude, it is sufficient to choose p‘ > p and y’ < y in such way that 0 i p ~ ,ispof Hilbert-Schmidt type and \ ~ i p ~ < , p1.~ ~ ~ s

$

4. Duality Theorems

Similarly as in the previous Section, we shall define the generalized q-Fock space Gi,7(N-p). Let p E N,y > 0 be given. For 6 = with

(@,)r=,,

an E N ! ; ~ , we put

Define

and

Gi(N’) :=

indlim

p - - r + m ; ,--roo

G:,,(N-p).

Then, we come t o the following duality relation.

99

Theorem 4.1. The strong dual FZ(N)* of F,Q(N)is identified Unth GBQ( N ' ) through the canonical bilinear form

n=O

where, for each n, (N@99' x N @ q n .

(an, fn)

is the canonical bilinear dual pairing on

Proof. Let 6 = ( @ n ) n E~ G ~Z (N'). Then (($, .)) is a continuous linear functional on F l ( N ) . In fact, for f= (fn)nEw E F i ( N ) , we have

Conversely, for T E F i ( N ) * , there exists such that, for any

!?

=

(Tn),EN with Tn E N@qn

00

we have

n=O

The linear functional T is continuous for a certain norm denote its norm by IITllp,p,rand we set *

=

with

an := q

w

((.Iq!)-l

~ ~ . ~ ~ q , ~ ,We p,7.

T,.

Let p' > p be such that iP/,, is of Hilbert-Schmidt type. For each n E N, we consider an orthonormal basis {@j,n}jEN for Then, the inequality

N7qn.

[ ( T n , o n , q ~ ' @ j , n )5[ ItTIIq,p,7IQij,nIp implies that

100

where Lemma 3.1 is taken into account. For any y' < y, t h e last estimation yields the following bound for 6

We conclude by choosing y, y' such t h a t

$ l l i p ~ , p <~ l1.~ s

References 1. S. Albeverio, Yu. G. Kondratiev and L. Streit, How to Generalize White Noise Analysis to non-Gaussian Spaces, in: Ph. Blanchard et al. (eds.), Dynamics of complex and irregular systems, World Scientific, Singapore, (1993). 2. N. Asai, I. Kubo and H. -H. Kuo, General Characterization Theorem and Intrinsic Topologies in White Noise Analysis, Hirochima Math. J. 31 (2001), 299-330. 3. A. Barhoumi and H. Ouerdiane, Infinite Dimensional q-Holomorphy and Duality Theorems, Preprint 2005. 4. M. Bozejko and R. Speicher, An example of generalized Brownian motion, Comm. Math. Phys. 137 (1991), 519-531. 5. M. Bozejko and R. Speicher, An example of generalized Brownian motion 11, Quantum probability and related topics VII (1992), 67-77. 6. M. Bozejko and R. Speicher, Completely positive maps o n Coxter groups, deformed commutation relations and operator spaces, Math. Ann. 300 (1994), 97-120. 7. M. Bozejko, B. Kummerer and R. Speicher, q-Gaussian Process: NonCommutative and Classical Aspect, Comm. Math. Phys. 185 (1997), 129154. 8. W. G. Cochran, H.-H. Kuo and A. Sengupta, A new class of white noise generalized functions, Infin. Dimens. Anal., Quantum Probab. Relat. Top. 1 (1998), 43-63. 9. R. Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui, U n the'ordme de dualite' entre espaces de fonctions holomorphes & croissance exponentielle, J. Funct. Anal., Vol. 171,No. 1 (2000), 1-14. 10. G. Gasper and M. Rahman, Basic hypergeometric functions, Cambridge University Press, (1990). 11. I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4, Academic Press., New York and London 1964. 12. T. Hida, H. -H. Kuo, J. Potthoff and L. Strait, White Noise: An Infinite Dimensional Calculus, Kluwer academic Publishers Group, 1993. 13. T. Hida, Brownian Motion, Spriger, Berlin-Heidelberg-New York 1980. 14. P.Kr&, Propri6te' de trace e n dimension infinie d'espace de type Sobolev, Bull. SOC.Math. France 105 (1977), 141-163.

101 15. P. Kree and H. Ouerdiane, Holomophy and Gaussian Analysis, Prepublication de l’institut de Mathematiques de Jussieu. C.N.R.S. Univ. Paris 6 (1995). 16. H. -H. Kuo, White Noise Distribution Theory, CRS. Press, 1996. 17. N.Obata, White Noise Calculus and Foclc spaces, Lecture Notes in Mathematics Vol. 1577, Heisenberg Springer-Verlag, 1994. 18. H. Ouerdiane, Fonctionnelles analytzques avec conditions de croissance et application c i l’analyse Gaussienne, Japan. J. Math. 20, No.1 (1994), 187-198. 19. K. R. P. Parthasaraty, A n Introduction to Quantum Stochastic Calculus, Monograph in Mathematics, Vol. 58, Birkauser Verlag, 1992. 20. H. Van Leeuwen and H. Maassen, A q-deformation of the Gauss Distribution, J. Math. Phys. Vol. 36,No. 9 (1995), 4743-4756.

COVARIANT QUANTUM STOCHASTIC FLOWS AND THEIR DILATION

VIACHESLAV BELAVKIN University of Nottingham, University Park, Nottingham NG7 2RD) England e-mail: uiacheslau. [email protected] LEE GREGORY University of Nottingham, University Park, Nottingham NG7 2RD) England e-mail: [email protected] A characterisation is made of not necessarily bounded quantum stochastic flows with bounded generator matrices in the general algebraic setting of von Neumann and It6 algebras. This characterisation is then used to construct dilations of filtering and contraction flows which are homomorphic and reduce to the original flow, giving stochastic generalisations of the Stinespring and Evans-Hudson dilations. Flows covariant with respect to a group action are also studied, along with their generators and covariant dilations.

1. Introduction

A very well known theorem of quantum mechanics, due to Stone, states that the generator of a strongly continuous group of unitary operators is antiHermitian. This result underpins the famous Schrodinger and Heisenberg equations describing the unitary evolution of a closed quantum system. Interest in irreversible open Markov quantum systems in the 1970s lead to the generalisation of Heisenberg’s equation, and to the study of semigroups of completely positive maps whose generators were classified by Lindblad [14]. Small revisions and generalisations followed [lo] until the discovery of quantum stochastic calculus in 1984 [13] which made it possible to consider stochastic evolutions. 102

103

We shall be concerned with time indexed completely positive maps cjt : A H B(9)giving the stochastic time evolution of all bounded system variables in a von Neumann algebra A. B(9)denotes the linear space of continuous kernels mapping a FkBchet space 53 into its topological dual 53*. In the bounded case $t has values in a von Neumann algebra U = A €3 M , M the noise von Neumann algebra. Among the properties that $t is assumed to have (detailed below) is that of quantum stochastic differentiability. Hence $t satisfies a quantum stochastic differential equation

with generator matrix X(A) and A the usual matrix of quantum stochastic integrators. The obvious project is then to classify the generators of these quantum stochastic flows to give a stochastic generalisation of Lindblad’s result. Work in this direction was published first by Belavkin [7] who characterised such flows in the case d = L(‘H), L(1-l) all continuous linear operators on a Hilbert space 1-l’ with B ( 9 ) generated by the commutant of an arbitrary It6 algebra. Finite dimensional characterisation was made by Lindsay and Parthasarathy [15] when A is a C*-algebra, with flows having values in the space of all unbounded operators densely defined on the exponential vectors of a Fock space. Later this was extended to infinite dimensional generator matrices with bounded elements by Lindsay and Wills [16]. Our treatment below will be for the case of bounded generators and relates to those mentioned above in the following way. Our algebra d is more general than in Ref. [7], but there unbounded generators were considered. Our It6 algebra is more general than in Refs. 115, 161, where flow values are in the space of all unbounded operators densely defined on Fock space, so the noise space is generated by the whole Hudson-Parthasarathy algebra as the commutant to a trivial It6 algebra. They considered a general C*-algebra A and Ref. [16] introduces some domain restrictions in order to work with unbounded generators with bounded matrix elements. Characterisation of the generator enables dilations to homomorphic flows, so that the original flow is recovered as the reduced dynamics. Two dilations will be presented, the first being of Stinespring type, giving a dilation independent of the original flow with reduction given by a quantum stochastic process. The second is of Evans-Hudson type, where the dilation now depends on the flow but the reduction is given by the vacuum conditional expectation. Flows which are covariant with respect to some group

104

action play an important role in physics, so it is important to consider the characterisations and dilations mentioned above within this context. This will be done at the end of the paper, and tied in to the previous results. 2. Quantum Stochastic Flows 2.1. Preliminaries

Here will be given an introduction to the definitions and motivations regarding quantum stochastic flows relevant to us, and some notations fixed. The subject of this paper is most often phrased in the language of unbounded operators densely defined on a Hilbert space. However, we find it much more natural to formulate everything in terms of continuous sesquilinear forms on Frkchet spaces and their associated kernels. This contains the usual formulation, but allows notions of continuity and composition to be more readily handled. Let 3 denote a Frkchet space dense in a Hilbert space 3with topological dual 5* 2 3 2 3, and let B(3)be the linear space of all continuous linear maps 3 H r ,the topology on 5* the weak-* topology induced by 3. Associated with every continuous (in the Frechet topology) sesquilinear form M on 3 8 3 is a continuous linear map called a kernel, also denoted by M , with M E B ( 3 ) (see Ref. [18]). The kernel is defined by M ( f , g ) = ( M f l g )for f , g E 3, (f*lf) the canonical pairing for f * E 5* given by llf112 if f * E 3. The converse, that every continuous kernel defines a unique continuous sesquilinear form, also holds, and this correspondence allows us to equip B(3)with the weak-* topology induced by 3 8 3. The conjugate kernel M* E B(3)is given by the form ( M * f l g )= ( M f l g ) * . This formulation generalises the usual setup of unbounded operators as linear maps 3 H 3 where 3 usually has no specific topology of its own, hence precluding notions of continuity. The algebra of bounded operators is a subalgebra of the linear *-subalgebra L(3) C B(3)of continuous (in the Frkchet topology) linear operators 9 H 9 from our framework. In particular, the bounded operators are those operators in C(3) which are continuous with respect to the Hilbert topology and hence are extendable to the whole of 3. That L(3) is a *-algebra is a vital fact since in the next section we will construct a *-subalgebra N C C(3) from a representation of the It6 *-semigroup in L(3). Further, it allows us to define the commutant of N in B ( 3 ) , by constructing the dual operator N t E L ( T ) of any operator N E L(3) by ( N t f * l g ) = ( f * I N g ) , and the unique weakly continuous

r,

105

extension of N to L ( r ) by N*t which will be denoted by N again. Then the formal definition of commutation N M = M N for an operator N and a kernel M makes sense, and can be written in terms of the sesquilinear forms: ( N M f I g ) = ( M f l N * g ) .Hence we can define the commutant M C B(3)of any *-subalgebra N of L(3). With the background assembled, let us clarify the object of our study. The von Neumann algebra A L(7-1)is the initial observable algebra for a quantum system under indirect observation, having separable Hilbert space 7-1. It is the commutant A = B' = {X E L(7-1): [X, 21 = 0,Z E B} of an involutive subalgebra B C L ( X ) which is not needed to be a von Neumann algebra, so 23 23'' = A'. The linear space U B(9)contains the results of the quantum stochastic evolution given by maps +t : A H U. Analogously to the initial space, U = 93' is the commutant of the *-subalgebra 23 L(9) where 9 G' c 9* is a F'r6chet pre-Hilbert subspace of the system plus environment Hilbert space G'. As alluded to by its name, G' is decomposable as = 7-1 @ 3 for an environment Hilbert space 3 which also has a F'r6chet pre-Hilbert subspace 3 5 3 C giving 9 = X @ 5. This results in a decomposition of U (respectively B) into A @ M (B@ N ) for a linear subspace M B(5)(a *-subalgebra N L(3)) with M N = N M . The *-algebra N will sometimes be referred to as the output noise algebra; when $t describes a measurement, N is the algebra of apparatus observables. The decomposability of U (and likewise 23) gives a useful way of thinking of its elements as A-valued continuous sesquilinear forms on 5, so that A ( f , g ) E A for A E 8,f , g E 3. This allows us to define the composition of two maps 4, 7 : A H by 407(A)(f, 9 ) = 4 ( ~ ( A ) ( fg, ) ) ( f ,9 ) . The maps we are interested in axe time indexed sets of time homogeneous Markov flows + t , so the composition will be a cocycle identity. This requires a filtration on U, defined as an increasing family of subspaces of U; Ut Usfor t s. Defining abstract shift maps ct : UsH Us+t,s , t E R+ allows us to make 4t into an abstract flow by requiring that 4s 0 0,o 4t = 4s+t. These preliminaries are sufficient to define a quantum stochastic flow in an abstract sense as a continuous, completely positive, adapted set of time indexed maps 4t : A H U satisfying the cocycle identity. However, to perform calculations with flows it is useful to consider a more concrete framework, in particular, using a quantum It6 algebra to generate the noise algebra N , and modelling the environment Hilbert space .F as a bosonic Fock space.

c

r

c

c

<

106 2.2. The Quantum It6 Algebm

This section shows how the output noise algebra N can be concretely realised. Turning to the physical interpretation for motivation, in particular to quantum measurement, N C M’ is a dense subalgebra in M’ of “nice” observables for a measurement apparatus (quantum or classical) coupled to the original quantum system. For a single instantaneous measurement described by a bounded Y E L ( k ) , k the Hilbert space of the apparatus, N is the polynomial algebra generated by Y , so is a subalgebra of the bounded operators L ( k ) on the environment (apparatus) Hilbert space k = 3. For multiple instantaneous measurements 7 a finite discrete index set, the Markovianity forces the use of several copies of the apparatus space k since the apparatus needs to be effectively reset every time a measurement is made (assuming finite time has passed between measurements). Choosing N as a subalgebra of L(k@IT1),with 3 = k@lTl in this case, will do the job. Analogously to before, the algebra generated by finite combinations Yt, 8 . . . @ Ytlr,is sufficient. For continuous time measurements we are forced toward some environment Hilbert space which has a continuous tensor product structure, and the natural choice is the bosonic (or symmetric) Fock space (L2(R+; k ) ) . In this Fock space representation, = ‘FI @ 3 k , and F k = the dense subspace 9 = ‘H 8 z k , where 5 k is a Frkchet pre-Hilbert subspace of F k , and is the natural domain on which the solutions of quantum stochastic differential equations are continuous [7, 21. The instantaneous apparatus observables become a time dependent observation process {Yt}tEw+in L ( $ k ) , adapted with respect to the Fock filtration: yt E L(zk,[O,t)) where g k , I = z k n 3 k , I , 3 k , 1 = r ( L ( ~ I ;k ) ) for any interval I C R+. The shifts ut : B(9)H B(Di)[t,,)), with 91= ‘FIB z k , I , are the natural Fock space shifts induced by the shifts on L2(R). Once more N is generated using the observation process, with Nt = N n L ( 5 k , [ o , t ) ) . The algebra of observables on a Fock space is usually taken to be generated by the Weyl operators. However we need to be a little more specific, since the algebra generated by yt will generally not be the same as that generated by all the Weyl operators. To this end, consider the infinitesimal forward increments dYt = Yt+dt - Yt. They form an algebra with multiplication rules defined via the quantum It6 formula [13]. This algebra is not a von Neumann, or even a C*-algebra, but when paired with a vector state can be abstractly defined as an It6 algebra:

{x}kET,

Definition 2.1. It8 Algebras. An It6 algebra is a pair ( 6 , l ) consisting

107

of a non-unital complex associative *-algebra b and a positive linear *functional 1 : b H C. The algebra b contains a “death” element d which annihilates all other elements; d b = {0}, and on which l ( d ) = 1.

It is not immediately clear from the definition of an It6 algebra that it encodes the multiplication rules for quantum stochastic process increments. However, the definition of an It6 algebra is abstracted from the properties of the algebra of differentials of operator-valued integrators on a Hilbert space, and the following representation shows this is in fact always the case, hence giving the expected multiplication table. First some notation. Let p, v E {-, 0 , +}, 0 be the indexing set {1,2, ..., dim(k)} whose elements index an orthonormal basis of the separable Hilbert space k , dA:(t) be the creation operator differential in Z k , dA’_(t) the annihilation, dA:(t) the preservation, dA+(t) = dt the time, and all other dAL = 0. Let M(k) denote the operator matrix *-algebra on a pseudopre-Hilbert space k = C @ t @ C where t is a locally convex topological pre-Hilbert space dense in k . The involution is given by b* = gb*g where * is the usual matrix adjoint of b E M(L) with metric g l = Sc”, and -(-, 0 , +) = (+,0 , -). Let BA(k) denote the upper triangular subalgebra of M(k) consisting of all continuous kernels b E M(k) such that (b(z,e , y)TI(z, e, y)T) = (b(0,e , y)T((z, e , 0 ) T ) for any e E t, z, y E C forming columns (5, e , y)’, and let LA&) be the usual *-subalgebra preserving k. Continuity is defined with respect to the restriction of the indefinite metric on k to 0 @ t @ C , which makes it positive definite and gives the topology coinciding with that on t. Theorem 2.1. The Belavkin Representation for It6 Algebras [S, 4 , 31. Every It8 algebra (b, I) has a canonical representation in LA@)f o r some Minkowski space k,given b y the *-homomorphism

[1

“1

[:111

k;b) i ( b ) = 0 r ( b ) k ( b ) , i(b*) = gi(b)*g, g = 0

0

,

(2)

where b E b and g is the metric defining the indefinite scalar product in terms of the standard Euclidean inner product, and l ( b ) = h*i(b)h with h* = (l,O,O). In Eq.(2), 7r is the GNS representation 7r(b)k(c)= k(bc) of b 3 b,c on the minimal pre-Hilbert space e = k ( b ) with respect to the functional 1 which has the Kolmogorov decomposition I(b*b) = k(b)*k(b). The locally convex topology on t is that induced by the the norm b H Ilk(b)l( and the

108

seminorms b H IIk(bc)ll for all c E 6, so that the representatives n ( b ) are continuous with respect to this topology, n ( b ) E ,C(e). We shall assume in the following that the It6 algebra is separable in the sense that there are at most a countable number of seminorms b H IIk(bc)JJ,so that e k is a l?rBchet pre-Hilbert space, bringing it in line with our considerations above. Note that 5 k is now generalised to 5e, a Frkchet space dense in r(L2(B; e)). One can also go further, and show that any representation of b by the quantum stochastic differential d A ( t , b ) of A(t, b) as independent increments on a Hilbert space can in turn be represented as it(b)AL(t)in the symmetric Fock space Fk. There are three canonical commutative It6 algebras which serve as guiding examples whilst gaining familiarity with this framework. Let us introduce the matrices

d=

000 000

1

d,=

001 000

[ 0 1 0 ] 1

d p = [0, l1o1] . 000

(3)

Firstly consider the almost trivial one dimensional It6 algebra n spanned by the death element d , so that n = C d . This can be called the Newtonian algebra since it has the same multiplication table as C d t in Newtonian calculus. The second example is the two dimensional algebra tw spanned by the death d and an idempotent element d,, d,d, = d . This is called the Wiener algebra for the obvious reason. Finally there is another two dimensional algebra p spanned by d and an element d,, with multiplication dpdp = d , + &. This is the Poisson algebra, as d, has the same multiplication as the forward differential of a compensated Poisson process. Higher dimensional versions of these algebras can be easily defined by taking the bases

:I,

[;;

Oe; 0

dk=

0 e; 0

d2p=

(4)

with ei the standard basis column vector having all zero entries except for the ith component, which is 1. The d dimensional Wiener and Poisson algebras will be denoted twd and p d respectively, and we also introduce the notation bp for the Hudson-Parthasarathy algebra of all upper triangular matrices of the form (2). These examples will be revisited in the sequel, and for some interesting results concerning these and other It6 algebras we refer to Ref. [8].

109

When manipulating the increments of a quantum stochastic process it is sufficient to work with the appropriate It6 algebra, where the components of the representing upper triangular matrix (2) are the coefficients of the standard integrators in differential equation defining the process A(t,b):

dA(t,b) = b:dA:

+ b;dA: + brdA'_ + bTdA?

(5)

where b: = r ( b ) , b; = k ( b ) , b; = k*(b) and bT = Z(b) from above. Returning to the general case, taking the It6 algebra b as that generated by d K identifies yt with A ( t , d K ) in (5). Now b is used to generate the algebra of observables on & by exponentiation, in a way analogous to the exponentiation used to define Weyl operators.

Definition 2.2. The Noise Algebra. The noise algebra N is the involutive algebra generated by solutions W ( t ,b) of the quantum stochastic differential equation

dW(t,b) = W ( t ,b) 8 dA(t,b),

W(0,b) = 1 ~ ~ .

(6)

These generalised Weyl operators are relatively bounded, W ( t ,b ) E C(Se), under the assumption of projective contractivity of b: ,:S and they give a representation of the unital *-semigroup 1+ b with respect to the product b* b = b b*b b*, see Ref. [7]. Note that W ( t ,b) is unitary if and only if (1 b)-' = (1 b)*. It will be a consequence of theorem 2.2 in the next section that since M is the commutant of N , the It6 algebra a generating M via equation (6) is exactly the commutant of 6. Before moving to the next section, let us give the rationale for using commutants in the way they have been above. This is mainly motivated by causality, but has important mathematical repercussions. The environment process Y t introduced above models the actual process observed by an experimenter seeking knowledge (necessarily indirectly) of a system process X t E A. Slightly generalising, we may consider Yt E !Bt = B 8Nt . By standard arguments of simultaneity of observables, information about X , can only be obtained from Yt,t s, if the following commutation conditions are fulfilled (which they are above).

+

+

+

+ +

<

Definition 2.3. The Non-Demolition Condition. The observation process Yt E !Bt is said to be non-demolition with respect to the generalised system process Xt E Ut if the non-demolition condition

[K,X,] = 0, is satisfied.

w <s

(7)

110

This is why we assume that the time evolution given by the maps 4, : A H B(D)has values in Ut = 23;. These commutativity conditions ensure that the generator of the flow commutes with all elements of 6, as will be proved in the next section. This will be needed in the characterisation of flows to construct a useful representation of 23. 2.3. The Calculus of Quantum Stochastic Flows

With these preliminaries dealt with, quantum stochastic flows and cocycles can now be defined.

Definition 2.4. Quantum Stochastic Cocycles. A quantum stochastic cocycle is a family of linear maps 4, : A H U indexed by t E R+ and satisfying the following conditions, (1) Complete positivity: Ci,j(q5t(A5Aj)qiIqj)2 0, for all t and any finite family (qi,Ai) E D x A. ( 2 ) Adaptedness: + , ( A ) E Ut VA E A. ( 3 ) Cocyclicity: +s+t = +s o (T, o +t. (4) Normality: Each map +t is continuous with respect to the ultraweak and weak-* topologies on A and U respectively.

We will be interested in cocycles satisfying quantum stochastic differential equations.

Definition 2.5. Quantum Stochastic Flows. A quantum stochastic cocycle is a quantum stochastic flow (equivalently, quantum stochastically differentiable) if it satisfies a quantum stochastic differential equation of the form d+t(A)=4toX;(A)@dAi, where the bounded normal generator map

+o(A) = A @ I F ~ ,

(8)

X has values X(A) E A @ LA@).

+

The family of flows : (R+, d) H U will be denoted Q(A,a), where a = 6’ generates M . Note that X is time independent due to the cocycle property of 4,. We will be only concerned here with the case when X is bounded (uniformly continuous) and normal (ultraweakly continuous). This means that t is identified with the Hilbert space k and each map E C(A) is normal. One can show [2] that uniform continuity implies

4 t ( 4 E W). To finish the definitions, the notion of martingale and hence of conditional expectation must be introduced. For the subspace Dt = 3-1 @ S k , [ o , t )

111

the vacuum conditional expectation E t : B(9)H B ( 9 t ) is defined as usual by the isometric embeddings Et : 9t H 9 ,Etqt = $ t @ f l k , [ t , m ) for .1Clt E 9t, flk,[t,,) the vacuum vector in 9poo). Hence E ~ ( A = )E,*AEt for A E U, where we have extended E,* to C(9'). A quantum stochastic process At is a submartingale if € , ( A t ) A,, with equality giving a martingale.

<

Definition 2.6. Filtering and Contraction Flows. A quantum stochastic flow $,(A) is called a contraction flow if IG 2 $t(I%) 2 ~ $ J ~ ( I N ) , t s, and a (sub)filtering flow if $ t ( I N ) is a (sub)martingale.

<

The question of what analytical conditions on a cocycle imply it is a flow (as well as many other questions) have been examined in a series of papers by Lindsay and Wills [ll, 17, 161. They considered the case of uniformly continuous maps A: (with further results for additional ultraweak continuity), with a C*-algebra as the initial algebra and none of the commutation considerations and It6 algebra generalisation given here. In particular, contraction cocycles whose Markov semigroup is norm continuous are the (unique) solution to an equation of the form (8) for some completely bounded A. Note that this question of when a cocycle is a flow can also be addressed by the representation theorem 2.1, which allows any quantum stochastic process t H $ t ( A ) having independent increments d$t(A) = $ t + d t ( A ) $,(A) forming an It6 algebra to be represented in the symmetric Fock space F k with the vacuum vector state. Focusing on flows from now on, the following corollary [5] of the representation theorem 2.1 gives a vital link between a flow and its generator. Let the map &'(A) = SEA be the ampliation of A to A @ Ik, used to define the quantum stochastic germ [7] y = X L of the flow &.

+

Theorem 2.2. The values $ ( A ) of a quantum stochastic flow $ E Q(A,a) satisfying equation (8) are algebraically isomorphic to $ o y(A). Proof. Consider the It6 algebra generated by the increments @ , ( A ) for all A in its upper triangular representation (2), where the matrix elements are the coefficients X:(A) in (8). Explicitly this is

Clearly the right hand side of flow equation (8) results from taking the trace of the matrices [A;] and [dh:]. The quantum It6 formula [13]

112

for quantum stochastic processes X t and yt, which have domains allowing multiplication, is

d (XtY,)= dXtyt

+ Xtdyt + d X t d y t ,

(10)

with the multiplication table

It is simple to see that the upper triangular representation combines the formula and table into the relation

d ( M A 1*t.)

(A211 = [(+t 0 Ag(A1))(G 0 @;(A;))* + (4t O A:(Al)) + ( n O @:(A;))*];€4 d q = [($t 0 y:(Al)) (Tt

O

P E W ) * - (4t

O

G(AI))(TtO ~x4;))*];

@d

q , (12)

where Tt satisfies the analogue of (8), dTt(A)= ~t o (P - L)E(A) @ dAL, and .:(A) = T ~ ( A * ) * . Hence the map $ ( A ) H (4 o y ) ( A )is multiplicative and preserves the involution, $ * ( A )H ($oy)*(A)with ($oy)*(A)= (+oy)(A*)*, and so is *homomorphic (linearity is obvious). Injectivity follows from the definition of the germs; if $oy(A)= 0 then +(A)= 0 since $ o y I ( A )= +oy$(A)= $ ( A ) . Surjectivity is also immediate since +(A)= 0 + (+ o y ) ( A )= 0. This theorem will be used repeatedly in the sequel to prove results about germ matrices concisely that would normally require complicated uses of the quantum It6 formula. As a first use, let us prove the assertion mentioned in the previous section, which will be used later in the characterisation of flow germs. Note whenever an It6 algebra is mentioned, it will be assumed to be in the canonical upper triangular representation in L A(k).

Lemma 2.1. The g e r m y of a flow

4

an Q ( d , a) has values in d @ a.

Proof. For y ( A )to be in d @ awe need that [ y ( A )I,x @ b ]= 0 for all b E 6, since yE(A) E A by the definition of a flow. We have that [+t(A), I x @ N t ]= 0 for any Nt E Nt, and in particular for Nt = W ( t ,6) since Nt is generated by b through equation (6). The isomorphism theorem 2.2 then asserts that

113 [+t 0 $A), Ix @ bW(t,b)] = 0, which can be rearranged using the original commutator to give

(Ix@ W ( t ,b))$t O [ y ( A )bl,

= 0,

since bc E @.

(13) 0

Finally for this section let us exhibit two canonical examples of equation (8) which are commonly studied, and which serve as paradigms for much of the subject. Take & ( A )= W J ( A @ IF~)W for~some quantum stochastic operator Wt E L ( 9 ; G ) having dual operator WJ E L(G;9'). The first example is when Wt satisfies the equation

dWt - KWtdt = LiWtdP,Z

(14)

where K,Li E d,and dP," = d$dAL is the dim(k)-dimensional compensated Poisson process with normalised intensities. It is clear that the germ of Wt is in the algebra d @ Pdim(k), and Wt is driven by a Poisson process. A simple application of theorem 2.2 gives

d+t(A)= &(L;ALi-K*A-AK)dt+4t((L;+I)A(Li+I)-A)dP;. (15) Secondly, take

dWt - KWtdt = LiWtdQf

(16)

where K , Li E d,and dQ1 = d%vdAL is the dim(lc)-dimensional Wiener process. This germ now belongs to d @ twdim(k), the evolution driven by the Wiener process. The flow is

d$t(A)= $t(LfALi - K*A - AK)dt + $t(LfA+ ALi)dQi.

(17)

These two examples will be unified in the characterisation of flows given below, and provide interesting examples for the accompanying dilation and covariance results. 3. Dilation of Quantum Stochastic Flows 3.1. Classification of Generators

Since flows have been defined as completely positive, it is possible to give an explicit classification of their germs y. The following theorem constitutes an addition to those appearing in Refs. [16] and [7] in the sense that general It6 algebra a and initial algebra d are used. Our result can be stated succinctly as y ( A ) = cp(A) K L ( A ) L(A)K*for some completely

+

+

114

positive map cp E L ( d ; d @ a )and operator K E C(d@a). The proof uses similar arguments to those applied in characterising generators of quantum dynamical semigroups, and constitutes an infinite dimensional (but bounded) generalisation of that appearing in Ref. [15],holding for general It6 algebras.

Theorem 3.1. Canonical form of flow generators. Associated with every q5 E Q(d,a) generated via ( 8 ) b y a normal germ y E C(d; d @ a) is a Hilbert space K: with basis indexing set o and operators K E L('FI), KL E L('FI@k;'FI),L; E L('FI;'FI@K:)and Lt E L('FI@k;'FI@K:).These define an operator L E L(31 @ k;'FI @ K),with K = C @ K: @ C, in the canonical dilation

y ( A ) = L*(A @ I K ) L= (18) with L; = L y , Lz = Lt* and K: = K;*, such that there exists a (not necessarily unital) *-representation j f of B @ b in LA(E@ K) giving the , @ b)] = 0. intertwining L ( B @ b) = j'(B @ b)L and [ ( A@ I K ) j'(B

Proof. Since our generators are normal and bounded we have [6]

A L;j(A)L; L ; j ( A ) L ; ] - [ A 0 0 1 OAO A OOA

[

0 K; K ] 0 0 0 0 0 0

for a normal representation j of d in C(K) and some separable Hilbert space K, and operators L i E L('FI;K), K,K,r E d with p E ( 0 , +} and i E 0 . Taking the normal decomposition j ( A ) = F * ( A @I K ) F for some Hilbert space K: and partial isometry F : K H 'FI @ K: and making the replacements LL H FLE gives (18) after a simple rearrangement. The representation j' is constructed on the lineal { ( A@ IK)Lx : A E A, x E 'FI @ k} by defining j f ( B@ b ) ( A@ 1 ~ ) L = x ( A @ k ) L ( B@ b ) ~ We . have

( ( A @ I K ) L X l j f ( B @ b ) ( A i @ I K ) L= x i )( ( A @ k ) L ( B * @b*)xI(Ai@ k ) L x i ) (20)

115

since lemma 2.1 gives y(A) E A @ a, where the pseudo scalar product ( I ) is given in theorem 2.1. Hence if the sum (Ai @ IK)Lx~vanishes, then j'(B @ b)(Ai @ 1 ~ ) L xalso ~ does, so j' is well defined on the given lineal. It follows easily from the definition that it is *-homomorphic, and the commutativity conditions follow from noting that j'(B @ b)LX = L ( B @ b)X by taking A = IN in the definition, so

j'(B @ b)(A @ k ) L x = (A @ IK)L(B@ b)x = (A @ k)j'(B @ b)LX.

0

There is a second decomposition of the germ of a flow which corresponds to taking the normal decomposition of the representation j'.

Theorem 3.2. Stochastic form of flow generators. Associated with every q5 E Q(A,a) generated via ( 8 ) by a germ y : A H A @ a i s a (not necessarily unital) *-representationj of A o n 7ia@Ka,where 'Ha = 'H @ 'Ho, Ka = C @ K" @ C, Ka = k @ ko for Halbert spaces 'Ho, ko, and a n operator La : 'H @ k H 'Ha @ K a giving the stochastic dilation y(A) = L"*j(A)L", (21) such that LaB = B"L" for all B E B@ 6, where Ba is the tensor embedding of B into 'Ha @ K", and [j(A),B"] = 0 o n the linear span of {Lax : x E

'H@k}. Proof. (Sketch) We shall assume that, analogously to the von Neumann algebra case, any normal *-representation j' of B @ b is pseudo-unitary equivalent to a subrepresentation of B" @ 6" on 'Ha @ K a ,where B" = B@10 is the amplification B @ 10 of B onto ' H a , and 6" is the amplification of b onto K" by the embedding k C K":

I by b;

I b; @ ,$

b,

[;;71 [;

b y o b y o ]

(22) 7

where [O E ko, llQll = 1. Therefore j'(B) = U*B"U, where U is a partial pseudo-isometry from 'H @ K to 'Ha @ Ka with U I = U$ the usual partial isometry appearing in the normal decomposition of j'(B@I),and UU* the pseudo-projector commuting with B" @ ba. The definition of j' in theorem 3.1 gives L = j'(1)L = U*UL, which allows us to write

L*(A @ IK)L = La*j(A)La, where j(A)

= U(A @ IK)U*,and

(23)

La = U L intertwines B @ b with B" @ ba:

LaB = Uj'(B)L = B"La.

(24)

116

Commutativity of j(d) with manipulation.

B" @ 6" on the given lineal follows by a similar 0

These two characterisations give a choice in how to represent a flow. The latter (of which Refs. [7] and [12] are examples) constructs the decomposition so that the representation j' of B @ b is a tensor embedding. In the former, which we will use in the sequel, the representation j is a tensor embedding, but j' is not. It can be shown that if the flow is majorised by a tracial map then both representations exist simultaneously, which in particular is always true in the finite dimensional case. The canonical representation 3.1 makes constructing a Stinespring type dilation [ll,71 of a quantum stochastic flow particularly simple, and this will be done in the next section. It is easy to see how the examples of flows driven by the canonical noises arise as examples of this characterisation. The representation space Ic is simply k , and for Poisson noise (15) -K*T = La+ -- La, L$ = (Li I)d$ (no sum), whilst for Wiener noise (17) Kt: = L; = La, L$ =Id!,3 i,jE = 0. The non-stochastic case requires a slightly different treatment since k = (0) so rows and columns in ? ( A ) and L mapping to or from k are deleted:

+

Hence the Lindblad form is recovered, and y(A) E d @ n where the representation of n as two dimensional matrices has been used. Of course it is possible to embed the semigroup into space with noise by placing appropriate columns and rows of zeros in (25). In the cases of filtering and contractive flows, the following lemma refines the characterisation above. Lemma 3.1. Canonical f o r m of filtering and contractive flow generators [7]. If the quantum stochastic flow in theorem 3.1 is subfiltering, the dilation space K: and operator L; can be chosen such that L,L; = K K*. If the flow is a contraction, Ic, L; and L: can be chosen such that

+

LYLI; = K -k K*

LzL: = Inmk

LiL; =K i .

(26)

117

contraction flow ( X ( I ~ ) $ ~ ~ $ O ) = lirnt+o+(($t(Ix) - Ig)$l$) 6 o for any $ = q 8 f E 9, where $O = q 8 (0 @ f(0) @ I). For a contraction flow, using our canonical form (18) for the germ gives

0

Kl-LrLZ

K+K*-L,L; K; - Lz L; 0

This positivity allows us to define the operator matrices

1

2 0.

(27)

0 - K i -K

0 0 0

(28)

0 L: 0 0

0 0 0 .

.

A

so that -X(Ix) = L ( I-~L*L ) = L*L, o the indexing set for a basis of an arbitrary Hilbert space. Hence is isometric, L * i = e L L*L, SO the new operators = LE @ L;, p E { o , + } , satisfy (26). If the flow is subfiltering, we only have K + K* - L;Lq 2 0, so defining the operators J!,; = L; @ Lq, J!,:= Lt @ 0 gives the required filtering condition. 0

+

3.2. Dilation

The dilation of quantum stochastic flows is an important topic with close links to quantum dynamics and measurement. Denote the dilated It6 algebra by c C CA(W) defined as the commutant to a dilated output It6 algebra b C: LA(W) having representation Fr6chet space 9 so that W = C @ 9 @ C (in fact will turn out to be a Hilbert space here). Let 0 C B(&,) be the dilated input noise algebra (which may or may not have M as a subalgebra) which is the commutant to the dilated output noise algebra P C C(&,) generated by b. A dilation of a flow 4 E Q(d, a) is a flow E Q(d, c), such that there exists a map lE : d @ 0 H A 8 M for which 4 = IE(4). The classical Stinespring dilation is well known; it deals with the representation of a completely positive map from a C*-algebra to the bounded operators on a Hilbert space. The following theorem can be viewed as a generalisation of this to our situation of a continuous time cocycle of maps on von Neumann algebras, with the image space being decomposable and having the probabilistic structure described. It is called canonical since it uses the canonical form of the flow generators.

4

Theorem 3.3. Canonical Stinespring Theorem. Given a quantum stochastic p o w 4 E Q(d, a) satisfying equation ( 8 ) , there exists a n It6 alge-

118

bra c with representation Hilbert space K and a quantum stochastic process Wt : 9 ++ 3.1 @ T K such that

4t (A) = W,t(A @ IFK ) Wt .

(29)

The process Wt is afiliated with Q in the sense that it intertwines B with representations jt of B in t? @ P G (d@ O)’, so Wt(B@Nt)= jt(B@Nt)Wt for B E t?, Nt E Nt. If 4 is a contraction then Wt can be chosen to be isometric. Proof. Firstly note that the dual to the kernel process Wt is a map WJ : 3.1 @ 8~ H D’, giving the correct action of 4 t ( A ) . From theorem (3.1) we know the general form of the germ of $, so that

&(A) = 4t

0

&*(A @ IK)L- (A @ I&)):@ dAL.

(30)

Consider the kernel process Wt generated by the germ L: dWt = (L - IK):Wt @ dAL,

Wo$J= $J,

(31)

where $J E 9. We have assumed that K = k, which can always be made true. If K C k then identify K with its embedding in k and insert zeros in the extra rows of L. If k C K: then identify k with its embedding in K, insert zeros in the extra columns of L, and compose Wt with the isometric embedding to recover the correct noise space for the flow. Alternatively, simply embed both K and k into the l 2 Hilbert space, since they are both separable. An application of theorem 2.2 to (29) with this Wt shows that $,(A) is of the appropriate form. The algebra d @ c is spanned by LK(A) for all A E d, so c is simply the unitisation of the trivial It6 algebra. Affiliation of Wt with Q in the above sense follows by constructing the representations j t analogously to j‘ in theorem 3.1. The same argument holds because &(A) is affiliated to U. Isometry of Wt in the contraction case follows from choosing L as in lemma 3.1; equation (27) with equality gives exactly the conditions for L 0 to generate an isometric operator by theorem 2.2. The canonical Stinespring theorem shows that in general flows have a nice structure. In particular, the *-multiplicativity of &(A) = ( A @ I - F ~ ) means that it is a homomorphic dilation, with the reduction to the original flow performed by conjugating with Wt and its adjoint. This dilation is obviously independent of the original flow and has a very simple form, which is counterpoint to the complexity of the reduction kernels Wt.

119

Ideally one would like the reduction to be a conditional expectation, that is a contractive, idempotent, surjective linear map lE : 0 I+ M , hence requiring that M C 0. This arises in a natural way if we have a distinguished vector on the difference. The conjugation by Wt and its adjoint is clearly not a conditional expectation, even in the contraction case. However, an extension of the canonical Stinespring construction will be made below which gives a homomorphic dilation, and recovery of the original flow is produced by a particularly nice conditional expectation. This extension is related to a type of quantum dynamical semigroup dilation known as Evans-Hudson dilation, which uses the canonical Fock space conditional expectation IEo with respect to the vacuum vector to recover the original semigroup. We can now formulate a canonical EvansHudson dilation for contraction flows (see also Ref. [12]), and a weak dilation (to be made precise below) for subfiltering flows. The canonical Stinespring type dilations have dilation space b = K,whereas the dilation space in the following Evans-Hudson type case will be b = k @ K. The conditional expectation from Fb to 3 k is given by the usual embedding E : 7 - t H 3 k ~ 7 - t H F b , d e f i n e d o n v e c t o r s ? , b i n ' H H 3 kby?,bI+?,b@QK, so that E o ( A ) = E'AE, A E d 8 0.

Theorem 3.4. Canonical Evans-Hudson Dilation. Consider a flow 4 E Q(d,a) satisfying Eq. ( 8 ) .

4 is a contraction flow then there exists a unitary operator cocycle Ut E L(7-t8 Fb)such that $(A) = U:(A 8 I.F,)U~E Q(d, c) is a n Evans-Hudson type dilation, with $t ( A )= lE0 (& ( A ) ) . (2) If 4 is a subfiltering flow then there exists a unitary operator cocycle Ut E L(7-t 8 Fb) such that $(A) = U:(A 8 IF,)Ut E Q(d,c) is a weak Evans-Hudson type dilation in the sense that (1) If

where +(t,b ) = v 8 W ( t ,b)Rk and

for any b E b and

v E 7-t.

Proof. Consider the operator matrices L, L,, ( A )E L(7-t H (C @ K @ k @ C)

120

;I.

given by

L- =

[;Jj-

I -Jo- -Je- -K 0 J," J," L;

;

A 0 0 i * I ) [ ;0 6,"(A)6p':]. 0

0 0

(34)

with K and L; as in theorem 3.1 above, J," = -(I," - J,"J,')i, J: = (I: - J,'J:)i, and J,' = J,"*. J: and J,' are contractions, but along with Je- and J; are otherwise undetermined. Using these to define the germ ?(A)= L * b b ( A ) L , this then generates a flow $(A)= U:ib(A)Ut, where Ut is generated by the germ L. Clearly $t is homomorphic if Ut is unitary. The isomorphism theorem 2.2 can be used to show that this is true if and only if L is unitary, PL = LL* = I . Calculating this explicitly gives the necessary and sufficient conditions

JL=L,J,",

J,-=L,J,",

K+K*=L,L:.

(35)

The It6 algebra c such that ?(A) E d@ c in the general case is the HudsonParthasarathy algebra hp of all quantum differentials and has representation space = K: @ k since

?(A) =

1

A (L,6,"(A) - AL,)J,"," L,6,"(A)L; - AK - K * A 0 J,",'d,"(A)J,"," + J,",'L(A)J,': J,",'(d,"(A)L; - L;A) [o 0 A

,

(36) where J,"," = J,",'* = (J,",J,"), and J,=' = J,":* = (J,', J:). For the contraction case, lemma 3.1 shows that without loss of generality we may assume that the operators in the characterisation (21) satisfy (26). Therefore, choosing J," = Lz, and defining J; = K; and JT through equation (35) gives a unitary L. Moreover, applying dUt = (L-I)EUt@dAL to a vector of the form II,@ O K ,II,E 7-i @ Fk,gives

[i 7

0 0 - K i -K

dUt('$ @ O K ) =

P

utdAL($' @ O K ) = (dwt'd') 8 Ok, (37) U

so the vacuum expectation delivers the original Wt. This can also be seen from (36) by sandwiching with II,@ OK, and this completes the proof for the contractive case. In the filtering case, again K + K* = L i L ; may always be assumed. For J: take an arbitrary contraction such that J,' is also a contraction

121

and define J; and J; through equation (35), again giving a unitary L. Applying dUt to a vector of the form $0 @ RK, $0 = r] @ R k , r] E 3-1, gives

000-K

dUt($O @ R K ) =

[:::

000 0

P

UtdAL($o @ RK)= (dWt$o) @ %. (38)

"

Along with the representations j, appearing in theorem 3.3, this gives

Hence the weak dilation property follows from the construction of Ut. 0 Thus, in contrast to the canonical Stinespring dilation, the canonical Evans-Hudson dilation gives a homomorphic flow with the original flow being recovered from the simple vacuum conditional expectation. If J: is isometric, which can always be made true if the flow is a contraction, then this dilation reduces to that sketched in Ref. [7] for A = B(3-1). Some examples now follow exhibiting this dilation. The first case to consider is the deterministic one, with the original flow having germ ?(A) E A @ n. The reversible case given by Hamiltonian evolution is trivial since the flow is already homomorphic. For the irreversible case the dilated noise space is 6 = K: due to the triviality of Ic, so the dilation is given by

[1

-f"

L=

0

(39)

J,"

1

A ( L , j ( A ) - AL,)J," L,j(A)LO, - AK - K * A r(A) = 0 J,"j(A)J," Jo"(j(A)L$ - L p ) [o 0 A

7

(40)

where J," = -I," since J,' = 0. One can easily see that in fact any unitary J," will suffice here by considering the conditions on the unitarity of L. The quantum stochastic differential equation having this germ is already known to be the dilation of a general quantum dynamical semigroup [13, 191. Note that for this non-stochastic case the definitions for subfilterjng and contraction quantum dynamical semigroups are the same. For a flow driven by Poisson noise (15), K: = Ic, so lj = Ic @ Ic = Ic 8 C?. A flow driven by Poisson noise can be thought of as a continuous quantum measurement, where at each of the jumps in the Poisson variabIe a reduction

122

+

Li I of the quantum system has taken place, dependent upon the type of jump labelled by i. We shall assume that the flow is isometric with Wt isometric, so that L:L. = K K * , LrLi + Lf Li = 0 (as per ( 2 6 ) ) . The dilated germ ( 3 6 ) becomes

+

+

I.

1

A (M: - I ) ( A ) M p( A) M:(A) - K ( A ) 0 6:M:(A) s:M,O(A) (M: - I ) ( A ) = 0 6:M,'(A) 6:(M: - M .f ) ( A ) M,'(A) 0 0 A

+ + +

(41)

1

+

where K ( A ) = A K K * A , M j ( A ) = M i A M j with M o = L. I and -M1 = L.L: L: L.. This describes a 2 x dim(lc)-dimensional Poisson noise, so isometric Poisson flows have Poisson dilations. This dilation corresponds in the one-dimensional case to that obtained in Ref. [l]. For an isometric Wiener flow the Evans-Hudson dilation becomes trivial since ( 2 6 ) gives, L. L: = 0, L:L. = K K* so that Wt is automatically unitary. In the next section these characterisations and dilations of quantum stochastic flows will be used to formulate covariance conditions on their generators ensuring the covariance of the flow itself.

+

+

4. Covariance 4.1. Covariant Quantum Stochastic Flows and Dynamical

Expectations This section introduces the covariance condition for quantum stochastic flows with respect to a group. Starting with a group G, denote a family of representations of G as automorphisms of Q by at : (9,A) H q ( g , A) E Q for (g,A) E G x Q. We consider covariance in the simplest case of an invariant evolution parameter t.

Definition 4.1. Covariant Quantum Stochastic Flow. A bounded flow $t E Q(A,a) is covariant with respect to a group G if there are nontrivial representations at of G in the adapted automorphisms of Q such that $t ( 4 9 1 A ) )= 4 g , $ t ( A ) ) ,

(42)

where a ( g , A) 8 IM = ao(g, A 8 I M ) ,A E A. Note that the initial condition aO(g)= a ( g ) 63 id^ is a direct consequence of the adapticity of at and equation (42). Taking t = 0 and using that

123

+o(A) = A @ I M in (42) gives the definition a(g, A) @ IM = ao(g, A €3 I N ) . Multiplying by an arbitrary 11.1@ M and using the adapticity ao(g, A @ I M ) ( I H€3 M ) = a o ( g , A @ M ) gives the initial condition. It will be assumed that a t ( g ) is implemented by ot(g, A) = v,*(g)AVt(g) for a unitary adapted cocycle representation vt of G in L(G). K ( g ) can be quite general, assumed to satisfy Hudson-Parthasarathy quantum stochastic differential equation of the form

d V t ( g ) = (sl(g) - b:)%(g) €3 dAL7 h ( 9 ) = v(g) €3 IM7

(43)

with S,”(g) E L(7f),and V(g) is a unitary representation of G in A. We say that the representation V,(g) is decomposable if &(g) = V(g) @ Vt(g) for some cocycle representation Vt(g) of G in L(F).This is equivalent to S,”(g) E @, so that Vt(g) E M if S(g) E a. The next section is devoted to calculating the consequences of this definition on the generators of the flow. 4.2. Covariant Flow Generators

Before giving the covariant dilation let us give the conditions for covariance in terms of generators of the flow. This is useful for seeing what kinds of covariance cocycle & ( g ) are possible for different flows.

Proposition 4.1. Differential covariance conditions f o r flows. A flow q5t E Q(d,a) is covariant with respect to unitary cocycle representation & ( g ) of a group G if and only if

4t 07(V*kJ)AV(S))= v,*(g)S*(g) [4t 07(A)1 S(g)Vt(g).

(44)

A suficient condition for this is that

Proof. The covariance condition (42) can be written W,*i,(V* (g)AV(g))Wt = v,*(g)W&(A)WtK(g)

(46)

for i,(A) = (A @ IF^). Differentiating this using theorem 2.2 gives W,*L*i,(V* (g)AV(g))LWt = v,*(g)S*(g)W,*L*iFC(A)LWtS(g)Vt( g ) , (47) from which (44) follows. The sufficient condition is immediate.

124

To be explicit, and for later use, below is presented the components of the matrix equality (45) as a system of equations: wtSl(g)Vt(g) - Klwts:(g)Vt(g) = -V(g)Klwt WSy(g)Vt(g) - KWtVt(S) - ~ l w t s ; ( g ) K ( g ) = -V(g)KWt L:wtS;(g)Vt(g)

LWts:(g)Vt(g) L:wtVt(g)

+

(48) (49)

= V(g)~:wt

(50)

= V(g)L$wt

(51)

WtV,(g) = V(g)Wt, (52) The final equation is simply the sufficient condition for covariance of the flow. Let us see what this entails for our standard examples of commutative and trivial noise algebras. Deterministic Flows. For reversible deterministic flows the flow generator in (25) is simply Z[H,A]&. Using this in (48)-(52) shows that for a deterministic reversible flow to be covariant with respect to a unitary representation &(g) generated by S(g) it is sufficient that S;(g) = S;(g) = 0, and [V(g), -iHIWt = wts;(g)Vt(g).

(53)

The commutator Hence Wt intertwines [V(g),-iH]with S;(g)Vt(g). [V(g),-iH]need not be zero since the representation of G is time dependent; this relation shows how the commutator changes to reflect this. If &(g) is decomposable (that is, Sf(g) E C ) then St(g) is arbitrary. Note that S,(g) is anti-hermitian due to the unitarity of S(g). For irreversible deterministic dynamics, with the generator given in Lindblad form (25) (K may always be taken to be of the form K = i H + i L $ L ; for some hermitian H), equations (48), (49) and (50) give the same conditions on S(g) as the reversible case (with ZH replaced by K in (53)). Additionally, using (52), equation (51) gives (V(g)L$ - L;v(g))w = 0,

(54)

so V(g) commutes with L:. Poisson Flows. For Poisson noise, equation (50) for decomposable S(g) gives

(Li-I-I)v(g)s;(g)wt

= V(g)(L

+I ) q %

(55)

(no sum), which clearly shows that St(g) must be diagonal. Equation (51) becomes

(Li+ I)v(g)s;(g)w

+ LiVk7)W = V(S)LiW

(56)

125

(no sum). Adding V ( g ) W t to both sides and using (55) gives

(Li+ I ) V ( g ) ( S $ ( g )- $ ( g )

+ I)Wt = 0,

(57)

so that S i ( g ) = S i ( g ) - I . Unitarity of S ( g ) implies that

S,T(g) = -S?(g)S{(g) = ( I - s , j * ( g ) ) s { ( g )= S $ ( g ) .

(58)

Hence & ( g ) is generated by the multidimensional Poisson algebra Pdim(k),

+

dVt(g) = S i ( g ) V t ( g ) d p ~ S T ( g ) V t ( g ) d t ,

(59)

where & ( g ) = S,:(g). Hence S ( g ) E A €3 Pdim(k). Equation (48) now gives us nothing new, and (49) relates S T ( g ) to Si(g). Wiener Noise. Finally, for the Weiner noise case (50) requires Sj ( 9 ) = 6: which in combination with (48) produces

[V(9), Lil Wt = V(9)S.Jg)Wt.

(60)

Equation (51) gives an identical relation for S $ ( g ) , so S $ ( g ) = St:(g), and analogously to the Poisson case % ( g ) is generated by the multidimensional Weiner algebra tudim(k),

dVt(9) = s i ( g ) V t ( g ) d Q f+ s ; ( g ) V t ( g ) d t ,

(61)

with Si ( 9 ) = St- ( 9 ) . Again, the S; ( 9 ) can be related to Si ( 9 )through (49). Hence, for the Poisson and Wiener examples & ( g ) E 9.l.

4.3. Covariant Dilations The previous results lead to the following proposition, giving covariant canonical Stinespring and Evans-Hudson dilations.

Proposition 4.2. Cova&ant Dilations of Quantum Stochastic Flows. Covariant quantum stochastic flows have canonical Stinespring and Evans-Hudson dilations covariant in the sense that there exists a cocycle representation of the group o n the dilated space with respect t o which the homomorphic dilation is covariant. Proof. Consider the quantum stochastic flow c#I~(A) = W;ix(A)Wt with canonical Stinespring dilation i x (A)and canonical Evans-Hudson dilation &(A) = U:i,(A)Ut, Ut given in theorem 3.4. Consider the representations of G defined by the unitaries i x ( V ( g ) )and & ( V ( g ) ) = V:i,(V(g))Ut. Then covariance follows immediately from the multiplicativity of ix and Hence both flows are respectively covariant. 0

6.

126

Hence the covariance cocycles R ( g ) with respect to which canonical Evans-Hudson dilations of covariant flows are covariant inherit all the properties of the dilation itself. In particular, if the original flow is driven by isometric Poisson or Wiener noise, the dilation and dilated group representation are also driven by Poisson or Wiener noise respectively. For irreversible deterministic flows, equation (54) immediately gives that the annihilation and creation coefficients of Y ( V ( g ) ) are zero (see Eq.(36)). Similarly, the time coefficient reduces to [K, V ( g ) ] ,so if the original group representation V , ( g ) is time independent, ST = 0, equation (53) shows that this entry also vanishes. Hence

for any unitary J,". This can be compared to the covariant dilation presented in Ref. 191.

References 1. V. P. Belavkin and P. Staszewski, Quantum stochastic differential equation f o r unstable systems, Journal of Mathematical Physics 41 (2000), no. 11, 7220-7233. 2. V. P. Belavkin, A quantum nonadapted ito formula and stochastic analysis in fock scale, Journal of Functional Analysis 102 (1991), no. 2, 414-447. 3. -, Chaotic states and stochastic integration in quantum systems, Russian Mathematical Surveys 47 (1992), no. 1, 47-106. 4. -, Kernel representations of star-semigroups associated with infinitely divisible states, Quantum Probability and Related Topics VII (1992), 31-50. 5. -, Quantum stochastic calculus and quantum n o n linear filtering, Journal of Multivariate Analysis 42 (1992), no. 2, 171-201. 6. -, O n the stochastic generators of completely positive cocycles, Russian Journal of Mathematical Physics 3 (1995), no. 4, 523-528. 7. -, Quantum stochastic positive evolutions: Characterization, construction, dilation, Communications in Mathematical Physics 184 (1997), 533566. 8. -, Quantum ito algebras: Axioms, representations, decompositions, Quantum Probability Communications XI (2003), 39-54. 9. P.S. Chakraborty, D. Goswami, and K.B. Sinha, A covariant quantum stochastic dilation theory, Stochastics in Finite and Infinite Dimensions (2001), 89-99. 10. E. Christensen and D. Evans, Cohomology of operator algebras and quantum dynamical semigroups, Journal of the London Mathematical Society 20 (1979), no. 2, 358-368. 11. D. Goswami, J. M. Lindsay, and S.J. Wills, A stochastic stinespring theorem, Mathematische Annalen 319 (2001), 647-673.

127 12. G. Goswami, J. M. Lindsay, K. B. Sinha, and S. J. Wills, Dilation of markovian cocycles on a von neumann algebra, Pacific Journal of Mathematics 211 (2003), no. 2, 221-247. 13. R. L. Hudson and K. R. Parthasarathy, Quantum itos formula and stochastic evolutions, Communications in Mathematical Physics 93 (1984), no. 3, 301323. 14. G. Lindblad, O n the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48 (1976), no. 2, 119-130. 15. J. M. Lindsay and K. R. Parthasarathy, O n the generators of quantum stochastic flows,Journal of Functional Analysis 158 (1998), no. 2, 521-549. 16. J. M. Lindsay and S. J. Wills, Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise, Probability Theory and Related Fields 116 (2000), 505-543. 17. -, Markovian cocycles on operator algebras adapted to a fock filtration, Journal of Functional Analysis 178 (2000), 269-305. 18. N. Obata, White noise calculus and fock space, Springer-Verlag, 1994. 19. K. R. Parthasarathy, A n introduction to quantum stochastic calculus, Birkhauser, 1992.

ALICKI-FANNES AND HUDSON-PARTHASARATHY EVOLUTION EQUATIONS

ALEXANDER C. R. BELTON* Institut Camille Jordan Universite' Claude Bernard Lyon 1 43 avenue d u 11 novembre 1918 69622 Villeurbanne cedex fiance E-mail: [email protected] r

The isomorphism between quantum semimartingale algebras is used to explain the relationship that exists between the evolution equation of Alicki and Fannes and that of Hudson and Parthasarathy.

1. Introduction

Alicki and Fannes [l]have demonstrated a method of dilating quantum dynamical semigroups via classical Brownian motion. Vincent-Smith [9] noticed that their technique sits naturally within the framework of quantum stochastic calculus, albeit with a non-standard form of adaptedness. This idea was extended [4,51 and it was shown that the original Alicki-Fannes equation is a special case of the following: =I,

dV = P d R

+ Q dA + RVdAt + S V d t ,

(1)

where the solution is unitary if and only if the coefficients satisfy the Hudson-Parthasarathy unitarity conditions, wiz. (P,Q, R , S ) = (W - I , L , -WL*,iK

-

iLL*),

(2)

with K self adjoint and W unitary. In (1) the coefficient processes (and so the process V - I ) are wacuurn adapted, ie., each X is such that EXE = X , where IE is the conditional expectation. *Work supported by the European Community's Human Potential Programme under contract HPRN-CT-2002-00279, QP-Applications.

128

129

The right-sided version of the evolution equation due to Hudson and Parthasarathy [7] has the following, more symmetrical form (in the case of one-dimensional noise):

Uo=I,

dU=EUdA+FUdA+GUdAt+HUdt,

(3)

where now the coefficient processes (and U ) are adapted, in the sense that Xt c) X t l ~ tIF[^ ) with respect to the decomposition 3 E Ft)@3p of Fock space, for all t 2 0. The unitarity conditions are necessary but now fail to be sufficient (in the case of coefficients which act on the entire Fock space: cf. Attal’s example of the LBvy transform [3, Section 11.2.41.) In this brief note, it is shown that the natural isomorphism [6] between the semimartingale algebras Sn and S produces a bijection between the sets of solutions to these evolution equations: if U is a unitary solution to (3) then I + ir(U - I ) is a unitary solution to (l),if V is a unitary solution to (1)then ii(V)= I + i i ( V - I ) is a unitary solution to (3) and these maps are mutually inverse, where ir and ii are the projections from non-anticipating to vacuum-adapted and adapted processes, respectively. The effect of this bijection on the driving coefficients is given explicitly.

1.1. Conventions We follow the usual conventions of quantum stochastic calculus and mostly adopt the modern notation of Lindsay [8], although the gauge process is denoted A, not N , and the differential notation dX=EdA+FdA+GdAt +Hdt is used for what Lindsay writes as

+

X t = XO N(E)t

+ A(F)t + A*(G)t + T(H)t

(t 2 0).

All processes considered herein consist of bounded operators. The LP norm of the function f is written IlfllP. 2. Results

Notation 2.1. Let 3:= h @r+ (L2(R+; k)) denote Boson Fock space over L2(R+;k), the space of square-integrable, k-valued functions on the half line, equipped with the separable initial space h, and, for all t 2 0, let 3 t ) := h @ and 3[, := so that j t : Ft) @ F[t 4 3;~ E ( u [ o , ~@ [ ) E ( u [ ~ , ~ ++ [)

UE(U)

V U E h, u E L2(R+;k)

130

is an isometric isomorphism, where E ( U ) denotes the exponential vector corresponding to u and the tensor-product sign between a and ~ ( uis, ) as usual, suppressed. The multiplicity space k is required to be separable.

Definition 2.1. Given Hilbert spaces H o and H I , an Ho-HI-process X is a family of bounded operators (Xt)t2o C B(H0 @ T ;H I €3 S)such that t H XtO is weakly measurable for all O E Ho @F. All processes are required to be non-anticipating: each X is such that EXIE = XIE. Definition 2.2. The conditional expectation is the C-C-process E such that Eta.(.) =a~(up~ for[ )all t 2 0, a E h and u E L2(R+;k). This extends, by ampliation with the identity, to an H-H-process which is denoted in the same manner: IEtf €3 ae(u) = f €3 a e ( u [ ~ , ~the [ ) ; choice of H will be clear from the context. Definition 2.3. An Ho-HI-process X is adapted if, for all t 2 0, a , b E h, f E H I , g E Ho and u , v E L2(R+; k), (f€34U),

X t g @ b d v ) )= ( f @ a + q o , t [ )X, t g € 3 W [ o , t [ (+[t,co[), ))

E(V[t,cO[)).

An Ho-HI-process X is vacuum adapted if, for all t 2 0, EtXtJEt = X t ; this is equivalent to requiring that

(f @ 4 U ) , x t g @ W v ) ) = (f €3 ae(u[o,t[),x t g @ b + J [ O , t [ ) ) for all a, b E h, f E H I , g E Ho and u , v E L2(R+; k).

Proposition 2.1. If X is an H o - H I process then there exists an adapted H o - H I process + ( X ) , the adapted projection of X , such that +(X)lE = E X E , + ( X ) = X i f X is adapted and the map X H + ( X ) is *-linear and multiplicative: ifY is an HI-H2-process then + ( Y X )= + ( Y ) + ( X ) . There also exists a vacuum-adapted Ho-HI-process i i ( X ) , called the vacuum-adapted projection of X , such that ir(X)E = EXE,+ ( X ) = X i f X is vacuum adapted and X H i r ( X ) is *-linear and multiplicative. Proof. For all t 2 0 let

+(x)t := (IH1@jt)(EtXtIHo@F,, €3 IFlt)(IHo@h)* and ir(X)t := EtXtEt.

0

Definition 2.4. A quadruple of processes (P,Q, R, S) is admissible if P is a k-k-process, Q is a k-C-process, R is a C-k-process, 5' is a C-C-process and, for all t 2 0,

( l l ~ o , t [ l l , IIQ[o,t[II, IIR[o,t[ll,ll~[o,t[ll) E LcOIO,t[x L 2 [ 0 , t [x L 2 [ 0 , t [x L1[O,t[.

131

Notation 2.2. The *-algebra of regular quantum semimartingales due to Attal [2] is denoted S; an element X E S is an adapted C-C process which admits the integral representation Xo = 0 ,

d X = P d h + QdA

+ RdAt + Sdt,

(4) with (P,Q, R, S) an admissible quadruple of processes. The analogous *algebra of vacuum-adapted quantum semimartingales [5] is denoted So. Theorem 2.1. If X E So has the representation

ii(X)o = 0,

dii(X) = ii(P- 2)dA

(4) then i i ( X ) E S and

+ ii(Q) dA + ?(R) dAt + ii(S) d t ,

where 2 := Ik €4 X for any process X . Conversely, if X E S is of the f o r m (4) then ?(X) E So and ir(X)o

= 0,

dir(X) = ir(P

+ X)d h + ir(Q) dA + ir(R) dAt + ir(S) dt.

Proof. See [6, Corollary 401, noting that ii(& €4 X )= Ik €4 ii(X).

0

Definition 2.5. An admissible quadruple of processes (P,Q, R, S) satisfies the unitarity conditions if there exist processes W , L and K such that (P,Q, R,S) = (W - I , L , -WL*,iK

-

iLL*),

W is unitary and K is self adjoint. Theorem 2.2. If (P,Q,R,S) is an admissible quadruple of vacuumadapted processes then there exists a unique solution to the quantum stochastic diflerential equation = I,

dV = P d h

+ Q dA + RVdAt + SVdt,

which is unitary if and only if (P,Q, R, S) satisfies the unitarity conditions. Proof. This result is stated in [5, Section 41 (with dim k < m) and may be proved by employing the techniques from that article, together with the following estimate [6, Proposition 371: if X E So has the representation (4) then

IIXtII

IIPro,t[llm + IlQ[o,t[ll2 + IIR[o,t[112+ IIS[o,t[ll1

vt 2 0.

Notation 2.3. If dim k > 1 then the Hudson-Parthasarathy evolution equation has the form

Uo = I, dU = E U d h -t F U d A + GU dAt -I- H U dt for an adapted, admissible quadruple ( E ,F, G, H ) .

(3’)

132

Theorem 2.3. Let V be a unitary solution to the Alicki-Fannes equation ( l ) , so (P,Q , R, S ) is a quadruple of vacuum-adapted processes such that

(P,Q , R, S ) = (W - I , L , -WL*, iK - i L L * ) , where W is unitary and K is self adjoint. The adapted projection of V is a unitary process U which satisfies the Hudson-Parthasarathy equation (3 ’), with driving coeficients ( E ,F, G ,H ) = (r;t.- I , i;,-r;t.i*,iK -

a,%*),

where I@ = ii(W?*) is unitary, i(, = ii(LV*)and K = ii(K) is self adjoint. Conversely, i f U is a unitary solution of the H P equation (37, so the quadruple of adapted processes (ElF,G, H ) = (W - I , L , -WL*,iK - i L L * ) , where W is unitary and K is self adjoint, then V = I+ir(U-I) is a unitary process which satisfies the A F equation ( l ) , with driving coefficients

(P,Q , R,S ) = (r;t. - I , E , where = 1 + ir(W6 - I ) is unitary, adjoint.

-ex*, ik - $Xi;*), = ir(LC) and

K

= i r ( K ) is self

Proof. By Theorem 2.1, if V is a unitary solution to (1) and U then

d(U - I ) = ii((W - I ) - (V - I ) ) dA =

=

ii(V)

+ % ( L dA ) - %(WL*)ii(V)dAt + ii(iK - @*)ii(V) d t

(+(WV*)- I)CdA + ii(LV*)CdA - ii(WV*).ir(LV*)*UdAt

+ (iii(K)- +ii(LV*)ii(LV*)*)Udt; the claim follows since the adapted projection preserves unitarity and selfadjointness. Conversely, if U is a unitary solution to (3’) and V = I ir(U - I ) then a little algebra shows that V is unitary and, by Theorem 2.1 again,

+

d(V - I) = ir((W - I ) C + 6 - I ) dA = ir(WC - I ) dA

+ ir(LC)dA + ir(-WL*U) dAt + i r ( ( i K - @*)U) d t

+ ii(LU)dA - ir(WL*)V dAt + ( i i r ( K ) - iir(LE)ir(LC)*)Vd t ,

133

since + ( X ) V = + ( X U )for any process X . The result follows: I++(W6-I) is unitary, working as for V , ( I + + ( W f i - - I ) ) + ( L 6 ) *= +(WL*)and + ( K ) is self adjoint. 0

Remark 2.1. It follows from Theorem 2.3 that V ++ ? ( V ) is a bijection from the set of unitary solutions to the AF equation (1)to the set of unitary solutions to the HP equation (37, such that, with the obvious notation,

(V-I,W-I,L,K) and the inverse U

(U - I ,

HI

H

(fi(V-I),fi(WV*-I),fi(LV*),fi(K))

+ +(U - I ) is such that

w - I , L , K ) H (+(U- I),+(W6- I ) , + ( L f i ) , ? ( K ) ) .

(To see this, it helps to observe that if X is adapted then ?(+(Xi’))= X and if X is vacuum-adapted then + ( f i ( X ) )= X . ) Acknowledgements The hospitality of the members of the Institut Camille Jordan, especially that of Professor Stkphane Attal and Dr Nadine Guillotin-Plantard, is most gratefully acknowledged.

References 1. R. ALICKI& M. FANNES, Dilations of quantum dynamical semigroups with classical Brownian motion, Comm. Math. Phys. 108 (1987), 353-361. 2. S. ATTAL,An algebra of non-commutative bounded semimartingales: square and angle quantum brackets, J. f i n c t . Anal. 124 (1994), 292-332. 3. S . ATTAL,Classical and quantum stochastic calculus, in Quantum probability communications X (R. L. Hudson and J . M. Lindsay, eds.), World Scientific, Singapore, 1998, 1-52. 4. A. C. R. BELTON,A matrix formulation of quantum stochastic calculus, D.Phil. thesis, University of Oxford, 1998. 5. A. C. R. BELTON,Quantum 0-semimartingales and stochastic evolutions, J. f i n c t . Anal. 187 (2001), 94-109. 6. A . C. R. BELTON,An isomorphism of quantum semimartingale algebras, Quart. J. Math. 55 (2004), 135-165. 7. R. L. HUDSON& K . R. PARTHASARATHY, Quantum Ito’s formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301-323. Quantum stochastic analysis - an introduction, in Quan8. J . M. LINDSAY, tum independent increment processes I (M. Schiirmann and U. Franz, eds.), Lecture Notes in Mathematics 1865,Springer, Berlin, 2005, 181-271. 9. G. F. VINCENT-SMITH, Classical and quantum dynamical propagators, preprint, Oxford, 1990.

BOSON COCYCLE AS THE SECOND QUANTIZATION OF THE BOOLEAN COCYCLE*

ANIS BEN GHORBAL Department of Mathematic, Politecnico di Milano, via Bonardi 9, IT-20133 Milano, I T A L Y E-mail:[email protected]. it FRANC0 FAGNOLA Department of Mathematic, Politecnico di Milano, via Bonardi 9, IT-20133 Milano, I T A L Y E-mail: [email protected]

The aim of this work is to prove that the second quantization of a solution of the Boolean quantum stochastic differential equation (B-QSDE) is solution of HudsonParthasarathy quantum stochastic differential equation (HP-QSDE).

1. Boolean quantum stochastic differential equation

Let ‘H be a Hilbert space and let ‘H@”be the n-fold tensor product of ‘H, where the 0-fold product is the one dimensional complex plane and the 1-fold product is ‘H itself. The Hilbert space

rB(x)=ce’H is called the Boolean Fock space over ‘H see [l]. The vector ( 1 , O ) is called the vacuum vector which we shall denote by R. For any h E 31 and T E L (3-1) we define B ( h ) ,T ( T )and B* ( h ) respectively the annihilation, preservation and creation operators on the Boolean Fock space on rB(‘H) as follows:

where a E C and hl E ‘H. B ( h ) and B* ( h ) are adjoint to each other on rB(‘H) and B ( h )B* ( h l )= ( h ,h l ) Pa, where Pa is the prcjection on CR. * AMS Subject Classification: 81325, 46L51, 47D06, 60H05

134

135

We put L2 (R; C ) = L2 (R). The Boolean quantum noises [l]are the operator processes on rB(L2(R)):

Bt = B ( t ):= B

(X[O,t)

1 , Tt = T-(4 :=

(%q,,,)

)

7

B f = B* ( t ):= B* ( x [ o , t ) ) 7

(1)

where xIo,,)is the indicator function of [ O , t ) and Mxro,,)denotes the multiplication operator by xto,,). Let us recall that the Hilbert space L 2 ( R ) can be looked up on as a continuous sum of Hilbert spaces, thus for each t E R+ we have the following orthogonal decomposition

L2 (R)

= L2((cY)-,0))63 L 2 ( [ O , t ) ) CB L 2 ( [ t , W ) ) = L2( [ O , t ) ) l CB

L2( [ O , t ) ) .

(2)

Introducing now another separable Hilbert space b, called the initial space, we denote by L2(R; b) the space of square integrable functions from R to b. Let ff = fj 8 rB (L2(R)), the Hilbert space tensor product of the initial Hilbert space 9 with the Boolean Fock space rB(L 2(R)) over L2 (R). Using the fact that b is a separable Hilbert space and (2) one obtain !2

=

=

b a3 L2 (R b) (b @ b [ O , t ) ) @ bi'o,t) (b 8 rB ( L 2(10, 4 1) ) 63 bi'o,t)

(3)

where Z means isometrically equivalent, $ [ O , t ) := L2( [ 0 ,t ) ;ti) is the subspace of R consisting of functions with support [ O , t ) and is the orthogonal complement of b [ ~ , ~in) ff. A bounded operator D on R is given by

D=

(:; ;;:)

7

B (b), D12 E B(L2 (R; f ~ ,) b), D21 E B($,L2 (R; $ ) ) and D22 E B(L2 (R; b)). An operator process is a family F = (Ft)t20 of operators on

where D" E

ff satisfying:

(1) for any t > 0 , Dt is the ampliation to (barrB ( L 2( [ O , t ) ) ) ) of an operator in b 8 rB( L ( [~0 ,t ) ) ) b CB $ [ O , t ) , i.e.

(2) for each

(:)

E ff the map

t H Dt

(5)

is measurable.

136

Families of operators satisfying (i) are called adapted, [l]. An operator process D = (Dt)t20is said to be locally square integrable if for each

fi we have

We denote by L2 (b) the space of locally square integrable operator processes. It is for these operator processes that the integral with respect to the Boolean quantum noises (l),for more details see [l]. For D = (Dt)t>oE L2 (b) we have

LDtdBt = where Jt : L2 (R; b) -+ tively by

(0 Kt ) , Jt

b and Kt

I"

1 t

DtdBf = 0 =

DtdTt

: L2 (R; b) -+ L2 (R; b) are defined respec-

We are interested in the solution of Boolean quantum stochastic differential equation (B-QSDE)

where L1, Lp, L3 and G E B ( b ) (where we identify L E B ( b ) with its embedding in B(ff)), dBt, dTt and dB: are the basic differentials in the Boolean Fock space. In order to simplify the notations we put

In the following we recall the appropriate result for more details see [l].

Theorem 1.1. T h e quantum stochastic differential equation (6) has a

137

where Zt is defined by

Furthermore, the process Y is unitary i f and only if 1 G = iH - -LL*, 2 where W , L , H E B ( b ) , with W unitary and H self-adjoint.

L1 = L ,

L2 = W

-

1, L3

= -WL*,

(11)

Proof. We begin by solving the B-QSDE (6) and by proving the unicity. Then we give the condition for the unitarity. If we put Dt = yt - 1,for any t 2 0, the B-QSDE (6) becomes

dDt = DtdM;1,L2-L3,6+

, Do = O .

(12)

Notice that we want to solve the B-QSDE (12) on the Hilbert space 2. The solution (if it exists!) has the form

where D = (Dt)t20 is now an operator process. Thus by applying the stochastic integration and by identification one obtain

138

The equation (13) gives 0:' = Tt - 1, where Tt := exp (tG). For the equation (14) we have for each f E L2 (R; f))

For the equation (15) we recall that F must be adapted, then [D;l ( E ) ] (5)= 0, for x 2 0. Therefore one obtain, for each E E 9 and each x E R,one have

[D?'

(a]

:= X [ O , t ) ( s ) Tt-sL3E =:

[Y2I(El]

.

(17)

Now using (5) and (17) one obtain

This prove the existence of the solution of the B-QSDE (12). Let us prove now that such solution is unique. Let E = (Et)t20be another solution (12). Thus, we obtain the following differential equation d (Dt - Et) = (Dt- Et) dMtLliL2.L3.G,

Do - Eo = 0 .

(18)

By using the same arguments as above and the initial condition, we conclude that a solution of (18) is 0. Therefore, the unique solution of (6) is Y = ( y t ) t z o given by (7)-(10). It is not difficult to see that for each t 2 0, Yt is a bounded operator and that its adjoint is given

where

with

139

Now by the same arguments used in [l],see also [ 5 ] , one find that Y is unitary if and only if (11) is satisfied where W , L , H E ~ ( I J with ) ,W unitary and H self-adjoint. 0

Remark 1.1. Suppose that we are in the special where (yt)t20is a solution of the B-QSDE 1 dx = (LdBt - L*dB; - -LL* P n d t ) x , Yo = 1. (19) .2 Then, the solution of the previous B-QSDE (19) is nothing else than the unitary operator process introduced by P.D.F. Ion, R.L. Hudson and K.R. Parthasarathy in their earlier paper [4]. We consider the operator, St : ff -+ ff on the Hilbert A, so that for t 2 0,

+

where st is the time shift on L2 (R) given by [st f ] (x):= f (Z t ) . Then, for a given Y = (yt)t20 solving the B-QSDE (6), we have the cocycle property, see [I],

Yr+t=YtS;YrSt, r , t E R + .

(20)

2. Second quantization

Let 7-1 be a Hilbert space. The Hilbert space m

r+(7-1) =

@ - P e n

n=O

is called Boson (or symmetric) [3] Fock space over 7-1. For any h E 7-1 denote by exp ( h ) = Cr20h@"/& (O! = 1) the exponential vector associated with h. For any hl, h2 E 7-1 we have the identity (exp ( h l ),exp (h2))= exp ((h, h2)).

(21)

The set {exp ( h ) I h E 7-1) of all exponential vectors is linearly independent and total in (7-1). Let 7-11, 7-12 be two Hilbert spaces and C a bounded linear mapping from 7-11 to 7-12. Then, from the universal property of the tensor product T@" is a bounded linear from 'H?" to 7-1fn. If now C is a contraction, the operator denoted by r(*)(C) defined from I'+ (7-11) to r+(7-1~2)~given respectively by

r+(C) (exp ( h ) ):= exp (Ch)

(22)

140

defines a contraction from I?+ ('HI) to ('Ha) called second quantization of C , for more details see [3, 71. Second quantization satisfies the functorial rules

r+(1)= 1, r+(c*) = r+(c)* , r+(clc2) = r+(cl) r+(c2), where C1 : 1-11 ---f 'H2 and Cz : ' H 2 -+ 'Hs are two contractions. For u E 'H and T E B ('H), define operators a ( u ) ,X ( T )and at ( u ) ,called Boson annihilation, preservation and creation operators respectively, on the dense domain of finite linear combinations of exponential vectors of ('H)

d

a (u) exp ( h ) := (u, h )exp ( h ), x ( T ):= - exp ( e E T h ) d

d&

a+(u) exp ( h ) := ;iz exp (exp ( h + 4)IEZO We consider the Boson Fock space 3+= t > 0 let

(23)

(L2(R; b)) over L2 (R; b). For

3; = r+( L 2((00-70) ; 9)) 3L.q = r+( L 2 (10, t ) ;b)) 7 3; = r+( L ( ~ [ t+OO) , ; b)) . 7

Then 3 3; €3 3;,t)€3 F:, via the continuous linear extension of the isometric map exp (f) exp (fl(.=-,o) ) €3 exp ( f l [ ~ ) €3 exp (flLt,+..)1. Let Y = (Y,),20 be solution of the B-QSDE (6) such that the unitarity condition (11) is satisfied. To simplify the notation we put X, = I'+ (Y,). Using Theorem 1.1, (21) and (22) one have

<

for all E IJand f E L2 (R; b). Then, the operator process X = (Xt)tZois called adapted processes, see [5, 7, 6, 21. The notion of adaptedness plays a crucial role in the theory of quantum stochastic calculus developed by Hudson and Parthasarathy [5]. Let

(:),

(8)

E A. Using (22) and (21) one have

141

Then, one obtains

We fix now an orthonormal basis (z,),~~of the separable Hilbert space fj. We make the identification

L2 (R x J) ,

L2 (R; fj)

(24)

where we identify the vector f in the former space with the function Rn+l defined by ( t , a ) H ( z a ,f ( t ) ) = f a (t). For simplicity we will restrict ourselves to the finite dimensional noise case J = (0,. . . ,n}. Thus, for t = 0 and by the previous notations one have

f (i) (')) (exp

,XtexP

f

It=O

142

3. Hudson-Parthasarathy quantum stochastic differential equation The Boson quantum noises

[2, 5 , 6, 71 are the operator processes on

r+( ~ (R; 2 fj)): A$ ( t )= A" ( t ):= a ( ~ [ o , t @ ) z a ) , AO,( t )= A: ( t ):= at ( ~ I o , t ) 4(27) A: ( t )= A 4 ( t ):= (Mx[o,t) @ Iza) ( Z P I) where a, /3 2 1, X[o,t) is the indicator function of [ O , t ) , M x I o , t@ ) 12), (zp I denotes the tensor product of the multiplication operator by X[o,t) and the operator z H (20, x) z a , and a, u t , X are the operators defined by (23). Let W = K @ r+( L 2(R; 9)) be the Hilbert space tensor product of a given Hilbert space K and J? ( L 2(R; fj)), the Boson Fock space over L2 (R; fj). The tensor product u @ exp (f),where u E K and f E L2 (R; fj), will be abbreviated to uexp(f). In this paper we are concerned with the Hudson-Parthasarathy quantum stochastic differential equation (HP@

QSW dUt =

C Fp"UtdAP,( t ) ,

Uo = 1

(28)

",Pa

where dh; ( t )= dt, dA; ( t ) are the fundamental quantum noise processes given in (27) and Ff are unbounded operators on the complex Hilbert space K,for more details see [2, 5, 6, 71. If U = (Ut)t20solves (28), then, for any u,w E K , f, g E L2(R;fj), t 2 0 , we have d -(tJ exp (9) ,ut u exP (f )

dt

=

C (vexp (9) FpaUtueXP (f))9" ( t )fP (4

(29)

7

"4

+

= ( v e x p ( g ) , ~ , O f ( t ) ~ t u e x p ( f ) )(wexp(g),gT ( t ) ~ : f ( t ) u t u e x p ( f ) )

+ (w exp (9) ,gT ( t )F , u t u exp (f))+ (v exp (g), FoOutu exp (f)) where ga ( t )= ga ( t ) ,f ( t )the "column" vector (fo ( t ), . . . ,f" ( t ) ) gT , (t) the 'kow" vector (go ( t ), . . . ,g" ( t ) ) ,F." : K @ fj -+ K defined by (u, F."v @ z a ) := (u, Fiw) and similarly for F,' and F:. In particular for t = 0, we have d -(vexp(g) ,Utuexp(f)

dt

= (vexp(g),F:f

(O)uexp(f)) + (veXP(g),ST(0)F:f

(O)uexP(f))

+ (wexp(g),gT( o ) ~ , u e x p ( f ) )+ ( v e x ~ ( g ) ~ ~ o 0 u e x ~ ( f ) )(30)

143

Theorem 3.1. The HP-QSDE

dUt =

[

n

+

(at (LIZ,)dA“ ( t ) a (Liz,) d A i ( t ) )

,=I

+ (Ir+(b)€Q

L2) d

h ( t )+ A (G)d t ] ut

(31)

with the initial condition UO= 1 on the space I?+ ( 5 ) €Q I?+ (L2(W;4)) has a unique solution given by the second quantization of the solution of the BQSDE ( 6 ) on ff, i.e. Ut = I?+ (K), where at and a are the bosonic creation and annihilation respectively acting o n (b), A (G) i s the diflerential second quantization, see (6, 71,and Ir+(b) is the identity operator acting on

r+ ( 5 ) . P r o o f . Let X t = I?+ (Y,)where Y = (K),>, solves the B-QSDE (6) on A. Comparing (25) and (29) it is clear that X t solves the (31). The conclusion follows since the unitary solution of (31) is unique. 0 4. An example

We consider the case [4]of the unitary operator process Y = (Yt)t20acting on the Hilbert space ff = r B (L2 (W)) i.e. the initial is 5 = C , and solution of the special B-QSDE 1

d K = ( - idBt - idB; - -Pndt)K, 2

YO= 1,

(32)

i.e. L = i , W = 1, H = 0. Then, the second quantization of the solution (32) is a solution of the HP-QSDE

dUt

=

[-iatdAt

- iadAf - -Ndt 2

l

l

Ut, UO= 1,

(33)

where at and a are the usual creation and annihilation operators on the initial space I?+ (C) = t2(N) (the quantum harmonic oscillator) and N := ata is the number operator on t2(N), defined by

N ( z o , z ~~ , 2 . .)~ = .( O , Z ~ 222,. , . .) . Acknowledgment. A.B.G. was supported by QP-Applications, European Research Training Network “Quantum Probability with Applications to Physics; Information Theory and Biology”, Contract No. HPRN-CT-2002-00279.

144

References 1. A. Ben Ghorbal and M. Schiirmann. Quantum stochastic calculus on Boolean Fock space. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7 (4):631-650, 2004. 2. F. Fagnola and S. J. Wills. Solving quantum stochastic dfferential equations with unbounded coefficients. J. Funct. Anal., 198 (2):279-310, 2003. 3. A. Guichardet. Symmetric Hilbert spaces and related topics. SpringerVerlag, Berlin, 1972. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes, Lecture Notes in Mathematics, Vol. 261. 4. R. L. Hudson, P. D. F. Ion, and K. R. Parthasarathy. Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae. Comm. Math. Phys., 83 (2):261-280, 1982. 5. R. L. Hudson and K. R. Parthasarathy. Quantum Ito’s formula and stochastic evolu- tions. Comm. Math. Phys., 93 (3):301-323, 1984. 6. P.A. Meyer. Quantum probability for probabilists, volume 1538 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1993. 7. K. R. Parthasarathy. An introduction to quantum stochastic calculus, volume 85 of Monographs in Mathematics. Birkhauser Verlag, Basel, 1992.

FUNCTIONAL INTEGRALS OVER SMOLYANOV SURFACE MEASURES FOR EVOLUTIONARY EQUATIONS ON A RIEMANNIAN MANIFOLD

YA. A. BUTKO M . V.Lomonosov Moscow State University Faculty of Mechanics and Mathematics Moscow, 11 9992 , Russia E-mail: [email protected] In this paper several results obtained for some evolutionary equations on a compact Riemannian manifold are presented. In particular, representations of the solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a manifold are obtained in the form of limits of finite-dimensional integrals. These limits coincide with integrals over Smolyanov surface measures on the set of trajectories in a manifold and over the Wiener measure, generated by Brownian motion in a domain with absorption on the boundary. Integrands are combinations of elementary functions of coefficients of the equation and geometric characteristics of the manifold. Also representations of the solution of the Cauchy problem for the Schroedinger equation on a compact Riemannian manifold are obtained in the form of functional integrals over Smolyanov surface measures. In the proof a substantial role is played by Smolyanov-Weizsaecker-Wittichasymptotic estimates for Gaussian integrals over a manifold, by the Chernoff theorem and by the method of transition from the Schroedinger to the heat equation going back to Doss.

1. Introduction

Many researches are devoted to diffusion processes in Riemannian manifolds and to representations of solutions of Kolmogorov equations by functional integrals over measures, generated by diffusion processes, corresponding to Hamiltonians of these equations. However, densities of transition probabilities of such processes can not be expressed with the help of elementary functions. In works of Smolyanov, Weizsaecker and their coauthors approximations of functional integrals over the Wiener measure, generated by Brownian motion in a manifold, were obtained in the form of finite-dimensional integrals of elementary functions that contain geometric characteristics of the manifold with coefficients and (As original Feynman definition of the functional integral is based on a limit of finite-dimensional integrals,

A.

i,

145

146

we call limits of finite-dimensional integrals representing solutions of evolutionary equations as Feynman formulas.) Functional integrals’ approximations that contain geometric characteristics of a manifold with different coefficients can be found in various literature. In the paper [6] authors reviewed a status quo and set the problem to make conditions of using of each particular coefficient more precise. For the first time in mathematical literature it was done in papers of Smolyanov, Weizsaecker and their coauthors [21, 22, 23, 26, 24, 191. In these works it was explained that different coefficients appear due to different ways of determination of a distance between objects in the manifold when one approximates transition probabilities of the Wiener measure on the set of trajectories in the manifold. The distance can be found by three different ways: with the help of the Riemannian metric of the manifold, with the help of the norm of the ambient Euclidean space or with the help of a metric of some ambient manifold. Independently on Smolyanov and Weizsaecker the coefficient was elucidated by Andersson and Driver in the paper [l] and a little bit later in [ll]F’roese and Herbst investigated the coefficient 1

s.

Approximations obtained in works of Smolyanov and Weizsaecker naturally lead to surface measures different from the Wiener measure, generated by Brownian motion in the manifold. Hence, in this paper we give also the representations of solutions of evolutionary equations on a Riemannian manifold by functional integrals over these (Smolyanov) surface measures. In the method of Smolyanov and Weizsaecker the Chernoff theorem plays the same role as the Trotter formula in the traditional approach. It significantly extends a field of applicability of the method. In the present paper we show that Smolyanov-Weizsaecker approach is applicable for boundary value problems and for Schroedinger type equations.

2. Preliminaries

For any Riemannian manifold K , denote by v o l ~the Bore1 measure on K and by p - a distance in K , generated by Riemannian structure of K . Due to Nash theorem [12] we can and shall assume that K is a smooth mdimensional manifold isometrically embedded into a Euclidean space RN and : K -+ RN is a smooth embedding. Let scal(x) = trRicci(x) be a scalar curvature of the manifold K at the point x E K . Denote by r2(x) dimension of the manifold times the square of the norm of the vectorvalued mean curvature of the manifold at the point x. We assume that functions

147

seal(.) and r2(.) - are continuous on K . For a domain G let dG denote its boundary. Denote as Co(G) the Banach space of continuous functions on G that vanish on dG equipped with the norm 11 . 11 given by l l f l l =

If(x)l.

SUP~~G Let the symbol D 3 ( G ) stand for the set of three times continuously differentiable on G functions with support in G. Denote by B([O,t ] K , ) the set of functions on [0,t]with range in K , that have jump discontinuities only. The number e2i is everywhere denoted as

&. Let K- be a smooth m-dimensional compact Riemannian manifold embedded into a Euclidean space RN.For any n E N K" = K x K x ... x K . Let us for any t 2 0 introduce a partition II = { t j : 0 = t o < tl < ... < t, = t } of the time interval. Assume that the diameter of II satisfies the condition lIIl = maxj I t j - tj-11 + 0 as n -+ 00. Let f : B([O,t], K ) -+ R - be a bounded continuous function. Then define fE(x1,..., 2), = f(cpg(x1,..., zn)(t)), where cpg(x1,...,zn)(t)- is the mapping of the set K" into B([O,t ] K , ) such that 'ps(z1,..., xn)(to) = z, cpg(x1,..., x n ) ( s ) = x j ,a s s 6 ( t j - l , t . j ] , j = l , ..., 72. For t 2 0, x,z E K write

P Y t , x,2) = q'(t,

5, 2)

( 2 4 4 2e

P2(.+)

- 2 t ,

=

-+ IR - be a Definition 1. Let t > 0, and x E K , and let f : B([O,t],K) bounded continuous function. By the integral of the function f over the measure W2r we mean the limit

148

Definition 2. Let t > 0, and II: E K , and let f : B([O,t ] ,K ) + IR - be a bounded continuous function. By the integral of the function f over the measure S2' we mean the limit

I

P'(t17 2, II:l)P1(t2-tl, 21,22)...P (tn-tn-1, %-I,

II:,)vOlK(c .,.i)...VOlK(dII:,).

If in definitions 1 and 2 we replace all symbols I by symbols E , we obtain definitions of integrals over measures Wz" and 5'2" respectively. Integrals in definitions above can be considered as integrals over the space c([o, t], K ) . As it was shown in works of Smolyanov, Weizsaecker and their coauthors [26, 19, 181, measures W z I and W"; coincide with the Wiener measure WE, generated by Brownian motion in the manifold; measures and S2" are equivalent to Wg, and their Radon-Nikodym densities are given by formulas:

5'2'

C([O,tl,K)

Further we call measures 5'2' and S2" as internal and external Smolyanov surface measures respectively. In our proofs we use the following version of the Chernoff theorem [4]. Let X - be a Banach space, and let L(X) be the space of all continuous linear operators on X equipped with the strong operator topology. For any linear operator A on X , denote by Dom(A) the domain of A . The derivative at the origin of a function F : [ O , E ) -+ L(X), E > 0 is a linear mapping F'(0) : Dom(F'(0)) + X such that F'(0)g = limt+ot-l(F(t)g - F(O)g),

149

where Dom(F’(0)) is the vector space of all elements g E X , for which the above limit exists.

Theorem 1. (Chernoff theorem). Let X - be a Banach space, let F : [O,w) 4 L(X) be a strongly continuous mapping, let F ( 0 ) = I be the identity operator, let IIF(t)II 5 eat for some a E R, and let D be a vector subspace of Dom(F’(0)) such that the restriction of the operator F’(0) to this subspace admits the closure. Let C be this closure. If C is a generator of a strongly continuous semigroup etC, then, for any T > 0, the sequence F ( t / n ) n , n E N, converges to etc as n -+ 00 in the strong operator topology, and this convergence is uniform with respect to t E [0,TI. Let us also present the definition (introduced in [23]) of Chernoff equivalence for one-parameter families of operators { F ( t ) , t > 0 ) and {etc, t > 0 ) . These two families of operators are said to be Chernoff equivalent if IIF(t)g - etcglJ = o ( t ) as t -+ 0 for any g E D1 C Dom(C), where D1-is the essential domain of the operator C. 3. Functional integrals corresponding to the Cauchy-Dirichlet problem for the heat equation

Let G - be a domain of a smooth m-dimensional compact Riemannian manifold K c RN with smooth boundary dG. Consider the Cauchy-Dirichlet boundary value problem in this domain for the heat equation with bounded continuous potential V : G -+ R (we can extend V to a continuous function on = G d G , we will denote this extension also as V ) .

c

U

g ( t , X )= ( - $ A K f ) ( t , x ) f(0,X) =fob) f ( t ,x ) = 0

+

V(X)f(t,X)

t 2 0,s E G xEG t 2 0,x E dG

(I)

Here A K stands for the Laplace-Beltrami operator -tr V2 on the manifold K . Assume that f and fo satisfy the conditions fo E C O ( ~f) ,: [O, m) x R, f ( t ,.) E CO(G),vt 2 0. Let an operator A in the space Co(c) be the generator of the semigroup resolving the problem (I). Then for any f E Dom(A):

c

150

Note, that D3(G)c Dom(A) is the essential domain for the self-adjoint operator A, and for any f E D 3( G )the operator A acts as following: ,

1 (Af)(z) = (--&f)(.) 2

+ V ( z ) f ( x ) , z E (7.

Let E ( . ) : [O,+m) -+ [O,+m) be a smooth function tending to zero when t 4 0. Suppose that ( P ~ ( ~ ) ( .is) a set of functions in D3 (G)which approximate the indicator of the domain G as t +. 0 with respect to the pointwise convergence topology. Consider the following operators acting in the Banach space CO(G):

T t ( t ): ( T , E ( t ) f > ( x=) ( P E ( t ) ( 4 j- e t v ( z ) f( Z ) Q E ( t , x , z)volddz), G

T f ( t ):

j- ,tV(

( q V ) f ) ( z=) ( P E ( t ) (

z) t s c a l ( z )e-

6 r2 (z) f ( 4 P E ( t ,z, z)vok(dz)

G

T,'(t) : T,'(t)fI(.)

=(PE(t)(4

j- et"(")f(z)q1(t,2,z)voMdz), G

T,'(t) : (T,'(t)f)(z)= ( ~ ~ ( ~ )e t(vx( z)) e ~ S C afl (z>pI(t, (z) z, z)volddz) G

One can check (see [2]), that these families of operators are Chernoff equivalent to the semigroup etA. Therefore, by the Chernoff theorem we obtain the following statement.

Theorem 2. Let etA be the semigroup of operators on Co(G), resolving the Cauchy-Dirichlet problem (I). Then 1) etA = s - lim (T,E(t/n)"), 2) etA = s - lim (T,E(t/n)"), n'w n+w 3) etA = s - lim (T,'(t/n)n), 4) etA = s - lim (T,'(t/n)n). n+w 71-00 where s-lim stands for the limit with respect to the strong operator topology on Co(G).

It can be shown (see [2]) that the limits of finite-dimensional integrals in the theorem (2) coincide with limits of finite-dimensional integrals which are of the same kind as those used in definitions of measures W z ' , W z E , S;' and S;". Hence, Feynman formulas obtained in the theorem (2) can be understood as functional integrals over surface measures:

Theorem 3. Let f ( t , x ) be the solution of the Cauchy-Dirichlet problem (I) with the initial condition fo E Co(G). Then the solution f (t,x ) can be

151

represented by a functional integral over the Wiener measure WE.

by a functional integral over the internal Smolyanov surface measure S2':

and by a functional integral over the external Smolyanov surface measure S;E:

4. Functional integrals representing solution of the Cauchy

Problem for the Schoedinger equation Consider the Cauchy problem for the Schroedinger equation on the manifold

K: i g ( t , X )= ( i A K f ) ( t , X ) 4- v ( X ) f ( t , X ) t 2 0, 5 E K (11) X E K f(0,z) = fO(5) Assume that f and fo satisfy the conditions fo E C(K), f : [ O , o o ) x K + C, f ( t , . )E C(K), W 2 0. Here C(K) for the Banach space of complex valued continuous functions on K equipped with the norm

152

11 11, l l f l l *

= SUPzEK

If(x)l.

+

Consider the set K = U x C ~ { x &(K - x)}. We say that a function g : K 4 C belongs to the class A, iff the following conditions are satisfied: 1) there is a domain 0, of some complex manifold such that the closure of 0, contains the set K; 2) there exists the unique analytical in 0, and continuous on 0, K function i j such that the restriction of B to K coincides with g. If in condition (2) the function i j is twice continuously differentiable on 0, U K , then we say that the function g belongs to the class Az. Let functions V and fo belong to the class A . Suppose that the Schroedinger equation with the potential V and the initial condition fo has in K the unique solution f ( t , z ) , which belongs to the class Az:

u

ig(t,z ) = ( i A & f ) (zt ), + V ( z ) f ( t z, ) t 2 0, t E K

{ f(0,

z ) = fob)

z E K.

The operator A& for functions from the class Az is defined as follows: the value ( A & B ) ( z ) equals the value of the analytical continuation of the function A K g at the point z. For functions fo, V of the class A and functions f ( t , .) of the class A2 for any y E K and any fixed x E K we can define following functions: cp"(t,Y) = f(t,a: &(Y - x)), cpti(Y) = f o b + J;l(Y - X I ) , V"(Y) = 4 V ( x &(y - x)). Let y in the symbol Ak mean that the Laplace-Beltrami operator AK acts on the variable y. Then ( A k c p " ) ( t , y ) = ( i A & f ) ( t , z ) for z = x &(Y - x). Hence, for any fixed x E K the function cp" solves the Cauchy Problem for the heat equation:

+

+

+

%(t,Y) = (-$Ak'P")(t,y) v"(0, Y) = cptib) Using results of works [21, 26,

+ V(Y)cp"(t,Y) t 2 0,YE K YEK

(1)

18, 191, we can represent solutions

9" of the Cauchy problem for the family of heat equations (1) with the

help of functional integrals over Smolyanov surface measures and over the Wiener measure, generated by Brownian motion in the manifold. If in obtained formulas we come back to functions f, fo, V and notice that f ( t , x) = cp"(t,x) then we get the following:

Theorem 4. Let K be a smooth m-dimensional compact Riemannian manifold isometrically embedded into the space RN c CN.Let functions V and

153

fo belong to the class A . Suppose that the Cauchy problem (11) for the Schroedinger equation with the potential V and the initial condition fo has in K the unique solution f ( t ,z ) , which belongs to the class Az. Then f ( t ,x) can be represented by a functional integral over the Wiener measure WE:

c(lo,tl, K ) by a functional integral over the external Smolyanov surface measure S 2 E :

and by a functional integral over the internal Smolyanov surface measure S2':

t

( c I ( t ,x ) ) - 1 =

.I

Q JSC~(E(T))~T e o

S2'( d c ) .

Acknowledgments Author expresses her deep gratitude to Prof. O.G. Smolyanov for useful discussions. References 1. L. Andersson, B.K. Driver, Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds, J. Funct. Anal., 65 (1999), no. 2, 430-498.

154 2. Butko Ya. A., Representations of the Solution of the Caushy-Dirichlet Problem for the Heat Equation in a Domain of a Compact Riemannian Manifold by Functional Integrals, Russian Journal of Mathematical Physics, 11 N 2 (2004), 1-9. 3. Butko Ya. A., Functional integrals for Schroedinger equation in a compact Riemannian manifold, Math. Zametki, 79 N 2 (2006), 194-200. 4. R. Chernoff, A Note on Product Formulas for Operator Semigroups, J . Funct. Anal., 2 (1968), 238-242. 5. R. Chernoff, Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators, Mem. Amer. Math. SOC.,140 (1974). 6. De Witt-Morette C., Elworthy K.D., Nelson B.L., Sammelman G.S., A stochastic scheme for constructing solutions of the Schroedinger equations, Ann. Ins. H. Poincare Sect. A (N. S.) 32 N 4 (1980), 327-341. 7. Doss H., Sur une Resolution Stochastique de 1'Equation de Schroedinger a Coefficients Analytiques, Communications in Math. Phys, V.73, N3, 1980, 247-264. 8. Eells J., Elworthy K.D., Wiener integration on certain manifolds, Problems in non-linear analysis (C.I.M.E., IV Ciclo, Varenna, 1970 ), Edizioni Cremonese, 1971. 9. R.P. Feynman, Space-time Approach to Nonrelativistic Quantum Mechanics, Rev. Mod. Phys., 20 (1948), 367-387. 10. R.P. Feynman, An Operation Calculus Having Application in Quantum Electrodynamics, Phys. Rev., 84 (1951), 108-128. 11. F'roese R., Herbst G., Realizing holonomic constrains in classical and quantum mechanics, Com. Math. Phys. 220 (2001), 489-535. 12. Nash J. F., The imbedding problem for Riemannian manifolds, Ann. Math., 63, (1956), 20-63. 13. 0.0.Obrezkov, The Proof of the Feynman-Kac Formula for Heat Equation on a Compact Riemannian Manifold, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6 (2003), no. 2, 311-320. 14. Sidorova N. A., The Smolyanov surface measure on trajectories in a Riemannian manifold, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 7, N3, September 2004, 461-472. 15. Sidorova N. A., Smolyanov O.G., Weizsaecker H. v., Wittich O., The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold, Journal of Functional Analysis, 206 (2004), 391-413. 16. Smolyanov O.G., Tokarev A.G., Truman A., Hamiltonian Feynman Path Integrals via the Chernoff Formula, J . Math. Phys., 43 (2002), no. 10, 51615171. 17. Smolyanov O.G., Truman A,, Hamiltonian Feynman Formulas for the Schrodinger Equation in Bounded Domains, Doklady Acad. Nauk, 70 (2004), 899-904. 18. 0. G. Smolyanov, H. von Weizsacker, 0. Wittich, and N. A. Sidorova, Surface Measures Generated by Diffusions on Paths in Riemannian Manifolds, Doklady Mathematics, 63 N 2 (2001), 203-208. 19. 0. G. Smolyanov, H. von Weizsacker, 0. Wittich, and N. A. Sidorova , Wiener

155

Surface Measures on Trajectories in Riemannian Manifolds, Doklady Mathematics, 65 N 2 (2002), 239-244. 20. 0. G. Smolyanov, Smooth measures on loop groups, Doklady Acad. Nauk, 345 N 4 (1995), 455-458. 21. Smolyanov O.G., Weizsiicker H. von, Wittich O., Brownian Motion on a Manifold as Limit of Stepwise Conditioned Standard Brownian Motions, Canadian Math. Society Conference Proceedings, 29 (2000), 589-602. 22. Smolyanov O.G., Weizsiicker H .v., Wittich 0. ”Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds” http : //arxiv.org/PS-cache/math/pdf /0409/0409155.pdf 23. Smolyanov O.G., Weizsiicker H. von, Wittich O., Chernoff’s Theorem and the Construction of Semigroups, Evolution Equations: Applications to Physics, Industry, Life sciences and Eqonomics - E V E Q 2000, M. Ianelli, G. Lumer, Birkhauser (2003), 355-364. 24. Smolyanov O.G., Weizsiicker H. von, Wittich O., The Feynman Formula for the Cauchy Problem in Domains with Boundary, Doklady Acad. Nauk, 69 N 2 (2004), 257-262. 25. Trotter H.F., On the Product of Semigroups of Operators, Proc. Amer. Math. SOC.,10 (1959), 545-551. 26. H. von Weizsacker, 0. G. Smolyanov, and 0. Wittich , Diffusion on Compact Riemannian Manifolds and Surface Measures, Doklady Mathematics, 61 N2 (2000), 230-235.

QUANTUM PROBABILISTIC MODEL FOR T H E FINANCIAL MARKET *

OLGA CHOUSTOVA International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vaxjio, S-35195, Sweden E-mail: Olga. ChoustovaQm.se

We use the formalism of quantum mechanics in the framework of the Bohmian (pilot wave) model to describe stochastisity of the financial market. We interpret nonclassical contribution to stochasticity as a psycho-financial (information) field $ ( q ) describing expectations of agents of the financial market.

1. Introduction

In economics and financial theory, analysts use random walk techniques to model behavior of asset prices, in particular share prices on stock markets, currency exchange rates and commodity prices. This practice has its basis in the presumption that investors act rationally and without bias, and that at any moment they estimate the value of an asset based on future expectations. Under these conditions, all existing information affects the price, which changes only when new information comes out. By definition, new information appears randomly and influences the asset price randomly. Corresponding continuous time models are based on stochastic processes (this approach was initiated in the thesis of L. Bachelier' in 1890). However, empirical studies have demonstrated that prices do not completely follow random walk. Low serial correlations (around 0.05) exist in the short term; and slightly stronger correlations over the longer term. Their sign and the strength depend on a variety of factors, but transaction costs and bid-ask spreads generally make it impossible to earn excess re*This work is supported by Profile Mathematical Modelling of Vkixjo university and EU-network on Quantum Probability and Applications

156

157 turns. Interestingly, researchers have found that some of the biggest prices deviations from random walk result from seasonal and temporal patterns. Therefore it would be natural to develop approaches which are not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. In particular, there are two well established (and closely related ) fields of research behavioral finance and behavioral economics which apply scientific research on human and social cognitive and emotional biasesa to better understand economic decisions and how they affect market prices, returns and the allocation of resources. The fields are primarily concerned with the rationality, or lack thereof, of economic agents. Behavioral models typically integrate insights from psychology with neo-classical economic theory. Behavioral analyses are mostly concerned with the effects of market decisions, but also those of public choice, another source of economic decisions with some similar biases. In physics, researchers are concerned with observer effects - which set clear limits on the process of observing as in Heisenberg's uncertainty principle. Quantum entanglement also introduces problems in some observational situations. These are all well-accepted foundations of 20th century philosophy of science and along with a few other such discoveries (like a universal maximum for the speed of light) form its core epistemology. Knowing these limits has helped develop a cognitive science by which humans might reasonably characterize the limits of their own perception. However, bias does not end with cognition. How to interpret the data on what humans 'can' observe becomes controversial when there are few individuals capable of reproducing experiments and compiling new models. Since the 1970s, the intensive exchange of information in the world of finances has become one of the main sources determining dynamics of prices. Electronic trading (that became the most important part of the environment of the major stock exchanges) induces huge information flows between traders (including foreign exchange market). Financial contracts Wognitive bias is any of a wide range of observer effects identified in cognitive science, including very basic statistical and memory errors that are common t o all human beings and drastically skew the reliability of anecdotal and legal evidence. They also significantly affect the scientific method which is deliberately designed to minimize such bias from any one observer. They were first identified by Amos Tversky and Daniel Kahneman as a foundation of behavioral economics. Bias arises from various life, loyalty and local risk and attention concerns that are difficult t o separate or codify. Tversky and Kahneman claim that they are at least partially the result of problem-solving using heuristics, including the availability heuristic and the representativeness.

158

are performed at a new time scale that differs essentially from the old ”hard” time scale that was determined by the development of the economic basis of the financial market. Prices at which traders are willing to buy (bid quotes) or sell (ask quotes) a financial asset are not more determined by the continuous development of industry, trade, services, situation at the market of natural resources and so on. Information (mental, marketpsychological) factors play very important (and in some situations crucial) role in price dynamics. Traders performing financial operations work as a huge collective cognitive system. Roughly speaking classical-like dynamics of prices (determined) by ”hard” economic factors is permanently perturbed by additional financial forces, mental (or market-psychological) forces, see the book of J. Soros [2]. In this paper we develop a new approach that is not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. Our approach can be considered as a special econophysical model in the domain of behavioral finance. In our approach information about financial market (including expectations of agents of the financial market) is described by an information field $(q) - financial wave. This field evolves deterministicallyb perturbing the dynamics of prices of stocks and options. Since psychology of agents of the financial market gives an important contribution into the financial wave $(q), our model can be considered as a special psycho-financial model. This paper can be also considered as a contribution into applications of quantum mechanics outside microworld, see also books [3,4]. The complete version of the present paper was submitted as the quant-preprint [5]. 2. Financial phase-space

Let us consider a mathematical model in that a huge number of agents of the financial market interact with one another and take into account external economic (as well as political, social and even meteorological) conditions in order to determine the price to buy or sell financial assets. We consider the trade with shares of some corporations (e.g., VOLVO, SAAB, IKEA, ...).‘ We consider a price system of coordinates. We enumerate corporations which did emissions of shares at the financial market under consideration: bDynamics is given by Schrodinger’s equation on the space of prices of shares. ‘Similar models can be developed for trade with options, see E. Haven [S]for the Bohmian financial wave model for portfolio.

159 j = 1 , 2 , ...., n (e.g., VOLV0:j = 1, SAAB:j = 2, 1KEA:j = 3,...). There

can be introduced the n-dimensional configuration space Q = Rn of prices, q = (q1,. . . ,q n ) , where qj is the price of a share of the j t h corporation. Here R is the real line. Dynamics of prices is described by the trajectory q ( t ) = ( q I ( t ) , . . . ,qn(t)) in the configuration price space Q. Another variable under the consideration is the price change variable: vj ( t )= q j (t)= limAt+o q j ( t t AAt t ) - q J ( t ) , see, for example, the book [7] on the role of the price change description. In real models we consider the discrete time scale At, 2At,. . . . Here we should use discrete price change variable zj(t) = qj(t +At) - qj(t). We denote the space of price changes by the symbol V ( = Rn),v = (211,. . . ,vn). As in classical physics, it is useful to introduce the phase space Q x V = R2n,namely the price phase space. A pair (q,v) = (price, price change) is called a state of the financial market. (Later we shall consider quantum-like states of the financial market. A state (q,v) is a classical state.) We now introduce an analogue m of mass as the number of items (i.e., in our case shares) that trader emited to the market.d We call m the financial mass. Thus each trader has its own financial mass mj (the size of the emission of its shares). The total price of the emission performed by the j t h trader is equal to Tj = m j q j . Of course, it depends on time: Tj(t) = mjqj(t).To simplify considerations we consider a market at that any emission of shares is of the fixed size, so mj does not depend on time. In principle, our model can be generalized to describe a market with timedependent financial masses, mj = mj(t). We also introduce financial energy of the market as a function H : Qx V R. If we use the analogue with classical mechanics. (Why not? In principle, there is not so much difference between motions in ”physical space” and ”price space”.), then we could consider (at least for mathematical modeling) the financial energy of the form: --f

4

Here K = C;==, mjv; is the kinetic financial energy and V(q1,.. . ,qn) is the potential financial energy, mj is the financial mass of j t h trader.e d‘Number’ is a natural number m = 0,1,. . . , - the price of share, e.g., in the US-dollars. However, in a mathematical model it can be convenient to consider real m. This can be useful for transitions from one currency to another. eThe kinetic financial energy represents efforts of agents of financial market to change

160

The potential financial energy V describes the interactions between traders j = 1,...., n (e.g., competition between NOKIA and EFUCSSON) as well as external economic conditions (e.g., the price of oil and gas) and meteorological conditions(e.g., the weather conditions in Louisiana and Florida). For example, we can consider the simplest interaction potential: V(q1,.. . ,qn) = z y = l ( q i - q j ) 2 . The difference Iq1 - q j l between prices is the most important condition for arbitrage. To describe dynamics of prices, it is natural to use the Hamiltonian dynamics on the price phase space. As in classical mechanics for material objects, it is useful to introduce a new variablep = mu, the price momentum variable. So, instead of the price change vector v = (211,. . . ,vn), we shall consider the price momentum vector p = ( P I , . . . ,p,), p j = mjvj. The space of price momentums is denoted by the symbol P. The space Q x P will be also called the price phase space. Hamiltonian equations of motion on the 8H price phase space have the form: q = =,@j = = 1,.. . , n .

-E,j

3. Financial Pilot W a v e We now consider a model in that dynamics of prices of shares is driven by an information field (or psycho-financial wave) representing the psychology of agents of the financial market. We represent such a wave in the same way as in the Bohmian mechanics for quantum particles. In fact, we need not develop a new mathematical formalism. We will just apply the standard pilot wave formalism to traders of the financial market. The fundamental postulate of the pilot wave theory is that the pilot wave (field) $(q1,. . . ,qn) induces a new (quantum) potential U ( q 1 , .. .,qn) which perturbs the classical equations of motion. A modified Newton equation has the form:

-=

where f = 89 and g = financial mental force.f

P=f+g,

-m. We call the additional financial force g a 89

prices: higher price changes induce higher kinetic financial energies. If the corporation jl has higher financial mass than the corporation j z , so mj, > mj,,then the same change of price, i.e., the same financial velocity vj, = vj2, is characterized by higher kinetic financial energy: K j , > Kj, . We also remark that high kinetic financial energy characterizes rapid changes of the financial situation at market. However, the kinetic financial energy does not give the attitude of these changes. It could be rapid economic growth as well as recession. fThis force g(q1,.. . ,qn) determines a kind of collective consciousness of the financial market. Of course, the g depends on economic and other ‘hard’ conditions given by the

161

By using the standard pilot wave formalism we obtain the following rule for computing the financial mental force. We represent the financial pilot wave $ ( q ) in the form: $(q) = R(q)eiS(q)

where

R(q) = I$(Q)I is the amplitude of $(q) and S(q) is the phase of $(q). Then the financial mental potential is computed as

and the financial mental force as gj(q1,. . . , qn) = = aU ( q l , . . . ,qn). These formulas imply that strong financial effects are produced by financial waves having essential variations of amplitudes. Example 1. (Financial waves with small variation have no effect). Let R = const. Then the financial mental force g E 0. There are no nonlocal effects which can be induced by nontrivial financial force. Thus if R = const, then it is impossible to perturb the psychological state of the whole financial market by varying the price of shares qj of the fixed trader j. The constant information field does not induce psychological financial effects at all. As we have already remarked the absolute value of this constant does not play any role. Waves of constant amplitude R = 1, as well as R = lo1'', produce no financial effect. Let R(q) = cq,c > 0. This is a linear function; variation is not so large. As the result g = 0 here also. No financial mental effects. Example 2. (Successive speculations) Let R(q) = c(q2 d ) , c, d > 0. Here U ( q ) = (it does not depend on the amplitude c !) and g(q) = The quadratic function varies essentially more strongly than the linear function, and, as a result, such a financial pilot wave induces a nontrivial financial mental force. In particular, there are nonlocal financial effects.

+

-&

.a.

financial potential V ( q 1 , .. . , q,,). However, this is not a direct dependence. In principle, a nonzero financial mental force can be induced by the financial pilot wave cp in the case of zero financial potential, V 0. So V = 0 does not imply that U 0. Market psychology i s not a totally determined by economic factors. Financial (psychological) waves of information need not be generated by some changes in a real economic situation. They are mixtures of mental and economic waves. Even in the absence of economic waves, mental financial waves can have a large influence to the financial market.

=

=

162

The only problem which we have still to solve is the description of the time-dynamics of the financial pilot wave, $(t,q). We follow the standard pilot wave theory. Here $ ( t , q ) is found as the solution of Schrodinger's

equation.

with the initial condition $(O, q1,. . . ,qn) = $(q1,. . . ,qn).g We underline two important features of the financial pilot wave model: a) all traders are coupled on the information level; b) reactions of the financial market do not depend on the amplitude of the financial pilot wave: financial waves $, 2$, lOOOOO$ will produce the same reactionsh. References 1. L. Bachelier, Theorie de la speculation, Ann. Sc. 1'Ecole Normale Superiere 111-17,21-86 (1890). 2. J. Soros, The alchemy of finance. Reading of mind of the market (J. Wiley and Sons, Inc.: New-York, 1987). 3. L. Accardi, Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica (I1 Saggiatore, Rome, 1997) 4. A. Yu. Khrennikov, Information dynamics in cognitive, psychological and anomalous phenomena (Kluwer, Dordreht, 2004). 5. 0. Choustova, Pilot wave quantum model for the stock market, http://www.arxiv.org/abs/quant-ph/O109122. 6. E. Haven, Bohmian mechanics in a macroscopic quantum system. Foundations of Probability and Physics-3, ed. A. Yu. Khrennikov (Melville, New York: AIP Conference Proceedings, 2006). 7. R. N. Mantegna and H. E. Stanley, Introduction to econophysics (Cambridge, Cambridge Univ. Press, 2000).

gWe make a remark on the role of the constant h in Schrodinger's equation. In quantum mechanics (which deals with microscopic objects) h is the Planck constant. This constant is assumed to play the fundamental role in all quantum considerations. However, originally h appeared as just a scaling numerical parameter for processes of energy exchange. Therefore in our financial model we can consider h as a price scaling parameter, namely, the unit in which we would like to measure price change. hThe amplitude of an information signal does not play so large role in the information exchange. The most important is the context of such a signal. The context is given by the shape of the signal, the form of the financial pilot wave function.

A NEW PROOF OF A QUANTUM CENTRAL LIMIT THEOREM FOR SYMMETRIC MEASURES

VITONOFRIO CRISMALE Dipartimento d i Matematica, Universitii d i Bari via E. Orabona 4, I-70125 Bari, I T A L Y e-mail: crisma1evQdm.uniba.it YUN GANG LU

Dipartimento di Matematica, Universitii d i Bari via E. Orabona 4, I-70125 Bari, I T A L Y e-mail: 1uQdm.uniba.it We present a new proof of the central limit theorem performed in [2] for symmetric measures based on a different approach.

1. Introduction In [2] a constructive quantum central limit is proved for any mean-zero real probability measure with moments of any order. The most important tool there used is interacting Fock space (IFS) (see and references therein for more details). More recently in the authors gave another proof of such result based on the realization that the convolution arising from addition of field operators in one-mode type IFS is the universal one of Accardi-Bozejko [l].In this note we give a new proof of such a theorem for symmetric measures, where, even in the framework of IFS, a new approach is privileged. Namely, after the introduction of a creation-annihilation process on a suitable IFS, we prove that the central limits of its even moments satisfy a system of equations whose unique solution is given by the (even) moments of the measure. It is worth to mention that, in [5] a similar central limit result is obtained. That result has been successively generalized in [8]. But between that result and our central limit theorem proved in the present paper, there 163

164

are two differences: i) the random variables considered here satisfy only the singleton condition and the uniform boundedness of the mixed moments (see [3]) and do not satisfy the weak independence used in [5]; ii) we explicitly realize both the approximating random variables and their limits as sums of creation and annihilation operators in suitable interacting Fock spaces.

2. Interacting Fock spaces

In this section we define interacting Fock spaces and give some properties about them that will be used in the following results.

Definition 2.1. Let ( X , X , p ) be a measure space and let family of functions with the following properties:

be a

(i) for any n E N, A, : ( X n , X n ) -+ R+ is bounded, positive, measurable; (ii) for any measurable function F, : (Xn, Xn)4 C if

then for any measurable function f : ( X ,X)-+ C,

J If (.>I2

IF,

2

(5%.

. .,z1)1

An+l(z,zn, . . . , 5 1 ) p ( d z ) p (dz,)

. . . p (dz1) = 0

We define, for each n E N, the (not necessarily finite) measure p, on X n by

and the associated L2-space:

H,

:= L2 (Rn,p,),V n 2

2

with pre-scalar product such that for any F,, G, E H ,

By taking the quotient and completing H , becomes an Hilbert space and with the convention that

Ho

:= C,

H := Hi

165

The space 00

F(H,{A,},) : = @ H , ,

@::=1@0@0@*..

(1)

n=O

is called the (standard) interacting Fock space with weight functions {A,},='. 00 In particular if the are constant, then the corresponding space r ( H ,{A}), is called a 1-mode type free interacting Fock space (1MT-IFS in short).

Definition 2.2. On the Interacting Free Fock space r ( H ,{A,},), f E H , for any n E N and for any F, E H , the creation operator

(A+ (f)Fn) (xn+1,xn,.. *

,21):= f (xn+1). F n (2n.r. . .

for any

21)

is well defined as a linear operator A + ( f ) : H, -+ H,+l and has an adjoint A ( f ) : H, HH,-1 (on an appropriate domain) called the annihilation operator

A ( f ) := ( A + ( f ) ) * Remark. The condition ii) in Definition 2.1 guarantees that the creation operator is well defined. The @-statistics of the operator stochastic process { A (f),A+ (9) : f , g E H } is coded into the mixed moments

(A"(")(f,) . . .

(f2)

A"(') (f~))

(2)

where (a)

:= (@, .@)

1

( A (f)A+ (9))= ( f , 9 > H ,

Remark. The following simple results are easy consequences of the definition of IFS. For any n E N and E = ( E (1), . . . ,E ( n ) )E (-1, l}, 0

if among { A"(,) (f,) ,. .. ,A"(1)( f l ) } there are same number of annihilators and creators, then

A"(,) (f,) . * A"(') ( f 1 ) @ = C@

(3)

166

if among { A E ( l()f i ) , . . . , than creators, then

(fn)}

~ d n(fn) ) . . . A'(')

there are more annihilators ( f i ) Q, = O

if n is odd

(Aacn)(fn). .

(f2)

(fl)) = 0

2N

if C c ( k )# N k=l

more generally, if there is i = 1 , 2 , . . , 2 N such that among { AE(i-1) (fi-1) , * * , (fi)} there are more annihilators than creators, then

By the above remarks, one knows that in order to calculate the mixed moments (2) it is sufficient to consider only the even mixed moments (f2N)*.

*

(f2)

(fl))

and only for those E E {-l,l}:N , i.e. E E {-l,l}2Nand such that the number of creators is equal to the number of annihilators, i.e. x k2=Nl & ( k ) = N and such that, counting from right to left, at each step the number of creators is larger than the number of annihilators, i.e. a

E

(k) 2 0, for all i = 1,2,. ,2N

k=l

The simplest class of standard IFS is that for which the functions (An)n in Definition 2.1 are constants. The condition (ii) of Definition 2.1 becomes in this case: c R+ and A, = 0 + A n + l = 0 Vn. For them the moments of the field operator A (f)+ A + (f)depend only on the L2-norm of the test function f , as the following result state.

Proposition 2.1. For any f , g E H the moments o f A ( f )+ A + ( f )and of A(g) + A+(g) are the same if and only zf llfll = 11g[[.

167

Proof. See [2]. Remark. There are many interacting Fock spaces in which the distribution of A ( f ) A + ( f ) depends not only on llfll but also on f itself. For example, if we take H := L2 ([0,1]) and A, ( z n ,... ,z1) := 5 2 2 ; .z;-' for any n, then the distribution of A(x[o,q) A + ( x [ ~ ,is~ the I ) arcsine law but A ( f i ~ [ o , ~ / A~ +] )( f i ~ [ ~ ,has ~ / a~different ]) distribution. For more example see [4,71 references within.

+

1

+

+

3. Central limit theorem for symmetric measures

In this section we present our main result. The notations and definitions are the same as in [2], but in this case we deal only with symmetric measures. We consider the following operators on the 1-MT-IFS (L2 (R+) { A n I L ) : 1

Ak := A(X[k,k+l)); A: := A+(X[k,k+l)), k = 071,. . .

(4)

and denote

w1

:=A1

;

wn :=

A ;V n L 1 An- 1

We give the following technical lemmata, whose proof can be obtained similarly as in [2].

Lemma 3.1. O n 1-MT-IFS r (L2 (It+), { A n } r = 1 ) , for any N E M, for 2N any E E {-1,1}+ and for any { f l , . - . ,f 2 ~ )C H , with the convenience that A0 := 1

where, E

{ l k , rk}:=1

E {-1, l}"," .

as the unique non-crossing pair partition determined by

Remark For details on non-crossing pair partition, see '. Lemma 3.2. The family { A k , with respect to the state (@,.@).

satisfies the singleton condition

168

Remark For details on the singleton condition, see [3]. Lem ma 3.3. (Uniform boundedness of the mixed moments). For any N E N , f o r any {kl,.. . ,k,} c N and f o r any E E {-1,1}+2N

I

I( with the conventions:

A0

. . .AfL!”) 5 [A (N)]2N := 1

(7)

and, f o r any m E N :

Theorem 3.1. Let be given

(W,t3) with moments of any order and with the sequences of Jacobi coeficients given by {wn}, ; A : the *-algebra generated by {Ak, A:}:, where these operators are defined over the 1-MT-IFS space r (L2 (W+), {A,}). T h e relationships among the space, operators and coeficients {w,,A,}, are given as in [l] and in particular for any n w, = &. p : a mean-zero symmetric probability measure o n

Then

k = 0 , 1 , 2 . . . , the distribution of Ak + A: with respect t o the state (@, 4) is exactly the measure p; ii) for any k = 0 , 1 , 2 . . - , a) for any

where Qk is defined in (5). Proof. The point i) follows from Proposition 2.1 and [l].We turn to prove ii). We have to compute the limit for N + 0;) of E

(( 5 flk=l

for any m E N. In fact

Qk)

m, I

169

If m is odd, then the vacuum state above is equal to zero. On the other hand, the measure p is symmetric, so all the odd moments vanish. Hence in this case, condition ii) is satisfied and we turn to the case in which m = 2n. (8) is equal to

jF'rom Lemma 3.2 and Lemma 3.3 we know that the sequence{Aj, AT};, satisfies the singleton condition and the boundedness of the mixed moments. As a consequence, from 3 , Lemma 2.4, it follows that (8) does not vanish only if it is equal to 1 kz, *'* ki (9) N"

c

c

(

k : { 1 , 2,...,2n}-r{l, ...,N } ~ ~ { 0 , 1 } ~ "

I R a n d k ) I=P Ik-'(k(j))l=2 V j = l ,

)

+

...2n

where 1.1 denotes the cardinality. Each map k defined as in the above summation induces a pair partition of the set { 1,2, . . . ,2n} and we will use the following notation k-l(k(j))=:{lj,rj},

lj > r j

j = l ,..., n

We recall that in the interacting Fock space structure only the non crossing pair partitions may give a non-zero contribution in the computations of vacuum expectations (see [4]) and from [4], Lemma 6, Section 22, any fixed E E (0, l}? uniquely determines one non-crossing pair partition { l j , rj}y=l on { 1 , 2 , . . . ,2n} and viceversa. As a consequence (9) becomes

where N.C.P.P. {2n} denotes the set of all non-crossing pair partitions of { 1 , 2 , . . . ,2n} 2nd &k is the element of the set {0,1}? which is uniquely determined by the map k inducing the pair partition { l j , r j } y = l . We denote by

170

and we prove the following relation by induction on n.

where c ( w ,m) is a function of w depending on m. In fact the case n = 1 is trivial. We suppose the assumption is true for any natural integer h 5 n - 1 and prove it for h = n . If we denote by ( { l j , ~ - j } y = ~ ) the subset of { Z j , ~ - j } y =such ~ that km is the position of the m

creator coupled with the first annihilator from the left (i. e. m = rn), then

{ Zj,

c

E N.C. P. P. { m - 1 }

~ j

(

km-i

. . . A;:')))

2n

c 2n

=

m=l

mEZN+l

where in the last equality we used the property up = 1. We now see that the limit for N -+ 00 of u? exists. In fact

uf = 1 NZpO 1.

171

Let us suppose that for any m < n lim uz does exist. Then N-tW

2n

It follows that the limit on the left hand side exists. Let us denote V m := lim uz for any m = 0,1,2,. . . (notice that uf = 1 for any N ) . The N+W

o the system of discussion above implies that the sequence { v m } ~ =satisfies equations 2n

j R o m another hand, if we consider Um

= /x2mdp,

Vm = 0,1,2,. . .

from i) it follows that p is the distribution of namely urn=

1

(Ak

+ A;)

for any k E N,

(

z 2 m d p = (AkA;)2m)

By using this fact, it is easy to prove that 2n

m=l

mE2N+l

The system (10) has a unique solution, then u,

= un

for any n, i.e.

References 1. L. Accardi, M. Bozejko: Interacting Fock Spaces and gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1,no. 4, pp. 663470 (1998). 2. L. Accardi, V. Crismale, Y.G. Lu: Constructive universal central limit theorems based o n interacting Fock spaces, to appear in Infin. Dimens. Anal. Quantum Probab. Relat. Top., preprint Volterra n. 591 (2005).

172 3. L. Accardi, Y. Hashimoto, N. Obata: Notions of independence related t o the free group, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, no. 2, pp. 201-220, (1998). 4. L. Accardi, Y.G. Lu, 1.Volovich: T h e &ED Hilbert module and interacting Fock spaces, International Institute for Advances Studies, Kyoto (1997). 5. T. Cabanal-Duvillard, V. Ionescu: U n the'ordme central limite pour des variables ale'atoires non-commutatives. ProbabilitBslProbability Theory, C.R.Acad. Sci. Paris, t. 325, SQrie 1, pp. 1117-1120 (1997) 6. A. Krystek, L. Wojakowski: Convolution and central limit theorem arising f r o m addition of field operators in one-mode type Interacting Fock spaces", Preprint (2005). 7. Y.G. Lu: O n the interacting free Fock space and the deformed Wigner law, J. Nagoya Math., 145,pp.1-28, (1997). 8. Mlotkowski W.: Fkee probability o n algebras with infinitely m a n y states, Probab. Theory Related Fields 115, no. 4,pp. 579-596, (1999).

ON THE MOST EFFICIENT UNITARY TRANSFORMATION FOR PROGRAMMING QUANTUM CHANNELS*

GIACOMO MAURO D’ARIANO~ QUIT group, INFM-CNR, Dipartimento d i Fisica “A. Volta”, via Bassi 6, 27100 Pavia, Italy PAOLO PERINOTTI~ QUIT group, INFM-CNR, Dipartimento di Fisica “A. Volta”, via Bassi 6, 271 00 Pavia, Italys

We address the problem of finding the optimal joint unitary transformation on system + ancilla which is the most efficient in programming any desired channel on the system by changing the state of the ancilla. We present a solution to the problem for dim(H) = 2 for both system and ancilla.

Keywords: Quantum information theory; channels; quantum computing; entanglement

1. Introduction

A fundamental problem in quantum computing and, more generally, in quantum information processing [l]is to experimentally achieve any theoretically designed quantum channel with a fixed device, being able to program the channel on the state of an ancilla. This problem is of relevance for example in proving the equivalence of cryptographic protocols, e. g. proving the equivalence between a multi-round and a single-round quantum *This work has been co-founded by the EC under the program ATESIT (contract no. ist-2000-29681), and the MIUR cofinanzzamento2003. tWork partially supported by the muri program administered by the U.S. Army Research Office under grant no. DAAD19-00-1-0177 Work partially supported by INFM under project PRA-2002-CLON. §Part of the work has been carried out at the Max Planck Institute for the Physics of Complex Systems in Dresden during the International School of Quantum Information, September 2005.

173

174

bit commitment [2]. What makes the problem of channel programmability non trivial is that exact universal programmability of channels is impossible, as a consequence of a no-go theorem for programmability of unitary transformations by Nielsen and Chuang [3]. A similar situation occurs for universal programmability of POVM's [4, 51. It is still possible to achieve programmability probabilistically [6], or even deterministically [7], though within some accuracy. Then, for the deterministic case, the problem is to determine the most efficient programmability, namely the optimal dimension of the program-ancilla for given accuracy. Recently, it has been shown [5] that a dimension increasing polynomially with precision is possible: however, even though this is a dramatical improvement compared to preliminary indications of an exponential grow [8], still it is not optimal. In establishing the theoretical limits to statcprogrammability of channels and POVM's the starting problem is to find the joint system-ancilla unitary which achieves the best accuracy for fixed dimension of the ancilla: this is exactly the problem that is addressed in the present paper. The problem turned out to be hard, even for low dimension, and here we will give a solution for the qubit case, for both system and ancilla. 2. Statement of the problem

We want to program the channel by a fixed device as follows

Pv,u(p) = TT2[V(Pc3 .)V+],

(1)

with the system in the state p interacting with an ancilla in the state u via the unitary operator V of the programmable device (the state of the ancilla is the program). For fixed V the above map can be regarded as a linear map from the convex set of the ancilla states d to the convex set of channels for the system V. We will denote by 9v,& the image of the ancilla states B under such linear map: these are the programmable channels. According to the well known no-go theorem by Nielsen and Chuang it is impossible to program all unitary channels on the system with a single V and a finitedimensional ancilla, namely the image convex 9v,d c %?' is a proper subset of the whole convex '& of channels. This opens the following problem:

Problem: For given dimension of the ancilla, find the unitary operators V that are the most eflcient in programming channels, namely which minimize the largest distance E ( V )of each channel C E '& from the programmable set 9 v , d :

E ( V ) max min 6(C, P ) 3 max min 6(C, Pv,,,). (2) C€V P€Pv,, CEV U E d

175

As a definition of distance it would be most appropriate to use the CBnorm distance IIC - P l l c ~ However, . this leads to a very hard problem. We will use instead the following distance

where F ( C , P ) denotes the Raginsky fidelity [9], which for unitary map C = U = U . Ut is equivalent to the channel fidelity [l]

xi

where C = Ci . C!. Such fidelity is also related to the input-output fidelity averaged over all pure states Fio(L4,P),by the formulaFio(U,P)= [l+dF(U, P)]/(d+ 1). Therefore, our optimal unitary V will maximize the fidelity

F ( V ) A UEU(H) min F(U,V ) , F ( U , V ) = maxF(U,Pv,u) U € d

(5)

3. Reducing the problem to an operator norm

In the following we will use the GNS representation I@)) = (Q €3 1)lI))of operators 9 E B(H), and denote by XT the transposed with respect to the cyclic vector II)), i. e. IQ)) = (Q €3 I ) l I ) ) = ( I €3 QT)lI)),and by X* the complex conjugated operator X* = (XT)t, and write lo*) for the vector such that (Iu)(uI€3 I) I I ) ) = Iv)lu*).Upon spectralizing the unitary V 51s follows

v=

c

eiekI Q k ) ) ( ( Q k 11

(6)

k

we obtain the Kraus operators for the map P V , ~ ( ~ ) k

nm

where Iun) denotes the eigenvector of (T corresponding to the eigenvalue A,. We then obtain

xI nm

ei(ek-eh) Tr[QiUtQk(TTQiUQh]

n[cAmU]12= kh

= Tr[aTS(U,V ) t S ( U V , )]

where

(8)

176

The fidelity (5) can then be rewritten as follows

4. Solution for the qubit case

The operator S(U,V) in Eq. (9) can be written as follows

S(U,V) = Tr,[(UT€3 I)V*].

(11)

Changing V by local unitary operators transforms S ( U , V) in the following fashion

S(u7 (Wl €3 W2)v(W3 €3 W4))= W,*s(W!uW$,v)W,*,

(12)

namely the local unitaries do not change the minimum fidelity, since the unitaries on the ancilla just imply a different program state, whereas the unitaries on the system just imply that the minimum fidelity is achieved for a different unitary-say WiUWJ instead of U . For system and ancilla both two-dimensional, one can parameterize all possible joint unitary operators as follows [lo]

V = (W1€3W2)exp [i(a1CTI€

3~1

+a2g2

@.a2

+a 3 0 3 €303 ')I

(W3€3 W4). (13)

A possible quantum circuit to achieve V in Eq. (13) can be designed using the identities [ga€3 ga7

go €3 go] = 0,

C(CZ €3 I ) C = CTZ €3 oz,

C ( I €3 gz)C= -c% (e-%cz

(14)

€3 g z ,

€3 e - -irr~ a z ) c(gZ€3 I>C( e % ~ z g % ~ = z )gg €3 gy,

where C denotes the controlled-NOT

c = lO)(Ol @ I + 11)(11 €3

gz.

(15)

This gives the quantum circuit in Fig. 1. The problem is now reduced to study only joint unitary operators of the form

+ (1202 €3 mzT + ~ 3 ~ €37 as')]. 3

V = exp[(i(alal@uiT This has eigenvectors

(16)

177

Figure 1. Quantum circuit scheme for the general joint unitary operator V in Eq. (13). Here we use the notation Go = exp(i4uG) with G = X , Y,Z.

where oj,j = 0 , 1 , 2 , 3 denote the Pauli matrices oo = I , o1 = o,, o2 = oy, 0 3 = oz.This means that we can rewrite S(U,V ) in Eq. (9) as follows 1 2

S(U,V )= -

3

C

epiej aj

uoj ,

j=O

with

eo = a1+ a2+ a 3 ,

ei = 2ai - eo .

(19)

The unitary U belongs to SU(2), and can be written in the Bloch form

U = noI with

nk

E R and n8

+ in

.CT

+ 1nl2= 1. Using the identity

we can rewrite

S ( U , V )=fioI+fi.., where

+

t3, --e-ieo

,-is

to, 1 5 j ~ 3 ,

tj=Itjlei@j,0

~ j 5 3 ,

It is now easy to evaluate the operator S ( U ,V)tS(Ul V ) . One has

S(U,V ) + S ( UV, )= v o l +

2).

+

6,

+ n* x f i ] . (24) Now, the maximum eigenvalue of S(U,V ) t S ( U ,V ) is vo + Iwl, and one has vo

=lfiiOl2

lfL12,

3 1VI2

i,j=O

= i [2S(fiOfi*)

3

1fiiI2)fijl2 - fiir26,:

=

2)

=2

C lfii121fij12sin2(4ii,j=O

$j),

(25)

178

whence the norm of S ( U ,V ) is given by 3

IIS(~,V)1I2=

C+jI2+

4& 2

n:npIti121tj12 sin2(q5i - q5j) .

(26)

i,j=O

j=O

Notice that the unitary U which is programmed with minimum fidelity in general will not not be unique, since the expression for the fidelity depends on {n;}. Notice also that using the decomposition in Eq. (13) the minimum fidelity just depends on the phases { O j } , and the local unitaries will appear only in the definitions of the optimal program state and of the worstly approximated unitary. It is convenient to write Eq. (26) as follows

IlS(U,V)1I2= u . t + &.i-%. (27) where u = (nz,n;,ng,ni), t = (l t 012,1t 112, 1t212, 1t312), and Tij = ltiI21tjl2sin2(q5i- q5j). One has the bounds u . t -t

JUTU 2 u . t 2 min I t j 12, 3

(28)

and the bound is achieved on one of the for extremal points u1 = Slj of the domain of u which is the convex set { u , uj 2 0, Cjuj = 1) (the positive octant of the unit four dimensional ball S:). Therefore, the fidelity minimized over all unitaries is given by

1 F ( V ) = - min l t j I 2 d2 3 The optimal unitary V is now obtained by maximizing F ( V ) . We need then to consider the decomposition Eq. (13), and then to maximize the minimum among the four eigenvalues of S(U,V)tS(U,V ) . Notice that t j = C , Hj,eiep, where H is the Hadamard matrix

H=f-"I 1 1

2

1

1

1-1 1 - 1 1-1-1 1

'

which is unitary, and consequently Cj l t j I 2 = Cj leiej l2 = 4. This implies that minj l t j l 5 1. We now provide a choice of phases 8, such that l t j l = 1 for all j , achieving the maximum fidelity allowed. For instance, we can take 80 = 0,81 = 7r/2,82 = 7 r , 83 = 7r/2, corresponding to the eigenvalues i, 1,-2, 1 for V. Another solution is B0 = 0 , O1 = -7r/2, Q2 = 7r, e3 = -7r/2.

179

Also one can set Oi -+ -&. The eigenvalues of S(U,V)tS(U, V ) are then 1 , 1 , 1 , 1 ,while for the fidelity we have

and the corresponding optimal V has the form (a, B a, f az 8 az)].

(32)

A possible circuit scheme for the optimal V is given in Fig. 2.

I

I

Figure 2. Quantum circuit scheme for the optimal unitary operator V in Eq. (31). For the notation see Fig. 1. For the derivation of the circuit see Eqs. (14).

We now show that such fidelity cannot be achieved by any V of the controlled-unitary form 2

v = C v k B l$k)($kl,

(+117/~2) = 0,

K , ~2 unitary on

H = c2.(33)

k=l 2 &) (k) (k) )(+j I the eigenvectors For spectral decomposition v k = &=l e k of v are I!ijjk)) = (k))I$k), and the corresponding operators are !ijjk =

\+:))($:I,

namely the operator S(U,V)is

C e+@ I $ ~ ) ( + ~ ) I U, ~ + ~ ) ) (34) (~~I with singular values xi=,e-iey)(q5y)lUl+~)) = Tr[ViU]. Then, the opS(U, V )=

j,k

timal program state is sponding fidelity is

l$h),

with h = argmaxk I Tr[ViU]I, and the corre-

F(U,V)= -ITr[V,tU]12,

1 4

(35)

F(V)= minF(U, V )= 0,

(36)

and one has U

180

u

since for any couple of unitaries v k there always exists a unitary such that Tr[V,U] = 0 for k = 1,2. Indeed, writing the unitaries in the Bloch form (20), their Hilbert-Schmidt scalar is equal to the euclidean scalar product in R4 of their corresponding vectors, whence it is always possible to find a vector orthogonal to any given couple in R4.The corresponding U is then orthogonal to both v k , and the minimum fidelity for any controlled-unitary is zero.

References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University press, Cambridge, 2000). 2. G. M. D’Ariano, D. Kretschmann, D. Schlingeman, R. F. Werner, unpub-

lished 3. M. A. Nielsen and I. L. Chuang, Programmable Quantum Gate Arrays, Phys. Rev. Lett. 79,321 (1997). 4. 3. Fiurbek and M. DuSek, Probabilistic quantum multimeters, Phys. Rev. A 69 032302 (2004). 5. G.M. D’Ariano, and P. Perinotti, Eficient Universal Programmable Quantum Measurements, Phys. Rev. Lett. 94 090401 (2005). 6. M.Hillery, V. Buiek, and M. Ziman, Probabilistic implementation of universal quantum processors, Phys. Rev. A 65 022301 (2002). 7. G.Vidal and J. I. Cirac, Storage of quantum dynamics on quantum states: a

quasi-perfect programmable quantum gate, quant-ph/0012067. 8. J. FiurGek, M. DuSek, and R. Filip, Universal Measurement Apparatus Controlled by Quantum Software, Phys. Rev. Lett. 89 190401 (2002). 9. M. Raginsky, A fidelity measure for quantum channels , Phys. Lett. A 290 11 (2001). 10. B. Kraus and I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A63 062309 (2001).

STABILITY ANALYSIS OF QUANTUM MECHANICAL FEEDBACK CONTROL SYSTEM

P.K. DAS Physics and Applied Mathematics Unit Indian Statistical Institute 203, B. T.Road, Kolkata-700108 e-mail:daspk@isical. ac. in

B.C. ROY The Institute of Radio Physics and Electronics Science College, Calcutta University 92, A . P. C. Road, Kolkata - 700009 e-mail:[email protected] In this paper we derive the state equation of the optical cavity in interacting Fock space as well as in boson Fock space. We design closed-loop feedback control system of a composite cavity QED in boson Fock space using beam splitter device and prove with the help of Nyquist stability criterion that the system is stable. The other physical characteristics, such as, phase margin and gain margin of the closed-loop feedback control system are also discussed.

1. Introduction The concepts and tools of control theory help us to understand the dynamics of complex networks. Extension of control theory to the quantum domain enables us to design complex quantum systems in a systematic way. A feedback control system is a system that maintains a relationship between the output and the input and their difference is a controller. The primary objective of the feedback control system is the elimination or reduction of error of the system output. The primary difference of quantum systems from the classical systems is that the input, output and state variables of the quantum system are operators rather than scalars acting on Hilbert spaces. The concept of interaction of single mode of quantized field in a cavity with a noisy external field has been utilized in this paper for finding the 181

182

state space model in interacting Fock space. The dynamics of the cavity in the interacting Fock space are obtained by utilizing the basic concepts of quantum stochastic process. The transfer function modelling of the closedloop system of a composite system of cavities utilizing a device of accepting two inputs and emitting two outputs of the beam splitter is used to design closed-loop feedback quantum control system. The state space modelling of quantum feedback control system in interacting Fock space is shown to be a generalization of description of quantum feedback control system in usual boson Fock space [l,21. The Nyquist stability analysis of the system in boson Fock space by using beam splitter has been discussed by constructing Nyquist plot of the transfer function of a composite system with a second cavity in the feedback path of the closed-loop control system. The Nyquist stability criterion along with the gain margin and phase margin of the composite cavity QED system is expressed in terms of the parameters of reflectivity and transmissivity of the beam splitter. The paper is organized as follows. In section 2, we discuss some basic facts which will be needed in the paper. In section 3, we model single QED system in interacting Fock space from which the state space modelling in bosonic mode can be derived easily as discussed there. In the remaining sections we are concerned in designing and analyzing the stability of feedback controlled cavity QED system in boson Fock space. In section 4,we design a single cavity QED feedback system with a second cavity in the feedback path using beam splitter. In section 5, we discuss in details the Nyquist stability analysis of quantum feedback system for composite QED system with a second cavity in the feedback loop. And finally in section 6 , we give a conclusion. 2. Preliminaries and Notations

In this section we discuss some basic preliminaries on interacting Fock space, interaction of optical cavity QED with the external field and the quantum stochastic process which will be needed throughout the paper.

2.1. Interacting Fock Space

As a vector space one mode interacting Fock space r(C) [3] is defined by

183

for any n E AT, where @In > is called the n-particle subspace. The norm of the vector In > is given by

< nln >= A,

(2)

where {A,} 2 0 and if for some n we have {A,} = 0, then A{}, = 0 for all m 2 n. The norm introduced in (2) makes r(@) a Hilbert space. We consider the following actions on I?(@) :

where A* is called the creation operator and its adjoint A is called the annihilation operator. In defining the annihilation operator we have taken the convention 010 = 0. The commutation relation takes the form [A,A*]= -- AN+1

AN

AN

AN-1

(4)

where N is the number operator defined by N J n>= n ) n >. In a recent paper [9]we have proved that the set { n = 0 , 1 , 2 , 3 , .. .} forms a complete orthonormal set and the solution of the following eigenvalue equation

5,

Afa = afa

(5)

is given by

where

y.

= C,"==,

We call fa a coherent vector in I?(@).

2 . 2 . Interaction of Cavity and the External Field

We consider the interaction of an interacting single-mode of quantized field confined in an optical cavity with a noisy external field. Let 7 - i ~and X B be Hilbert spaces of the cavity and the external field respectively. The composite system is expressed by the tensor product space 7 - i ~@ 7 - i ~ The . total Hamiltonian is given by

Htotal = H A 8 IB

+ IA 8 H B + Hint

(7) where H A describes the Hamiltonian of the cavity mode. This Hamiltonian may be further decomposed into two parts

H A = Hca,

+H.

(8)

184

Here H is the residual Hamiltonian determined by the optical medium in the cavity, referred to as a free Humiltoniun. HB is the Hamiltonian of the external field. The interaction Hamiltonian Hint consists of four terms. We drop the energy non conserving terms corresponding to the rotating-wave approximation and obtain the simplified Hamiltonian as

Hint(t) = i&[U(t)b+(t)

- U+(t)b(t)l

(9)

with

[b(t),b+(t’)] = q t - t’)

(10)

and y is a coupling constant. Here u is the annihilation operator of the cavity and b is the annihilation operator of the external field. The operator b ( t ) is a driving field at time t and we interpret the parameter t to mean the time at which the initial incoming field will interact with the system and not that b ( t ) is a time-dependent operator at time t.

2.3. Quantum Stochastic Process In order to describe quantum stochastic process we define first an operator

lo t

Bi,(t,tO) =

bin(s)ds

(11)

where bin@) satisfies the commutation relation (10).The operator bi,(t) represents the field immediately before it interacts with the system and we regard it as an input to the system. Now, from (11)we get

Then we write down the increments

dBin(t) = Bi,(t

+ d t ) - Bi,(t),

dBk(t)= B,?-,(t+ d t ) - B z ( t )

(13)

From (12) and (13) we get

[dBi,(t),dB&(t)]= dt.

(14)

This leads to the natural definition of quantum stochastic process as

+

dBi,(t)dBi+,(t) = ( N ’ 1)dt dBk(t)dBi,(t) = N’dt dBZ,(t)dBi,(t) = Mdt d B L ( t ) d B z ( t )= M*dt

(15)

185

and all other products higher than the second order in dBi, are equal to zero. N’ and M are real and complex numbers satisfying

Nt(N’

+ 1 ) 2 [M12.

(16)

The evolution of an arbitrary operator X is given by

X ( t )= U+(t)XU(t)

(17)

in which the unitary operator U ( t ) is generated by the Hamiltonians ( 7 ) and (8). H,,, and HE drive the cavity and the external field respectively. We shall assume here H to be zero. The unitary operator of the system is then given by

U ( & ) = e&(adBL-a+dBin)

(18)

U+(dt) = eJ;/(a+dBin-adBk)

(19)

Also we have

The increment of an arbitrary operator r of the system driven by the stochastic input bin is given by

dr(t) = r(t

+ d t ) - r ( t )= U+(dt)r(t)U(dt)- r ( t )

(20)

Now

U+(dt)r(t)U(dt) - e J ; i ( a + d B i n - - a d B ~ ) r ( t ) e ~ ( a d B -a+dBim) L =

r ( t )+ f i [ U + d B i , - a d B 2 , r(t)]+ +Z{(N’ + 1)(2a+ra - a+ar - ?-.+a) +”(2ara+ - aa+r - Tau+) +M(a+(a+r - ?-a+)- (a+r - ra+)a+) +M*(a(ar - ra) - (ar - ra)a)}dt

Hence we get

3. Modelling of Single QED System

To describe the state space model of open loop quantum system we must describe the state equation of the system along with the input-output relation of the system.

186

3.1. State Equation of the Cavity To describe the dynamics of the operator a ( t ) in the open quantum system we replace T in (22) by a to get da = ~

+

( td t ) - ~ ( t ) = f i [ a + d B i , - a d B 2 , a]+ +${(" 1)(2a+aa - a+aa - aa+a)+ +"(2aaa+ - aa+a - aaa+)+

+

+

+M[a+,[a+,a ] ] M*[a,[a,a]]}dt = +(hAN kAN-1 )bz,(t)dt :{-(" l)(* - &)a +"a( - &)}dt = { - 2 ( h - &)a - f i ( h- k ) b i n ( t ) } d t

*

2

AN

+

+

AN

AN-1

(23)

AN-1

This implies

The equation (24) defines the state equation of the single optical cavity QED in the interacting Fock space. The dynamics of the cavity in the bosonic mode can be obtained from (24) by using the operation Nln >= nln >. The state equation of the cavity then reduces to U(t)

Y

= --a(t) - &bzn(t)

2 which is the usual quantum Langevin equation.

(25)

3.2. Input-Output Relation of the Open loop System

Due to the interaction of the evolving incoming field with the cavity an outgoing field is produced. To describe this we need to define an operator

lo t

Bout(t,to) =

bout(s)ds

(26)

where

bout(t) = U+(dt)bi,(t)U(dt)

(27)

The input-output relation after the interaction at time t is given simply by the following derivation. We have dBout( t )= (dt)dBi, ( t )U ( d t ) - , ~ ( a d ~ ~ - - a + d ~ ; , ) + d ~ ~ , ( t ) , ~ ( a d ~ ~ - a (28) + d ~

u+

+ f i U [ d B i , , dB&]

= dBin(t)

. , )

187

G

Figure 1. The configuration of input-output relation of single cavity QED in bosonic mode.

Now using (14) the above relation gives us

+fiudt

(29)

fim + bin (t).

(30)

bout(t)dt = bi,(t)dt and hence we have the required relation bout ( t )=

This gives the input-output relation of the single cavity QED system. The fig.1 shows the input-output configuration of a single cavity QED through the system operators which defines the open-loop single cavity QED system. 3.3. Transfer Function of the Open-Loop Quantum System

We have seen that the cavity dynamics may be thought of as a single input and a single output(SIS0) system. The equations (24) and (25) give the state equation of a single cavity in different modes. The operator bi,(t) is the input and the operator bOut(t)is the output of the cavity. The state equation of the cavity dynamics along with the output equation can be rewritten as

u(t)= A'a(t)+ B'bi,(t) bout(t) = C'a(t) D'bi,(t)

+

where

(31)

188

When A h = 1 we get the bosonic mode of the cavity system as given by (25). The dynamics of the cavity in interacting Fock space is a first order differential equation with operators as coefficients whereas in bosonic mode the equation is of first order with constant coefficient. Taking Laplace transform of equations (31) and assuming zero initial state the QED system can be represented by the gain G(s) of the cavity from the input to the output as bout(s) = G(s)bin(s)

(33)

where the gain G(s) of the cavity in the interacting Fock space is given by G ( s )= -YA;(SI

+ -A:)-' Y +1 2

(34)

In bosonic mode, the gain of the single cavity is simply described as

As the cavity dynamics in boson Fock space is linear with constant coefficient, one can successfully apply the Nyquist stability criterion for analyzing feedback controlled problems of the system in boson Fock space. Henceforth, we shall mainly concentrate our discussion in bosonic mode to the problem of modelling optical cavity QED feedback control system and shall study their stability. 4. Mathematical Model of the Feedback Control of the

Cavity QED Using Beam Splitter

A beam splitter is an optical device, a partially silvered piece of glass, which allows us to perform manipulations of two input signals and two output signals of a feedback control system. The input field bin(t) is sent to one port of the beam splitter which is chosen to have reflectivity a and transmissivity p, and the feedback operator with negative sign evolve from the feedback path is sent to the other port of the beam splitter. In this way the beam splitter may be used to split the input fields into two operators of which one may be taken to define the feedback controller of the feedback system. We now utilize the concept of beam splitter to design the quantum feedback control system. The input-output configuration of the composite QED system is shown in fig.2. The input signals bin and bz to the beam splitter are related to the

189

outputs bo and

where cy and

b3

by

p are real and satisfy a2+ p2 = 1. Hence we get b3 = a b i n

+ pb2

bo = Pbin - ab?. From the input-output relation of each cavity, we get

bi = f b2 = f

i a A

+ bo +

i a ~ bi

Each signal in the feedback loop can now be written as bo = &bin bl= A b i b2 = &bin b3 = bin

n

-e ( f i a A +f i a B ) +& f i a A -e f i a B + h ( f i a A +f i a B )

(39)

+& ( f i a A +f i a B )

Figure 2. The design of the composite system in which the second cavity in the feedback path by using beam splitter.

Assuming the non-interaction of the cavity A with the cavity B , the gain H ( s ) of the bath B in the feedback path is described, as in subsection 3.3, by equation (38) as

H(s)= -YB(s

+ -)2

Y B -1

+1

(40)

Then in the case of the composite system with a second cavity in the feedback path, the closed-loop transfer function is written as

M ( s )=

+

Ws)

1 aG(s)H(s)

190

The stability of the closed-loop system is characterized by the zeros of 1 aG(s)H(s),or equivalently, by the roots of

+

It then follows that R e s < 0 which implies that the closed-loop system is asymptotically stable. The stability of the composite system is characterized by computing the poles of the characteristic equation of the system. So there arises a difficulty in calculating the characteristic equation. However, the stability of a composite system can be easily analyzed by considering open-loop transfer function with the help of the Nyquist stability criterion. 5. Nyquist Stability Analysis of the Quantum Feedback Control System

So far, we have concerned with the construction of transfer functions of quantum feedback control systems which provide some information about the absolute stability of a composite system with a second cavity in the feedback path. This representation experiences system gain function that provide various valuable insights into the problems such as, stability, gain margin and phase margin of the QED system. This problem of the quantum mechanical system can be solved by simply analyzing the open-loop transfer function of the system using the Nyquist criterion. The Nyquist stability analysis is a graphical method that determines the stability of closed-loop system by analyzing the property of the frequency domain plot called Nyquist plot of the open-loop transfer function L ( s ) = a G ( s ) H ( s )of the system. Specifically, the Nyquist plot of L ( s ) is a plot of L ( j w ) in polar coordinates or in Cartesian coordinates of I r n L ( j w ) versus R e L ( j w ) as w varies from 00 to 0 and from 0 to -m. This gives also the information about the relative stability of an unstable system. It also gives the indication on how the system stability may be improved (if needed) for getting a desired output. A simple Nyquist contour rS in s-plane having no pole and zero on the imaginary axis is depicted in the adjoining figures 5 . The closed Nyquist contour consists of four parts: C1, C2,C3 and C4. The section Cl is defined by s = jw,O 5 w < 00; section C2 is defined by s = j w , -00 < w I 0; section C3 is defined by s = Rej', R -+ 00, -$ 5 9 5 0, and section C4 is defined by s = Rej', R -+ 00,0 5 B 5 f.

191

We now discuss in some details the Nyquist stability analysis of the cavity QED system. The closed-loop transfer function M ( s ) shown in fig.2 with a gain H ( s ) in the feedback path has been described in section 4 by equation (41) in the form

M(s)=

PG(s) 1 aG(s)H(s)'

+

The transfer function relating the feedback variable bz(s) to the input bo(s) is the open-loop transfer function L ( s ) = oG(s)H(s),shown in the fig.2. As the open-loop transfer function L ( s ) is in the factor form, the Nyquist plot in L(s)-plane can be easily obtained. We express L ( s ) as

The open-loop transfer function L ( s ) has two poles to the left half and two zeros to the right half of the s-plane. There are no zero or pole on the imaginary axis. So the Nyquist contour, as described in fig.5, may be used to describe the characteristic properties, such as, stability, phase margin, gain margin of the closed-loop feedback control system from the Nyquist plot in L(s)-plane. To describe the Nyquist plot in L(s)-plane, we put s = j w in L ( s ) for the portions Cl, C2 of the closed-loop contour rs. Then

L(jw)=

a ( j w - ?)(jw - ?f) ( j w ?)(jw y)

+

+

(44)

Hence IL(jw)l = Q for all values of w. Expressing the function L ( j w ) as L ( j w ) = u(w)+jw(w), the Nyquist plot of the portions Cl, C2 of the Nyquist contour rs can be easily programmed and the corresponding results are shown in fig.3 and fig.4. Again, to find the Nyquist plot in L(s)-plane of the curves C3,Cq, let us put s = Reje,-.rr/2 5 9 5 ~ / in 2 (44) we then get

192

where E’ =a-+ 0 as R - t M. The Nyquist plot of the portions C3, C4 in L ( s ) are shown in fig.3 and fig.4 by the small circular arcs about the centre a j 0 . These circular arcs coincide at the point a j 0 when R -+ 00. We observe that the Nyquist plot in L(s)-plane of the closed contour rs encircles the origin twice, and hence the number of encirclement in the anticlockwise direction about the origin is N = Z - P where P = 0 and Z = 2, and hence N = 2. Thus, the Nyquist criterion of the number of encirclement about the origin satisfies N = Z - P and passes through the points E = a j 0 and F = -a j 0 , where 0 < a < 1. According to the Nyquist criterion the closed-loop feedback composite QED system designed by using a beam splitter is asymptotically stable. The phase margin(PM) is zero. The gain margin(GM) is given by

+

+

+

+

In the special case of 50/50 beam splitter, a = it is expressed in decibel, then

GM

20Z0glo-

1

1

lOEl

5.And so, GM = a.If

= 3.01 > 3 d B

Note that the measurements of GM and PM of a closed-loop feedback control system indicate the information about the degree of stability of the control system.

6. Conclusion

The problem of controlling a composite cavity system with a second cavity in the feedback path can be utilized to study the experimental problem of coupling a single photon with an atom within an optical cavity. The Nyquist stability analysis of a system with a feedback gain evolved by using a second cavity in the feedback path described in this paper is a general study in phase plane. The process described in this paper can be easily extended to the case of cascaded system with a finite number of optical cavities.

193

References 1. Yanagisawa, M. and Kimura, H.: Transfer Function Approach t o Quantum Control- Part I: Dynamics of Quantum Feedback Systems, IEEE Transactions

on Automatic Control, Vol. 48, no. 12, 2107,(2003). 2. Yanagisawa, M. and Kimura, H.: Transfer Function Approach t o Quantum Control- Part 11: Control Concepts and Applications, IEEE Transactions on Automatic Control, Vol. 48, no. 12, 2121,(2003). 3. Accardi, L. and Bozejko, M.: Interacting Fock Space and Gaussianization of Probability Measures, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(4), 663,(1998). 4. Wiseman, H. M., and Milburn, G. J.: Quantum Theory of Optical Feedback via Homodyne Detection, Phy. Rev. Letts, Vol. 70, no. 5, 548, (1993). 5. Doherty, A. C., Habib, S., Jacobs, K., Mabuchi, H. and Tan, S. M.: Quantum Feedback Control and Classical Control Theory, Phy. Rev. A, Vol. 62, 012105, (2000). 6. Ogata, K.: Modern Control Engineering, Prentice-Hall of India, (2004). 7. Kuo, Benjamin C.: Automatic Control Systems, Prentice-Hall of India, (2003). 8. Gopal, M.: Control Systems, Tata McGraw-Hill, (2003) 9. Das, P. K.: Coherent states and squeezed states in interacting Fock space. International Journal of Theoretical Physics. vol. 41, no. 06, 1099-1106, (2002), MR. No. 2003e: 81091 (2003).

194

TRlO Data

TRll Data 1.5

~

0

4.5

1

-1.5 -15 -1

-0.5

0.89 - (1.00

0 0.5 1.11

1

15

- 0.00

Figure 3. Nyquist plot of G ( s ) H ( s )as s traverses C4 and C1

I

Figure 4. Nyquist plot of G ( s ) H ( s ) as s traverses CZ and C3

TR9 Data

jm

-IS'..... -15 -1

'

-0.5

'

0

0.88 -ow

Figure 5.

Nyquist contour rs which is di-

and an arc

&,

C4

of infinite radius.

'

0.5

'

1

'

15

195

5 OPEN "O", #12, "tr9.out" 10 FOR I = 0 TO 1.57 STEP .01 15 X = 1 - .125 * COS(1) 20 Y = .125 * SIN(1) 25 PRINT #12, USING " # # # . # # " ; X; 30 PRINT 112, " . 35 PRINT #12, USING " # # # . # # " ; Y 40 NEXT I 45 FOR I = 8 TO 0 STEP -.01 50 X = (I 4 - .37 * I 2 t .004) / ((I 2 t .06) (I 2 - .06)) / ((I 2 t .06) 2) 55 Y = (I 60 PRINT #12, USING " # # # . # # " ; X; 65 PRINT #12, " . 70 PRINT #12, USING " # # # . # # " ; Y I5 NEXT I 80 FOR I = 0 TO -8 STEP -.01 85 X = (I 4 - .37 * I 2 t .004) / ((I 2 t .06) 90 Y = (I * (I 2 - .06)) / ((I 2 t .06) 2) 95 PRINT #12, USING " # # # . # # " ; X; 100 PRINT #12, " . 105 PRINT #12, USING " # # # . # # " ; Y 110 NEXT I 115 FOR I = 4.71 TO 6.28 STEP .01 120 X = 1 - .125 * COS(1) 125 Y = .125 SIN(1) X; 130 PRINT 812, USING " # # # . # # " ; . 135 PRINT #12, " 140 PRINT #12, USING " # # # . # # " ; Y 145 NEXT I 150 CLOSE #12 155 END IT

2)

PI

2)

A

*I

11

Figure 7. The basic programme to generate data to draw Nyquist plot in the G ( s ) H ( s ) plane.

MARKOV STATES O N QUASI-LOCAL ALGEBRAS *

FRANCESCO FIDALEO Dipartimento di Matematica Uniuersith d i Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy E-mail: fidaleoOmat.uniroma2.it

We review the definition of Markov states on quasi-local algebras, their structure and the main properties on known models.

1. Preliminaries Since Refs. [l,51, the investigation of the Markov property had a inpetuous growth, in view also to natural applications to various fields such as quantum statistical mechanics and information theory. Recently, a sistematic investigation of Markov chains and states was extended to typically quantum models such as quasi-local algebras describing Fermions, e.g. the CAR algebra. Yet, there is not a satisfactory general theory of the Markov property in quantum setting. The present note is devoted to quote the known results about the structure, and the main properties on known models of the quantum Markov states. The reader is referreed to Refs. [2, 4,91 for the proofs and further details. A quasi-local algebra associated to the set I equipped with a Boolean structure, and an orthogonality relation Ibetween pairs of elements, is a C*-algebra U with an isotonic family {Ua},c~of local C*-algebras such that

I

(i) U {U. cy E I } is dense in U; (ii) the algebras U, have a common identity I; ~~

*The author is grateful to Italian CIRM and INDAM for hospitality and financial support.

196

197

(iii) there exists an automorphism a of U with a2 = L and a(U,) such that

=

a,

(AB - €(A,B)BA) = 0 ,

w h e n e v e r A E U ~ U U , , B E U p + U U p , a I p ,a n d E ( A , B ) = - l i f AEU ,; B E U i , E(A,B)= 1 in the three remaining possibilities.

is the decomposition of A w.r.t. a in the even and odd part, see Sec. 2.6 of Ref. [8] for further details. Let p 4 a , and E : U, H Up be a completely positive identity preserving linear map. We call such a map a transition expectation. We say that

E (i) is even if aE = Ea; (ii) is a quasi-conditional expectation if there exists y + ,B such that E ( X Y ) = X E ( Y ) whenever X E U,;” (iii) has the Markow property if there exists y 4 p such that E(U,\p) c UP\,.

Let cp E S(U), and J

c 1 such that

u

U, = U. The state cp is called a

aEJ

Markov state w.r.t. the filtration

J ,

if a,,B E J and

p 4 a implies

cpra,o~a,p = cpla,

for some transition expectation E,,p : U, H Up. We refer the reader to Sec. 2 of Ref. [4]for further details. The previous set up naturally applies to quasi-local algebras based on classes of subsets of a fixed set (e.g. spin systems living on standard lattices Zd). It is not sufficient in order to understand the fine structure of Markov states. Yet, it is explicit enough in order to establish natural connections with the KMS boundary condition, as well as phenomena of phase transitions and symmetry breaking for quantum Markov fields on Zd [3]. In the case of linearly ordered lattices, and quasi-local algebras on them arising from infinite tensor product or Canonical Anticommutation Relations (CAR for short [8]), we are able to exhibit the explicit structure of Markov states. say that E is a quasi-onditional

expectation w.r.t. the triplet

C 2lp

C K.

198

2. Markov states on linearly ordered sets

We specialize the situation to the linearly ordered countable sets I containing, possibly a smallest element j - and/or a greatest element j+. In other words, I is order-isomorphic to Z,Z-, Z+or to a finite interval [ j - - l,j+]c Z, the case 11)< +co being almost trivial. We consider the cases when C'

(i)

:= @ M d j

(c)

(non homogeneous infinite tensor product), in

j€l

this situation (ii) U :=

v %ti)

(T

C'

= L;

, where U is the CAR algebra generated

by anni-

j€I

the mentioned d j annihilators and creators generate the local algebra %ii} (non homogeneous CAR algebra). In view of physical applications, we deal without further mention with locally faithful even states (e.g. locally faithful states cp such that y = p a ) , and with even transition expectations. The following definition specializes the matter in Sec. 1 to the present situation.

Definition 2.1. (Ref. [4], Definition 4.1) A state cp on U is called a Markov state if, for each n < j + , there exists a quasi-conditional expectation En w.r.t. the triplet U,-l~ c U,] c U,+11 satisfying

Let cp E S(U) be a Markov state, the ergodic limit

E,

C(e,)h

1 k-l

:= lim -

k k

h=O

of e, := E,[a[n,n+l, plays a crucial r61e. Indeed, it uniquely determines, and is determined by the conditional expectation &, : %,+I] H %,I, given for x E %-l], y E z'1[,+1] by

199

see Sec. 4 of Ref. [4]. In addition, cp is uniquely determined, for every k by all the marginals

cp(Xk.. . X,)=cp(Ek(XkEk+l(Xk+l . . .cl-l(XZ-lXd. . )>) -cp(Ek(XkEk+l(Xk+l . . . & Z-l(Xl-lQ(Xd) ...I))

< 1, (1)

*

7

where the Xk,. . . ,Xi linearly generate all of Up,,]. Let cp be a Markov state, together with the sequence {E,}~<,+ of even two-point transition expectations canonically associated to cp as previously explained. I general, the parity automorphism 0 acts nontrivially on all ) )the ranges of the c,. Consider the Abelian the centres 3, := ~ ( R ( E ,of C*-subalgebra C c U+ generated by the projections

{ q I q = p v a ( p ) ,p minimal projection of 3, ,j E I } .

- -C' C

v C{,)

N

@C{3)

has a natural local structure inherited by that

3EJ

of U. Let p be the Markov measure on

R := spec(C) =

TJI R, = TJI spec(C{,)). 3EJ

3€1

associated to c p r ~ . The main result about the structure of Markov states on quasi-local algebras is contained in the following Theorem 2.1. Let cp E S(U) be a Markov state. T h e n for each w E R there exist a quasi-local algebra B, a completely positive identity preserving m a p E, : U H B, and a mznimal Markov state $, E S ( B w )such that the field of states {qWo E,},Es2 is *-weakly measurable, and cp(A) =

/

R

$w(Ew(A))p(dw) ,

AE

u.

(2)

In the previous theorem, minimal means that the parity automorphism of B, acts transitively on all the centres of the ranges of the two-point transition expectations E , , ~ : B , J ~ , ~ H+B,,J~} ~ ] associated t o the Markov state,,$I Theorem 2.1 provides a split of the Markov state cp into a classical component (e.g. the classical process on spec(C) = R determined by the state cpre, or equivalently the Markov measure p on R), and purely quantum processes (e.g. the minimal Markov states living on the fibres of 52. Relatively to the purely quantum component, they cannot be further decomposed. Indeed, the corresponding spaces of classical trajectories of

200

such processes consist of one-point spaces. Furthermore, it is possible to establish a reconstruction result for Markov states on quasi-local algebras considered here. The reader is referred to Refs. [2, 91 for the proof of Theorem 2.1, and for further details. We end the present section by collecting the main properties of Markov states.

-

(i) Suppose that I contains a smallest element or that CT acts trivially on 3j infinitely often as j -00. Then the support in a**of the Markov state cp is central,b and cp is faithful. (ii) We have for the translation invariant Markov state 'p, S(cp)

= ~(cpr~~,,,,) -

s(9in{,)

s(cp) being the mean entropy of cp, and S(cp[g,,,,,) the von Neumann

entropy of cp r B ( k , I ] . The proof of (i) appeared in Ref. [ll]for translation invariant Markov states C'

on @Md(C)

. The reader is referred to Ref. [2] for non homogeneous

N

Markov states on infinite tensor product algebras, and to Ref. [9]for Markov states on non homogeneous CAR algebras. (ii) is quoted in Ref. [12] for the case of infinite tensor product algebras, and in Ref. [9] for the case of CAR algebras.

3. Diagonalizable Markov states In the present section we focus our attention on the class of Markov states such that the parity automorphism CT acts trivially on all the centres 3j 3 ~(R(E~ Such ) ) .states are called diagonalizable in Ref. [4].They exhibit larger structure. In this situation, the 3j generate an Abelian algebra, and

c = V 3j

C'

. Furthermore, let w = (. . . ,Wk-1,Wk,Wk+l,.. . )

E R be a

j€I

trajectory,

qWin (2) can be symbolically written as $W

=

n

4;-

7711Wj,Wj+,

.

(3)

j<j+-l

In Eq. (3), the product is the usual tensor product of states in the tensor product algebra cases, and is understood in the terminology of Ref. [7], as a bThis simply means that R, is cyclic for associated to 9.

~ ~ ( ( u )( ' T ,

~ , N ~ , being R ~ )the GNS triplet

201

product state extension for CAR algebras. Even if the $ Jare ~ product states, the even states q$j,wj+lare localized in the twepoint interval [ j , j 11. Namely, nontrivial diagonalizable Markov states exhibit an interaction. We can explicitely write the local densities associated to the diagonalizable Markov state. Namely, consider the Radon-Nikodym derivatives (e.g. the densities) T[k,$] w.r.t. the unnormalized trace of %[k,l],

+

‘ p T q k , l ] = n q k , l ] (T[k,lI

or equivalently, the potentials

.)

7

{ h [ k , ~ ] } k obtained g

~ [ k , l ]= e - h [ k , l l

as

.

Then h[k,$ has the nice decomposition I- 1

h[k,I] = Hk

+

Hj,j+l + El

(4)

*

j=k

Here, the selfadjoint operators { H j } j E 1 , {i?j}jE~ are localized in UFj,, and {Hj,j+l}j<j+in U6,j+11respectively, and satisfy the commutation relations h

A

[Hjl Hj,j+lI = [Hj,j+l,Hj+lI

= W j ,Hjl = [Hj,j+l,Hj+l,j+21= 0 .

(5)

We pointed out in Sec. 2 that, under suitable conditions, the extension to all of 7rv(U)” of a (locally faithful) Markov state ‘p is a KMS state for an appropriate time evolution. The fine structure of the potentials of a diagonalizable Markov state (Eqs. (4), (5)) allows us to exhibit such a time evolution as a one parameter group of automorphisms of U. Namely, the diagonalizable Markov state ‘p is a KMS state for the one parameter group of automomorphisms ot given, for X E U, by g t ( z ) :=

lim

e-ith[k3~lXeithlk911

.

klj-

Itj+

The previous considerations clarify the deep connections between the Markov property and the quantum statistical mechanics. Another remarkable result is that the diagonalizable Markov states are likings of classical Markov states w.r.t the same localization of their quantum counterpart. Namely, the Markov states considered in the present section are indeed diagonalizable. Let R; := R ( E j ) ’

A

U{j)>

n[k,l] := R i

v

U[k+l,l-l]

v%.

(6)

202

Notice that U= (

u

%[k,l]

%[k,l])

differs from

%[k,l]

by boundary terms.

Thus,

as well. For each interval [k,I ] , take an even maximal

[Wid

Abelian subalgebra Di)[k,l]c

containing the leading part

of the Hamiltonian (4). Put 9 := (

u

a[k,l]).

[k,llCI

Theorem 3.1. Let cp E S(Q) be a diagonalizable Marlcov state. Then there exists a conditional expectation C : U I--+ 9 such that cp = c p r ~ o e . In addition, the measure p o n spec(9) associated to c p [ ~ is a Marlcov measure w.r.t. the natural filtration of 9. The proof of Thm. 3.1 appeared firstly in Ref. [ll]for the translation invariant Markov states on @ Md(C)

C'

. The general cases relative to tensor

N

products, and non homogeneous CAR algebras are quoted in Ref. [lo, 91 respectively. In our generic situation, the spectral resolution of the two-point block of the Hamiltonian (4) has the form ~ , , ~ + = 1

CK; e(:. . nn+l n n + l

w)(x,j)

'

(8)

i,j

where { e;,yGi,l)} c Md, (C) @I (C) is a suitable system of matrixunits for Md, (C) @ Md,+l (@). It is in general impossible, for any choice of the system of matrix-units { e z } C Md,(C), to write (8) as nn+l n

eii 8 en+l jj

Hn,n+~=

.

(9)

i,j

The generic case when the spectral projections of two-point block of the Hamiltonian cannot be factorizable as above, has the meaning of a local entanglement effect. Taking into account the above considerations, one can assert that each quantum Markov state on Z arises from some underlying (non trivial) classical Markov process. But, due to this entanglement phenomenon, it is not of king type, (9) being the general form of a king type interaction.

203

The quantum character of such states manifests itself in the following way. In order to construct (or recover) such states, one should take into account various nontrivial local filtrations of U (like those described by the % [ ~ , J I ) ,together with various (commuting) boundary terms in the nearest neighbour Hamiltonian. Conversely, if one chooses to investigate quantum Markov states by considering only the natural filtration { U [ k , l ] } k g of U, one obtains a leading term as that in (7). But non commuting boundary terms could naturally arise in (4),see the examples in Sec. 6 of Ref. [2]. In the consrtuctive approach, the appearance of such non commuting boundary terms cannot be disregarded in order to obtain general infinite volume Gibbs states for a fixed commuting nearest neighbour interaction. However, it should be noted that, if the nearest neighbour model is translation invariant or periodic, then according to Thm. 1 of Ref. [6], the construction of quantum Markov states does not depend on boundary terms. We end the present section by mentioning the generalization to CAR systems of the following result (see e.g. Ref. [12] for the case of tensor product algebras). For translation invariant diagonalizable Markov states cp we get

4Cp)

= M a )7

where a is the one-step shift on U, and &(a) is the Connes-NarnhoferThirring dynamical entropy of 'p w.r.t the shift a. For the proof of this result, see Ref. [9]. 4. Non diagonalizable examples of Markov states

As the parity automorphism is trivial for infinite tensor products, all the Markov states are diagonalizable in this situation. Yet, nondiagonalizable Markov states naturally arise in the CAR situation. Examples of nondiagonalizable Markov states firsly appeared in Ref. [4]already in the most simple situation of U{n} N Mz(C). They are constructed as follows. Define, for a fixed x in the unit circle T, 1 qx := - (I 2

+ xao + za,+)

where ao,a$ are the annihilator and the creator generating a faithful state r] E S(qxU[o,l~qx). Put

E(X)= r](QxXqx)qx + r](4x4X)qx)q-,,z

UIO}.

E fl[O,l] .

Choose

204

With r the normalized trace on M,(C), E, := E o a-n, and xk E U { k ) ,. . . ,xl E u{l), the marginals (1) with ‘p[n,,,= r , uniquely determine a shift-invariant locally faithful Markov state cp on the CAR algebra U :=

v

C‘

U{j)

, satisfying

all the required properties. Sec. 6 of Ref. [9]

jEZ

contains the proof that the above mentioned states are nontrivial (e.g. not product state extensions), as well as other examples of nondiagonalizable Markov states on Fermion algebras. I t unclear to the author if nondiagonalizable Fermi Markov states are KMS states. However, it is expected that their Hamiltonians cannot arise from nearest neighbour commuting inter actions.

References 1. Accardi L, O n noncommutative Markow property, Funct. Anal. Appl. 8 (1975), 1-8. 2. Accardi L., Fidaleo F. Non homogeneous quantum Markow states and quantum Markow fields, J. Funct. Anal. 200 (2003), 324-347. 3. Accardi L., Fidaleo F. Quantum Markow fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 123-138. 4. Accardi L., Fidaleo F., Mukhamedov F. Markow states and chains on the CAR algebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top., to appear. 5. Accardi L., Frigerio A. Markovian cocycles, Proc. R. Ir. Acad. 83 (1983), 251-263. 6. Araki H. O n uniqueness of KMS states of one-dimensional quantum lattice systems, Commun. Math. Phys. 44 (1975), 1-7. 7. Araki H, Moriya H. Joint eztension of states of subsystems, Commun. Math. Phys. 237 (2003), 105-122. 8. 0. Bratteli and D. W. Robinson Operator algebras and quantum statistical mechanics, Vols. I, 11,Springer, Berlin-Heidelberg-New York, 1981. 9. F. Fidaleo, Non homogeneous Fermi Markow states, preprint (2005). 10. Fidaleo F., Mukhamedov F. Diagonalizability of non homogeneous quantum Markov states and associated won Neumann algebras, Probab. Math. Stat. 24 (2004), 401-418. 11. Golodets Ya V., Zholtkevich G. N. Markovian KMS states, Theor. Math. Phys. 56 (1983), 686-690. 12. M. Ohya and D. Petz Quantum entropy and its use, Springer, BerlinHeidelberg-New York, 1993.

SOME OPEN PROBLEMS IN INFORMATION GEOMETRY

P. GIBILISCO Dipartimento SEFEMEQ and Centro V . Volterra, ~ Facoltd di Econornia, Universitd d i Roma L L T oVergata”, Via Columbia 2, 00133 Rome, Italy E-mail: gibiliscoOvolterra.mat.uniroma2.it

T. ISOLA Dipartimento d i Matematica, Universitci di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy E-mail: isolaOmat.uniroma2.it

In this paper we recall some open problems in Information Geometry.

1. Orlicz geometry and statistical manifolds

Let ( X ,F,p ) be a measure space and let

be the associated maximal statistical model. To have an infinitedimensional version of Information Geometry one has to solve the following problems: i) to give a differentiable manifold structure to M,; ii) to equip M , with the a-geometries, namely the family of geometries containing exponential, mixture, and Fisher-Rao geometry. These two problems where solved in Ref. [14, 7, 21. The Pistone-Sempi solution is particularly appealing also from a physical point of view. Indeed in the M , manifold points are close if the Kullback-Leibler relative entropy is not too big, while in LP topologies the situation is quite different; therefore if one has to take into account entropy then the right topology should be an Orlicz one (see Ref. [IS]). In Ref. [14, 7, 21 there are two main ideas: i) use Orlicz geometry to give a manifold structure to M,; ii) use natural geometry (Levi-Civita con205

206

nection) of L p spheres to costruct a-connections. The quantum version of these ideas do not share the same fate. Indeed the second idea has an immediate quantum version because of the similarity between commutative and non-commutative LP spaces (see Ref. [2]). The Pistone-Sempi costruction does not appear so easy to "quantize", because of intricacy of the theory of quantum Orlicz spaces. Some important steps in this direction have been done by Grasselli, Streater and recently also by Jencova. Maybe the situation may change if we are able to present the PistoneSempi costruction in a more natural way using an embedding (similarly to the a-connection case). Actually the exponential geometry is a limit case of a-geometries, namely LP geometries. Nevertheless, while the exponential parallel transport is simple, the exponential geometry does not come from a natural embedding. Maybe a new look to the commutative case can be of help in the quantum one. We suggest to consider an idea already contained in Refs. [7, 81. Given a Young function @, we denote by L' = L'(p) the associated Orlicz space. Suppose the Young function @ is invertible when restricted to the positive axis (this is not always the case: consider the @ relative to L"). We define the Amari @-embedding A' : M , 4 L'(p) by A'(p)

:= @ - ' ( p )

.

The standard example is p ---t ppa (we would like to give a similar treatment to p -+ log(p)). In Ref. [7] we proved the following result.

Proposition 1.1. Let S' = {v E L' : IIvIIe Banach space L'. Then

=

1) be the unit sphere ofthe

A @ ( M , )c s@. In general, the space L' is not uniformly convex, so the LP approach cannot be imitated directly. Nevertheless, is quite possible that for a suitable Young function @ the sphere S' has a "good" behaviour in the region A @ ( M , ) .In this case we may try to deduce the Pistone-Sempi structure as a pulback of the natural geometry of the S' sphere by the @-embedding. 2. Quantum Fisher information and uncertainty principle

The Heisenberg uncertainty principle

207

is an immediate consequence of a stronger inequality involving covariance proved by Schrodinger that is

where

with p density matrix and A, B self-adjoint matrices. It is natural to ask if a similar bound exists as a function of the commutators [A,p ] , [B,p] instead of the commutator [A,B ] ; to this respect one should consider the Wigner-Araki-Yanase theorem for quantum measurement, which states that observables not commuting with a conserved quantity cannot be measured exactly (see Ref. [ll]). It can be surprising that only in very recent times an inequality of this type has been proved. To present it in an expressive form we need to introduce the machinery of monotone metrics (the quantum counterpart of Fisher information). Let M , be the space of complex n x n matrices and let VA be the set of density matrices namely

v; = { p E M,ITrp

= 1, p

> 0).

A monotone metric (or quantum Fisher information) is a family of riemannian metrics g = { g n } on {VA},n E N,such that

holds for every Markov morphism T : M , 4 M , (completely positive, trace preserving map) and all p E VA and X E T,VA. With each operator monotone function f one associates the so-called Chentsov-Morotzova function 1 for x,y > 0. Cf(X,Y) := Y f (xy-') Define L,(A) := pA, and R,(A) := Ap. Now we can state the fundamental theorem about monotone metrics (classification is up to scalars). Theorem 2.1. (Petz 1996) There exists a bijective correspondence between monotone metrics on VA and symmetric operator monotone functions. This correspondence is given by the formula

9f(AB , ) := 9f,p(A,B ) := W A . Cf(&

qJ(B)).

208

Let V be a finite dimensional real vector space with a scalar product g ( . , .). We define, for v, w E V , Area,(v, w):= d g ( v , v) . g ( w , w)- 1g(v, w)12. Let

f&)

:=

(x - 1 ) 2

(d- l)(xl-P

- 1)

One can prove that the functions f p are operator monotone. Since i[p,A] is traceless and selfadjoint, then i[p,A] E T,DA. Let go(., .) := g f p ( . ,.) be the quantum Fisher information associated to f p , and Areao(., .) the corresponding area functional on the tangent space. The monotone metric g p is known as Wigner-Yanase-Dyson monotone metric of parameter P. We are ready for the main result Theorem 2.2. The inequality 4

is true for

PE

( 0 , l ) and is false for ,B E [-1,0) U (1,2].

The case P = has been conjectured in Ref. [ll]by S. Luo and Z. Zhang and proved in Ref. [lo] by S. Luo himself and Q. Zhang. The general case P E (0,l) has been proved independently by H. Kosaki in Ref. [9] and by K.Yanagi, S. Furuichi and K. Kuriyama in Ref. [17]. The counterexample for P E [-1,O) u (1,2] is due to the present authors and can be found in Ref. [6]. Let us make some more comments on this result. The proof by YanagiFuruichi-Kuriyama appears simpler than Kosaki proof. Nevertheless Kosaki was also able to establish necessary and sufficient conditions to have equality and moreover he showed that the function

i).

is increasing on (0, Note that the inequality respects the ordering of the associated operator monotone functions. Let us underline that the W Y D metrics come from LP-geometry (see Ref. [4]) where p = Therefore the inequality of theorem 2.2 is true for p E ( l , + m ) that is when LP spaces are well behaved Banach spaces. A natural question is the following: are there other operator monotone functions f such that for the associated quantum Fisher information g := g f

b.

209

it is true that 1 S,(A, B ) 2 ZArea,(i[~,PI, i [ ~pI12 ,

for any p , A, B? Another problem has been suggested by Kosaki in Ref. [9]: can Theorem 2.2 be generalized in the setting of arbitrary von Neumann algebras? 3. Schur-convexity of curvature for statistical models

Let us recall that a function f : DA -+ IR is Schur-convex (Schur-increasing) if A + B =+ f(A) 2 f ( B ) ,where the symbol stays for the ”more mixed” relation. A Schur-convex function behaves like entropy, namely it increases with mixing. Let ( M ,g) be a riemannian manifold. The scalar curvature at the point p , denoted by Scal,(p), is (up to normalizing factor) the “average curvature” at p . Because of its relation with volume of geodesic balls it has been suggested by Petz that, in Information Geometry, the scalar curvature should have the meaning of average statistical uncertainty and therefore the Schurconvexity of scalar curvature would be a desirable property (see Ref. [13]). In what follows jj is defined by l / p l/@= 1. Recall that the function f(z)= (z - 1)/ log(z) is operator monotone and the associated monotone metric is known as the Bogoliubov-Kubo-Mori metric. The W Y D ( p ) metric is the quantum Fisher information associated to

+

+

1

fp(.>

(z- 1)2

:= I

PP (& - l ) ( d - 1)

(here we use a different parameter, p = l/p, and a different normalization with respect to Section 2). The LP-geometries on the state space DA are the geometries given by pull-back of the embeddings p ++ pp5 for p E [l,+m) and p ++ log(p) for p = +m (these are simply the a-geometries quoted in Section 1). A number of conjectures have been formulated in this field.

Conjecture 3.1 (Petz conjecture) The scalar curvature of BKM-metric is Schur-convex. Conjecture 3.2 (WYD-conjecture) The scalar curvature of WYD(p)-metric is Schur-convex for p near 1 . Conjecture 3.3 (LP-conjecture) The scalar curvature of LP-geometry is i) Schur-convex for p E (2, +m]; ii) Schur-concave for p E (1,2).

210

Using a continuity argument one can prove that if the WYD-conjecture is true then the Petz conjecture is true (see Ref. [5]). At first sight the WYD-conjecture does not seem any easier to prove than Petz conjecture. But as explained in Ref. [5] the LP-conjecture “almost” implies the W Y D conjecture and the geometric content of the LP-conjecture is self-evident when one looks at a picture of the unit sphere in LP spaces. Up to now the Petz conjecture has been proved only in the 2 x 2 case (see Ref. [12]) while the LP-conjecture has been proved in the commutative case for n = 2 (Ref. [5]). Further interest in this area derives from the geometrical approach to statistical mechanics where it is postulated that the scalar curvature is proportional to free energy density: a recent account of this subject can be found in Refs. [15, 11.

References 1. D.Brody and A. Ritz. Information geometry of finite king models, J . Geom Phys., 4 7 207-220, (2003). 2. P.Gibilisco and T. Isola. Connections on Statistical manifolds of Density Operators by Geometry of Noncommutative LP-Spaces, Inf. Dim. Anal. Quant Prob. €4 Rel. Top., 2 169-178, (1999). 3. P.Gibilisco and T. Isola. Wigner-Yanase information on quantum state space: the geometric approach, J. Math. Phys., 44(9): 3752-3762, (2003). 4. P.Gibilisco and T. Isola. , On the characterization of paired monotone metrics, Ann. Ins. Stat. Math., 56(2): 369-381, (2004). 5. P.Gibilisco and T. Isola. On the monotonicity of scalar curvature in classical and quantum information geometry, J . Math. Phys., 46(2): 023501,14, (2005). 6. P.Gibilisco and T. Isola. Uncertainty Principle and Quantum Fisher Information, Preprint arXiv:math-ph/0509046vl, (2005). 7. P.Gibilisco and G. Pistone. Connections on nonparametric statistical manifolds by Orlicz space geometry, Inf. Dim. Anal. Quant Prob. €4 Rel. Top., 1 325-347, (1998). 8. P.Gibilisco and G. Pistone. Analytical and geometrical properties of statistical connections in Information Geometry, in “Mathematical Theory of Networks and Systems”, A. Beghi, L. Finesso, G. Picci (eds.), p. 811-814, I1 Poligrafo, Padova (1999). 9. H. Kosaki. Matrix trace inequalities related t o uncertainty principle, Inter. Jour. Math., 6 629-645, (2005). 10. S. Luo and Q. Zhang. On skew information, IEEE Trans. Infor. Theory, 50(8), 1778-1782, (2004). 11. S . Luo and Z. Zhang. An informational characterization of Schrodinger’s uncertainty relations, J. Stat. Phys., 114, 1557-1576, (2004). 12. D. Petz. Geometry of canonical correlation on the state space of a quantum system, J. Math. Phys., 35, 780-795, (1994).

21 1 13. D.Petz. Covariance and Fisher information in quantum mechanics, J . Phys. A: Math. Gen, 35,929-939,(2002). 14. G.Pistone and C. Sempi. An infinite-dimensional geometric structure on the space of all probability measures equivalent to a given one, Ann. Stat., 33, 1543-1561,(1995). 15. G. Ruppeiner, Riemannian geometry of thermodynamics and systems with repulsive power-law interactions, Phys. Rev. E 72,016120, (2005). 16. R.F. Streater, Quantum Orlicz Spaces in Information Geometry, Open. Sys. & Inf. Dyn. 11, 359-375,(2004). 17. K. Yanagi, S. Furuichi and K . Kuryama. A generalized skew information and uncertainty relation, to apper on IEEE Trans. Infor. Theory, Preprint arXiv:quant-ph/O501152~2, (2005).

INTRODUCTION TO DETERMINANTAL POINT PROCESSES FROM A QUANTUM PROBABILITY VIEWPOINT

ALEX D. GOTTLIEB Wolfgang Pauli Institute, Nordbergstrasse 15, A-1 090 Wien, Austria alexOalexgottlieb. corn Determinantal point processes on a measure space (X, C, p ) whose kernels represent trace class Hermitian operators on L 2 ( X ) are associated to “quasifree” density operators on the Fock space over L 2 ( X ) .

1. Introduction

This contribution has been informed and inspired by several surveys of the topic of determinantal point processes that have appeared in recent years. [l, 2, 31 The first of these, Soshnikov (2000), is inspired by the determinantal point processes that arise in random matrix theory: the set of eigenvalues of a random matrix is a realization of a determinantal point process, if the random matrix is sampled from any of the unitary-invariant ensembles of Hermitian matrices (e.g., GUE), or from uniform measure on the classical (orthogonal, unitary, or symplectic) matrix groups, or from the Ginibre Ensemble. The review by Lyons (2003) is inspired by the Transfer Current Theorem [4], which implies that the edges occurring in a randomly (unifcrmly) sampled spanning tree of a given finite graph G are a determinantal random subset of the edge set of G. Lyons’s review concentrates on random subsets of countable sets, while Soshnikov’s review is oriented to treat discrete subsets of a continuum. A very recent survey of determinantal processes (Hough et al. (2005)) includes the following newly-found example: the zero set of a power series with i.i.d. gaussian coefficients is a determinantal point process [5] (the radius of convergence equals 1 almost surely). Hough et al. (2005) explain how a simple insight gives one a handle on number fluctuations in determinantal point processes. [6, 7, 8, 91 The insight is that, in a determinantal point process with finite expected number 212

213

of points, the distribution of the number of points is equal to the distribution of the sum of independent Bernoulli(Xj) random variables, where 0 < X j 5 1 are the nonzero eigenvalues of the “kernel” of the determinantal process. For example, consider the number of eigenvalues of a random n x n unitary matrix that lie in a given arc A of the unit circle. Denote this number by #nA. If the length of A is positive but less than 27r, then

is asymptotically normal with unit variance. [7, 10, 111 The subset of eigenvalues that lie in A forms a determinantal point process on A , for it is the restriction of a determinantal point process on the whole circle, hence #nA is distributed as a sum of independent Bernoulli random variables. Thus, once one knows that the variance of #nA is (lnn)/7r2 +o(n) [12, 131, the asymptotic normality of (1) follows from the Lindeberg-Feller Central Limit Theorem. Determinantal point processes have a physical interpretation: they give the joint statistics of noninteracting fermions in a “quasifree” state. Indeed, this motivated the introduction of the concept of determinantal (or “fermion”) point processes in the first place. [14] Analogously defined “boson” point processes arise in physics and are called “permanental” point processes in probabilistic writing. [14, 31 Recently, too, researchers have continued to investigate determinantal point fields from a quantum probabilistic point of view. [15, 161 We adopt this viewpoint here, and realize that the satistics of a determinantal point process with trace class Hermitian kernel K on L 2 ( X )are those of observables on the Fock space F0(L2(X)) with respect to the density operator on F0(L2(X))that determines the gaugeinvariant quasifree state with symbol K on the CAR subalgebra. However, we do not dwell below on the physical interpretation, nor do we discuss states on the CAR algebra in the following. Our main objective will be to construct the determinantal point process on X with kernel K,when K is the integral kernel of a Hermitian trace class operator on L 2 ( X ) with 0 5 IlrCll 5 1. Once the construction is understood, the fact that the number of points in a measurable subset of X is distributed as a sum of independent Bernoulli random variables becomes obvious. Finally, let us remark that determinantal/permanental processes have a couple of different interesting generalizations. [17,18]And another rich survey of determinantal processes has just appeared in the electronic archive!

1191

214

2. Determinantal probability measures on finite sets

Let X be a finite set, and let 2% denote the set of all subsets of X. Let P denote a probability measure on 2%, and let X be a random subset of X distributed as P. Then P(X 1 E ) denotes the measure of the class of all subsets of X that contain the subset E. If there exists a complex-valued function K on X x X such that

~j))~y&

for all subsets ( ~ 1 ~ ~. .2, zm} , . of X,where (Ic(zi, denotes the m x m matrix whose ( i ) j ) t hentry is Ic(zi,zj),then P is said to be a determinantal probability measure [2] with kernel Ic. The probabilities (2) determine the probabilities P(E) by inclusion-exclusion,hence there can be at most one determinantal probability measure with a given kernel Ic. A very basic example of a determinantal probability on 2% is the law of the random set produced by independent Bernoulli trials for the membership of each element of X ;in this case the kernel K ( z / , z ) = Sx~,P(zE X ) . Suppose P is determinantal with kernel Ic. Then the complementary probability measure

Pyx = S ) = P(X = x \ S ) is determinantal with kernel Z - K , where Z ( d , z ) = 6,!,. use the identity

TOprove this,

The determinants on the right-hand side of (3) are probabilities according

215

to (2), therefore det (Z(zi,zj) - K(zi,~j))~:=, m

=1 -

CP({2j} cX )

+

c

P(bj17Xj2}

cX )

1 9 1< j z I m

j=1

+

+ ... (-l)nLP({21,.. . , x m } c X ) =P(XCX\{Z~,...,Z~}) [by inclusion-exclusion] = P ( ( X \ X ) II{s~,...,xm}) =

Pyx 3 ( 2 1 , . . .>2,}) .

(4)

Suppose that K = {x1,22,...,xn} is an n-member set. Define the n matrix Kij = ( K ( ~ i , z j ) ) ~ If~ =K ~is. a Hermitian matrix, then both K and 1 - K must be nonnegative matrices, since all of their submatrices have nonnegative determinants by (2) and (3,4). Hence, if K: is the kernel of a determinantal random set and K is Hermitian, then K must be the matrix of a nonnegative contraction on @", i.e., necessarily O 5 ll.Kll 5 1. Conversely, if K is the matrix of a nonnegative contraction on C", then we will show that there exists a determinantal probability measure on with kernel K:(i,j) = Kij. The rest of this section is devoted to the construction of a determinantal probability measure whose kernel is a nonnegative contraction. Our point of view is that there exists a density operator on the Fock space over en whose diagonal elements in the standard Fock basis give the desired probabilities. A density operator is a nonnegative Hermitian operator of trace 1. Let F ( V ) denote the exterior algebra over @", i.e., 2{1*...1n)

F(@") = @ @ Cn @ /I2@"@

*

*. @

,

@ A"@"

(5)

where /Im@" denotes the mth exterior power of P. The exterior algebra F ( C n )is spanned by vectors of the form v1 A 212 A . . . A V m , where v1,. . . ,vm are any m vectors in and m is any number between 1 and n (together with an extra "vacuum vector" R to span the first summand). The expression v1 A 212 A . . . A v, for vectors is formally multilinear in v1,. . . ,om and satisfies

en

Vj A . ' ' A v A ~' . ' A

v = ~ - VI A ' . . A v j A * ' . A

v

~

for j = 2 , . . . n. The exterior algebra F ( P )is 2n dimensional and supports the inner product m

( v ~ A . . . A v ~w t ,l A . . . A w m ) = Smtm d e t ( ( v i , ~ j ) ) ~ ~ = ~

216

(the vacuum vector is orthogonal to all v1 A . . .A v, and has unit norm). It can be shown that F(Cn)is isomorphic to a subspace of the Fock space

Fo(C") = C @ Cn a3 (C" via the map that assigns 1@ Ocn

. . . @ OBm-lcn

OC

to

211

A 212 A

.. . A. ,12

@ C,) @ . .

. @ (@"en)

. . @ Og,n@n to R and CBS C [ V ~ . .,. , w,] CBOg,m+lcn . . . @ OCZYC~ +

In (6), SC[vl,. . . ,v,]

(6)

denotes the Slater determinant

where Sm denotes the group of permutations of (1,.. . ,m} and U, is the unitary operator defined on gmCn when 7r E S , by the condition that UT(W1

C3 w2 @ * . . @ W m ) =

w , - I ( ~ ) C3 w , - I ( ~ ) @

. . . C3 w,-I(,)

(8)

for all ~ 1 ,. .. ,W m E Cn.Henceforth, we identify the exterior algebra F(C") with this subspace of FO(Cn), and call it the "fermion Fock space." An orthonormal basis of F(Cn), called a Fock basis or "occupation number" basis, can be built using any ordered orthonormal basis v = (211,. . . , v,) of C". The vectors of the Fock basis can be conveniently indexed by subsets of { 1,.. . ,n}: the empty subset of { 1,.. .,n} corresponds to the vacuum vector R and a nonempty subset {jl,. . . ,j,} c { 1,.. . ,n } with jl < . . . < j, corresponds to the vector vjl A .. . A vj,,,. That is, I S c (1,. . . ,n } } is a basis for F(Cn), where the orthonormal set {fv(S) fv({}) = R and fv(S)= vjl A A vj, when S = {jl,... , j m } with jl

< . . . < j,.

Suppose K is a nonnegative contraction on Cn and let v = (vl, . . . ,v,) be an ordered orthonormal basis of Cn such that Kvj = X j v j for all j. Let DK denote the density operator

on F ( C n ) ,where (fv(S),. )fv(S)denotes the rank-one orthogonal projector onto the span of fv(S). Proposition 2.1. Let K be a nonnegative contraction o n C n and let DK denote the associated density operator (9) o n the Fock space F(Cn). Then, for all ordered orthonormal bases w = ( ~ 1 ,... ,w,) of C",

s

(fw(s),DKfw(S))

(10)

217

is a determinantal probability measure on 2{19.-3n) with kernel K ( i , j ) = (Kwi,wj). Proof: We first define the "second quantization" maps from operators A on @, to operators r , [ A ] on the Fock space &(@,), and the dual maps from density operators D on &,(en) to m-particle ''correlation operators" K,[D] on @T". Let J ( m ,k ) denote the set of injections of (1,.. . ,m} into (1,.. . ,k } . The cardinality of J ( m ,k ) is k[,] = k ( k - 1 ) . . . (k-m+l), the mthfactorial power of k . For any operator A on @Tn, and any injection j E J ( m ,k ) with k 2 m, we define the operator A ( j ) = U(1j1)(2j2)...(mj,) ( A @ I @ . . . @ I)U(lj1)(2j2)...(mjm) on @%, where U(lj1)(2jz).,.(mj,) denotes the permutation operator (8) for the product of disjoint transpositions (lj1)(2j2).. (mj,). Define r , [ A ] on TO(@") by +

rm[A] = O@@...@O@m-l@n

C

@

C

A(') @ . . . @

j€J(m,m)

A(') .

j€J(m,n)

-

A density operator D on F ( P ) extends to a density operator D @ 0 on &(en), which we will denote by D as well. The map A Tr(DI',[A]) is a linear functional on the space of linear operators on @."@". there exists a unique operator K,[D] on @@ ," such that

Therefore

Tr(rm[AID)= ~ ( A K , [ q for all linear operators A on @T". In physics language, K,[D] is the m-particle correlation operator for the state with density operator D . If DK is defined as in (9), the key identity

m times

K,[DK]

=

K

@K @.

.. @ K

sgn(x)U,

(11)

T€S,

may verified by comparing matrix elements of both sides with respect to the basis {vjl 63 . . . @ vj, 1 j1,.. . ,, j E (1,.. . ,n}}. Given an ordered orthonormal basis w = (201, . . . , w,), let P; denote the projector (wj,.)wjfor j = 1,.. . , n. For distinct 21,.. . , x m , the operator r,[P; @ . . . @ P;,] is diagonal in the Fock basis {fw(S) 1 S c { 1,. . . , n } } , and

218

3. Determinantal finite point processes

A finite point process on X is a random finite subset of a space X. Let C be a a-field of measurable subsets of X. [20] A finite point process on (X, C) is specified by the probabilities po,p1, p ~. .,. that there are 0 , 1 , 2 , . . . points in the configuration, and, for each n such that p n # 0, a symmetrical conditional probability measure pn on ( X n , WE). [21] Now let p be any positive “reference” measure on ( X , C). A finite point process is determinantal on (X, C, p ) with kernel K : X x X C if

-

(12) for all disjoint, measurable E l , . . . ,Em, m 2 1. [3] If K (z,y) is the standard version of the integral kernel of a nonnegative trace class contraction K on L 2 ( X ,C, p ) , then there exists a unique [22] determinantal point process on (X, C, p ) with kernel K(z,y). Conversely, if the kernel of a determinantal point process on X is the integral kernel of a trace class Hermitian operator K on L 2 ( X ) ,then K must be a nonnegative contraction. [23] In this section we construct the determinantal point process on X whose kernel is the standard kernel (17) of a given trace class operator K on L 2 ( X ) with 0 5 (IK((5 1. This is accomplished by constructing a density operator on 3 ( L 2 ( X ) )as we have done in the preceding section - our quantum probabilistic point of view. There are many other ways to accomplish the same end, with or without our point of view. The original approach of Macchi (1975) was to start with a formula for the Janossy densities [24] of the

219

desired point process, and then to verify (12) for that process. Soshnikov (2000) attacks the problem by first showing that certain Fredholm determinants involving K define factorial moment generating functions for finite families of random variables {#(Xn Ej)lEj E C}, and then constructing the desired determinantal point process via Kolmogorov extension from its finite dimensional distributions. Lyons (2003) uses the geometry of Fock space, but only in the case where K is a finite rank projector, then dilates nonnegative contractions to projections on a larger space to handle the general case. Hough et al. (2005) verify directly that kernels of finite rank projectors yield determinantal point processes, then treat the general case as a mixture, in the probabilistic sense, of determinantal processes with projector kernels. Any density operator on F ( L 2 ( X C, , p ) ) of the form D = @Dn defines a finite point process on X as follows. p , = Tr(D,) is the probability of the event the configuration has exactly n points. The measure p, is absolutely continuous with respect to B n p and it is defined by way of the isomorphism

, p,(E) = Regarding D, as an operator on L 2 ( X n , @ P C , @ P p ) define p ; ' T r ( D n M ~ ) ,where M E denotes the operator on L 2 ( X n )of multiplication by the indicator function of E E BnC. Given a trace class nonnegative contraction K on L 2 ( X ) , a density operator DK on F ( L 2 ( X ) )may be defined using the spectral information in K as was done in (9) above:

where A1 2 A2 2 ... are the eigenvalues of K and {f(S)}is the Fock basis constructed from the eigenvectors of K (really, any extension of an orthonormal system of eigenvectors of K to an orthonormal basis of L 2 ( X ) ) . D K has the form @(OK),and it can be verified [25] that the rn-particle correlation operator K m [ D ~exists ] and satisfies the key identity (11). Let p , and pn be as defined above, for D K . Let lE denote the expectation with respect to random point process defined by these p , and p,. Then

220

if K ( z , y) is the usual version of the integral kernel of K , i.e.,

K(Z,Y) = X X j 4 j W r n

(17)

j

where Kdj = X j 4 j and CXj = TrK. Equations (14) - (16) imply (12) holds; the finite point process defined through DK is determinantal with kernel K. Now that we have constructed the process, we can see immediately from (13) that the total number of points in a random configuration is distributed as the sum of Bernoulli(Xj) random variables. In particular, - A), is the probability that there are no points at all. This equals the Fredholm determinant Det(l - K ) . Let E be a measurable subset of X.It is not difficult to check via (12) that the determinantal point process on ( E ,CIE,P I E ) with kernel KE 3 M c K M c l , is the restriction to E of the determinantal point process with kernel K on X . Hence the probability that there are no points in E equals the Fredholm determinant Det(I-KE). In the context of random matrix theory, this yields formulas for the spacing distributions of eigenvalues. [27, 281 In case IlKll < 1, set L = (I- K ) - l K . It is easy to check from (13) that

n(l

by comparing matrix elements of both sides of this identity with respect to an eigenbasis of K . This identity yields the determinantal formulas for the Janossy densities. [14, 211

221 4. Determinantal processes of infinitely many points

Suppose that X is a locally compact Hausdorff space satisfying the second axiom of countability [29], and let C denote the Borel field of X. In this context, a point process is a random nonnegative integer-valued Radon measure (a Radon measure is a Borel measure which is finite on any compact set). [3] Let p be a cT-finite Radon measure on X. [9] A point process on (X, C, p ) is determinantal with kernel K: if (12) holds. Most work on determinantal processes with infinitely many points has been done for the cases where X is a countable set with the discrete topology and p is counting measure, or X is a connected open subset of Rd and p is Lebesgue measure, or X is a finite disjoint union or Cartesian product of said spaces. If K is a locally trace class Hermitian operator on L 2 ( X )such that 0 5 llKll 5 1, then (a version of) its integral kernel is the kernel of a determinantal point process on X. [l]This point process is the limit in distribution of the determinantal processes with kernels lc(z)K:(z,y)lc(y), where C ranges over an increasing family of compact subsets of X. Conversely, if the kernel of a locally trace class Hermitian operator K defines a determinantal point process, then K must be a nonnegative contraction. 11,141 The case of a countably infinite set X with counting measure is treated in detail in Lyons (2003). In this pleasant special case, Hermitian operators on L 2 ( X ,2 x , #) are automatically locally trace class. On the other hand, equations (2) and (3)-(4) readily imply that K and I - K are both nonnegative operators. Therefore, the kernel of a Hermitian operator K on L 2 ( X ,2x, #) is the kernel of a determinantal point process on (X, 2x, #) if and only if K is a nonnegative contraction. Acknowledgments This work was supported by the Austrian Ministry of Science (BM:BWK) via its grant for the Wolfgang Pauli Institute and by the Austrian Science Foundation (FWF) via the START Project (Y-137-TEC) of N.J. Mauser. I dedicate this to Steve Evans who introduced me to determinantal point processes.

References 1. A. Soshnikov. Determinantal random point fields, Russian Math. Surveys 55, 923 - 975 (2000).

2. R. Lyons. Determinantal probability measures, Publ. Math. Inst. Hautes Etudes Sci. 98,167 - 212 (2003). 3. J. Ben Hough, M. Krishnapur, Y . Peres, and B. VirAg. Determinantal processes and independence, Preprint arXiv:math.PR/O503110 (2005). 4. R. M. Burton and R. Pemantle. Local characteristics, entropy, and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21, 1329 - 1371 (1993). 5. Y . Peres and B. VirBg. Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Mathematica 194,1 -35 (2005). 6. 0. Costin and J. Lebowitz. Gaussian fluctuation in random matrices, Phys. Rev. Lett. 75,69 - 72 (1995). 7. K. L. Wieand. Eigenvalue distributions of random unitary matrices, Probability Theory and Related Fields 123,202 - 224 (2002). 8. A. Soshnikov. The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Annals of Probability 28, 1353-1370 (2000). 9. A. Soshnikov. Gaussian limit for determinantal random point fields, Annals of Probability 30, 171 - 187 (2002). 10. P. Diaconis and S. N. Evans. Linear functionals of eigenvalues of random matrices, iPrans. A m . Math. SOC.353, 2615 - 2633 (2001). 11. P. Diaconis. Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture, Bull. Am. Math. SOC.40,155 - 178 (2003). 12. E. M. Rains. High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107,219 - 241 (1997). 13. Note that the asymptotic variance of the number of eigenvalues in an arc is independent of the length of the arc! Another astounding fact is that the numbers # I and # J of eigenvalues in two intervals I and J are asymptotically uncorrelated if I and J have no endpoints in common. See Wieand (2002) and Diaconis (2003). 14. 0. Macchi. The coincidence approach t o stochastic point processes, Adv. Appl. Prob. 7,83 - 122 (1975). 15. E. Lytvynov. Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density, Rev. Math. Phys. 14,1073-1098 (2002). 16. H. Tamura and K. R. Ito. A canonical ensemble approach t o the fermion/boson random point processes and its applications. Preprint arXiv:math-ph/0501053 (2005). 17. P. Diaconis and S. N. Evans. Immanents and finite point processes, J . of Combinatorial Theory A91, 305 - 321 (2000). 18. T. Shirai and Y. Takahashi. Random point fields associated with certain F'redholm determinants I: fermion, Poisson and boson point processes, J. Functional Analysis 205,414 - 463 (2003). 19. K . Johansson. Random matrices and determinantal processes. Preprint arXiv:math. PR/0510038 v l (2005). 20. The space K is ordinarily assumed t o be a nice topological space: in Daley and Vere-Jones (2003) it is a complete separable metric space and in other

223 works 18, 181 it is locally compact and second countable. But for our purposes C, p ) . in this section, we only need a measure space (X, 21. D. J. Daley and D. Vere-Jones. A n Introduction to the Theory of Point Processes, Volume I. Springer-Verlag, New York, 2003. 22. Uniqueness follows from a sort of inclusion-exclusion formula for the measures pnpn in terms of all “m-point correlation functions” det ( K ( z i ,~ j ) ) ~ with ~ : ~ m 2 n. 23. The hypothesis that K is trace class is used to prove that JJKll5 1 in general. [I, 31 However, this hypothesis is not required if X is a locally compact space and K is continuous (and p is a Radon measure on the Bore1 field). In such cases, if the (continuous) kernel of a determinantal point process on X is the integral kernel of a bounded - but not necessarily trace class - Hermitian operator K on L 2 ( X ) ,then necessarily 0 5 IlKll 5 1. 24. The Janossy densities are the densities of the measures n!p,p,. 25. See Proposition 2.2 of archived manuscript math-ph/0303070 . 26. Equation (16) is the same as (1.27) of Soshnikov (ZOOO), but it can be confirmed directly by substituting (17) in (16). 27. C. A. Tracy and H. Widom. Introduction to random matrices, in Springer Lecture Notes in Physics 424, 103 - 130 (1993). 28. C. A. Tracy and H. Widom. Correlation functions, cluster functions, and spacing distributions for random matrices, J . Stat. Phys. 92, 809 - 835 (1998). 29. Such are the spaces considered in the work of A. Lenard on correlation densities for infinitely many particles (viz. Theorem 1 in Soshnikov (2000)).

NOTE ON THE TIME OPERATOR*

TAKEYUKI HIDA Meijo University Nagoya, Japan

There are many ways of understanding the time operator. In this report the time operator will be discussed from the view point of stochastic analysiss. First, a stationary stochastic process is taken to be a representation of the ordinary time t , which is an abelian group and is linearly ordered set. Together with the one parameter group of the time shift we can find the well known commutation relations in terms of generators. Second, we observe a semi-group that defines the propagation of a diffusion process. Its generator satisfies another interesting commutation relations with the differential operators acting on the space of functions on the configuration space. There one can see the transversal relations among the generators, so thatthe theory of dynamical systems can be applied.

1. Introduction

There are many directions of white noise theory to be developed; not only within mathematical science, but in the fields of quantum dynamics, molecular biology and sociology. Those fields are not simply applications, and indeed they suggest to us lot of interesting problems that can be discussed in white noise analysis and they lead us to propose a more general setup of the analysis. Some of those directions are traditional and some others are new. The main part of this report is devoted to the time operator from the view point of white noise analysis, in particular in a connection with transformation group.

'AMS 2000 Mathematics Subject Classification: 60H40

224

225

2. Time operator The theory of time operator has been investigated since many years ago, in the study of random evolutional systems, where the time development is described by a one-parameter unitary group of operators depending on the time variable t. In the study of a flow (or a dynamical system) of one-parameter unitary group we needed some additional one-parameter group of operators satisfying suitable relationship with the given flow that plays important roles. The relationship may be expressed in terms of commutation relations among the associated infinitesimal generators of the flows as members of the complex Lie algebra.

A general algebraic observation on commutation relations between two members may be stated as follows. Let a and p together with the identity I form a complex Lie algebra under the Lie product [., .]. We can consider three typical cases for a pair a and p. 1) commutative:

[a,p]= 0,

2) one is transversal to the other: [ c x , ~=] a, 3) canonical commutation relation: [a,p] = iI. (See PI.)

Recalling this fact, we take a generator, let it now be denoted by a, of the one-parameter group describing the time evolution. We shall be in search of another possible one p such that introducing a suitable Lie product of a and ,8 is one of the above three. We will see, in this section, that the relationships 2) and 3) naturally arise in the study of stationary stochastic processes, based on the time shift. This should be noted in the study of a stationary stochastic process.

3. Second order process.

With this observation, we are going to setup the problem on time operator. To this end, it is necessary to prepare some background on stochastic

226

process. It seems better to start with elementary and in fact classical examples rather than abstract approach. Let X ( t ) = X ( t , w ) , t E R , w E R(P), be a stationary second order stochastic process. Assume that X ( t ) is mean continuous and is purely nondeterministic. Define M t ( X ) to be a closed subspace of L2 = L2(R, P ) spanned by the X ( s ) , s 5 t. Note that we do not care the probability distribution of the X ( t ) in question. The M t ( X ) is an increasing sequence of subspaces, and V t E R M t ( X )is denoted by M ( X ) which is a Hilbert space (C L 2 ) . The assumption that X ( t ) is purely nondeterministic is now expressible as

nM ~ ( x )

= (01.

tER

With these background, a projection operator E ( t ) is defined in such a manner that :

E(t): M ( X )

+

Mt(X).

Continuity of X ( t ) implies continuity of E ( t ) in t. The collection { E ( t ) t; E R } turns out a resolution of the identity I . Namely,

E ( + m )= I ,

E ( t ) E ( s )= E(t v s), E(-m) = 0. Definition A self-adjoint operator T on the space M ( X ) given by

T

=

s

tdE(t)

is well-defined and is called the time operator. On the other hand, there is a mapping Ut such that

ut :

U t X ( s )= X ( s + t ) .

It is an isometry, so that it extends, by linearity, to a unitary operator acting on the Hilbert space M ( X ) ;still denoted by the same notation

utus = ut+s.

227

By the mean continuity of X(t), we can prove that

ut

-+

I, as t

-+

0.

Thus, we are given a continuous oneparameter group Ut,t E R, of unitary operators on M(X). Now one may be interested in the commutation relations between the time operator and the unitary group Ut. It is easy to show the following

Proposition It holds that

TUt = UtT

+ tUt.

Recall Stone’s Theorem for continuous one-parameter unitary group. Actually, we have a self-adjoint operator H such that

Ut = exp[itH]. Hence,

Corollary There is a commutation relation [H,T]= iI, which is useful in the study of the stationary stochastic process X(t).

4. Gaussian case

If, in particular, the second order process X(t) is Gaussian, one can speak of interesting connections with the canonical representation theory of X(t). Established theory is ready to be applied. Regarding canonical representation of a Gaussian process, we have to note an interesting result. If we are given a multiple Markov (say, N-ple Markov) stationary Gaussian process which is purely nondeterministic, then the canonical representation exists and expressed in the form (see [2]) N

228

where {fi(t)} is a fundamental system of solutions of ordinary differential equation with constant coefficients, and where Vi(t)’s are additive Gaussian processes such that {Ui(t)} are linearly independent system for every t. In addition, we can prove that a vector-valued process

is additive, hence it is Markov. The above discussion can be extended to the case of a simple Markov vector-valued Gaussian process. There is a multiple Markov Gaussian process that can be reduced to the above case. Suppose an N-ple Markov Gaussian process X ( t ) , t 2 0 , is dilation quasi-invariant,namely X ( a t ) for a > 0 , is the same process as a’/’X(t). Then, by changing the time parameter t to e t , we are given a stationary N-ple Markov process. So the above trick can be applied.

5. Nonlinear case

A nonlinear version of the time operator theory can also be discussed in somewhat more detail. The basic Hilbert space is now taken to be the one involving all square integrable functions which are measurable with respect to the sigma-fields generated by the given stationary process X ( t ) . In this case, we do not need to assume existence of moment of any order of the stationary process X ( t ) . Let B t ( X ) be the sigma-field generated by measurable subsets of R determined by the X ( s ) ,s 5 t. It is understood that B t ( X ) is the smallest sigma-field with respect to which all the X ( s ) , s 5 t , are measurable. Set B ( X ) = V B t ( X ) . Then, we have Hilbert spaces L : ( X ) = L’(R,Bt,P) and L 2 ( X ) = L2(w,B ( X ) ,P ) , respectively. There is naturally defined an orthogonal projection

E’(t) : P ( X ) -+ L ? ( X ) . Set E ( t ) = E’(t+). The collection { E ( t ) ;t E R} forms a resolution of the identity on the Hilbert space L 2 ( X ) . The oprtator T given by

T=

s

tdE(t)

229

is called the time operator on L 2 ( X ) . The same notations are used as in the case of M ( X ) , if no confusion occurs. On the other hand, we can define a unitary group of shift operators {Vt, t E R}. Starting with a mapping V, :

V t X ( s )= X ( t

+ s),

implies a flow (one-parameter group of measure preserving transThen, formations) {Tt,t E R,} on the measure space ( w , B ( X ) , P )such that

Tt : B s ( X )

+

Bs+t(X).

More precisely, for a cylinder set w-set

A=

(W:

( X ( S l , w ) , . . . , X ( S n , w ) )E Bk),

with Bk being a Bore1 subset in Rn, we define TtA by

TtA= ( w : ( & X ( S ~ , W ) , .,V,X(Sn,W)) .. E Bk). Obviously, Tt is a measurable transformation and

P(TtA) = P(A) holds. Then, Tt extends to a measure preserving transformation on the algebra A ( X ) generated by all the cylinder sets of the form A above, where the operations union and intersection of cylinder sets commute with Tt. Since the sigma-field B ( X ) is generated by A ( X ) , the Tt can be extended to a measure preserving transformation on (0,B ( X ) ,P). It is easy to prove that

namely {Tt,t E R} forms a one-parameter group with TO= I . To proceed to the next step we assume that Assumption The measure space (Q, B ( X ) ,P ) is an abstract Lebesgue space. With this assumption the measure space (Q, B ( X ) ,P ) given above may be said to be isomorphic to the Lebesgue measure space (without atoms).

230

Examples. 1) If X(t) is a continuous stationary Gaussian process with canonical representation, then the space (R, B(X), P ) is an abstract Lebesgue space without atoms. 2) For a linear process expressed as a linear functional of a white noise and a Poisson noise the associated measure space is also an abstract Lebesgue space without atoms.

With this assumption, the family of set transformations {Ti}turns into a family of point transformations. In addition, the family forms a oneparameter group of measure preserving transformations on the measure space (R, B(X),P ) except a null set. Also, by assumption of the continuity of X ( t ) , it can be proved that

Ttw is measurable in (t, w ) . Hence, by the usual argument we can prove the following assertion. Proposition Set

(UtcP)(~)= cP(Ttz). Then, {Ut, t E R} forms a continuous one-parameter group of unitary operators acting on the Hilbert space L 2 ( X ) :

Ut

4

I ast+O.

Having obtained the unitary group {Ut}, we can prove the commutation relation with the time operator T established before. Note that the commutation relation is exactly the same in expression as in the linear case where the entire Hilbert space is taken to be M(X). Theorem It holds that

T U t = U t T itUt. There, more profound results are included. Apply Stone’s theorem to Ut to have the infinitesimal generator which is self-adjoint and is denoted by H :

231

Then, we have the commutation relation

[ H , T ]= il Thus, similar results are obtained as in M ( X ) , however, there is a short note to be added. Since M ( X ) c L 2 ( X )holds and two spaces have been introduced consistently so far as X ( t ) has second order moment, all the operators representing statements commute with the projection operator

P :P(X)

-+

M(X).

It is our hope that the above relationships between operators will give some help to the profound investigation of the evolutional phenomena described by X ( t ) .

6. Diffusion process

We now turn our eyes to a stationary Marlcov process X ( t ) with t 2 0. assume that the transition probability density p ( t , 5,y) exists and smooth in (t,u,w). Define an operator &, t 2 0, by

Then, we have a semi-group of operators & acting on C:

VtVs = &+s,

&

-+

I,(t+O).

We can therefore appeal to the Yosida-Hille theorem for one-parameter semi-group to obtain a generator A such that: -I w - t+O lim t f=Af

v,

for f in a dense subset of C.

To fix the idea, we consider the case where X ( t ) satisfies the Langevin requaton:

d X ( t ) = -AX(t)dt

+ dB(t).

Then, A has an explicit expression of the form 1 d2 A=---Au-. 2 du2

d du

232

With this operator A we establish an interesting commutation relation. Namely

Proposition Let A be as above and let S =

2. Then, we have

[A,S] = AS. Thus, a transversal relation appears. This is a typical relationship in the theory of dynamical systems, which tell us some properties from the viewpoint of a dynamical system that is determined by the Langevin equation.

In addition, the operator S is the generator of the shift (the oneparameter group of translations of space variable), so that we can use harmonic analysis arising from the shift and the semi-group of the phase transition. Some more details on these topics will be seen in the forthcoming paper.

References 1. L. Accardi et al. (eds.), Selected Papers of Takeyuki Hida. World Scientific Pub. Co. Ltd. 2001. 2. T. Hida, Canionical representation of Gaussian processes and their applications. Memoires Coll. of Sci. Uniuv. of Kyoto, A33 (1960), 109-155. 3. T. Hida, Stationary stochastic processes. Mathematical Notes. Princeton University Press. 1970. 4. T. Hida, Complex white noise and infinite dimensional unitary group. Lecture Notes in Math. Mathematics, Nagoya Univ. 1971. 5. T. Hida, Brownian motion. Springer-Verlag. 1980. Japanese Original: Buraun Unidou. Iwanami Pub. Co. 1975. 6. T. Hida, Analysis of Brownian functionals. Carleton Math. Notes no.13, 1975. 7. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White noise. An infinite dimensional calculus. Kluwer Academic Pub. Co. 1993. 8. T. Hida and Si Si, Lectures on white noise functionals. World Scientific Pub. Co. 2005. to appear 9. H. -H. Kuo, White noise distribution theory. CRC Press Inc. 1996. 10. R. LBandre, Theory of distribution in the sense of Connes-Hida and Feynman path integral on a manifold. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 6 (2003) 505-517. 11. R. LBandre and H. Ouerdiane, Connes-Hida calculus and Bismut-Quillen superconnections. Stochastic analysis: Classical and Quantum, Perspectives of White noise theory, ed . T. HIda, World Scientific Publ. 2005, 72-85. 12. K . Yosida, Functional analysis. Springer-Verlag, 6th ed. 1980.

ON THE DYNAMICAL SYMMETRIC ALGEBRA OF AGEING: LIE STRUCTURE, REPRESENTATIONS AND APPELL SYSTEMS

:

MALTE HENKEL RENE SCHOTT~STOIMEN STOIMENOV JEREMIE UNTERBERGER

I:

The study of ageing phenomena leads to the investigation of a maximal parabolic subalgebra of conf3 which we call a1t We investigate its Lie structure, prove some results concerning its representations and characterize the related Appell systems.

1. Introduction

Ageing phenomena occur widely in physics: glasses, granular systems or phase-ordering kinetics are just a few examples. While it is well-accepted that they display some sort of dynamical scaling, the question has been raised whether their non-equilibrium dynamics might posses larger symmetries than merely scale-invariance. At first sight, the noisy terms in the Langevin equations usually employed to model these systems might appear to exclude any non-trivial answer, but it was understood recently that provided the deterministic part of a Langevin equation is Galilei-invariant, then all observables can be exactly expressed in terms of multipoint correlation functions calculable from the deterministic part only [8].It is therefore of interest to study the dynamical symmetries of non-linear partial differential equations which extend dynamical scaling. In this context, the so-called Schrodinger algebra 5cg has been shown to play an important r61e in phaseordering kinetics. In what follows we shall restrict to one space dimension and we recall in figure 1 through a root diagram the definition of 5 c I ~as a parabolic subalgebra of the conformal algebra confs [7]. * LPM, Universite Henri Poincar6, BP 239, 54506 Vandoeuvre-lb-Nancy, France t IECN and LORIA, Universite Henri Poincare, BP 239, 54506 Vandoeuvre-lb-Nancy, France LPM, Universit6 Henri Poincar6, BP 239, 54506 Vandoeuvre-16s-Nancy, France and Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria SIECN, Universite Henri Poincarb, BP 239, 54506 Vandoeuvre-lkNancy, France

233

234

0

0

0

Figure 1. (a) Root diagram of the complexified conformal Lie algebra (conf,)c and the labelling of its generators. The double circle in the center denotes the Cartan subalgebra. The generators of the maximal parabolic subalgebras (b) sc5 and (c) alt are indicated by the filled dots.

2. A brief perspective on the algebra alf

There is a classification of semi-linear partial differential equations with a parabolic subalgebra of conf3 as a symmetry [lo]. Here we shall study the abstract Lie algebra aKt, which is the other maximal parabolic subalgebra of conf3 (see figure 1) and its representations; we shall also see that, like the algebra 5 4 , it can be embedded naturally in an infinite-dimensional Lie algebra W which is an extension of the algebra Vect(S') of vector fields on the circle. Quite strikingly, we shall find on our way a 'no-go theorem' that proves the impossibility of a conventional extension of the embedding alt C ~ 0 n f 3 . 2.1. The abstract Lie algebra alt

Elementary computations make it clear that (see figure 1and D = 2x0-N) aKt=(V+,D,Y-g)K ( X I , Y + , M O ) : = ~ K $

(1)

is a semi-direct product of g N sK(2, R) by a three-dimensional commutative Lie algebra 5; the vector space $ is the irreducible spin-1 real representation of 5[(2,R), which can be identified with 51(2, R) itself with the adjoint action. So one has the following Proposition 2.1: (1) aIt N 5[(2, R) @ R [ E ] / Ewhere ~ , E is a 'Grassmann' variable; (2) aKt N p3 where p3 z11 50(2,1) K R3 is the relativistic Poincare' algebra in (2+l)-dimensions.

235

Proof : The linear map CP : alt t d ( 2 , IR) €9 R [ E ] / Edefined ~ by

CP(V+)= Ll, @(D)= Lo, CP(Y-+) = L-1 1

qx,)= -Lq, 2

CP(Y+)= LZ, CP(M0) = L t 1

is easily checked to be a Lie isomorphism. In particular, the representations of a[t Wigner studied them in the 30’es.

N

p3

0 are well-known since

2.2. Central extensions: an introduction

Consider any Lie algebra g and an antisymmetric real two-form a on g. Suppose that its Lie bracket [ , ] can be ’deformed’ into a new Lie bracket

-

[ , ] on

5

-

x IRK, where [K,g] = 0, by putting [(X,O),(Y,O)] = ([X, Y], a ( X , Y)). Then 6 is called a central extension of g. The Jacobi identity is equivalent with the nullity of the totally antisymmetric threeform d a : A3(g) -+ IR defined by := g

da(X, Y, 2 ) = Q!([X,Y], 2 )

+ a([Y,21,X) + 412,XI, Y).

Now we say that two central extensions g1,gz of g defined by a1,a2 are equivalent if (12 can be gotten from g1 by substituting (X, c) H (X, c X(X)) (X E g) for a certain 1-form X E g*, that is, by changing the nonintrinsic embedding of g into 61. In other words, a1 and a2 are equivalent if a2 - a1 = dX, where dX(X, Y) = (A, [X,Y]). The operator d can be made into the differential of a complex (called Chevalley-Eilenbergcomplex), and the preceding considerations make it clear that the classes of equivalence of central extensions of g make up a vector space H 2 ( g ) = Z 2 ( g ) / B 2 ( g ) , where 2’ is the space of cocycles Q! E A2(g*) verifying d a = 0, and B2 is the space of coboundaries dX, X E g*. We have the well-known Proposition 2.2: The Lie algebra aIt has no non-trivial central extension: H2(aIt) = 0. All this becomes very different when one embeds ah into an infinitedimensional Lie algebra.

+

2.3. Infinite-dimensional eztension of alf

The Lie algebra Vect(S1) of vector fields on the circle has a long story in mathematical physics. It was discovered by Virasoro in the 70’es (see [Ill)

236

that Vect(S1) has a one-parameter family of central extensions which yield the so-called Virasoro algebra ~ i := r Vect(S1) B RK = ( ( L , ) , ~ Z , K )

with Lie brackets

[K,Lnl = 0,

[Ln,J5ml = .( - m)Ln+m + 6n+m,0 c n(n2 - 1)K ( c E R)

When c = 0, one retrieves Vect(S1) by identifying the (L,) with the usual Fourier basis (ei"edO),Ez of periodic vector fields on [0,27r],or with -zn+l a d t with z := eie. Note in particular that (L-1, Lo, L1) is isomorphic to sI(2, R), and that the Virasoro cocycle restricted to sI(2,R) is 0, as should be (since sI(2, R) has no non-trivial central extensions). It is tempting to embed aIt N d(2, R) @ R [ E ] / E into ~ the Lie algebra

w := vect(S1) @ rw[e]/e2= (L,),Ez

!x (Li)nEz,

with Lie brackets

[L,, Lml = .( - m)Ln+rn, [L,, L l l = .( - m)Li+,,

[G, -qnI = 0.

These brackets come out naturally putting W in the 2 x 2-matrix form

leading to straightforward generalizations. Note in particular that there exists a deformation of the so-called 'Schrodinger-Virasoro algebra', introduced in [6, 71 as an infinite-dimensional extension of the Schrodinger Lie algebra S C ~ that , can be represented as upper-triangular 3 x 3 Virasoro matrices (see [9]for more details). In terms of the standard representations of Vect(S') as modules of Qdensities Fa = {u(z)(dz)"} with the action d

f(4z(u(z)(dz)") = (fu' + (.f'4(4(dz)", we have Proposition 2.3: W N Vect(S1) !x F-1. There are two linearly independent central extensions of W : 1. the natural extension to W of the Virasoro cocycle on Vect(S1), namely [ , ] = [ , ] except for [L,, L-,I = n(n2-1)K+2nLo. In other words, Vect(S1) is centrally extended, but its action on F-1 remains unchanged; 2. the cocycle w which is zero on A2(Vect(S1)) and A2(3-1), and defined by w(L,, L L ) = 6 n + m , ~n(n2- l ) K on Vect(S1) x 3 - 1 .

-

-

237

A natural related question is: can one deform the extension of Vect(S1) by the Vect(S1)-module 3-1 ? The answer is: no, thanks to the triviality of the cohomology space H2(Vect(S1),3-1) (see [3], or [5]). Hence, any Lie algebra structure [ , ] on the vector space Vect(S1)$3-1 such that

-

I v

[(X,4), (Y, $)I

=

([X,YIVect(Sl),advect(sl)X.$ - advect(sl)Y.4+ B(X,Y))

is isomorphic to the Lie structure of W (where B is an antisymmetric twoform on Vect(S')). So one may say that W and its central extensions are natural objects to look at. 2.4. S o m e results on r e p r e s e n t a t i o n s of

W

We now state two results which may deserve deeper thoughts and will be developed in the future. Proposition 2.4 ('no-go theorem'): There is n o way t o extend the usual representation of aIt as conformal vector fields into a n embedding of W into the Lie algebra of vector fields on EX3. Proposition 2.5: The infinite-dimensional extension W of the algebra a h is a contraction of a pair of commuting Virasoro algebras bit $air ---f W . I n particular, we have the explicit diflerential operator representation

L, = -tn+lat + ( n+ l)t"ra, - ( n + l)ztn - n(n+ 1)ytn-'r L€ n = -tn+la, - ( n + 1)yt" where x and y are parameters and n E Z. 3. Appell systems

Definition 3.1. Appell polynomials {h,(x); n E N } on IR are usually characterized by the two conditions 0 0

h,(z) are polynomials of degree n, Dh,(x) = nh,-l(z), where D is the usual derivation opertor.

Interesting examples are furnished by the shifted moment sequences hn(x) =

s_(,.

00

+ Y)nP(dY)

where p is a probability measure on R with all moments finite. This definition generalizes to higher dimensions. On non-commutative algebraic structures, the shifting corresponds to left or right multiplication

238

and, in general, {h,} is not a family of polynomials. We shall call it AppeZZ systems (see [2]for details). Appell systems of the Schrodinger algebra scg have been investigated [I] but the algebra aIt requires a specific study. aIt has the following Cartan decomposition:

a

a[t = !$ CBI CB C = { Y1,X I } CB {Yo,X o } CB { Y-1, X - I }

(2)

and there is a one to one correspondence between the subalgebras J!3 and = aibi, where {bi, i = 1,.. . , 6 } is a basis of ak. The ai are called coordinates of the first kind. Here we use the basis bl = Y1, b2 = X I , b3 = Yo, bq = Xo, b5 = Y-1, bs = Y I . Let ALT be the simply connected Lie group corresponding to aKt. Group ellements in a neighborhood of the identity can be expressed as

2. Write X E aIt in the form X

,x

x!=l

= eAibi

. . . eA6b6

The Ai are called coordinates of the second kind. Referring to decomposition ( 2 ) , we specialize variables, writing V1,V2,B1, B2 for A l , A2, As, A6 respectively. Basic for our approach is to establish the partial group law: e B 1 Y - 1 + B 2 X - ~ e V ~ Y ~ +=?. V~X We ~ get

Proposition 3.1. In Coordinates of the second kind, we have the Leibnitz formula, 9(0,0,0 , 0, Bi, B2)g(Vi,V2,0,0,0,0) = g(Al,442,A37 A47 A59 A6) =

239

Now we are ready to construct the representation space and basis-the canonical Appell system. To start, define a vacuum state R. The elements Yi, Xi of p can be used to form basis elements

Ijk) = Y,jXl"R,j, k 2 0

(6)

of a Fock space 5 = span{ I j k ) } on which Y1 ,Xi act as raising operators, Y-1, X-1 as lowering operator and Yo, XOas multiplication with the constants y, x (up to the sign) correspondingly. That is, Y1R = [lo),X1R = 101) Y-1R

= 0 ,X - l R =

0

(7)

YOR = -ylOO),XoR = -2100)

(8)

The goal is to find an abelian subalgebra spanned by some selfadjoint operators acting on representation space, just constructed. Such a twodimensional subalgebra can be obtained by an appropriate "turn" of the plane p in the Lie algebra, namely via the adjoint action of the group element formed by exponentiating X-1. The resulting plane, !Qp say, is abelian and is spanned by

y1

~

,PX-lyl,-BX-

= Yi

-

2PYo + P2Y-1

Xl = ePx-lxle-Px- = x1- 2 0 x 0 + p2x-1

(9)

Next we determine our canonical Appell systems. We apply the Leibniz formula ( 5 ) with B1 = 0 , B2 = P, V1 = 21, V2 = 22 and (7). This yields ,zlFleZzXl~ = ePX-le"lYlezzX1e-Px-l~ = e P X - l e z l Y l e z z x ' ~ =

+e ~p m(l

= e(1-P.z)

1-Pzz)e

1 - 0 ~ ~

-Pz2)-2ZZ;2

(10)

To get the generating function for the basis (jk)set in equation (10)

Substituting throughout, we have Proposition 3.2. The generating function for the canonical Appell system, Ijk) = Y,jX,kR is

+

(1 Pv2)-2Zf2

where we identify of Y11Y2.

(12)

= y1.l and X1R = y2.1 in the realization as function

240

With v1 = 0, we recognize the generating function for the Laguerre polynomials, while 212 = 0 reduces to the generating function for Hermite polynomials. References 1. P. Feinsilver, Y. Kocik and R. Schott, Representations of the Schrodinger Algebra and Appell Systems, Progress of Physics, 52 (2004) 343-359. 2. P. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus, Vol.3: Representations of Lie Groups. Kluwer Academic Publishers, Dordrecht, 1993. 3. D. B. Fuks, Cohohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. 4. M. Henkel, R. Schott, S. Stoimenov and J. Unterberger, O n the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems. Prepublication Institut Elie Cartan, 2005. 5. L. Guieu, C. Roger, preprint, available on http://www.math.univ-montp2.fr/~guieu/The~Virasoro~Project/Phasel/. 6. M. Henkel, J. Stat. Phys. 75, 1023 (1994). 7. M. Henkel, J. Unterberger, Schrodinger-invariance and space-time symmetries, Nuclear Physics B660 (2003) 407-435. 8. A. Picone and M. Henkel, Local scale-invariance and ageing in noisy systems, Nuclear Physics B688 (2004) 217-265. 9. C. Roger and J . Unterberger, in progress (2005). 10. S. Stoimenov and M. Henkel, Dynamical symmetries of semi-linear Schrodinger and diffusion equations, Nuclear Physics B 7 2 3 (2005) 205-233. 11. M. Virasoro, Phys. Rev. D1, 2933-2936 (1970).

AN ANALYTIC DOUBLE PRODUCT INTEGRAL

R L HUDSON School of Mathematical Sciences, University of Loughborough, Loughborough, Leicestershire LE11 3TU, Great Britain. -++

+

The double product integral fl (1 X(dAt C3 d A - d A @ d A t ) ) is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients.

1. Introduction --re

n

+

A purely algebraic theory of double product integrals such as (1 X(dAt@dA-dA@dAt))[4, 91 has been developed based on formal power series in the formal parameter A, so that questions of convergence are avoided. This is not mere laziness; the resulting theory is useful in the construction of algebraic quantum groups [4,91. Moreover, if X is replaced by a complex parameter, it is known that the power series defining the matrix elements between nonvacuum exponential vectors of the double product in-

n

--tt

(1+ XdA @ dA) have zero radius of convergence, thus precluding tegral a meaningful analytic theory in this case. In the algebraic theory it is known [9] that, for an arbitrary formal power series dr[X]with coefficients in the tensor product Z @ Z with itself

n

-+t

of the algebra Z of It6 differentials,

n

t+

+

(1

+ Xdr[X])has a multiplicative

(1 Xdr'[X])) where the formal power series Xdr'[X] is inverse given by the quasiinverse of Xdr[h]defined by

+

+ X2dr[h]r"h]= Xdr"h] + Xdr[h]+ X2dr"h]r[h]= 0. (1) n (1+ X(dAt @ dA - dA @ dAt))it may be verified using

Xdr[h] Xdr"h]

--tt

In the case of the usual It6 multiplication table

dAdAt = dT, all other products vanish,

(2)

that the quasiinverse of X(dAt@dA-dA@dAt)is -X(dAt@dA-dA@dAt),

n

-it

and since the formal adjoint of

+

(1 X(dAt @ dA - dA @ dAt)) is 24 1

242

n

t+

n

t+

+

( 1 ~ ( d@ ~d~ t- d~

n

+t

the double product

d ~ t ) t=)

( 1 - ~ ( d@ ~d~ t- d~ 8 d ~ t ) ) ,

( 1 + X(dAt @ d A - d A @ d A t ) ) is formally unitary.

n

4-

+

One might therefore expect that there is an analytic theory of (1 X(dAt @ d A - dA @ d A t ) ) for a real parameter X in which it consists of unitary operators in Fock space. This paper provides such a theory. 2. The double time orthogonal dilation [3]

Interpreting ( l + X ( d A t @ d A - d A @ d A t ) ) as a second quantised infinitesimal rotation and recalling the functoriality [ l o ]and continuity of the second -++

n

( l + X ( d A t @ d A - d A @ d A t ) ) is quantisation map, one might expect that itself the second quantisation of a double product of infinitesimal rotations. Such a double product has been constructed [3].It consists of a family of operators w = w,":; in the Hilbert space L2(R+)@L2(IR+)

(

)la,bl,l~,tlc~+

enjoying the following properties.

Each W,";,"is a unitary operator, acting non trivially on the subspace L2(]a,b ] )@ L 2 ( ] st,] )and as the identity on its orthogonal complement. 0

For fixed Is, t ] ,W,";,"is a reverse evolution in ] a ,b] and, for fixed ] a ,b], W,";,"is a forward evolution in ] s , t ] that , is, for a < b < c and T < s < t , 0

w,";,"wbc,I& = w,;,",w;;gPWb+ a,t = Wb,' a,t . 0

(3)

W is covariant under shifts and time reversal; for arbitrary p, q E R+

(S, @s*)*w,~~~:;;(S,@s*) = w,"::,(R: @R:)*W:;,"(R: m

i ) = w,";;

where S, denotes the isometric shift through p and Ri the time reversal operator for the interval ] a ,b ] ;

The matrix operator W,";,"is given explicitly in two equivalent forms: (4)

(5)

243

where the notation is as follows.

and K { K } denotes the integral operator on L2(R+) whose kernel is K,.: (z,y))= 1 if a < z < y < b (resp. b > z > y > a ) and 0 otherwise, xt is the indicator function of the interval Is,t ] ,and At, A: are the bounded mutually adjoint operators K { >;} and K { <:} respectively. We call W the double time orthogonal dilation. In fact it is possible to regard W as itself an avatar of the double product integral -+

n

+

(1 X(dAt @ dA - d A 8 d A t ) ) based on the Boolean version of quantum stochastic calculus of [l]. Second quantisation converts it into the corresponding product integral based on the usual Fock space quantum stochastic calculus as will now be shown. 3. Stochastic differential equations -+t

n

+

(1 Xdr[X]) is defined in the algebraic theory 1a,bl x 18 $1 through the solution of algebraic quantum stochastic differential equations [9, 81. We first consider the algebraic differential equation in which t is the time variable

Let us recall first how

+

(idz @ d ) Q[X](t) = Xdri13[X](Q[A](t)1i2 12), Q[X](s) = 0

(6)

where the first copy of the It6 algebra in Z @ Z is taken as the (left) system algebra so that the solution is a formal power series with coefficients in the tensor product Z @ P: where P," is the algebra of iterated stochastic integrals based on the time interval [s, t [ in the single Fock space F,and

244

the superscripts 1 , 2 , 3 label places in the tensor product Z @ P:@Z.Thus Q [ X ] ( t ) is given explicitly by the series of iterated integrals 00

Q[X] =

C AN ( i d 1 @ I i ( N ) )d r [ X ] 1 9 N + 1 d r [ X ] 1 " .

. .d r [ X ] 1 9 2

(7)

N=l

where now the superscripts 1 , 2 , . . .,N + 1label places in the tensor product Z@BNZ, the iterated integral map I,"(N)is the linear map from B N Z to P: such that

1

It(N)(dLl@dL2@'" @ d L N ) =

dLl(tl)dL2(t2)

' ' *

dLN(tN)

8
and we use multiplication in the first copy of the It6 algebra. Since Q [ X ] is of form X q [ X ] with q [ X ] E (1@ P:) [ [ A ] ] it can be used as driving term in a second algebraic differential equation in which b is the time variable: (d @ i d p : ( ~ )P )[ X ] ( b )

= XP[X]113(b)q[X]2'3,

P[X](U) = 1

(8)

where the system algebra P,"(N)is on the right and the superscripts label places in P,"@Z@P:. This yields a solution 00

P[X](b) =

C AN ( I t ( N )

@idp)

Q [ X ] ~ ~ ~ .~. q[[ XX] N ]1 m~ , ~ ~ .

N=O

(where when N = 0, I,"(N)@ i d p t acts on the empty product to give 1) using multiplication in the algebra P,", which is a well defined formal power series with coefficients in P,"@ P,". Alternatively we may consider first the algebraic differential equation with b as time variable (d @ i d z )Q ' [ X ] ( b ) = (Q'[X](b)113

+ 11)X d r 2 T 3 [ X ] , Q ' [ X ] ( u )

=0

(9)

in which the second copy of Z in Z @ Z is regarded as a right system algebra, followed by ( i d p @ d)P ' [ X ] ( t ) = XQ'[X]1~3P'[X](t)1,2, P ' [ X ] ( s ) = 1

(10)

in which P: is a left system algebra and Q'[X] = Xq'[X]. The algebraic multiplicative Fubini theorem [9] states that these two procedures yield the same formal power series, P [ X ] = P ' [ X ] , which is then defined to be -++

n

(1

la,blxls,tl

+Xdr[X]).

245

When dr[X]= X(dAt@dA-dA@dAt)the series (7) terminates at N = 2 in view of (2);

c c 00

AN (idz c3 It;(N))dr[X]1>N+1dr[X]1”. . . d r [ X ] 1 , 2

N=l 2

=

AN (idz 8 I , ( N ) ) ( i d z 8 It;”’) dr[X]1”+1dr[X]11N.. .dr[X]1>2

N=l

=

(dA‘

1

8

-X2dT 8

d ~ ( t-~dA ) 8

S
/ 1

1

dAt(tl))

S
dAt(tl)dA(ta)

S
=

(dAt 8 ( A ( s )- A ( t ) )- dA 8 (At(s) - A t ( t ) ) ) -X2dT &I

( A t ( t l )- At(s))dA(t1)

S
so that, for real X (8) is a well formulated quantum stochastic differential equation (qsde) with unbounded operator coefficients (d 8 i d p : ) P(X,b) = P(X,b)173(X(dAt ( t ~ ) ~ ( A-(A(t))3 s) - dA(b)2(At (s) - At (t))3

( A t ( t l )- At(s))dA(tl))3},

--X2db2(/ S
P(X,.) = 1.

(11)

Similarly (10) becomes the qsde

(id.:

8 d ) P y x ,t ) = {X((At(b) - A t ( ~ ) ) 1 d A (t )(A(b) 3 -A(~))’dAt(t)~)

-x2

X’(X,s) = 1.

(/

1

( ~ t (t ~ t)( U ) ) d ~ ( t d, )t )3 } ~ yt p~ 2, ,

a
(12)

Theorem 1 below shows that the second quantised double time orthogonal unitary dilation r(W,b,’f) satisfies both of these qsdes in the weak sense, thus simultaneously justifying its identification with the double product integral ++

n

+

(1 X(dAt 8 dA - dA 8 dAt )) and verifying an analytic F’ubini theorem in this case. The second quantisation r(W;,’f) is the unitary operator on the Fock space F(L2(IR+)CB L ~ ( ~ R +=) .T(L~(IR+)) ) B F ( L ~ ( I R +acting )) on expo-

246

in view of the general commutation relations

d(,)r(v) = r(v)a+(v*u), r(v)a(u) = a(uu)r(v) between creation and annihilation operators and second quantistions.

Theorem 3.1. As a process in b (resp. in t), I'(W:$)

satisfies the qsde

(11) (resp.(l2).

Proof. By definition of second quantisation

(4.f)@ e(g), r(w,;Mf') @ 4 7 ' ) ) Using both (4)and (5)

w;:

= exP ((f,g),

(;;))

.

247

Hence by the fundamental theorem of calculus

from which it follows that

Using (14) and the first fundamental formula of quantum stochastic calculus [Ill the first two terms become

and, again by the fundamental formula, the third term becomes

e ( f )63 e(g>,-XJbr (W:,’:) dA(b)2(At(s)- A t ( t ) ) 3 e ( f ’ @e(g’) ) a

Hence the proof that r satisfies (11) is completed by observing that, using (3), the functoriality of second quantisation, and (13), the re-

248

maining term

(12) is proved similarly.

249 4.

Epilogue

One of the obstructions to proving index theorems using quantum probability is the absence of a canonical quantum Brownian motion on a Riemann manifold. Indeed, while there is a canonical classical Brownian motion X,any decomposition X = At + A of X into creation and annihilation processes suffers from the non-uniqueness that the pair ( A t ,A ) can be replaced by the pair (eieAt,e-ieA) related to it by a gauge transformation. In this context it is interesting and perhaps significant to observe that the combination dAt 63 dA- dA 63 dAt is invariant under gauge transformation. In a later paper we shall show that the double product integral ++

n

+ X(dAt 63 dA - dA++ 63 dAt)) may be used to define an ordered triangular product integral n (1+ X(dAt(s)dA(t)- dA(.~)dAt(t))) living (1

a<s
by decomposing the triangular region in a single Fock space F (L2(R+)) { ( s , t ) : a < s < t < b } into an infinite union of disjoint rectangles each having a vertex on the diagonal line, and transfering each double product over each such rectangle into the single Fock space by using splitting at the diagonal time value in the latter. It is conjectured that such iterated product integrals and their natural multidimensional generalisations can be made to live canonically on manifolds. In this connection the use [2] of classical iterated integrals to onstruct manifold invariants is suggestive.

References 1. A Ben Ghorbal and M Schurmann, Quantum stochastic calculus on Boolean Fock space, Greifswald preprint (2003). 2. B Harris, Iterated integrals and cycles on algebraic manafolds, World Scientific (2004) 3. R L Hudson, A double dilation constructed from a double product of rotations, submitted t o Markov processes and Applications, J T Lewis memorial volume, Loughborough preprint (2005). 4. R L Hudson, It6 calculus and quantisation of Lie bialgebras, Ann. I H PoincarC-PR 41,375-390 , P A Meyer Memorial Volume (1995) 5. R L Hudson, P D F Ion and K R Parthasarathy, Time orthogonal unitary dilations and noncommutative Feynmann-Kac formulae, Commun. Math. Phys. 83,261-280 (1982). 6. R L Hudson, P D F Ion and K R Parthasarathy, Time orthogonal unitary dilations and noncommutative Feynmann-Kac formulae 11, PRIMS Kyoto 20, 607-633 (1984). 7. R L Hudson and J M Lindsay, A noncommutative martingale representation theorem for non-Fock quantum Brownian motion, J . Funct. Anal. 61,202221 (1985).

250

8. R L Hudson, K R Parthasarathy and S Pulmannovd, Method of formal power series in quantum stochastic calculus, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3,no. 3, 387-402 (2000). 9. R L Hudson and S Pulmannovd, Double product integrals and Enriquez quantisation of Lie bialgebras,Letters in Mathematical Physics 72, 211-224 (2005). 10. E Nelson, The free Markov field, J. Funct. Anal. 12,211-227 (1973). 11. K R Parthasarathy, An introduction to quantum stochastic calculus, Birkhauser (1992). 12. I E Segal, Mathematical characterisation of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math. 6,500-537 (1962).

ON GENERALIZED QUANTUM TURING MACHINE AND ITS APPLICATION

SATOSHI IRIYAMA AND MASANORI OHYA Tokyo University of Science Depertment of Information Science Yamazaki 2641, Noda city, Chiba, Japan E-mail: [email protected] Ohya and Volovich have proposed a new quantum computation model with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine by rewriting usual quantum Turing machine in terms of channel transformation. Moreover, we define some computational classes of generalized quantum Turing machine and show that we can treat the Ohya-Volovich (OV) SAT algorithm.

1. Introduction The problem whether NP-complete problems can be P problem has been considered as one of the most important problems in theory of computational complexity. Various studies have been done for many years. Ohya and Volovich [l, 21 proposed a new quantum computation model with chaotic amplification process to solve the SAT problem, which went beyond usual quantum algorithm. This quantum chaos algorithm enabled to solve the SAT problem in a polynomial time [l,2, 31. In this paper we generalize quantum Turing machine so that it enables to describe non-unitary evolution of states, we show GQTM for the OV SAT algorithm referring to the paper [9] and calculate the computational complexity of GQTM for the OV SAT algorithm. This study is based on mathematical studies of quantum communication channels [4, 51. 2. Generalized Quantum Turing Machine

Classical Turing machine(TM or CTM) M,l is defined by a triplet (Q, C , 6), where C is a finite alphabets with an identified blank symbol #, Q is a finite set of states (with an initial state qo and a set of final states 45) and 6 : Q x C 4 Q x C x { -1,O, 1) is a transition function. Note that { - 1 , O , 1) 251

252

indicates moving direction of the tape head of TM. The deterministic TM has a deterministic transition function 6 : Q x C -+ 2Q x C x {-1, 0,1} , that is, 6 is a non-branching map, in other words, the range of 6 for each (qla) E Q x C is unique. A TM M is called non-deterministic if it is not deterministic. In this section, we introduce a generalized quantum Turing machine (GQTM), which contains QTM as a special case.

Definition 2.1. Usual Quantum Turing machine Mp is defined by a quadruplet Mq = (Q1C, 'H, U ) , where 'H is a Hilbert space described below in (1)and U is a unitary operator on the space 'H of the special form described below in (2). Let C = Q x C x Zbe the set of all classical configurations of the Turing machine M,l, where Z is the set of all integers. It is a countable set and one has

Since the configuration of TM can be written as C = (q, A , i ) one can say that the set of functions {I q, A , i >} is a basis in the Hilbert space 'H. Here q E Q, i E Z and A is a function A : Z--f +. We will call this basis the computational basis. By using the computational basis we now state the conditions to the unitary operator U . We denote the set I' = {1,0, -1}. One requires that N

there is a function 6 : Q x C x Q x C x r 4 C which takes values in the field N

of computable numbers C and such that the following relation is satisfied:

u 14,A , i) =

A ( i ) ,P, b, ). IP, A f , i

+ ).

.

(2)

P>bva

Here the sum runs over the states p E Q, the symbols b E C and the elements (T E I?. Actually this is a finite sum. The function Af : Z -+ 4, is defined as

Af(j)

=

b if j = i , A ( j ) if j # i .

Note that if, for some integer t E N = { 1 , 2 , ...}, the quantum state U t ( 4 0 , A , 0) is a final quantum state, i.e. llEQ(qF)Us(40,A , 0) 11 = 1 and for

253

any s < t , s E N one has l\EQ(qF)US140, A, 0)ll = 0 , then one says that the quantum Turing machine halts with running time t on input A . Now we define the generalized quantum Turing machine (GQTM) by using of a channel A (see below) instead of a unitary operator U .

Definition 2.2. Generalized Quantum Turing machine M,, (GQTM) is defined by a quadruplet M,, = ( Q ,C, E , A ) , where Q and C are two alphabets, 7f is a Hilbert space and A is a channel on the space of states on 7-l of the special form described below. Q and C are represented by a density operator on Hilbert space 'HQ and 'HZ,which are spanned by canonical basis { Iq) ;q E Q} and { la) ; a E C } , respectively. A tape configuration A is a sequence of elements of C represented by a density operator on Hilbert space 'HZ spanned by a canonical basis { IA) ; A E C'} , where C* is the set of sequences of alphabets in C. A position of tape head is represented by a density operator on Hilbert space 7 - l ~spanned by a canonical basis { li) ;i E Z}. Then a configuration p of GQTM Mgq is described by a density operator on 'HE 7 f @'HZ ~ 87-l~. Let B ( 7 f ) be the set of all density operator on Hilbert space 7-l. A quantum transition function A is given by a completely positive (CP) channel

A : 6 (7f) + 6 (a).

ck

For instance, given a configuration p = X k l$k) (&I , where XI, = 1,Xk 2 0 and $k = I q k ) 8 IAk) 8 l i k ) ( q k E Q,Ak E C * , i k E Z) is a vector in a basis of 7-l. This configuration changes to a new configuration p' by one step transition as p' = A ( p ) = pk I&) ($kl with C pk = 1,pk 2 0. For any configuration p, GQTM Mgq is called UQTM Muq if the quantum transition function 6 of GQTM M,, is given by

Ck

A6

(PI = UbPIJ6*,

where Us is a unitary operator in 'H. Obviously Mu, = Mq. Several studies have been done on QTM whose transition function is represented by unitary operator. A transition of GQTM is regarded as a transition of amplitude of each configuration vector. We categorize GQTMs by a property of CP channel A as below.

Definition 2.3. A GQTM M,, is called unitary QTM (UQTM, i.e., usual QTM), if all of quantum transition function A in M are unitary CP channel.

254

c,

For all configuration p = A,p, (&A, = 1,A, 2 0), a GQTM M,, is called LQTM Mlq if A is affine ; A (C,A,p,) = C , A,A (p,) . Since a measurement defined by AMP = CPkpPk with a PVM {Pk} on 3-1 is a k

linear CP channel, LQTM may include a measurement process. For a more general channel the state change is expressed as

Nl4,A

(4

7

i) ( 4 , A (i) ,4)=

c

6(4,A(i),p,b,a,p’,b’,o’)

p,b,u,p’ ,b‘ P’

Ip,A!,i+o) (p,Af,i+oI with some function 6(q,A ( i ) , p ,b, o,p’, b’, d)such that the RHS of this relation is a state. Thus we define two more classes of GQTM for non-unitary CP channels.

Definition 2.4. A GQTM M,, is called a linear QTM(LQTM) if its quantum transition function A is a linear quantum channel. Unitary operator is linear, hence UQTM is a sub-class of LQTM. moreover, classical TM is a special class of LQTM.

Definition 2.5. A GQTM M,, is called non-linearQTM(NLQTM) if its quantum transition function A contains non-linear CP channel.

A chaos amplifier used in [l,21 is a non-linear CP channel, the details of this channel and its application to the SAT problem will be discussed in the sequel. 2.1. Computational class f o r GQTM

Given a GQTM M,, = (Q, C, 6) and an input configuration po = /win)(uinl, (lwin) = 140) 8 IT) 8 [0)),a computation process is described as the following product of channels

A1 0 . .. O At ( P O ) = P j

Iwj)

(~fl

where A 1 , . . . ,At are CP channels. Applying the CP channels to an initial state, we obtain a final state p j and we measure this state by a projection (or PVM) pj

I

= 14f) ( Q f 8 Ic €3 Iz,

255

where Ix,IZ are identity operators on a halting probability such that

Nc,N z ,respectively. Let p 2 0 be

tr‘F1x@uz(PfPf) = P l4f) (4fl * Then, we define the acceptance (rejection) of GQTM and some classes of languages.

Definition 2.6. Given GQTM M,, and a language L, if there exists t steps when we obtain the configuration of acceptance (or rejedion)by the probability p , we say that the GQTM M,, accepts (or rejects)L by the probability p , and its computational complexity is t. Definition 2.7. A language L is bounded quantum probability polynomial time GQTM(BGQPP) if there is a polynomial time GQTM M,, which accepts L with probability p 1 f. If NLQTM accepts the SAT OV algorithm in polynomial time with probability p 2 then we may have the inclusion

i,

N P C BGQPP. where N P is a language class that a deterministic Turing machine, which recognize with some informations in polynomial time of input size exists.

3. SAT Problem Let = ( 2 1 , . . . ,x,} ,n E N be a set. xk and its negation zk (k = 1,.. . ,n) are called literals Let 3 {q,. . . ,G}be a set, then the set of all literals is denoted by X’ f X U x = ( 2 1 , . . . ,z n , q , .. . ,G}.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 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 -1, false -0”. Then Cis 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 z in X . Then the truth value of a clause C is written as t (C) = VzEct(x).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) = Ay=n=,t (Cj)= 1. Thus the SAT problem is written as follows:

x

x

256

Definition 3.1. SAT Problem: Given a Boolean set X E {XI, . . . ,z,}and a set C = {Cl,. . . , Cm} of clauses, determine whether C is satisfiable or not. 4. SAT algorithm in GQTM

In this section, we construct a GQTM for the OV SAT algorithm. OV SAT algorithm is a quantum algorithm with the chaos amplifier explained in the paper [l,2, 61. The GQTM with the chaos amplifier belongs to NLQTM because the chaos amplifier is represented by non-linear CP channel. The OV algorithm runs from an initial state po = 10.) (vol to & through p E Ivf) (vfl.The computation from po = )0.1 (vol to p = Ivf)(vfl is due to unitary channel A c = Uc 0 U c , and that from p = Ivf) (vflto pf is due to a non-unitary channel o AI, so that all computation can be done by A$A o AI o A c , which is a completely positive, so the whole computation process is deterministic (see [9]). It is a multi-track (actually 4 tracks) GQTM that represents this whole computation process. A multi-track GQTM has some workspaces for calculation, whose tracks are independent each other. This independence means that the TM can operate only one track at one step and all tracks do not affect each other. Let us explain our computation by a multi-track GQTM. The first track stores the input data and the second track stores the value of literals. The third track is used for the computation of t (Ci), (i = 1, . ,rn) described by unitary operators. The fourth track is used for the computation o f t (C) denoting the result. The work of GQTM is represented by the following 8 steps:

-

0

0

0 0 0

0

0

0

3

+

Step 1 : Store the counter c = 0 in Track 1. Calculate [? ( n - l)] 1, we take this value as the maximum value of the counter. Then, store it in Track 4. Step 2 : Calculate c 1 and store it in Track 4. Step 3 : Apply the Hadamard transform to n a c k 2. Step 4 : Calculate t (C1) ,.. . t (Cm)and store them in Track 3. Step 5 : Calculate t (C) by using the value of the third track, and store t (C) in Track 4. Step 6 : Empty the first, second and third Tracks. Step 7 : Apply the chaos amplifier to the result state obtained up to the step 6. Step 8 : If c = [$ (n - l)] 1or GQTM is in the final state, GQTM halts. If GQTM is not in the final state, GQTM runs the step 2 to

+

+

257

the step 8 again. T h e detail of this quantum algorithm is explained in the paper [9].

4.1. Computational complexity of the SAT algorithm We define the computational complexity of the OV SAT algorithm as the

0

product of TQ ( U p ' ) and TCA( n ),where TQ U p ) is the complexity of unitary computation and TCA( n ) is that of chaos amplification. T h e following theorem is essentially discussed in [7, 2, 31.

Theorem 4.1. For a set of clauses C and n Boolean variables, the computational complexity of the OV SAT algorithm including the chaos amplifier, denoted by T ( C , n), is obtained as follows.

where poly ( n ) denotes a polynomial of n. T h e computational complexity of quantum computer is determined by the total number of logical quantum gates. This inequality implies t h a t the computational complexity of SAT algorithm is bounded by 0 ( n ) for the size of input n while a classical algorithm is bounded by 0 (2") .

References 1. M.Ohya and I.V.Volovich, Quantum computing and chaotic amplification, J.

opt. B, 5,N0.6 639-642, 2003. 2. M.Ohya and I.V.Volovich, New quantum algorithm f o r studying NP-complete problems, Rep.Math.Phys., 52, No.1,25-33 2003. 3. M.Ohya and N.Masuda, N P problem in Quantum Algorithm, Open Systems and Information Dynamics, 7 No.1 33-39, 2000. 4. M.Ohya, Complexities and Their Applications to Characterization of Chaos, Int. Journ. of Theoret. Physics, 37 495, 1998. 5. LAccardi and M.Ohya, Compound channels, transition expectations, and liftings, Appl. Math. Optim., Vo1.39, 33-59, 1999. 6. M.Ohya and I.V.Volovich, Quantum information, computation, cryptography and teleportation, Springer (to appear). 7. S.Akashi and S.Iriyama, Estimation of Complexity for the Ohya-MasudaVolovich SAT Algorithm (to appear). 8. C.H.Bennett, E.Bernstein, G.Brassard, U.Vazirani, Strengths and Weaknesses of Quantum Computing, SICOMP Vol. 26 Number 5 pp. 1510-1523. 1997. 9. S.Iriyama, M.Ohya and I.V.Volovich, Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm, TUS preprint.

DYNAMICS WITH INFINITE NUMBER OF DERIVATIVES FOR LEVEL TRUNCATED NON-COMMUTATIVE INTERACTION

LIUDMILA JOUKOVSKAYA Steklov Mathematical Institute Russian Academy of Science, Moscow MSI, Vaxjo University, Sweden E-mail: 1joukovQmi.ras.m We study dynamics in field models appearing in a level-truncation scheme for non-commutatively interacting string field theory. Distinguishing property of such models is that the corresponding equations of motion contain infinite number of derivatives. We study existence of physically interesting solutions of these equations in special approximation for interacting open-closed string model. We also present a general relation for stress tensor as well as energy conservation law for the case of arbitrary (finite) number of levels. Recent applications of such models include cosmological inflation and dark energy problems.

1. Introduction

Recent cosmological observations such as data from the WMAP satellite [l] suggest that the Universe is presently accelerating [2, 31. It is widely accepted that a new physics beyond the Standard Model is required to explain this phenomenon. It appears that the bulk of energy density in the Universe is gravitationally repulsive and can be treated as an unknown form of matter - Dark Energy - with negative pressure. Different models of the dark energy are usually described with the state parameter w = p / ~where , E is the energy density and p is pressure of the Dark Energy. The latest observational data show that w lies in the range -1.61 < w < -0.78 [l, 4,51. The most intriguing question here is whether w lies before or after -1. Various theoretical models such as quintessence scalar field model, Dirac-Born-Infeld action, and other were constructed to describe the case -1 < w < 0. There are much more questions with the models describing the case w < -1. One of a possible model with w < -1 is a phantom model. In this case we get a branch of problems as instability on the classical and 258

259

quantum level. One can say that modern cosmology is faced up by experimental data with two questions - what is the fundamental theory which leads to Dark Energy, and whether it is possible to provide satisfactory description for the case w < -1. Recently it has been shown [7] that a D-brane decay [6] described in terms of string field theory leads at large times to an effective phantom model, that can approximate this decay. Moreover, as a result of this decay a stable spectrum appears and the model on the quantum level does not suffer from instability of ordinary phantom models. The central role in this development is played by the construction of special solutions of classical equations of motion which come from the underlying string field theory [6]. One of the distinguishing properties of these equations is that while being covariant they are nonlocal in the sense that they contain infinite number of derivatives [13]. The aim of this paper is to present some results concerning the study of these types of equations of motion. The paper is organized as follows. In section 2 we present level-truncated action and equations of motion for the lowest excitation on the non-BPS D-brane. In section 3 we present the construction of the stress tensor and demonstrate the energy conservation law. In section 4 we study the model of interacting open and closed strings, we present a special approximation for which it is possible to construct physically interesting nontrivial solution. Finally, in section 5 we present the relation for the stress tensor for the case of arbitrary (finite) number of fields, we also provide explicit relations for energy and pressure for the case of spatially homogeneous configurations. 2. Level-truncated action and equations of motion

To describe the open string states living on a single non-BPS D-brane one has to consider GSOf states [lo]. GSO- states are Grassmann even, while GSO+ states are Grassmann odd. The unique (up to rescaling of the fields) gauge invariant cubic action unifying GSO+ and GSO- sectors is

1 +-((Y-~IA-,QBA-)) - ((Y-2JA+,A-, A-)). 2 For A- = 0 one gets the action [ll, 121. The low level action contains the tachyon field 4 and one auxiliary field u, then in the approximation of

260

slowly varying axillary field one gets the following action [16]

SL41 =

1 [%J2(4 C h

2

- ,K2 a,4(.)a"4(.)

- ;1 4 -4

(4 >

(1)

where K, is a parameter, &x) = exp(kO)#(s), and the differential operator elco could be understood as a series expansion O0

eko=c-, n=O

k"O" n!

k=-

1 8

and

O=-EJ2+A.

Here and below we denote the time derivative as d. Equation of motion corresponding to the action (1) takes the form (K2u

+ 1)e-ao4

=43.

(3)

There is a direct physical interpretation of spatially homogeneous time dependent configurations [6]. For spatially homogeneous configurations, +(x) = $(t),the equation of motion (3) takes the following form

+ l ) e P 2 4= 4 3 ,

(4)

4

where = $(t)= e - i a 2 $ ( t ) . We will consider equation (4) as an equation for the function &t). The function 4(t) is related with $(t) by means of la2 the well-defined transformation 4(t)= ex 4(t). This equation for the case K = 0 has been recently investigated in (241, in such a form it appears in padic string theory, where it has been rigorously proved that there exists a solution interpolating between two vacua. For small values of K. the existence of such solutions was numerically established in [15]. 3. Stress tensor, energy conservation, pressure

In order to obtain the stress tensor it is convenient to use the following formula from the general relativity theory 113, 14, 161

Let us write the expression for stress tensor for the systems with the action

261

where W = W ( 4 )- is a differentiable function of stress tensor takes the form

6,$(x) = e k n 4 ( z ) . The

note that here we used the Feynman formula [14]

In this note we are interested in the case of space homogeneous configurations - configurations depending only on time, 4(z) = 4(t),such configurations are already of great interest in physical applications [6]. The energy of the system is given by the zero component of the stress tensor E ( t ) = Too. Let us prove the energy conservation theorem for our type of action (6). A similar theorem incorporating infinite series is presented in [13].

Proposition 3.1. The energy

E = T(dq5)2 IE2 - 5q52 1

+ W ( $ )+ k

1

l dp ( e - k p a z ~ ) ~ ( e k p a z d $ ) ,

86

is conserved o n the solutions of equation of motion

H

where A d B

= AdB - BdA,

4 = eaa24.

Proof Let us prove the energy conservation directly, we have

Now using the identity 1201

262

the equation of motion, and the definition of the field $ we get

7

E.O.M.

The pressure is defined in terms of the energy-momentum tensor P(t)i = -Tj (no summation). For space homogeneous configuration the pressure takes the form

P ( t ) = -E(t)

+~

1

~ ( 3- 24k )/ ~d p (de-kpa2dW)(ekpa2d$). (8) 0

One can see that for bounded configurations interpolating between stationary points we have P ( t ) -+ -E as t -+fco. 4. Interacting open-closed SFT model

Let us now consider closely related model which incorporates two fields. This is so-called toy model of interacting open-closed string tachyons [18, 191 1 K2 -q52 L $ U $ 2G2 + g& - J2$ , (9) 2 2 3 where fields Cp and $ are interpreted as open and closed tachyon fields respectively. Studying the structure of the potential one can show that physically interesting solutions interpolating between vacua are possible only for the following values of the coupling constant g: g = 413 and g = 13/6 [19]. One defines $ and as $(z) = ek10q5(z),4(z) = ekzn$(x), where and kl, k2 are positive constants. In the case of spatially homogeneous configurations we have the following equations of motion

+

+

+

-1

Ti3

4

(-K:d2 (-Kid2

6

+ l)e2kla2$ - $2 + g?z,- 2 4 4 = o + 4)e2'za24 + g$ - $2 = 0,

(10)

4

where = $(t), = $(t). For the case g = 13/6 the solution of the system (10) was constructed in [18] using the following iterative procedure (-":a2

(-Kid2

+ l ) e 2 k q n - 4; + g&+1 - 2&jn + 4)e'k~a'4~+ g&+l- 4; = o

=0

(11)

263

It is interesting that for the case g = 4/3 the iterative procedure (11)does not converge [19]. Existence of solution in this case is a rather difficult open question. The study [15] suggests to try to analyze the case K: = K: = 0 - in this case the corresponding nonlocal operators have positive kernels. Although our numerical investigations suggest that even in this case the iterative procedure (11)does not converge. On the other hand following [15, 161 one may study the approximation of slowly varying field $, i.e. the case k2 = 0 - the interaction is linear in and thus approximation $ M might be reasonable. In the described approximation the system (10) for the case g = 413 reduces to the following equation

4

4

for which one may prove the following constructive theorem stating the existence of solution.

Theorem 4.1. For the equation eaaZ

a ( t )= a a 3 ( t )+ (1- a ) @ @ ) ,

(13)

there exists a solution satisfying the following boundary conditions lim @(t)= f l ,

(14)

t-tfm

Moreover, it is given by the limit of the following (implicit) iterative procedure sign t (15) a@:+, (1- a)a.,+I = eaa2a.,, ao(t)= -(12 One can find the proof of this theorem in [26]. Equation (12) reduces to canonical form (13) under the transformation

+

5 9

Figure 1. a) Iterations @o(t), @i(t),@ z ( t ) , @ 3 ( t ) , @4(t), @so@)(bottom to top) and b) Numerical limit @ l 5 o ( t ) for equation (13) obtained with iterative procedure (15).

264

+

Note that equation (13) is invariant under shifts @(t)-+ @(t A) and reflections @(t)-+ -@(t).One can also proof [26]that any bounded solution of equation (13) satisfies the bound I@(t)l 1.

<

5. Arbitrary number of interacting fields

Here we study the generalization of the model ( 9 ) , we consider the action

where W = W ( & ,. . . ,&) is a function differentiable in 4 1 , . . . )& and & ( x ) = ekio+i(x). Equations of motion corresponding to this action have the form

For this type of systems the stress tensor takes the form N

~ c , p ( x= ) -gorp

-

c{

KPac,+iaB+i

i= 1

For spatially homogeneous configurations the energy takes the form

In this case we can also formulate the energy conservation theorem

Proposition 5.1. The energy (19) is conserved o n the equations of motion (17) corresponding t o the action (16). The pressure corresponding to our action (16) takes the form

265

As in the case of a single field on configurations interpolating between vacua we have limt-+fmP ( t ) = -E. The explicit expression for pressure allows the study of pressure dynamics which is important from the point of view of applications in cosmology [9, 161. Acknowledgments I would like to thank I.Ya. Arefeva, M. Bozejko, J. Toft, V.S. Vladimirov, and Ya. Volovich for fruitful discussions. The author would like to thank the organizers of the 26-th Conference “Quantum Probability and Infinite Dimensional Analysis” for the opportunity to give a talk on this conference and for partial support of my participation in the conference. The author would also like to thank Profile of Mathematical Modeling of Vaxjo University (Sweden) for partial support of my participation in the conference. This work is supported in part by personal fellowship of the Swedish Institute under Visby Programme, by the “Dynasty” Foundation (awarded by the Scientific board of ICFPM), by the “Russian Science Support Foundation”, by RFBR grant 05-01-00758, and by INTAS grant 03-51-6346. References 1. D. N. Spergel et al., First Year Wilkinson Microwave Anisotropy Probe ( W M A P ) Observations: Determination of Cosmological Parameters, Astroph. J. Suppl. 148 175, 2003. 2. S.J. Perlmutter et al., Measurements of Omega and Lambda from 42 HighRedshijl Supernovae,Astroph. J . 517 565, 1999. 3. A. Riess et al., Observational Evidence f r o m Supernovae f o r an Accelerating Universe and a Cosmological Constant, Astron. J . 116 1009, 1998. 4. R.A.Knop et al., New constraints on wm, w x , and w f r o m an independent set of eleven high - redshijl supernovae observed with HST, astro-ph/0309368. 5. M. Tegmark al., The 3-d power spectrum of galaxies from the SDSS, Astroph. J . 606 (2004) 702-740. 6. A. Sen, Tachyon Dynamics in Open String Theory, hep-th/0410103. 7. I.Ya.Aref’eva, Nonlocal String Tachyon as a Model f o r Cosmological Dark Energy, astro-ph/0410443. 8. V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, P-adic Analysis and Mathematical Physics, World Sci. 1994. 9. A. Sen, Non-BPS States and Branes in String Theory, heplth/9904207, A. Sen, Rolling Tachyon, JHEP 2002, 0204, 048, A. Sen, Time Evolution in Open String Theory, JHEP 2002, 0210, 003. 10. A. Sen, B. Zwiebach, Tachyon condensation in string field theory, JHEP 003 (2000) 002, hep-th/9912249;

266

11.

12. 13.

14. 15. 16.

17. 18.

19. 20. 21. 22. 23. 24. 25.

26.

N.Berkovits, A.Sen, B.Zwiebach, Tachyon condensation in superstring field theory, hep-th/0002211; I.Ya. Aref’eva, P.B. Medvedev and A.P. Zubarev, New representation f o r string field solves the consistency problem for open superstring field, Nucl.Phys. B341 (1990) 464.; C.R. Preitschopf, C.B. Thorn and S.A. Yost, Superstring Field Theory, Nucl.Phys. B337 (1990) 363. N. Moeller and B. Zwiebach, Dynamics with Infinitely Many Derivatives and Rolling Tachyons, JHEP 2002, 0210, 034. H. Yang, Stress tensors in p-adic string theory and truncated OSFT, JHEP 0211 (2002) 007, hep-th/0209197. Yaroslav Volovich, Numerical Study of Nonlinear Equations with Infinite Number of Derivatives, J.Phys.A: Math. Gen. 2003, 36,8685-8701. I.Ya. Aref’eva, L.V. Joukovskaya and A.S. Koshelev, Time Evolution in Superstring Field Theory on non-BPS brane. Z. Rolling Tachyon and EnergyMomentum Conservation, JHEP (2003) 0309 012. Nicolas Moeller, Martin Schnabl, Tachyon condensation in open-closed p-adic string theory, JHEP 2004, 0401, 011. Kazuki Ohmori, Toward Open-Closed String Theoretical Description of Rolling Tachyon, Phys.Rev. D 2004, 69, 026008. L.Joukovskaya, Ya. Volovich, Energy Flow from Open to Closed Strings in a Toy Model of Rolling Tachyon, math-ph/0308034. L.Joukovskaya, Energy Conservation for p-Adic and S F T String Equations, Proceedings of Steklov Mathematical Institute, 2004, 245, 98. L. Brekke, P.G. Freund, M. Olson and E. Witten, Nonarchimedean String Dynamics, Nucl. Phys., 1988, B302, p. 365. P.H. Frampton and Y. Okada, Effective Scalar Field Theory of p-Adic String, Phys. Rev. D, 1988, v. 37, N 10, p.3077-3079. L. Brekke and P.G.O. Fkeund, p-Adic Numbers Physics, Phys. Rep. (Rev. Sct. Phys. Lett.), 1993, 233, N 1, p.1-66. V.S. Vladimirov, Ya.1. Volovich, O n the Nonlinear Dynamical Equation in the p-adic String Theory , Theor. and Math. Phys. 2004, math-ph/0306018 . V.S. Vladimirov, O n the nonlinear equation of p-adic open string f o r scalar field, Talk at the conference dedicated t o 100-years anniversary of S.M. Nikolsky, 2005. L. Joukovskaya, Iterative method for solution of integral equations describing rolling type solutions in string theory, Teor. Math. Phys. 2005 (in press).

A LOGARITHMIC SOBOLEV INEQUALITY FOR AN INTERACTING SPIN SYSTEM UNDER A GEOMETRIC REFERENCE MEASURE

ALDERIC JOULIN Laboratoire de Mathimatiques et Applications Universite' de La Rochelle Avenue Michel Cripeau 17042 La Rochelle Cedex E-mail: ajoulinOuniv-lr.f r NICOLAS PRIVAULT Laboratoire de Mathimatiques et Applications Universite' de La Rochelle Avenue Michel Cripeau 17042 La Rochelle Cedex E-mail: nprivaulOuniv-lr.f r Logarithmic Sobolev inequalities are an essential tool in the study of interacting particle systems, cf. e.g. (4, 51. In this note we show that the logarithmic Sobolev inequality proved on the configuration space MZd under Poisson reference measures in [I]can be extended to geometric reference measures using the results of [2]. As a corollary we obtain a deviation estimate for an interacting particle system.

1. Logarithmic Sobolev inequality for the geometric

distribution Consider the forward and backward gradient operators d+f(k)

= f(k

+ 1)- f ( k ) ,

d-f(k)

1

l{k>i)(f(k - 1)- f ( k ) ) ,

k

E N,

and the Laplacian

2 = -d:*d+

= d+

1 + -d-

P which generates a Markov process on N whose invariant measure is the geometric distribution 7r on N with parameter p E (0, l),i.e.

7 r ( { k } )= (1 - p ) p k , 267

k

E

N.

268

Denote by E, the expectation under defined as Ent,

T

and by Ent, the entropy under

T,

[fl = E7r [f1% fl - E,[fllog E7r [fl.

We recall the modified logarithmic Sobolev inequality proved in [2] for the geometric distribution 7r.

Theorem 1.1. Let 0 < c < - logp and let f : N + R such that Id+fl 5 c. W e have

In higher dimensions the multi-dimensional gradient is defined as d:f(k)

=f(k

+ ei) - f ( k ) ,

i = 1 , . . . ,n,

where f is a function on Nn, k = (kl,...,kn) E canonical basis of Rn, and the gradient norm is lld+f(k)l12 =

c

Id?f(k)I2 =

cIf(k +

N", ( e l , . . . , e n ) is the ei) - f(k)I2.

(1.2)

i=l

i=l

From the tensorization property of entropy, (1.1) still holds with respect to 7rBn in any finite dimension n:

provided Idi f 1 5 c, i = 1,.. . ,n. As a consequence the following deviation inequality for functions of several variables under rBnhas been proved in [2] using (1.1) and the Herbst method.

Corollary 1.2. Let 0 < c < -1ogp and let f such that ld:fl 5 p, i = 1 , . . . ,n, and lld+f1I2 5 a2 for some a,P > 0. Then f o r all r > 0,

where ap,c

=

PeC (1 - P I P - @I

denotes the logarithmic Sobolev constant in (1.1). Our goal in the next section will be to extend these results to interacting spin systems under a geometric reference measure.

269

2. Logarithmic Sobolev inequality for an interacting spin

system Given a bounded finite range interaction potential GJ = { ( a , i.e.

:

R c Zd},

let the Hamiltonian H A be defined as

c

R f l A#@

where 7~ denotes the restriction of 77 to N R , R c NZd. The Gibbs measure 7rX on N" associated to a N-valued spin system on a finite lattice A c Zd with boundary condition w E NZd\" is defined by its density with respect to 71" := "'r7 as:

where IT is the geometric reference distribution on N,2; is a normalization factor, and

H , w ( ~= I )H A ( v A w A ~ ) ,

rl E N'~,

where qw is defined as

+

(rlw)rc = r l k l ~ ( k ) ~ k l ~ ( k ) , k E Zd,

whenever 7 E NA, w E again

NB,and A , B c Zd are such that A n B = 0 . Let

270 7]k

> 0 , cf. [ l ]We . assume that there exists a constant C > 0 depending on

11@11

only, with 1

C < - c",k,v,+) For f : E 4 R we let: 8 x ( e f )= kEh

v E NA,

5 C,

-

/

A

c Z d , k E A.

(2.1)

c",k, a, +)ef(')Id~f(a)12d.lrX(a),

and

/

8 A ( e f )=

ef(u)Id~f(a)12d.lrh(a).

kEh

Next we consider the family of rectangles of the form

R = R ( k ,11, ...,Id) = k -k ([I,111 where k E Zd and

11,.

X

* * *

X

[I,I d ] ) I I Z d ,

. . ,l d E N, with size(R) = max l k . k = l , ...,d

Let 9~ denote the set of rectangles such that size(R) 2 L

and size(R) 5 10 min l k . k=l,...,d

Definition 2.1. We say that .lrx satisfies the mixing condition if there exists constants C1 and Cz, depending on d and \\@I1only, such that:

5

cl e - C z d ( A , B ) ,

for all L 2 1, A E 9~ and A , B

(2.2)

c A such that A , B

E92~ with

AnB

=

0.

We refer to [l]and [4]for conditions on Q, under which (2.2) holds under a geometric reference measure. Our goal is to prove the following logarithmic Sobolev inequality under the Gibbs measure .lrx.

Theorem 2.2. Assume that the mixing condition (2.2) holds, and let c < - logp. Then there exists a constant T~ > 0, independent of A and w,such that Ent,X [ e f ]5 r c 8 x ( e f ) ,

(2.3)

271

for every f : E 4R such that lldff

Ilp(h)

I C , 7rA-a.e.

In particular we have

which implies, as in Corollary 1.2, a deviation inequality under Gibbs measures. Corollary 2.3. Assume that the mixing condition (2.2) holds, and let c - logp. Let f be such that [Id+f l l p ( ~I)P and

<

for some a l p> 0. Then for all r > 0, 7rx

(f - ET; [f]2 r ) 5 exp

c2r2

rc

4yca2P2

P

- -a2yc)) .

(2.5)

Due to Hypothesis (2.1), condition (2.4) can be replaced by

IId+f

(v)II?Z(A)

I C-la2,

~ ~ ( d-qa.e. )

Denoting by rI denote the infinite volume Gibbs measure associated to for some ro > 0 we get the Ruelle type bound: ~ { r El N~~ : 17121 2~IAII > >~ X (-(cr P - C~,)IAJ

7rx,

+ c ~ n [ 1 r ] n ,1 1 )r > ro,

for all finite subset A of Zd, under the mixing condition (2.2). Indeed, it suffices to apply the uniform bound (2.5) with f ( r ] ) = 1 ~ ~ 1a' , = CJAl, /3 = 1, and the compatibility condition

This shows in particular that II satisfies the (RPB)' condition in [3].

272

3. Proof of Theorem 2.2

Recall that for c < - logp, by tensorization, Theorem 1.1 yields as in (1.3) the logarithmic Sobolev inequality ~nt,,[ef] I ScgA(ef),

(3.1)

R such that ~ ~ d + f ~ ~ I p (c,A.rrA-a.e., ) with an optimal which is independent of A c NZd.Let now S A , ~ , ,denote the optimal constant in the inequality

f : NZd constant s, I for all

--f

Ent,;[ef]

I SA,,,,g:(ef),

I l d + f I l ~ (I ~ )c.

Lemma 3.1. For every A C Zd, there exists a constant A := Ce41Allloll > 0 depending only on IAl, c and independent of w E NZd, such that SC

A -< SA,W,C 5 A s , . Proof. We follow the proof of Proposition 3.1 in [l].From (2.1) we obtain: -

C-1e-21AIII*II

g A ( e f )< g x ( e f > I Ce21~lIl'll g A ( e f ) .

(3.2)

From the relation Ent,[f]

= %IE,[flogf

- flogt

-f

+ t]

and the bound

we have e-21A111*11Ent,,

[ef] 5 Ent,;

[ef]

I e21AlllollEnt,,

from which the conclusion follows using (3.1) and (3.2).

[ef] ,

0

Let for L 2 1:

which is finite by Lemma 3.1. Prop 3.1. Assume the mixing condition (2.2) is satisfied. Then there exists a constant K. depending on (I@((, such that S2L,C

for L large enough.

5 (1

-

5)

-l SL,C

(3.3)

273

Pro08 The proof of this proposition is identical to that of Proposition 4.1, pp. 1970-1972 and Proposition 5.1, p. 1975 in [l],replacing the Dirichlet form used in [l]with 8;. 0 Finally, Theorem 2.2 is proved by taking from Proposition 3.1.

"yc

=

S U P ~ Swhich L , ~ ,is finite

References

1. P. Dai Pra, A.M. Paganoni, and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab., 30(4):1959-1976, 2002. 2. A. Joulin and N. Privault. Functional inequalities for discrete gradients and application t o the geometric distribution. ESAIM Probab. Stat., 8:87-101 (electronic) , 2004. 3. Y. Kondratiev, T. Kuna, and 0.Kutoviy. On relations between a priori bounds for measures on configuration spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7(2):195-213, 2004. 4. F. Martinelli. Lectures on Glauber dynamics for discrete spin models. In Lectures on probability theory and statistics (Saint-Flour, 1997), volume 1717 of Lecture Notes in Math., pages 93-191. Springer, Berlin, 1999. 5. B. Zegarlinski. Analysis of classical and quantum interacting particle systems. In L. Accardi and F. Fagnola, editors, Quantum interacting particle systems (Trento, 2000), volume 14 of QP-PQ: Quantum Probab. White Noise Anal., pages 241-336. World Sci. Publishing, River Edge, NJ, 2002.

TO QUANTUM MECHANICS THROUGH GAUSSIAN INTEGRATION AND THE TAYLOR EXPANSION OF FUNCTIONALS OF CLASSICAL FIELDS *

A.YU. KHRENNIKOV International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vaxjo, S-35195, Sweden E-mail:Andrei.KhrennikovOmsi.urn.se

We propose a solution of the problem of hidden variables. In our approach QM can be reconstructed as an asymptotic projection of statistical mechanics of classical fields. Determinism can be reestablished in QM, but the price is the infinite dimension of the phase space. Our classical+ quantum projection is based on Gaussian integration on the Hilbert space and the Taylor expansion (up to the second order term) of functionals of classical fields. Our solution of the problem of hidden variables is given in the framework which differs essentially from the conventional one (cf. Einstein-Polosky-Rosen, von Neumann, Kochen-Specker,..., Bell). The crucial point is that quantum mechanics is just an asymptotic image of prequantum classical statistical field theory (PCSFT).

1. Introduction

The problem of reduction QM to classical statistical mechanics (problem of hidden variables) has been discussed in a huge number of articles, books, and conferences.a At the moment there is more of less general opinion that QM could not be reduced to classical statistical mechanics. The main reason for such a position are various ”NO-GO” theorems (e.g., von Neumann’s theorem or Bell’s theorem). However, a possible physical impact of ‘This work is supported by Profile Mathematical Modelling of V k j o university and EU-network on Quantum Probability and Applications are not able to present here a review on this problem. We just mention the book of von Neumann [l] and some recent publications [2, 31, see especially papers of L. Accardi, G. Adenier, L. Ballentine, W. De Muynck, W. De B a r e , Th. Nieuwenhuizen.

274

275

the "NO-GO"-activity was extremely overestimated.b Every "NO-GO" theorem is based on a list of physical and mathematical assumptions. Such assumptions are typically not justified [5, 61. One of such assumptions is the coincidence of classical (prequantum) and quantum statistical averages and the coincidence of ranges of values of classical physical variables and corresponding quantum observables. These conditions are violated in our approach. We consider a small parameter Q -+ 0 - dispersion of fluctuations of a (classical) background field $(z) :

s

s

b2(4+ q2(z)1da:dP(Q,P) = Q,

Lz (R3) x Lz (R3)R3

+

0.

(1)

Here $(z) is a vector field with two components $(z) = ( q ( z ) , p ( z ) ) and p is a Gaussian measure on the infinite-dimensional phase space R = L2(R3) x L2(R3).Quantum mechanics is obtained as the lim,,o of classical statistical mechanics with the infinite-dimensional phase-space. 2. Quantum mechanics as a projection of a classical model with the infinite-dimensional state space

A classical statistical model is described in the following way: a) physical states w are represented by points of some set R (state space); b) physical variables are represented by functions f : 52 + R belonging to some functional space V(R); c) statistical states are represented by probability measures on R belonging to some class S(R); d) the average of a physical variable (which is represented by a function f E V(R)) with respect to a statistical state (which is represented by a probability measure p E S(R)) is given by < f >,- Jaf(w)dp(w). A classical statistical model is a pair M = (S(R), V(R)). This is a measure-theoretic statistical model. We recall that classical statistical mechanics on the phase space R2n = R" x Rn gives an example of a classical statistical model. The quantum model (in the Dirac-von Neumann formalism [l]in the complex Hilbert space H,) is described in the following way: a) physical observables are represented by operators A : H , + H , belonging to the class of continuous self-adjoint operators L, (IT,);b) statistical states are represented particular, there was shown [4] that all distinguishing features of the quantum probabilistic model (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model.

276

by density operators, see von Neumann [l](the class of such operators is denoted by D(H,)); d) the average of a physical observable (which is represented by the operator A E L,(H,)) with respect to a statistical state (which is represented by the density operator D E D(Hc))is given by von Neumann’s formula: < A >D= Tr DA. The quantum statistical model is the pair Nquant = (D(H,),Ls(Hc)).This is an algebraic statistical model. To simplify considerations, in this paper we shall consider the case of the real (separable) Hilbert space H . Thus in the definition of Nquant the complex Hilbert space H, should be changed to the real Hilbert space H . Everywhere below a Gaussian measure on H with the zero mean value and the covariation operator B is denoted by the symbol P B . Let us consider a classical statistical model in that the state space R = H (in physical applications H = L2(R3)is the space of classical fields on R3) and the space of statistical states consists of Gaussian measures with zero mean value and dispersion (2)

where a > 0 is a small real parameter. Denote such a class of Gaussian measures by the symbol S,%(R). We remark that scaling preserves the class of Gaussian measures. Let us make the scaling of the classical background field:

(we emphasize that this is a scaling not in the physical space R3, but in the space of fields on it). To find the covariation operator D of the image p~ of the Gaussian measure p ~ we, compute its Fourier transform: pD(<) = ei(eiY)dpD(y)= ei(e’%)dpB($) = e - k ( B e * t ) .Thus

1,

1,

B D=-

(4)

Q

We shall use this formula later. We remark that by definition:


>ps=

s,

f($)&B($)

=

s,

f(&WPD($).

Let us consider a functional space V(R) which consists of analytic functions of exponential growth preserving the state of vacuum:

f(0) = 0 and there exist

CO, CI 2 0 : If($)I

I CoeC1llG1l.

277

We remark that any function f E V(R) is integrable with respect to any Gaussian measure on R. Let us consider the family of the classical statistical models

M a = (SZ(R), V(R)). Let a variable f E V(R) and let a statistical state p~ E Sz(R). We find the asymptotic expansion of the (classical) average < f >Pe= JQf($)dp~($) with respect to the small parameter a. In this Gaussian integral we make the scaling (3):

where the covariation operator D is given by (4). We remark that JQ(f’(0),$)dp~($)= 0, because the mean value of p o is equal zero. Since p~ E Sg(R), we have Tr D = 1. The change of variables in (5) can be considered as scaling of the magnitude of statistical (Gaussian) fluctuations.c By (5) we have: < f > p = $ Ja(f”(0)y,y) d p o ( y ) o(a), a -+ 0, or

+

a

< f >p -- Tr D f”(0) + o ( a ) , 2

0.

4

(6)

We see that the classical average (computed in the model M a = (S,%(R),V(s2))by using the measure-theoretic approach) is coupled through (6) to the quantum average (computed in the model N q u a n t = (D(R), L,(O)) by the von Neumann trace-formula). The equality (6) can be used as the motivation for defining the following classical + quantum map T from the classical statistical model M a = (Sz,V) onto the quantum statistical model Nquant= (D,L,) :

B a

T : S,%(R)-+ D(R), D = T ( ~ B=)-

(7)

(the Gaussian measure p~ is represented by the density matrix D which is equal to the covariation operator B of this measure normalized by a ) ;

T : V(R) + Ls(R),

Aquant

1 2

= T ( f )= -f”(O).

(8)

=Negligibly small random fluctuations a ( p ) = fi (where Q is a small parameter) are considered in the new scale as standard normal fluctuations. If we use the language of probability theory and consider a Gaussian random variables c(X), then the transformation (3) is nothing else than the standard normalization of this random variable: (in our case EE = 0).

278

Our previous considerations can be presented as Theorem 1. T h e m a p T : SE(R) -+ D(R) i s one-to-one; the m a p T : V(R) -+ L,(R) is linear and the classical and quantum averages are coupled by the asymptotic equality (6). We emphasize that the correspondence between physical variables f E V(R) and physical observables A E L,(R) is not one-to-one.d Example. Let fi($) = (A$,$) and f2($) = sin(A$,$), where A E L,(R). Both these functions belong to the space of variables V(R). In the classical statistical model these variables have different averages: J,(A$, $)@($) # sin(A$1 $)dP($). But J,[(A1CI,1CI) sin(A$,$)]dp(s) = o ( ~ ) , Q+ 0. Therefore by using QM we cannot distinguish these classical physical variables. Moreover, nontrivial classical observables can disappear without any trace in the process of transition from the prequantum classical statistical model to QM. For example, let f($) = cos(A$,$) - 1. This is nontrivial function on 0. But, for any p E S z ( R ) , we have < f >p= o(a),a + 0. Thus in quantum theory f is identified with g E 0. Physical conclusions. Our approach is based on considering the dispersion a of fluctuations of classical fields as a small parameter. For any classical physical variable f($) , there is produced its amplification fa($) = Then the quantum average is defined as

s,

if($).

< A >quantum= Or-0 lim < fa >classical, where A = frf”(0). So QM is a statistical approximation of an amplification of the classical field model (for very small fluctuations of vacuum).

3. Pure quantum states as Gaussian statistical mixtures In QM a pure quantum state is given by a normalized vector $ E H : 11$11 = 1. The corresponding statistical state is represented by the density operator: D+ = $ 8 $. In particular, the von Neumann’s trace-formula for expectation has the form: Tr D+A = (A$,4). Let us consider the correspondence map T for statistical states for the classical statistical model M a = (S8,V). A pure quantum state $ (i.e., the state with the density operator D+)is the image of the Gaussian statistical mixture p+ of states dA large class of physical variables is mapped into one physical observable. We can say that the quantum observational model Nquantdoes not distinguish physical variables of the classical statistical model M a . The space V(C2) is split into equivalence classes of physical variables: f N g c) f”(0) = g”(0).

279

$ E H . Here the measure p$ has the covariation operator B$ = ~ D Q . This implies that the Fourier transform of the measure p$ has the form: & ( y ) = e-T(Yi*)2, y E H . This means that the measure p+ is concentrated on the one-dimensional subspace H$ = {z E H : z = s$, s E R}. Conclusion. Quantumpure states $ E H , 11$11 = 1, represent Gaussian statistical mixtures of classical states 4 E H . Therefore, quantum randomness is ordinary Gaussian randomness (so it is reducible to the classical ensemble randomness, cf. with von Neumann’s idea 111 about so called “irreducible quantum randomness”).

4. Pure states as one-dimension projections of spatial

white-noise In section 3 we showed that so called pure states of QM have the natural classical statistical interpretation as Gaussian measures concentrated on one-dimensional subspaces of the Hilbert space H . On the other hand, it is well known that any Gaussian measure o n H is determined by its onedimensional projections. To determine a Gaussian random variable [ ( w ) € H , it is sufficient to determine all its one-dimensional projections: & ( w ) = ( 1 1 , , [ ( ~ ) ) , $ E H . The covariation operator B of E (having the zero mean value) is defined by ( B $ ,$) = E[$. We are interested in the following problem: Is it possible to construct a Gaussian distribution o n H such that its one-dimensional projections will give us all pure quantum states, $ E H , II$lI = 1” We recall that in our approach a pure quantum state $ is just the label for a Gaussian random variable &, such that E[$ = c~11$11~. Thus the answer to our question is positive and pure quantum states can be considered as one-dimensional projections of the &-scaling of the standard Gaussian distribution on H . The standard Gaussian distribution p on H (so the average of p is equal to zero and cov p = I , where I is the unit operator) is nothing else than the white noise on R3 (if one chooses H = L2(R3)). Thus pure quantum states are simply one-dimensional projections of the spatial white noise. It is well known that the p is not a-additive o n the (Tfield of Bore1 subsets of H . To escape mathematical difficulties, we consider the finite-dimensional case. We consider the family of Gaussian random variables [$, 11, E R”, E& = 0, E[$ = all$111~. This family can be realized as & ( w ) = ($, [ ( w ) ) where [ ( w ) = &q(w) and ~ ( w E) R” is standard Gaussian random variable (so Eq = 0,cov q = I ) . For any $ E R”, we define the projection P$ to this

280

vector: P$(k)= ($, k)$. Let f : R” + R be a real analytic function of the exponential growth. Then we have:

W(f’l(O)$,$) +

Thus Ef(P+<(w)) = o(a),a-+ 0. If ll$ll = 1 (a pure quantum state), then we get: Ef(P+<(w)) = $(f”(O)$, $) o ( a ) ,a --f 0. Here A = f”(0) is a symmetric linear operator. We ”quantize” the classical variable f(z),z E R”, by mapping it to the operator A = +f”(O), see Theorem 1. The Gaussian random variable [+, II$JII = 1 is quantized by mapping it into the pure quantum state $. Theorem 2. There exists a Kolmogorov probability space such that all pure quantum states can be represented by Gaussian random variables o n this space. The correspondence $ -+ [+(w) is linear: X1$1 X2$2 + h<$,,(w) + k&~,,,(w), where h , h E R. We pay attention that physical variables <+(w),$ E R”, ( o n e dimensional projections of the scaling [(w)of the standard Gaussian random variable r](w) € R”) cannot be mapped onto nontrivial quantum obseruables. Prequantum classical physical variables <+ (w)= ($,w)are linear functionals of w.Therefore T ( & ) = <$(O) = 0. Nevertheless, quantum mechanics contains images of <+ given by quantum states $, but only for $ with 11$11 = l ! We call [(w)a background random field. All pure states could be extracted from the the background random field by projecting it to one dimensional subspaces. Our approach explains the origin of the scalar product on the set of pure quantum states. We consider the l/a-amplification of the covariation of two Gaussian (prequantum) random variables <$,, (w)and

+

+

E$, (w): $%bl

(w><+z (w)= ($1

7

$2).

Conclusion. The Halbert space structure of QM is induced a by the (prequantum) Gaussian random field (the background field <(w))through the a + 0 asymptotic. References 1. J. von Neumann, Mathematical foundations of quantum mechanics (Princeton Univ. Press, Princeton, N. J., 1955). 2. A. Yu. Khrennikov, ed., Foundations of Probability and Physics, Q. Prob. White Noise Anal. 13 (WSP, Singapore, 2001). 3. A. Yu. Khrennikov, ed., Quantum Theory: Reconsideration of Foundations2, Ser. Math. Modeling 10 ( V k j o Univ. Press, Viixjo, 2004). 4. A. Yu. Khrennikov, Phys. Lett. A 316, 279-296 (2003).

281

5. A. Yu. Khrennikov, Interpretations of Probability, (VSP Int. Sc. Publishers, Utrecht/Tokyo, 1999; second edition, 2004). 6. L. Accardi, Some loopholes to save quantum nonlocality. Foundations of Probability and Physics-3, ed. A. Yu. Khrennikov (Melville, New York: AIP Conference Proceedings, 1-20, 2005).

HYPERBOLIC QUANTIZATION *

ANDRE1 KHRENNIKOV AND GAVRIEL SEGRE International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vaxjo, S-35195, Sweden E-mail: Gavriel.SegreOvxu.se

We study quantization in the hyperbolic Hilbert space: harmonic oscillator and electromagnetic field.

1. Introduction

Let us define the hyperbolic algebra as the ring G of numbers of the form x j y , where x , y E R while j, called the hyperbolic imaginary unit, is such that j 2 = +l. The elements of such an algebra has been called in the mathematical literature with different names (cfr. and references therein): hyperbolic numbers, double numbers, split complex numbers, perplex numbers, and duplex numbers. We will call them hyperbolic numbers and we will refer to j as to the hyperbolic imaginary unit. The complex field C and the hyperbolic ring G are the two bidimensional Clifford algebras 3: C = C l o , ~ ,G = C l l , ~Given . z = x j y E G let us define its conjugate as f := x - j y . Hyperbolic numbers emerged in the research of one of the authors as the underlying number system of a mathematical theory, ”Hyperbolic Quantum Mechanics.” The aim of this note is to study in more details how stable is the structure of quantum mechanics with respect to the transition from one bidimensional Clifford algebra to another: from C to 6.

+

+

415,6

2. Hyperbolic Hilbert space and hyperbolic Fock space

A hyperbolic linear space we define as a module over the algebra G. A hyperbolic inner product space is a pair (V,(., .)) such that: V is a hyperbolic *This work is supported by Profile Mathematical Modelling of V k j o university and EU-network on Quantum Probability and Applications

282

283

linear space and (., .) : V x V +-+6 is such that: a). (u,v w) = (u,v) (u,w)Vu,v,w E V ; b). (.,Xu) = X(u,?J) vu,w E v,vx E 6 ;c). (u,v) = (v,u) vu,v E v; Example 1. Let 6"denote the set of all n-ples of hyperbolic numbers; for x = (XI,...,xn),y = ( y l , . . . , y n ) E 6" define: ( x , y ) := Cy=13.iyi. (G", (., .)) is an hyperbolic inner product space Example 2. Define Li(R) to be the set of hyperbolic valued measurable functions on R that satisfy d x f ( x )f (x) < 00. Let us introduce:

+

s_',"

+

s_',"

(f,9 ) := d x f ( x ) g ( x ) .The (Li(R), (., .)) is an hyperbolic inner product space A hyperbolic Hilbert space is a triple (V,(., .), 11 . II), where (V,(., .)) is a hyperbolic inner product space, (V,11 . 11) is a Banach module over 6 and the structures of the inner product space and the Banach module are coupled through the Cauchy-Bunyakovsky inequality: 1(u, v)I I llull IIvII. This is a special case of Hilbert spaces over supercommutative Banach superalgebras which where introduced in the book7 in the framework of so called functional superanalysis of Vladimirov-Volovichs~g. 3. Hyperbolic Quantum Mechanics

Axiom 1. The pure states of a n hyperbolic quantum systems are rays o n a hyperbolic Hilbert space H. Axiom 2. Hyperbolic quantum mechanical observables are self-adjoint operators o n H having the real spectrum. The expected value of the hyperbolic observable 0 in a state $ € H such that ($, $) # 0 is given by:

Axiom 2. The evolution of the pure state is described by the hyperbolic analogue of Schrodinger's equation:

where H is a n observable representing the Hamiltonian. 4. Harmonic oscillator in Hyperbolic Quantum Mechanics

Let us define the quantum harmonic oscillator as the hyperbolic-quantum dynamical system with hamiltonian:

H := W N

(3)

284

where:

N

:= a t a

(4)

[U,Ut] = 1 Equations (4)and (5) imply that:

",a] = -a " , a t ] = at Equations (6) and (7) imply that:

Naln > = ( n - 1)a)n>

where:

< mln > := 6m,n

(11)

so that we can infer that a J n>E E I G E N S P A C E ( N , n - 1) and atln >E E I G E N S P A C E ( N ,n 1). It follows that it there exist an hyperbolic number c(n) = c,(n) jcy(n)E G such that:

+

+

aln > = c(n)ln- 1 >

(12)

and hence:

< nlat

= < n - ~le(n)

n = = < nlataln >= c(n)c(n)< 12 - lln - 1 > = c(n)c(n)= (CZ(.N2

- (C?4(.N2

(14)

In the same way it follows that it there exist an hyperbolic number d ( n ) = d,(n) j d y ( n ) E G such that:

+

+

atin > = d(n)ln 1 > and hence:

< nla

=

< n + 11J(n)

(15)

285

n + 1 = < n + l I N J n +1 > = < n + lJataln+1 > = c ( n + l)E(n + 1) < 72 + lln 1 > = c(n + l)E(n + 1) =

+

(cz(n

+ 1)12 - (cy(72. +

(17)

Let us observe also that:

n = < nlNln > = < nlatuln >= < nlaat - 11n >

+ 1IZ(n)d(n)ln+ 1 > - < nln > = Z(n)d(n) < n + lln + 1 > - < > = Z ( n ) d ( n )- 1 =
71172

and hence:

c ( n ) c ( n ) = ( c z ( n ) ) 2- (

~ ~ ( 1 2 = ) ) Z(n)d(n) ~ -1 =

( d , ( n ) ) 2- (dY(n)l2- 1 (18)

The key point is, now, to observe that:

(H, (., .)) hyperbolic inner product space i+ (x,x) L 0 Vx E H

(19)

and hence in particular:

( ~ l >n= nln >)

i+

( n = < nlNln > = < nlataln > 2 0)

(20)

Thus one cannot guarantee that the spectrum consists of natural numbers. In principle, it may contain a number of series of the form n k,k = 0, f l ,f 2 , ..., where n is in general a real number - the generator of the series. This problem needs more studies.

+

5. Hyperbolic-quantization of the electromagnetic field

Let us consider the Lagrangian density of electromagnetism: L := := a,A, - &A,. Introduced the mo- ~ F , w F ~-wJ,AP, where: F,, mentum conjugated to A,: IF := = -FPO one has the primary a& first-class constraint lo :

and the secondary first-class constraint:

din' = Jo (22) Canonical quantization in Hyperbolic Quantum Mechanics (that we will call canonical hyperbolic-quantization from here and beyond) is given by the ansatz: {a,

')Poisson

+

j [ * *I ,

(23)

286

In particular the equation: {Ap(z,t ) ,~

~ t)}Poisson ( 2 ,

=

dLJ(2 - 2)

(24)

gives, under canonical hyperbolic-quantization, the relation:

[Ap(2, t ) ,~ ~ t )(] = 3 j6;6(2 , -2)

(25)

The existence of the constraints (21) and (22) complicates significantly the situation. In particular (25) is not consistent with the imposition of the Coulomb gauge condition . A’ = 0, since it implies that:

e [e X(2,t ) ) Ti@’, , t ) ] = j 6 W 6 ( 2 - 2’) # 0 *

(26)

Instead of embarking ourselves in the complex machinery of quantization in presence of constraints, let us follow the trick l 1 of replacing 6 i j with a suitable tensor Aij such that:

[e. A(?,t ) ,~ ~ (t2) ] ’=, j A i j @ 6 ( 2- 2’) = 0

(27)

A trivial computation implies that:

One arrives, consequentially, to the commutation relations:

[Ai(Z,t),Aj(Z’,t)] = [ ~ i ( 2 , t ) , d ( z ’ , t )= ] 0

(30)

As in the complex case, it is more convenient to work in the momentum represent ation: [uT(z),a!(V)]

=

6r,S6z,zl

[ a , ( i ) , a , ( V > ] = [a:(z),a!(R)] = 0

(31) (32)

One has furthermore that: (33)

P

= -pV(lc,r)

(34)

287 where:

N ( & r ) := a;(i)a,(i)

(35)

The hyperbolic-quantum radiation field may be seen as an infinite collection of uncoupled hyperbolic-quantum harmonic oscillators. B u t at the moment we are not able t o provide a n analogue of the conventional corpuscular interpretation of the quantized electromagnetic field through introducing a hyperbolic analogue of photon (see our study of the hyperbolic harmonic oscillator).

References 1. B. Jancewicz, The extended Grassmann algebra of R3, in: Clifford (geometric) algebras with applications in physics, mathematics and engineering, W.E. Baylis, editor, p.389-421 (Birkhauser, Boston, 1996). 2. G. Sobczyk, Introduction to geometric Algebras, in: Clifford (geometric) algebras with applications in physics, mathematics and engineering, W.E. Baylis, editor, p. 37-43 (Birkhauser, Boston, 1996). 3. A. Yu. Khrennikov and G. Segre, An introduction to hyperbolic analysis, math-ph/0507053 (2005). 4. A. Yu. Khrennikov, Contextual approach to quantum mechanics and the theory of the fundamental prespace, J. Math. Phys., 45, 902-921 (2004). 5. A. Yu. Khrennikov, Interference of probabilities and number field structure of quantum models, Annalen der Physik, 12,575-585 (2003). 6. A. Yu. Khrennikov, Hyperbolic quantum mechanics, Advances in Applied Clifford Algebras, 13,1-9 (2003). 7. A. Yu. Khrennikov, Supernalysis (Nauka, Fizmatlit, Moscow, 1997, second edition - 2005, in Russian; English translation: Kluwer, Dordreht, 1999). 8. V. S. Vladimirov and I. V. Volovich, Superanalysis, 1. Differential Calculus, Theor. and Mathem. Physics, 59, 3-27 (1984). 9. V. S. Vladimirov and I. V. Volovich , Superanalysis, 2. Integral Calculus, Theor. and Mathem. Physics, 60, 16%198 (1984). 10. M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992). 11. L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1996).

DISCRETE E N E R G Y SPECTRUM I N DISCRETE TIME DYNAMICS*

A. KHRENNIKOV International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vixjo, S-35195, Sweden E-mail: Andrei. KhrennikovQmsi.mu.se

Ya. VOLOVICH International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vixjo, S-35195, Sweden Yaroslav. VolovichQmsi.m u s e

We present the classical-type dynamical model which poses some distinguishing properties of quantum mechanics. The model is based on a conjecture of existence of fundamental indivisible time quantum. The resulting formalism - discrete time dynamics - is constructed and used to study the energy spectrum for motion in central potential. We show that the spectrum is discrete and present a method of reconstruction of the corresponding potential for which the discrete time dynamics leads to the experimentally observed spectra.

1. Introduction

In this paper we try to obtain a classical-type dynamical model which resembles some distinguishing properties of quantum mechanics. We show how starting from modifying classical physics with the following natural postulate - time is discrete - one obtains discrete energy spectrum. In our previous investigations we have shown how the resulting formalism leads to quantum-like interference of particles (see [lo, 111 for more details and motivations, for discussions of distinguished properties of quantum mechanics, its probabilistic structure, and motivations for construction of classical-type models see [l,2, 3, 4,5 , 61). 'This work is supported by Profile for Mathematical Modeling of VriXjo University and EU-network on QP and Applications.

288

289

Let us start from considering the well known dynamical equation for an observable A = A(p,q )

DtA = { A , H } where H = H(p,q ) is a Hamiltonian of the system and in the right hand side is a Poisson bracket. The left hand side of (1) is the same in both classical and quantum dynamics, it contains a continuous time derivative DtA = The construction of descrete dynamics is done with the help of discrete derivative which is postulated to be D,(')A = $ [ A ( t T ) - A ( t ) ] ,where T is the discreteness parameter. This parameter is finite and is treated in the same way as Plank constant in quantum mechanical formalism. The discrete time dynamics could be solved in the sense of the following relation

g.

+

A(t

+

T)

= A(t)

+ T { A ,H }

(2)

which provides the evolution of any dynamical function A = A(p,q ) . Note that in our model the coordinate space is continuous. As we will see discreteness of time enriches classical mechanics with some new properties which are usually thought as having quantum nature. In particular as it will be shown below in discrete time mechanics stationary orbits (i.e. finite motion) have discrete energy spectrum. We point that the phase space is assumed here to be a continuous real manifold.

2. Motion in central field with constant radius

Let us study the motion in central field U = U ( r ) with constant radius. Following the general approach described in the previous section we start from the classical Hamiltonian and then write the dynamical equations. In polar coordinates (r,'p) the Hamiltonian of the system in central potential U ( r ) is given by P,2

H =2m

P; ++ U(T), 2mr2

(3)

where p , and p , denote momenta corresponding to r and 'p - radial and angular coordinates respectively. Using (2) let us write the dynamical equa-

290

tions. We obtain T(t

+

T)

=T(t)

+ T Pr m

(4)

+ 7 P, 2

(6)

d t + 7) = cp(4

P,(t + 7) = P,(t) (7) We are interested in the periodic trajectories with a period which is multiple of our time quant

Tn

N

(8)

nT

As it was shown in [ll]equations (4)-(8) lead to discrete energy spectrum. Our task here is to answer the following question: which potential U ( r ) leads to a given energy spectra of the system. The concept is that the energy spectrum is observable and the internal potentials are to be determined from the experimental data. Thus our main aim is to find potentials such that classical discrete time dynamics would produce experimental energy spectrum, in particular, those predicted by quantum mechanics. Let us study the case of circular orbits, i.e. the case when pr = 0. jFrom (5) we find that

Now, the condition for the period (8) gives us being combined with (6) and (9) gives

where the constant rameter T )

(p(nT)

= cp(0)

+ 27r which

c is given by (note that < depends on discreteness pa-

The relation (10) contains the radius of the n-th orbit, we underline the dependence on n writing r = r(n). Finding p , from (9) and substituting it to (3) we find the expression for energy spectrum

E(n) =

1 r(n)2 -c+ U(r(n)), 2 n2

291

where the subscript n = 1 , 2 , . . . denotes that the quantities correspond to the n-th orbit. jF’rom (10) and (12) we want to find U as a function of T in terms of a given energy spectrum E(n). We proceed as follows (see also [ l l ] ) , assuming the continuous parameter n we take derivative of both parts of (12) in n,

where prime denotes the derivative in n. The use of (10) allows us to get a differential equation for r(n)in terms of only known quantities. We have 1 n2 2r(n)r’(n)- -T(n)2= --E’(n) I n

(14)

Introducing a new variable p = r2 we obtain a linear differential equation which could be rewritten as

which can be integrated to obtain r(n)= { i n (nE(n)-

ln

E(k)dk)

Equation (15) expresses n-th radius in terms of n, i.e. it has the form T = f ( n ) now if we invert it we relate n in terms of T, n = f-’(r), which if substituted to (12) gives an equation for U in terms of T only (actually in terms of r(n),but we perform interpolation effectively ignoring the fact that the relation strictly holds only for orbit radii). 3. Energy spectrum for various models 3.1. Physical energy spectra and corresponding potentials

The procedure described in the previous section allows us to find potentials which lead to known spectra. Here we show the reconstructed potentials for some interesting physical spectra, the computational details could be found in[ll]. a). Spectrum of unperturbed hydrogen atom

En=--, I n2

n=1,2,...,

292

corresponds to the following potential U(T) = -

1

r2t

+ 2y

Note that this is not Coulomb potential, although it is interesting that unlike Coulomb one it is nonsingular for all T . b). Spectrum of harmonic oscillator (in 2 0 with frequency w and mass m)

EA = h ( n + l), n

..

=O,l,.

corresponds to the potential U(T) = -3T 2 / 3 2

(+tiwlrJm

2/3

Here one may note the Ti constant in the potential, it appears due to its presence in the original energy levels. 3.2. Energy spectrum for given potentials

For a given potential it is straightforward to compute corresponding energy levels. Indeed, from (10) we find T = rn and upon substitution to (12) we get En.Let us consider several common potentials which result in rather simple expressions for energy spectrum. a). Polynomial potential n (y>l U(T) = m u , Tn = -

, E

n - 21

- -a(2+u)

r23* -

The case u = -1 corresponds to Coulomb potential, for which we have

b). Logarithmic potential U ( T )= a l n r , rn = n

8 ,

En = a

[i

+ln ( n n ]

This potential is interesting since it gives the linear dependence of n-th radius on n.

293 4. Discussion and conclusions

There is an open problem of the quantitative value of the discreteness parameter T . One might speculate its relation with Plank time constant [7] the smallest measurable time interval in ordinary QM and gravity - which is given by x 5.3910-44 (sec.)

tpl =

Although currently there is no direct relation with this quantity. It is interesting to note that the fact that presented model requires potentials which are different from classical ones is not surprising. Here there is a similarity with the Bohmian mechanics[l] in which potentials are also very different from original classical ones. In the presented approach we start from the measurable quantities - energy spectrum - and construct the internals of the model - the potentials. A more interesting question would be to study the “classical” limit in order to see whether the effective classical potentials are obtained in this limit. As we already noted in[ll] there might be a deep interrelation between the energy-time uncertainty relations [8] and Bohr-Somerfeld quantization rules [9] in quantum mechanics and our discrete time model. Indeed, if one writes the Bohr-Somerfeld quantization rules for energy and time as canonical variables, then for a system with conserved energy one might get

EnTn nti N

This relation is similar to period quantization condition (8) which we use in our model. In fact one may argue that if we make the r in equations of motion depend on the energy of the system as 7-

E = To-,

E where 7-0 is the “fundamental” time quantum and E a “fuqdamental” energy quantum, we get precisely the semiclassical quantization rules. The question arise how to treat the energy E here and what will happen with the dynamics. Acknowledgments The authors would like to thank B. Hiley, A. Plotnitsky, G. ‘t Hook, H. Gustafson, and K. Gustafson for discussions on quantum-like models with discrete time.

294

References 1. D. Bohm, B.J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory, London: Routledge & Kegan Paul. 2. L. E. Ballentine, Rev. Mod. Phys., 42, 358-381 (1970). 3. R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGrawHill, New-York, 1965. 4. G. ’t Hooft, Determinism beneath Quantum Mechanics, Proceedings of “QUO Vadis Quantum Mechanics”, Philadelphia, 2002, quant-ph/0212095 -, Quantum Mechanics and Determinism, hep-th/0105105 5. L. Accardi, “The probabilistic roots of the quantum mechanical paradoxes” in The wave-particle dualism. A tribute to Louis de Broglie on his 90th Birthday, edited by S. Diner, D. Fargue, G. Lochak and F. Selleri, D. Reidel Publ. Company, Dordrecht, 1984, pp. 297-330. 6. A. Yu. Khrennikov, J. Phys.A: Math. Gen., 34, 9965-9981 (2001) -, J . Math. Phys., 44, 2471- 2478 (2003) -, Phys. Lett. A , 316, 279-296 (2003) -, Annalen der Physik, 12,575-585 (2003) 7. C. Callender, N. Huggett (edditors), Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Grauity, Cambridge Univ. Press, 2001. C. Rovelli, Quantum Gravity, Cambridge Univ. Press, 2004. 8. L. Mandelstam, I.E. Tamm, J. Phys. (Moscow) 9, 249, (1945). D. Ruelle, Nuovo Cimento A 61, 655, (1969). P. Pfeifer, J. Frohlich, Rev. Mod. Phys., 67 759, (1995). 9. V.P. Maslov, M.V Fedoriuk, Semi-classical Approximations in Quantum Mechanics, Boston: Reidel, 1981 10. A.Yu. Khrennikov, Ya.1. Volovich, Discrete Time Leads to Quantum-Like Interference of Deterministic Particles, in: Quantum Theory: Reconsiderations of Foundations, ed. A. Khrennikov, Viixjo University Press, pp. 455 -, Interference as a statistical consequence of conjecture on time quant, quant-ph/0309012 -, Discrete Time Dynamical Models and Their Quantum Context Dependant Properties, J.Mod.Opt. Vo1.51, No.6-7, pp. 1113, (2004). 11. A, Khrennikov, Ya. Volovich, Energy Levels of “Hydrogen Atom” in Discrete Time Dynamics, in: Quantum Theory: Reconsiderations of Foundations-3, ed. A. Khrennikov, AIP Conference Proceedings, 2005 (in press).

CONVOLUTION ASSOCIATED WITH THE FREE cash-LAW

ANNA DOROTA KRYSTEK * Mathematical Institute University of Wroctaw pl.Grunwaldzki 2/4 SO-384 Wroctaw, Poland Anna. [email protected] LUKASZ JAN WOJAKOWSKI* Mathematical Institute University of Wroctaw pl.Grunwaldzki 2/4 SO-384 Wroctaw, Poland [email protected] In this paper we consider the deformation of conditionally free convolution, connected with the free cosh-law. Because that measure is freely infinitely divisible, we can define a new, associative convolution using the theory from [5]. We calculate the central and Poisson measure for that convolution and show that the coefficients of the continued fraction form of the Cauchy transform for the central and Poisson limit measures of that convolution are equal to the respective coefficients of the underlying measure starting from the third level.

1. Definitions In this paper we continue the investigations on the convolutions arising from the conditionally free convolution of Bozejko, Leinert and Speicher [2] through deformations of the second measure. Those investigations started with the tdeformation studied in the papers of Bozejko and Wysoczaiiski [3, 41 and Wojakowski [lo]. This was generalized by Krystek and Yoshida [6]. Further examples were provided by Oravecz [7, 81 and Krystek and Wojakowski [ 5 ] . In the latter paper, the authors introduced a family of deformations depending on a selected compactly supported freely infinitely divisible measure, and proved limit theorems in terms of properties of the R-transforms of the limiting measures. In this work we calculate the central and Poisson measures for the convolutiondriven "Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-(JT-2002-00279.

295

296

by the free cosh-law, that is by the measure with density

The measure C is infinitely divisible with respect to the free convolution and is a particular case of the central measures for conditionally free convolution [2]. Those measures were also considered by [I] and called free Meixner laws. It can be calculated that Voiculescu’s RB-transform and Cauchy transform of the measure C are equal to

Gc ( z ) =

dC(x) 3 = Z-X

z - d m 2(22

+ 1)

’

where w is the Wigner law. Let us start with the definition of the infinitely divisible deformation and of the respective convolution connected with the free cosh-law, see [5] for details, definitions and theorems in more general cases:

Definition 1.1. Let C be the free cosh-law and p any measure with compact support. Let us consider the following map p H ’Dip:

which depends on the second free cumulant of the measure p and on the nonnegative parameter t. For s 2 0 we understand by Cs the s-fold free convolution power of C having the following RB-transform:

RE ( z ) = s . R F ( z ) . Definition 1.2. For compactly supported probability measures p, v we define their L o n v o l u t i o n by p m v = (p,Dofp)rn(v,’Dofv).

By a result of [ 5 ] we have that the convolution

is associative.

2. Central limit theorem Theorem 2.1. Let p be a compactly supported probability measure on the real line with mean Zero and variance equal to I . Let C be the free cosh-law. Then the

297

sequence

q/fim. . O B l / f i P *

= (D1/fip7 v ! D l / f i p ) "' '

is * - weakly convergent as N

+ 00

'

(D1/mp7 v!D1/fip)

to the measure &, such that & = V!& and

qt,Ct)(4

= z.

The measure & is absolutely continuous with density f t t ( x )

t J(4

1

+ 4 t - x2)

for x E [ - 2 & T i 7 2 d T i 7 .

f d X=)G x 4 + ( t 2 - t - 2 ) x 2 + t + 1

Proof. Using the general central limit theorem from [5] we obtain that the limiting measure Et satisfies the relation

R g , c t ) ( 4= z. Because of

2 t Get ( z )

- = z1 Get

1+

(z)

Jm7

we have the following expression for the Cauchy transform of the measure Ct

GEt

=

(2

+ t )z - J t 2

+

(-4 - 4 t 2 2 ) 2 (t2 9 ) Using the relation between conditional transform and Cauchy transforms we obtain

Get ( z ) =

+

+2) + Jt2

2 (t2 ( 2 t 2 - t - 2 ) z + 223

( 9- 4 t - 4 ) ( 2 t 2 - t - 2 ) + 2 z3 - J.t 2 (.2 2 - 4 t - 4 ) - , 2 (z4+ ( t 2 - t - 2 ) 2 2 + 1 + t )

The appropriate choice of the branch of a square root gives in limit for real z = x

Jt2

(z2 - 4 t - 4 ) =

tJx2-4t-4 -tJx2 - 4 t - 4 itJ4t+4-x2

forx>2&Ti7 forx 5 - 2 & T i , for - 2 2 < x < 2 2 .

298

We would like to find the atoms of the measure &, that is the zeros of the denominator of the Cauchy transform of the measure & . There is no atoms if either

(t2- t - 2 ) - 4 (1 + t ) < 0, 2

that is for

t

E

(0,3)

or

(t2 - t - 2 ) 2 - 4 (1+ t ) 2 0,

and

(t2- t - 2 ) > 0 and (1+ t) > 0,

that is when

t 2 3. Thus we have atoms only for t = 0. By Proposition in chapter X111.6 in [9] one can find that there is no singular parts of the measure &. It follows from regularity properties of Gtt (z). The density fct (x),x E R of the absolute continuous part of the measure & can now be calculated by the Stieltjes formula 1

fcL(x) = -- lim SGc, ( x + i ~ ) . 7r

€+o+

Thus we obtain

Moreover we can calculate that the measure & has the following continued fraction expansion 1

GEt ( z ) =

1

z-

L

z-

l+t l+t

z2--

z

-

..

A diagram of this measure for t = 1is presented on the following figure.

3. Poisson limit theorem Theorem 3.1. For X > 0 define for all N

0

299 Figure 1 . Density of the central limit measure for the cosh-convolution

where

The absolutely continuous part of the measure px is given by

and where = 2 ~ - - 5 3 (4+4x+2tx)

+ z2 (2 + 4X + 2 t X + 2X2 + 2 t X 2 + 2 t 2 A')

- 4 t 2 X 3z

+ 2t2X4.

The measure px may have at most four atoms.

Proof. Using the fact that the limiting measure px is determined by the relation [51

and because of 22

Gcxt ( z ) =

+ t z x - t Ad22 - 4 -4 t x 2 2 2 + 2 t 2 A2

7

300

we obtain 2X

1

--

GPX ( z ) -

- 222 - z (2

(2+tZX2)

+ t A) + t X

+ dz2- 4 - 4 t X )

(2tX

'

Hence

G,,

2 2 - z (2+tX)+tX ( (2)=

223 - 2 t 2 X3 - z2 (2

Z

t

X

+ (2 + t ) A) + t z X

+

(2tX

d

W

+ dz2- 4 - 4 t A)

where

Q ( ~= )2 z 4 - z 3 ( 4 + 4 X + 2 t X )

+ z 2 (2 + 4 X + 2 t X + 2 X2 + 2 t X2 + 2 t2A')

- 4 t 2 X3 z + 2 t 2 X4.

The denominator of the Cauchy transform of the measure p~ may have four simple real roots x j , thus the measure p~ has at most four atoms dZj with respective weights. The actual dependence of atoms and weights on X and t is rather complicated. The density f p , (x), x E R of the absolutely continuous part of the measure p~ can now be calculated by the Stieltjes formula 1

f p , ( x ) = -- lim SG,, ( x + i ~ ) 7r €+Of

- X JtW -

(4

+ 4 t X - x2)

7r Q ( x )

for x E [ - 2 4 X i i , 2 J l T I 4 . Moreover, we obtain the following continued fraction form of the Cauchy transform of the measure p x : 1

GP, (2) =

X

z-A-

t X

z-1-

1+tX

z-

l+tX 2-2

-

-.

30 1

References 1. M. Bozejko and W. Bryc, On a class of free Le'vy laws related to a regression problem, accepted for publication in JFA, 2005 2. M. Bozejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditinallyfree random variables, Pac. J . Math., 175no.2, (1996). 357-388 3. M. Boiejko and J. Wysoczaiski, New examples of convolutions and noncommutative central limit theorems, Banach Cent. Publ., 43, (1998), 95-103 4. M. Bozejko and J. Wysoczaiski, Remarks on t-transformations of measures and convolutions, Ann. Inst. Henri Poincare Probab. Stat., 37 (6), (2001), 737-761 5. A. D. Krystek and t.J. Wojakowski, Associative convolutions arising from conditionallyfree convolution, IDAQP, 8 no.3,(2005), 515-545 6. A. D. Krystek, and H. Yoshida, Generalized t-transformatonsof probability measures and deformed convolution, Probability and Mathematical Statistics, 24 no. l, (2004), 97-1 19 7. E Oravecz, The number of pure convolutions arising from conditionallyfree convolution, IDAQP, 8 no.3,(2005), 327-355 8. F. Oravecz, Pure convolutions arising from conditionally free convolution, preprint, 2004 9. M. Reed and B. Simon, Methods of Modem Mathematical Physics I y Analysis of Operators,AcademicPress, New York, 1978 10. t.J. Wojakowski, Probability Interpolating between Free and Boolean, Ph.D. Thesis, University of Wrodaw, 2004

A THEOREM ON LIFTINGS OF STATISTICAL OPERATORS

J. KUPSCH Fachbereich Physik, T U Kaiserslautern 0-67653 Kaiserslautern, Germany In this note we investigate the problem of calculating the state of a system if the state of a subsystem is given.

1. Introduction

In quantum mechanics the states of a physical system are given by the statistical operators or density matrices in the Hilbert space associated to this system. It is well known that the state of a subsystem is uniquely calculated as the reduced statistical operator by the partial trace. But the inverse problem: to define an affine linear mapping from the set of states of a subsystem into the set of states of an enlarged system such that the reduced state coincides with the original state, has been studied in the literature only recently. These notes present the main results on this lifting problem on the basis of Refs. [l,21 and [3]. We use the following mathematical notations. Let 3-1 be a separable complex Hilbert space, then C(3-1) is the real Banach space of bounded selfadjoint operators, and Cl(1-I) is the real Banach space of selfadjoint nuclear operators. The respective norms are the operator norm and the trace norm. The quantum mechanical state space is the convex set D ( 7 f )of positive trace one operators on 3-1. The subset of rank one projectors on 7 f is denoted by p (3-1). 2. The problem of lifting

Let 7f and G be two complex Hilbert spaces, then F : D ( 7 f ) afine linear mapping if for W1, W2 E D(3-1)

F(X1Wl + X2W2) = XlF(W1) 302

--f

+ X2F(W2) E D(G)

D(G) is an

303

is true for all XI, XZ 2 0 , XI

+ Xz = 1.

Remark 2.1. For the importance of affine linearity in quantum mechanics see Ref. 4. Any affine linear mapping F : D(X)4 D ( E ) can be extended to a contraction mapping Ll(7-l)+. Ll(C7). See e.g. Ref. [5]. Now we consider a total system with Hilbert space 'H = 'Hs '8 ' H E . The system we are interested in has the Hilbert space 'Hs, the environment of this system has the Hilbert space ' H E . The main statement of this note is

-

Theorem 2.1. Let F : D('Hs) D(7-l~@ ' H E ) be a n a f i n e linear mapping such that trwEF(WS) = Ws f o r all Ws E D ( 7 - t ~ ) .Then there exists a n element WE E D ( X E ) such that F(Ws) = ws '8 WE.

(1)

The product ansatz (1) has been used as an obvious solution of the problem since a long time, see e. g. Refs. [6] or 7. The new result is that there are no other solutions. The Theorem has first been derived by Pechukas [l]for the two dimensional Hilbert space 3-1 = U?. A general proof was given in Ref. [2] (without knowing the publication of Pechukas) and then in Ref. [3]. The proof presented here borrows some ideas from Refs. [l]and [3]. The proof of Theorem 2.1 is given in three steps. The first step is Lemma 2.1. Let W E D ( 7 - l~ @'HE) with trEW = Ps E P('Hs) then there such ) that W = Ps @WE. exists a statistical operator WE E V ( 7 - t ~

Proof. Let W E D ( ' H s ' ~ ' H Ebe ) the statistical operator of the total system with the reduced state Ws = trEW = PS E ?(as) c D('Hs). Then tr?-ls@%~(PS '8 I E ) w ( P s '8 I E ) = t r X s (trE(Ps '8 IE)w) = t r ~ , P=~1

(2)

follows. The identity (2) implies that the range of the projector Ps '8 I E includes the range of the statistical operator W, and we have (PS @ IE)w(pS '8 I E ) =

w.

(3)

Since Ps has rank one, the space Ps'Hs is spanned by a single normalized eigenvector f s = Ps fs E 'Hs. All vectors in the range of Ps '8 IE have the structure $ = fs '8 g with some vector g E 'HE. As a consequence of (3) all eigenvectors of W with a non-vanishing (i.e. positive) eigenvalue must

304

have this form fs €3 g. The selfadjoint nuclear operator W has therefore the structure W = Ps @WE with WE = t r s W E ~ ( X E ) . 0

Remark 2.2. Generalizations of Lemma 2.1 can be found in Ref. [8]. Let F be the affine linear mapping of Theorem 2.1, then Lemma 2.1 implies that F ( P s ) = Ps C3 WE'

(4)

for all Ps E P('Hs),where WE' E D ( X s ) may depend on Ps. In the next step we exclude this dependence.

-

Lemma 2.2. I f F : D(lHs) D ( X s €3 ' H E ) is a n a f t n e linear mapping with the property F ( P s ) = Ps €3 WE' f o r all PS E P ( X s ) , t h e n WE' = WE does n o t depend o n P s . Proof. Take two projection operators PI and P2 of P ( X s ) , PI # P2, then there exist two normalized linearly independent vectors fj E Xs,11 = 1, j = 1 , 2 , such that Pj fj = fj. We can choose the phases such that the imaginary part of the inner product (f1 1 fz) vanishes. Defining the normalized orthogonal vectors f3 = c3( f~ f2) and f 4 = C4(f1 - f2) with constants c3,4 > 0 we have

fjII

+

fi

= af3

+ Pf4, f2 = of3 - Pf4 with a,P 2 0, 'a + P2 = 1.

The projection operators P k onto f k , k = 1,..., 4, satisfy the identity PI

+ P 2 = a2P3 + @P4.

(5) By assumption the mapping F is affine linear and can be extended to a linear mapping Cl(7-l~)-+ C1('Hs €3 ' H E ) , which we also denote with F, see Remark 2.1. Hence we have F(P1 + Pz) = F(P1) + F(P2) and F(a2P3+ P2P4) = a2F(P3) + P2F(P4). Then the assumption of Lemma 2.2 and the identity (5) imply PI

C3 WI

+ P 2 €3 w, = a 2 P 3 8 w3 +PZP4€3 w4

with some statistical operators

w k

(6)

E D(XE).

Let P(6) be the projection operator onto the unit vector af3 + Pei* f4, 6 E R. As a consequence of (6) the partial traces trs(P(6)gIE) ( P I €3 WI ~2 ~ 2 =) \a2+ p2ei* WI+la2 - p2ei.9 wZ and trs(P(6) C3 IE) (a2&C3 W3 P2P4 €3 W4) = a4W3 P4W4 define the

+

+

l2

+

l2

305

same operator. But that leads to the identity (W1 - Wz)cos.9 = 0 for 19 E R, and W1 = Wz follows. Hence we have derived

F(PS) = PS €3 WE

(7)

with a statistical operator W E ,which does not depend on Ps.

In the last step of the proof of Theorem 2.1 the identity (7) is extended to (1) by affine linearity. Remark 2.3. To derive the theorem only af€ine linearity is used; continuity and complete positivity are consequences. 3. Restriction mapping for observables

The Theorem 2.1 is equivalent to a statement for observables. To formulate this statement we first extend F to a linear mapping on C,('Hs), see Remark 2.1. Let F be a continuous linear mapping Ll('Hs) +Ll('Hs@"HE), then there exists a unique linear mapping F* : L('Hs €3 ' H E ) -+ L('Hs), continuous in the uniform and in the ultraweak topology, such that I W)N, = (A I ~(W)),,@N,for all A E C(7-f~€3 ' H E ) and all W E C l ( H s ) . Thereby ( B I W ) , = tr,BW is the standard bilinear pairing C ( H ) x C I ( H ) -+R.

P*W

-

Lemma 3.1. Let F : Ll('Hs) Ll(7-l~ €3 XE)be a continuous linear mapping and let F* : C('Hs @ ' H E ) + C('Hs) be its adjoint mapping, then F * ( B €3 I E ) = B f o r all B E L('Hs) i f and only i f tr,,F(W) = W f o r all

w E .Cl('HS).

Proof. The proof is straightforward, see Ref. [2].

0

Theorem 2.1 and Lemma 3.1 imply the following theorem for the restriction of an observable [2].

Theorem 3.1. If R : C('Hs €3 ' H E ) -+ L ( X s ) is a linear mapping, continuous in the ultraweak topology, and i f R ( B @ I E ) = B is true f o r all B E L ( X s ) then there exists a n element WE E D ( 7 - l ~such ) that R(A) = trx,A(Is €3 W E )

f o r all A

E

L ( X s €3 'HE).

Remark 3.1. The restriction R is completely positive.

(8)

306

4. Remarks about Zwanzig projectors

There is a more general notion of defining a subsystem based on the projection operators of Nakajima 101 and Zwanzig [ll]. These projection operators P are defined on the Banach space of trace class operators, P : Lx(3-1) -+ L1(3-1),such that P(D(3-1))c D(3-1). But one can also define adjoint Zwanzig projectors Q on the Banach space of observables, Q : L(3-1)4 L(3-1),which satisfy, see Sect. 7.2 of Ref. [12],

Q is a continuous linear operator onC(x), Q2=Q, QA>OifA>O, Q I = I .

(9)

Since QL(3-1)= B is a closed subspace of the Banach space L(3-1),the definition of an operator Q solves a restriction problem, which is more general than the problem of Theorem 3.1, as can be seen from the following example.

Example 4.1. The Hilbert space of a composite system is 3-1 = 3-1s@ 3-1~. Let WE E D(3-1~) be a reference state of the environment, then

A

E

L(3-1~ @%HE)-+ QA := (trEA(Is @ W E ) @ ) IE E L(3-1~ @ 3-13) (10)

is an operator on L(3-1),which has the properties (9). The non-trivial factor of the projection operator (10) A -+ trnA(I1 @W2) is exactly the restriction mapping (8). The restriction mapping of Theorem 3.1 is always completely positive. But there are Zwanzig projectors (9) which are not completely positive, see Sect. 7.7.2 of Ref. [12]. Hence using Zwanzig projectors the dynamics of subsystems might not be completely positive.

References 1. P. Pechukas. Reduced dynamics need not be completely positive. Phys. Rev. Lett., 73:1060-1062, 1994. 2. J. Kupsch, 0. G. Smolyanov, and N. A. Sidorova. States of quantum systems and their liftings. J . Math. Phys., 42:1026-1037, 2001. 3. T. F. Jordan, A. Shaji, and E. C. G. Sudarshan. Dynamics of initially entangled open quantum systems. Phys. Rev. A , 70:052110-1-14, 2004. 4. G. W. Mackey. The Mathematical Foundations of Quantum Mechanics. Benjamin, New York, 1963. 5. W. Guz. On quantum dynamical semigroups. Rep. Math. Phys., 6:455-464, 1974. 6. C. Favre and P. A. Martin. Dynamique quantique des systhmes amortis <non markoviens>>. Helv. Phys. Acta, 41:333-361, 1968.

307 7. E. B. Davies. Quantum Theory of Open Systems. Academic Press, London, 1976. 8. D. Giulini. Elementary properties of composite systems in quantum mechanics. In Ref. [9], pages 407-414. 9. E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I. 0. Stamatescu. Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin, 2nd edition, 2003. Corr. 2nd printing 2005. 10. S. Nakajima. On quantum theory of transport phenomena. Prog. Theor. Phys., 20~948-959, 1958. 11. R. Zwanzig. Statistical mechanics of irreversibility. In W. E. Brittin, B. W. Downs, and J. Downs, editors, Boulder Lectures in Theoretical Physics, Vol. 3 (1960), pages 106-141, New York, 1961. Interscience. 12. J. Kupsch. Open quantum systems. In Ref. [9], pages 317-356.

POSITIVE MAPS BETWEEN M z ( C ) AND M n ( C ) . ON DECOMPOSABILITY OF POSITIVE MAPS BETWEEN M z ( C ) AND M,(C)*

WlADYSlAW A. MAJEWSKI Institute of Theoretical Physics and Astrophysics, Gdarisk University, W i t a Stwosza 57, 80-952 Gdarisk, Poland [email protected] MARCIN MARCINIAK Institute of Theoretical Physics and Astrophysics, Gdalisk University, W i t a Stwosza 57, 80-952 Gdarisk, Poland [email protected]

+

A map 'p : Mm(@)+ Mn(C) is decomposable if it is of the form 'p = 'pi 9 2 where 'pi is a C P map while 'p2 is a cc-CP map. A partial characterization of decomposability for maps 'p : M z ( @ )4 M3(@)is given.

1. Introduction

Let cp : M,(C) -+ Mn(C) be a linear map. We say that cp is positive if p(A) is a positive element in Mn((c) for every positive matrix from Mm(C). If k E W, then cp is said to be Ic-positive (respectively k-copositive) whenever [cp(Aij)]$=l (respectively [cp(Aji)]!,j=I) is positive in Mk(M,(C)) for every positive element [Aij]f,j=l of Mk(M,(C)). If cp is k-positive (respectively Ic-copositive) for every k E N then we say that cp is completely positive (respectively completely copositive). Finally, we say that the map cp is decomposable if it has the form cp = ( ~ 1 9 2 where cp1 is a completely positive map while cp2 is a completely copositive one. By P(m,n) we denote the set of all positive maps acting between Mm(C) and M,((c) and by PI(,, n) - the subset of P(m,n) composed of all positive

+

*Supported by KBN grant PB/1490/P03/2003/25. The authors would like also thank the support of EU RTN HPRN-CT-2002-00729.

308

309

unital maps (i.e. such that 'p(1) = I). Recall that P (m,n) has the structure of a convex cone while PI(,, n) is its convex subset. In the sequel we will use the notion of a face of a convex cone.

Definition 1.1. Let C be a convex cone. We say that a convex subcone F c C is a face of C if for every c1, c2 E C the condition c1+ c2 E F implies C I , C ~E F . A face F is said to be a maximal face if F is a proper subcone of C and for every face G such that F C G we have G = F or G = C. The following theorem of Kye gives a nice characterization of maximal faces in P ( m ,n).

Theorem 1.1. [6] A convex subset F c P ( m , n ) is a maximal face of P(m,n) if and only zf there are vectors E E Cm and r] E Cn such that F = F C , where ~

and Pc denotes the one-dimensional orthogonal projection in Mm((C) onto the subspace generated by the vector <. The aim of this paper is to discuss the problem whether it is possible to find concrete examples of the decomposition cp = ' p l ' p 2 onto completely positive and completely copositive parts where 'p E P(m,h). It is well known (see [ll,141) that every elements of P ( 2 , 2 ) , P ( 2 , 3 ) and P ( 3 , 2 ) are decomposable. In [lo] we proved that if 'p is extremal element of P l ( 2 , 2 ) then its decomposition is unique. Moreover, we provided a full description of this decomposition. In the case m > 2 or n > 2 the problem of finding decomposition is still unsolved. In this paper we consider the next step for solving this problem, namely for the case m = 2 and n = 3 . Our approach will be based on the method of the so called Choi matrix. Recall (see [3, l o ] ) for details) that if cp : Mm -+ Mn is a linear map and {Eij}$j.=l is a system of matrix units in Mm(C),then the matrix

+

Hq =

[~(Eij)]$=E l Mm(Mn(c)),

(1.2)

is called the Choi matrix of cp with respect to the system {Eij}. Complete positivity of 'p is equivalent to positivity of H, while positivity of 'p is equivalent to block-positivity of H, (see [3, lo]). Recall (see Lemma 2.3 in [lo]) that in the case m = n = 2 the general form of the Choi matrix of a

310

positive map cp is the following

where a, b, u 2 0 , c, y , z , t E CC and the following inequalities are satisfied:

(1) Icl2 I ab, (2)

PI2 I bU,

(3) IYI

+

IZI

I(aUy2.

It will turn out that in the case n = 3 the Choi matrix has the form which similar to (1.3) but some of the coefficients have to be matrices. The main result of our paper is the generalization of Lemma 2.3 from [lo] in the language of some matrix inequalities. It is worth to pointing out that technical lemmas leading to this generalization are formulated and proved for more general case, i.e. for cp : M2(C) -+ Mn+l(@)where n 2 2. 2. Main results

In this section we will make one step further in the analysis of positive maps and we will examine maps cp in Pl(2,n+ 1) where n 2 1. Let { e l , e2} and { f l ,f 2 , . . . ,f n + l } denote the standard orthonormal bases of the spaces CC2 and CCn respectively, and let {Eij}& and {Fkl};:Il be systems of matrix matrix units in Adz(@)and Mn+l(C) associated with these bases. We assume that cp E F C ,for ~ some E E CC2 and 7 E By taking the map A ++ V*cp(WAW*)Vfor suitable W E U ( 2 ) and V E U ( n 1) we can assume without loss of generality that [ = e2 and 7 = f l . Then the Choi matrix of cp has the form

+

. . .. ..

. ..

..

.. .

H=

. . ..

311

We introduce the following notations:

c = [ C l c 2 . .. c n ] ,

Y=

[y1

y2

.. . y n ] ,

2 = [z1 z2

. .. z n ] ,

.Uln

u11 u12

*.

U n l un2

. . . unn

The matrix (2.1) can be rewritten in the following form

The symbol 0 in the right-bottom block has three different meanings. It

roi

Proposition 2.1. Let cp : Adz(@) -+Mn+l(@) be a positive map with the Choi matrix of the form (2.2). Then the following relations hold: (1) a 2 0 and B , U are positive matrices, (2) if a = 0 then C = 0 , and if a > 0 then C*C 5 aB, (3) 2 = 0,

(4) the matrix

[TI T* U

E

M z ( M n ( @ ) )is block-positive.

[:]:

Proof. It follows from positivity of cp that blocks on main diagonal, i.e. a C cp(E11)= c* and cp(E22) = , must be positive matrices. This

[

312

immediately implies (1) and ( 2 ) (cf. [14]). From block-positivity of H we conclude that the matrix

I;:[

=

[ (fly

1

cp(E1l)fl) ( f l l cp(E12)fl)

(fll cp(E21)fd (fll cp(E22)fd

is a positive element of Mz(C). So IC = 0, and ( 3 ) is proved. The statement of point ( 4 ) is an obvious consequence of block-positivity of H. 0

For X =

. . . 5,]

[ X I5 2

E M I , ~ ( Cwe ) define llXll =

(CZ, lzi12)1/2.

By 1 x1 we denote the square ( nx n)-matrix (X*X)1/2.Let us observe that for any X E Ml,,(C) we have

1x1 = IlXll% where = IIXII-lX* and Pc denotes the orthogonal projection onto the one-dimensional subspace in Cn generated by a vector 5 E C" (we identify elements of Mn,1(C)with vectors from en). We have the following

Lemma 2.1. A map cp with the Choi matrix of the form

is positive if and only i f the inequality

+ (z*,rT)+ n(nrqI2

I(y*,rT) < [Qu

[

+ n( K B )+ 2 x ( c * , r ~ n ) ](AW)

(2.4)

holds for every cr E C, matrices r = [ y1 79 . . . m ] and A11 A12 . . . A1" A = A21 A22 . . * A2n ] , ~ ~ ~ C , A i ~ € C f o r 2 ,,..., ~ =n,l 1suchthat 2 ~

An1

An2

.

* *

Ann

2 0 and A 2 0 ,

(1)

Q

(2)

r*r5

The superscript r denotes the transposition of matrices. Proof. Obviously, the map cp is positive if and only if w o cp is a positive functional on M2(C) for every positive functional w on Mn+l(C).

313

Let w be a linear functional on Mn+l(C). Recall that positivity of w is equivalent to its complete positivity. Hence, the Choi matrix H, = [ ~ ( F i j ) ] : ; : ~of w is a positive element of Mn+l(C). Let us denote a = w(F11), yj = w ( F l j + l ) for j = 1,2,. . . ,n,X i j = w(Fi+l,j+l) for i , j = 1,2,. . . ,n and and A are defined as in the statement of the lemma. Then, we came to the conclusion that w is a positive functional if and only if the matrix

r

*; [

I;]

is a positive element of Mn+l(C). This is equivalent to the conditions (1) and (2) from the statement of the lemma. Similarly, if w’ is a linear functional on M z ( C ) , then its positivity is equivalent to positivity of the matrix [ ~ ’ ( E i j ) ] & =Consequently, ~. w’ is positive if and only if w’(Eii) 2 0 for i = 1,2, w’(E21) = w’(E12) and IW’(E12)I2

I W’(Ell)W’(E22)

(2.6)

(cf. Corollary 8.4 in [ l l ] ) . Now, assume that a, r and A are given and the conditions (1) and (2) are fulfilled. In described above way it corresponds to some positive functional w on Mn+l(C). Let w’ = w o p. Then n i,j=l

= aa

+ Tr(A‘B) + 2%(C*,r T )

c n

w’(E22) =

XijUij

= Tr(ATU)

i,j=l n

=

n

n

(Y*, r T+ ) ( Z * ,rT)+ Tr(A7T).

(2.9)

It follows from the above equalities that (2.6) is equivalent to the inequality (2.4). Hence, the statement of the lemma follows from the remark contained in the first paragraph of the proof. 0

Proposition 2.2. If the assumptions of Propositions 2.1 are fulfilled, then

I Y+I 1215 a1/2U1/2.

(2.10)

314

Proof. Firstly, let us observe that the inequality (2.4) can be written in the form

+

I ( Y * , P ) (z',rr)12+/Tr(A~T)12+2X[ ( ( Y * J T ) + ( Z * , P ) )Tr ( A T ) ]

5 [aa+ Tr (A'B)

-r

+ 2X(C*,r')]Tr ( A v . l )

r

Putting instead of we preserve the positivity of the matrix (2.5) and the above inequality takes the form

12+

I ( Y * r') , + (Z*, P)

[(

+

ITr (ATT)12-2% (Y*,rT) (Z*, r.,) Tr ( A T ) ]

5 [a0+ Tr (ATB) - 2X(C*,r')]Tr ( A V ) If we add both above inequalities and divide the result by 2, then we get

and consequently

+ +

+

I(Y*,r y 2 I(z*,r7)12 2 X ( Y * ,T')(Z*, F') 5 aaTr(A7U) Tr(A'B)Tr(A'U) - p?r(A'T)12

(2.11)

Now, let rl be an arbitrary unit vector from C". Then

hence

( Y *F , ) ( Z * ,r') = I(Y*,r~)lI(z*, 171. Put also A = Er*r.Then from (2.11) and (2.12) we have

ll(lYl+1z1)q112I a T r ( ( r * r ) ' U ) + E 2 (Tr(A'B)Tr(ATU) - /Tr(ATT)12) Since E is arbitrary then we have

Lemma 2.2. Let cp : Mz(C) 4 Mn+l(C) be a linear m a p with the Choi matrix of the form (2.3). Then

315

(1) the map 'p is completely positive i f and only if the following conditions hold: ( A l ) Z = 0, (A2) the matrix

[;*,.':I

C* B T

is a positive element of Mzn+l(@).

I n particular, the condition (A2) implies:

(A3) i f B is a n invertible matrix, then T*B-lT 5 U , (Ad) C ' C I a B , (AS) Y*Y 5 aU, (2) the map cp is completely copositive i f and only if ( B l ) Y = 0, (B2) the matrix

:*[ :I

C* B T* is a positive element of M2n+l(C).

I n particular, the condition (B2) implies:

(B3) if B is invertible, then TB-lT* 5 U , (B4) C * C I a B , (B5) Z*Z 5 a U , Proof. It is rather obvious that the conditions (Al) and (A2) imply positivity of the matrix (2.3), and consequently the complete positivity of the map cp. On the other hand it is easy to see that (A2) is a necessary condition for positivity of the matrix (2.3). In order to finish the proof of the first part of point (1)one should show that positivity of (2.3) implies Z = 0. Let L1 be a linear subspace generated by the vector f 1 and let L2 be a subspace spanned by f 2 , f 3 r . . . ,f n + l , so Cn+' = L1 @ L2. Any vector wE can be uniquely decomposed onto the sum w = dl) d 2 )where , dz) E Li, i = 1 , 2 . Blocks of the matrix (2.3) are interpreted as operators. Namely: B,T, U : L2 4 L2, C,Y,Z : L2 + L1, and a : L1 -+ L1. Recall (cf. [13]), that the positivity of the matrix (2.3) is equivalent to the following inequality

+

en+'

+ (w2,

[;;I

.,)

2 0

316

for any vl,212 E (Cn+l. This is equivalent to

( u p7 a v y )

+ (up B v p ) + ( v p

U?p)

+ 2~(vj'),C~1~))+2X(v~~),Y~~))+2X(v~), Zviz))+2X (v$2)lTvp)) 2 0

(2.13)

+

vjl) vj2) for j = 1 , 2 , and ~ i ' ) ~ v ; ) E L1 and vr),v p ) E L2. Assume that v;') = 0, v p ) = 0, vi2) is an arbitrary element of L2, and = -rZv?) for some T > 0. Then (2.13) reduces to

where

vj =

ZIP)

( v p 7 B v j 2 ) ) - T I j Z v q 2 2 0.

(2.14)

This inequality holds for any vi2) E L2 and T > 0. It is possible only for = 0. To show the second part of the point (1) one needs to notice that posi-

z

implies the positivity of the matrices

tivity of the matrix

B T [T* U ] *' : [

i] [Y"*i]* and

Recall that the map is completely copositive if and only if the partial transposition of the matrix (2.3) is positive. The partial transposition (cf. [lo)) of (2.3) is equal to

p;$p Z*TO U

So, to prove the point (2) we can use the arguments as in proof of the point (1). Now, assume that cp : M2(@)+ A&(@) is a unital positive map, and cp E F e z , f l Hence . its Choi matrix has the form

(2.15)

317

+

where B and U are positive matrices such that B U = 1 and conditions listed in Propositions 2.1 and 2.2 are satisfied. From the theorem of Woronowicz (cf. [14]) it follows that there are maps cp1,cpz : Mz(C) --+ M3(Qj)such that cp = + cpz, and cp1 is a completely positive map while cp2 is a completely copositive one. From Definition 1.1 we conclude that both and cp2 are contained in the face F e , , f l . So, from Proposition 2.2 it follows that their Choi matrices H1 and H2 are of the form

+

where ai, Bi, Ui 2 0 for i = 1 , 2 , and the following equalities hold: a1 a2 = a , TI T2 = T , B1 B2 = B and U1 U2 = U . In [lo] we proved that if cp : Adz(@.)+ M2(C) is from a large class of extremal positive unital maps, then the maps cp1 and cp2 are uniquely determined (cf. Theorem 2.7 in [lo]). Motivated by this type of decomposition and the results given in this section (we ‘quantized’ the relations (1)-(3) given at the end of Section 1) we wish to formulate the following conjecture: Assume that cp : M2(C) + M3(C) is a positive unital map such that U # 0 , Y # 0, 2 # 0 and (Y( (21= U1/’. Then the decomposition cp = cp1 cp2 onto completely positive and completely copositive parts is uniquely determined.

+

+

+

+

+

References 1. E. M. Alfsen and F. W. Shultz, State spaces of operator algebras, Birkhauser, Boston, 2001 2. 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer Verlag, New York-Heidelberg-Berlin, vol. I (1979), vol. I1 (1980) 3. M.-D., Choi, Completely Positive Maps on Complex Matrices, Linear Algebra and its Applications 10 (1975), 285-290. 4. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras, Illinois J. Math. 18(4) (1974), 565-574. 5. M.-D. Choi, Some assorted inequalities for positive maps on C*-algebras, Journal of Operator Theory 4 (1980), 271-285. 6. S-H. Kye, Facial structures for the positive linearmaps between matrix algebras, Canad. Math. Bull. 39 (1996), 74-82. 7. L. E. Labuschagne, W. A. Majewski and M. Marciniak, O n kdecomposability of positive maps, t o appear in Expositiones Math, e-print: math-phyd0306017.

318 8. W. A. Majewski and M. Marciniak, On a characterization of positive maps, J . Phys. A : Math. Gen. 34 (2001), 5863-5874. 9. W. A. Majewski and M. Marciniak, k-Decomposability of positive maps, in: Quantum probability and Infinite Dimensional Analysis, Eds.: M. Schurmann and U. Franz, QP-PQ, vol. XVIII, World Scientific 2005, pp. 362-374; e-print: quant-ph/0411035. 10. W. A. Majewski and M. Marciniak, Decomposability of extremal positive maps on M 2 ( @ ) , t o appear in Banach Center Publications; e-print: math.FA/0510005 11. E. Stormer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. 12. E. Stormer, Decomposable positive maps on C*-algebras, Proc. Amer. Math. SOC.86 (1980), 402-404. 13. M. Takesaki, Theory of operator algebra I, Springer-Verlag, Berlin 2002. 14. S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), 165-183.

THERMODYNAMICAL FORMALISM FOR QUASI-LOCAL C*-SYSTEMS AND FERMION GRADING SYMMETRY

HAJIME MORIYA Department of Mathematics, Graduate School of Science, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan. This paper is a summary of my talk on 26th conference of QP and IDA in Levico (Trento), February, 2005 with some supplemental explanation. Thermodynamical formulations such as characterizations of equilibrium states and spontaneous symmetry breaking are given for general quasi-local systems. We show that the unbroken symmetry of fermion grading (the univalence super-selection rule) follows essentially only from the local structure that satisfies the graded commutation relations.

In [l,21 a characterization of local thermal stability for temperature states of quantum lattice systems is given. (The former is for quantum spin lattice systems, and the latter is for fermion lattice systems.) Let 5lOc be a set of all finite subsets of the lattice Z”, v being an arbitrary positive integer. Assume that there is a finite number of degrees of freedom on each site of the lattice. A typical example of the models under consideration is the following: The algebra on a site i, d { i } ,is generated by fermion operators ai, a;, and bosonic objects that are given by Pauli matrices u?, a:, af commuting with all fermion operators and all spin operators with different indexes. The interaction among sites is determined by the potential @, a map to A satisfying the following conditions: from 3lOc (@-a) @(I)E AI, @(0) = 0. (@-b) @(I)*= @(I). (@-c)@(@(I)) = @(I). (@-d)EJ (@(I)) = 0 if J c I and J # I. (@-e) For each fixed I E slot, the net (Hj(1))j with HJ(I) := C,{ @(K); K n I # 0, K c J} is a Cauchy net for J E 5lOcin the norm topology converging to a local Hamiltonian H(1) E A.

319

320

The above EJ denotes the conditional expectation of the tracial state (or any product state such as the Fock state) from A onto AJ for J E Slot. Let P denote the real vector space of all @ satisfying the above set of conditions. The set of all *-derivations with their domain Aloe commuting with 0 is denoted D(dloc).There exists a bijective real linear map from @ E P to 6 E D(dloc). The connection between 6 E D(AI,,) and @ E P is given by

6(A)= Z[H(I),A], A E A1 for every I E &,, where the local Hamiltonian H(1) is determined by (@-e) for this @. Let at (t E R) be a one-parameter group of *-automorphisms of A. A state ‘p is called an (at,P)-KMS state if it satisfies

‘p(AQiL4B)) = ‘p(BA) for every A E A and B E dent, where dentdenotes the set of all B E A for which at ( B ) has an analytic extension to A-valued entire function az( B ) as a function of z E C. Our dynamics at is assumed to be even, namely at 0 = 0 at. We also assume the following in order to relate at with some 6 E D(dloc). (I) The domain of the generator 6, of at includes Aloe. (11) Aloe is a core of 6,. Let I, denotes the complementary of I, i.e. IUI, = Z”. Let w be a state of (A, { A I } I E Z l o c ) . For I E 51,,, the conditional entropy of w is defined in terms of the relative entropy by

3I(w) := -S(trI

0 WIAI,,

w ) = -s(w.

E I ~ ,w )

I 0,

where EI, is the conditional expectation onto AI, and w . EI=(A):= w(EI,( A ) )for A E A. Let @ E P. The conditional free energy of w for I E &, is given by

F;p,,(w) := 31(4- P w ( H ( I ) ) , where H(1) is the local Hamiltonian for I with respect to @.

Definition 1. A state ‘p of A is said to satisfy the local thermal stability condition for @ E P at inverse temperature /3 or (@,P)-LTS condition if for each I E Slot

F;pp(‘p)2 F;pP(w) for every State w such that wIdI, = ‘p(dI,.

321

There is another definition of local thermal stability [2] that has the same variational formula as above but takes the commutant algebra di as the complementary outside system of a local region I instead of dI,.We shall call this alternative version LTS’ condition, where the superscript ‘1’ may stand for the commutant. Also by ‘I’ we indicate that this formalism is not natural compared to Definition 1. The formalism of LTS’ using commutants makes it possible to exploit the known arguments for quantum spin lattice systems [l]where local commutativity holds. In fact we easily obtained several similar results for fermion lattice systems [2] to those known for quantum spin lattice systems. Among them, the equivalence of KMS and LTS’ conditions for the fermion lattice systems holds without assuming the evenness on states. For even states, LTS’ and LTS are shown to be equivalent. Hence we also obtained the equivalence of KMS and LTS conditions under the assumption that the states are even. We, however, think that LTS’ is not so natural, even though it is mathematically useful, because there is no physically good reason in twisting the given CAR local structure into the tensor-product one by Jordan-Wigner transformations. On the contrary, our LTS obviously respects the given quasi-local structure and seems to be natural. We have shown that LTS condition does not permit the broken symmetry of fermion grading [3]. If there would exist a KMS state that breaks fermion grading symmetry for some even dynamics, then there are noneven states (made of it by perturbation of a local Hamiltonian multiplied by the inverse temperature) that inevitably satisfy the KMS condition, however, violate the LTS. In other words, if there would be breaking of the univalence superselection rule for temperature states, then the equivalence of KMS and LTS conditions should be violated. We conjecture that such breaking will never happen for temperature states and our LTS and LTS’ turn to be equivalent. We note that non-factor quasi-free states, which are non-physical examples of breaking of the fermion grading [6], have all type-I representations. From now on, we consider general graded quasi-local systems that encompass lattice and continuous, also fermion and fermion-boson systems. For such systems, a criterion named macroscopic spontaneously symmetry breaking, MSSB, is defined. It is formulated based on the idea that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every complementary outside system of a local region specified by the given quasi-local structure.

322

For bosonic (tensor product) systems, this MSSB reduces to the usual one, i.e. non-triviality of centers solely for the total system. But the former is in general stonger than the latter. We show the absence of MSSB for fermion grading symmetry in a model independent setting. For its proof, we make use of some observations noted in [4,51. The point is that no extension is possible for a pair of prepared states on disjoint regions if they are both noneven.

Acknowledgements. The author is a COE post-doctoral fellow of Department of Mathematics, Graduate School of Science, Hokkaido University and acknowledges this fellowship. The author is grateful to the organizer of this conference.

References 1. Araki, H., Sewell, G.L.: KMS conditions and local thermodynamical stability of quantum lattice systems. Commun. Math. Phys. 52, 103-109 (1977). Sewell, G.L.: KMS conditions and local thermodynamical stability of quantum lattice systems 11. Commun. Math. Phys. 55, 53-61 (1977). 2. Araki, H., Moriya, H.: Local thermodynamical stability of fermion lattice systems. Lett. Math. Phys. 60, 109-121 (2002). 3. Moriya, H.: Macroscopic spontaneous symmetry breaking and its absence for Fermion grading symmetry, preprint. 4. Moriya, H.: Separability condition for the states of fermion lattice systems and its characterization, preprint. 5. Moriya, H.: On a state having pure-state restrictions for a pair of regions. Interdisciplinary Information Sciences. 10, 31-40 (2004). 6. Manuceau, J., Verbeure, A,: Non-factor quasi-free states of the CAR-algebra. Commun. Math. Phys. 18, 319-326 (1970).

MICRO-MACRO DUALITY AND AN ATTEMPT TOWARDS MEASUREMENT SCHEME OF QUANTUM FIELDS

IZUMI OJIMA RIMS, Kyoto University, Kyoto 606-8502, Japan

1. Micro-quantum systems vs. macro-classical systems

Here I consider the problem of consolidating the well-known heuristic idea of “quantum-classical correspondence”, according to which macroscopic classical objects are to be regarded as condensates of infinite quanta emerging from a microscopic quantum system. This attempt is for the purpose of establishing effective mathematical methods or guiding principles for controlling bi-directional transitions between microscopic quantum systems and macroscopic classical levels, whose essence can be boiled down into such an expression as “Micro-Macro duality”, mathematically formulated as categorical adjunctions found at many different levels in physical theories. This will be seen just a mathematically polished version of the above old heuristics, quantum-classical correspondence (q-c correspondence, for short), in physics. As exemplified by such interesting phenomena as “macroscopic quantum effects”, the contrasts between [Micro vs. Macro] (according to length scales) and between [Quantum vs. Classical] (due to the essential differences of structures) are, precisely speaking, independent of each other to certain extent, owing to the absence of an intrinsic length scale to separate quantum and classical domains. Since this kind of mixtures can be taken as exceptional, however, I restrict my consideration here to such generic situations that processes taking place at microscopic levels are of quantum nature and that the macroscopic levels are described in the standard framework of classical physics, unless the considerations on the last point become crucial. On this premise of the parallelisms among micro/quantum/noncommutative and macro/classical/commutative, respectively, the essential contents of q-c correspondence involve the following two levels: 323

324

1) Superselection sectors and intersectorial structures detected by a centre: the major gap between the microscopic levels described by noncommutative algebras of physical variables and the macroscopic ones by commutative algebras can be clearly formulated and understood in terms of the notion of a (superselection) sector structure consisting of a family of sectors (or pure phases) described mathematically by factor states and representations, the totality of which describes physically mixed phase situations involving both classical and quantum aspects. Sectors or pure phases are faithfully parametrized by the spectrum of the centre of a relevant representation of the C*-algebra of microscopic quantum observables describing a physical system under consideration. Physically speaking, elements of the centre are mutually commutative classical observables which can be interpreted as macroscopic order parameters. 2 ) Intrasectorial structures detected by a MASA: while the above intersectorial structure describes and controls the coexistence of and the gap between quantum( =intrasectorial) and classical(=intersectorial) aspects, we need to detect the intrinsic quantum structures within a given sector, not only theoretically but also operationally (up to the resolution limits imposed by quantum theory itself). In the usual discussions in quantum theory with finite degrees of freedom, a maximal abelian subalgebra (MASA, for short) plays canonical roles in specifying a quantum state according to measured data, in place of the centre trivialized by Stone-von Neumann uniqueness theorem. As seen below, this notion of MASA need be reformulated in such a quantum system as quantum fields with infinite degrees of freedom, whose algebras of observables may have non type I representations. Under certain conditions, this formulation will be seen also to determine the precise form of the coupling between the object system and the apparatus required for the implementing a measurement process. In the attempt to utilize the obtained results for attaining a satisfactory measurement scheme of quantum fields, we encounter some subtle points caused by non-type I algebras like local subalgebras of type III in QFT, in close connection with the absence of maximal partition of unity consisting of minimal projections. This problem necessitates meaningful approximation schemes for bridging the gaps between continuous spectra inherent to type 111 structures at the mathematical level and the discrete spectra unavoidable in the actual measurements at the operational level. Once this point is resolved, it would suggest, at the same time, the possibility of reconstruction scheme of a microscopic quantum algebra from macroscopic classical observables combined with their data structure in duality.

325 2. Sectors and order parameters as q-c correspondence

To attain a clear-cut separation between quantum and classical aspects in terms of sectors and order parameters, we first recall the standard notion of quasi-equivalence [I] 7r1 M 7r2 of representations 7 r 1 , 7 ~ 2of an abstract C*-algebra U describing the observables of a given microscopic quantum system: it is defined as unitary equivalence up to multiplicity by 7r1

M 7r2

4% 7r1(U)”

N

7r2(U)”

In the universal representation [l],(7r,

* c(7r1) = c(7r2). :=

@ 7r,,Ej,

:=

WEEN

El,),

@ WEEX

U** =: U”, of U consisting of all t h e GNS representations (T,, fj,, R,) for states w E En, the central support c(7r) of a representation (7r,fi, = P,fj,) with support projection P, E 7ru(U)‘ is the smallest projection in the centre := IU”n7r,(U)’ to pick up all the representations quasi-equivalent to 7r: c(7r) =projection onto 7r(U)‘fi, c 4,. On this basis, we introduce a basic scheme for q-c correspondence in terms of sectors and order parameters: the Gel’fand spectrum Spec(S(U”)) of the centre 7r,(U)”

21

s(%”)

h

h

can be identified with the factor spectrum U of U: Spec(3(U”)) N U := Fa/ M, defined by all quasi-equivalenceclasses of factor states w E Fa with trivial centres 3(7r,(U)”) = 7r,(U)” n 7r,(U)’ = (cI~j,.

Definition 2.1. A sector (or physically, pure phase) of an observable algebra is defined by a quasi-equivalence class of factor states. In view of the commutativity of 3(U”) and of the role of its spectrum, we can regard [2] h

0

Spec(%”)) N U as the classifying space of sectors to distinguish

0

3(U”) as the algebra of macroscopic order parameters to spec-

among different sectors, and ify sectors. h

Then the dual of embedding map 3(U”) N Lbo(U)~f U”, h

Micro:

U* 3 Ea

--H

Prob(U) C L”(G)* : Macro,

can be interpreted as a universal q(uantum)+c(lassical) channel, transforming microscopic quantum states E En to macroscopic classical states h

E

-

Prob(U) identified with probabilities [2]. This basic q + c channel,

En 3 4

h

p+ = 4” l 3 ( n ) t E ) ~3 ( ~ p=) M 1 ( S p e c ( 3 ( U ” ) ) )= Prob(U) ,

326

gives the probability distribution p4 of sectors contained in a mixed-phase state 4 of U describing a quantum-classical composite system:

,-.

B3A

-

~ " ( x A )= p4(A) = Prob(sector E

A 1 4),

where 4" denotes the normal extension of 4 E Ea to a". While it tells us as to which sectors appear in 4, it cannot specify precisely which representative factor state appears within each sector component of 4. 3. Intrasectorial structure & MASA as q-c correspondence

To detect operationally the intrasectorial structures inside of a sector w given by a factor representation (T,, 4,, Q,), we need to choose a maximal by the abelian subalgebra (MASA) A of a factor algebra M := 7r,(Iu)", condition A' n M = A L"(Spec(A)) [3]. Note that, if we adopt such a definition of MASA as A = A' found in the usual discussions of quantum systems with finite degrees of freedom, the relation A' = A c M implies M' c A' = A c M , and hence, M' = M ' n M = 3 ( M )is of type I, which does not fit to the general context of infinite systems involving non-type I algebras. Since a tensor product M 8 d (acting on the Hilbert-space tensor product fi, @I L2(Spec(A))) has a centre given by

3 ( M 8 A) = 3 ( M )8 A = 1 8 Lm(Spec(A)), we see that the spectrum Spec(A) of a MASA A can be understood as parametrizing a conditional sector structure of the coupled system of the object system M and A, the latter of which can be identified with the measuring apparatus d in my simplified version [2] of Ozawa's measurement scheme4. This picture of conditional sector structure is consistent with the physical essence of a measurement process as "classicalization" of some restricted aspects A(c M ) of a quantum system, conditional on the coupling M 8 A of M with the apparatus identified with A. In addition to the choice of relevant algebras of observables, what is important in the mathematical description of a measurement process is to specify coupling terms between algebras of observables, M and d,of the object system and of the apparatus so that a microscopic quantum state of M can be uniquely determined from the macroscopic data of the pointer position Spec(A) of the measuring apparatus. To solve this problem we note that the algebra A is generated by its unitary elements which constitute an abelian unitary group U ( A )=: U. Since this group is infinite-dimensional in general, the existence of an invariant Haar measure is not guaranteed.

327

Just for simplicity, however, we assume here the existence of an invariant Haar measure du on U ,which requires U to be locally compact. Rewriting the condition A = A' n M for A to be a MASA of M into such a form as A = M n A' = M n U(A)' = M A d ( U we ) , see that A is the fixed-point subalgebra of the adjoint action Ad(u)X = uXu* ( X E M ) of u E U on M [5]. From this viewpoint, the relevance of the group duality and of the Galois extension can naturally be expected. Through a formulation in terms of a multiplicative unitary [6], the universal essence of the problem can be exhibited as follows. In the context of a Hopf-von Neumann algebra M ( c B ( f i ) ) with a coproduct r : M --t M @ M and a Haar weight, a multiplicative unitary V E U ( ( M@ M,)-) c U(4@ fi) is so defined as to implement r, r(x) = V * ( l@ x ) V , and is characterized by the pentagonal relation, V12V13V23 = v23v12,on rj @ Ej @ 4,expressing the coassociativity of r. Here subscripts i, j of &j indicate the places in Ej@fi@fi on which the operator V acts. It plays fundamental roles as an intertwiner, V(X '8 L ) = (X@X)V,showing the quasi-equivalenceamong tensor powers of the regular X(w) := (i @ w ) ( V ) E h;r, defined by a representation X : M , 3 w generalized Fourier transform, X(w1 * w2) = X ( w l ) X ( w 2 ) ,of the convolution algebra M,, w1 * w2 := w1 @ w2 o I?. On these bases, a generalization of group duality can be formulated for Kac algebras 161. In the case of M = L"(G,dg) with a locally compact group G equipped with a Haar measure d g , the multiplicative unitary V is explicitly given on L2(G x G ) by

-

for t E L2(G x G ) ,s,t E G,

( V t ) ( s t, ) := t ( s , s-lt)

or symbolically, V ( s t, ) = Is,s t ) , in the Dirac-type notation. Identifying the above M with L"(u^) = U(A)" = A for G := u^, the character group of our abelian group U = U(d),we adapt this machinery to the context of the MASA A. In terms of the group homomorphism E :U M associated with A M , the spectral decomposition of U is given via SNAG theorem by E ( u ) = &,ay(u)dE(y) (u E U ) , with dE an

-

-

M-valued spectral measure E ( A ) = E ( x A ) on u^ (for Bore1 sets A c @. Then V is represented as E,(V) = &a d E ( y )'8 A, on L 2 ( M )@ L2(u^)by

E*(V)(CA@ Ix)) =

1

dE(y)tA'8 Irx),

-YEA

for Y , X E 6,

[A

E E(A)L2(M),

(1)

satisfying the modified pentagonal relation, E, (V)12E*(v)13v23 =

328

V23E*(V)12. (Here the Hilbert space L 2 ( M ) provides M with its standard form.) By the equality A = L m ( S p e c ( A ) ) = Loo(fi),the same algebra A of observables to be measured maximally consistently can be viewed in two ways, respectively, as a function algebra on S p e c ( d ) consisting of characters on the commtutative algebra d , and as the corresponding one on the commutative group with product only: in contrast to the absence of an intrinsic “base point” in S p e c ( d ) (regarded as a Hausdorff space), the identity character L E 6, L ( U ) 3 1 (Vu E U ) present in the context of group duality U can be distinguished physically by its important role as the neutral position of measuring pointer. In the usual approaches, the importance of neutral positions remarked earlier by Ozawa has been overlooked for lack of the suitable place to accommodate it in an intrinsic way. It can also be related to the breakdown, AR # L 2 ( M ) ,of A-cyclicity of the cyclic and separating vector R of M in L 2 ( M )= P[dn]l E A’\(MUM’) = ( AV M’)\(MUM’), when A = A ’ n M # A’ [7]. The important operational meaning of the equality (1) and the role of the neutral position L can clearly be seen, especially when is a discrete group which is equivalent to the compactness of the group U (or the almost periodicity of functions on it): choosing x = L, we have the equality, E*(V)(&@ ( L ) ) = Er @ Iy) (Vy E which gives the required correlation (“perfect correlation” due to Ozawa8) between the states & of microscopic system M to be observed and those Iy) of the measuring probe system A coupled to the former. With a generic state [ = C,,a c7& of M , an initial uncomlated state E @ I L) is transformed by E, ( V )to a correlated one:

fi

-

fi

m,

fi

fi),

E*(V)(J@ 14) =

c

C7‘5-Y @

17).

7€ii The created perfect correlation establishes a one-to-one correspondence between the state & of the system M and the measured data y on the pointer, which would not hold without the maximality of d as an abelian subalgebra of M . On these bases, we can define the notion of an instrument 3 unifying all the ingredients relevant to a measurement as follows:

J(Alwc)(B):=(wC@ 1 =

L)(

LI)(E*(V)*(B @ XA)E*(V))

(( tI @ ( LI)E*(V)*(B @ xA)E*(v)(lC) @I

L)).

In the situation with a state wc = ( tI ( - ) E ) of M as an initial state of the system, the instrument describes simultaneously the probability p(A Iwc) =

329

3 ( A l q ) ( l ) for measured values of observables in A to be found in a Bore1 set A and the final state J(Alwt)/p(Alwt)realized through the detection of measured values [4]. In this way, we have attained two important improvements, to remove the restriction inherent to the usual characterization of MASA and to specify the coupling term necessary for the measurement scheme due to Ozawa originally formulated in systemd of finite degrees of freedom with type I algebras; these results constitute the crucial step in extending his scheme to such general quantum systems with infinite degrees of freedom as QFT. To be fair, however, we note that there remain some unclear points, such as the possibility for the Haar measure du to be absent and the non-uniqueness of MASA A = A' n M , both of which are the difficulties caused by the infinite dimensionality. As remarked at the beginning, the consistency problem should be taken serious between the mathematically universal structures of type 111 and the finite discrete spectra inevitable at the operational level, which is closely related with such type of criteria as the nuclearity condition in algebraic QFT to select the most relevant states and observables. By starting from a suitable choice of an operationally accessible subalgebra, a scheme is examined to recover a full local subalgebra of observables from the former, within which the standard determination of a semisiple Lie algebra by its root system can be regarded as a special case of the scheme [7]. References 1. Dixmier, J., C*-Algebras, North-Holland, 1977; Pedersen, G., C*-Algebras and Their Automorphism Groups, Academic Press, 1979. 2. Ojima, I., A unified scheme for generalized sectors based on selection criteria, Open Systems and Information Dynamics, 10 (2003), 235-279; Temparature as order parameter of broken scale invariance, Publ. RIMS 40,731-756 (2004). 3. Dixmier, J., Von Neumann Algebras, North-Holland, 1981. 4. Ozawa, M., J. Math. Phys. 2 5 , 79-87 (1984); Publ. RIMS, Kyoto Univ. 21, 279-295 (1985); Ann. Phys. (N.Y.) 259, 121-137 (1997). 5. Ojima, I., Micro-macro duality in quantum physics, t o appear in Proc. Intern. Conf. on Stochastic Analysis, Classical ?nd Quantum; math-ph/0502038. 6. Baaj,S. and G. Skandalis, Ann. Scient. Ecole Norm. Sup. 26 (1993), 425-488; Enock, M. and Schwartz, J.-M., Kac Algebras and Duality of Locally Compact Groups, Springer, 1992. 7. Ojima, I. and Takeori, M., work in progress. 8. Ozawa, M., Perfect correlations between noncommuting observables, Phys. Lett. A, 335, 11-19 (2005).

THE LEVY LAPLACIAN ACTING ON SOME CLASS OF LEVY FUNCTIONALS

KIMIAKI SAITO Department of Mathematics, Meijo University Nagoya 468-8502, Japan E-mail: [email protected] ALLANUS H. TSOI Department of Mathematics, University of Missouri Columbia, M O 65211, U.S.A. E-mai1:tsoiQmath.missouri. edu In this paper we shall discuss the Ldvy Laplacian as an operator acting on some class of the Ldvy functionals. The Laplacian acts on Gaussian and Poisson functionals in the class as a scalar operator. Based on this result, we introduce some domain of the Laplacian on which is a self-adjoint operator. We also discuss associated semigroups and associated stochastic processes.

Introduction An infinite dimensional Laplacian was introduced by P. LBvy in his famous book [12]. Since then this exotic Lapiacian has been studied by many authors from various aspects see [l-6,9,10,13,15,16,17,20] and references cited therein. In this paper, we study recent results on stochastic process associated with the LBvy Laplacian generalizing the methods developed in the former works [11,14,18,22-261. We construct a new domain of the LBvy Laplacian consisting of some LBvy functionals and associated infinite dimensional stochastic processes. This paper is organized as follows. In Section 1 we summarize basic elements of white noise theory based on a stochastic process given as a difference of two independent LBvy processes. In Section 2, following the recent works Kuo-Obata-Sait6 [ll],Obata-SaitB [18], SaitB [23, 241 and SaitbTsoi [25], we formulate the LBvy Laplacian acting on a Hilbert space consisting of some LBvy functionals and give an equi-continuous semigroup 330

331

of class (Co) generated by the Laplacian. In Section 3, we generalize this situation by means of a direct integral of Hilbert spaces. In Section 4, based on infinitely many Cauchy processes, we give an infinite dimensional stochastic process associated with the LBvy Laplacian.

1. Preliminaries Let E = S(R) be the Schwartz space of rapidly decreasing R-valued functions on R. There exists an orthonormal basis {ev},20 of L2(R) contained d2 in E such that Ae, = 2(v+l)e,, Y = 0,1,2,.. ., with A = -=+u2+1. For p E R define a norm I . , 1 by lfl, = ( A p f l t 2 ( ~for ) f E E and let E, be the completion of E with respect to the norm I . Ip. Then E, becomes a real separable Hilbert space with the norm I ., 1 and the dual space EL is identified with E-, by extending the inner product (., .) of L2(R) to a bilinear form on E-, x E p . It is known that E = proj limp-+, Ep, and E* = indlim,,, E-,. The canonical bilinear form on E* x E is also denoted by .). We denote the complexifications of L2(R), E and Ep by Lg(R), E c and Ec,,, respectively. - be independent LBvy processes of Let {L;,,(t)}t>o - and {L;,,(t)}t>o which the characteristic functions are given by (a,

02

+ (ezxz- I ) , 2 where m E R , c 2 0 and X E R. Set h,,x(t) = Li,,(t) - 15:,~(t) for all t 2 0. Then we have E[eizLL.x(t)] = eth('), t

2 0,

j = 1 , 2 , h ( z )= i m z - - z 2

+

E[eiz"~,A(t)] = e t ( h ( z ) + h ( - z ) ) = exp { --ta2z2 t(eixz + e-ixz - 2 ) ) 7 t 2 0.

s

Set C(S) = ""P{JR(h(&(4) + h(-
J

E* x E*

s

e x P { i ( z , s ) ) ~ P , x ( 4= C(<), = ( t 1 , < 2 ) E E x E ,

+

where ( 2 , s )= ( Z 1 , h ) ( 2 2 , & ) , 2 = ( 2 1 , ~E~E* ) x E*, E = (ti,&)E E x E. Let (L2),,,x= L2(E*x E * ,pu,x) be the Hilbert space of C-valued squareintegrable functions on E* x E* with L2-norm 11 . Ilo,x with respect to p,,x. We call an element of (L2),,,xthe Lkvy functional. The Wiener-It6 decomposition theorem says that: 00

@ Hn,

( ~ ~ > u= , x

n=O

332

where H, is the space of multiple Wiener integrals of order n E N and HO= C . The U-transform of 'p E (L2),,x is defined by

W E ) = C(t)-l/ E * x E * ' p ( 5 )exp{i(x, E))dP(X),

E

E E x E.

Theorem 1.1. [19] (see also [7, 9, 161) Let F be a complex-valued function defined on E x E . Then F is a U-transform of some Lkvy functional in (L2)m,xif and only if there exists a complex-valued function G defined on EC x EC such that

E and r] in E c x E c , the function G(zJ + r ] ) is an entire function of z E C , 2) there exist nonnegative constants K and a such that 1) f o r any

IG(E)I 5 KexP [aIEl;] 7 YE E

Ec x E c ,

3) F(E) = G(ia2E1+ia2E2++X(eiXE1-e-iXEz)) for allE

= (
ExE.

2. The LQvyLaplacian acting on the Levy functionals

Consider F = Up with the functions z H F(6

'p

+

E (L2),,x. By Theorem 1.1, for any <,7 E E zr]) admits the Taylor series expansions:

xE

where F ( n ) ( J :) E x . . . x E + C is a continuous n-linear functional. Fixing a finite interval T of R, we take an orthonormal basis {
exists for any E E E x E and i ~ ( U ' pis) in U [ ( L 2 ) , , ~The ] . L6vy Laplacian AL is defined by ALP = U - l i ~ U c p ,cp E 'DL. Given 2 0 , X E R,n E N and f E L&(R)6n, we consider 'p E (L2>,,x of the form: 'p=L., f ( ~ 1 , .. ., un)dAm,X(ul)*

The U-transform Up of

'p

is given by

. . dAm,X(un).

(2.1)

333

+

+

where Zu,x(E)(uJ)= ia2[1(uJ) ia2&(u,) X(ezxf1(u3) - e-zx~z(uJ)). For any (T 2 0, X E R and n E N let E , , A , denote ~ the space of 'p which admits an expression as in (2.1), where f belongs to J ~ & ( R and )~" supp f c T". Set E,,x,o = C for any o 2 0, X E R. Then E,,x," is a closed linear subspace of (L2),,x. If any element of E,,x," is an eigenfunction of A,, then (T = 0 or X = 0. Using a similar method as in [25], we get the following

Theorem 2.1. For each (T 2 0 , n E N and X E R the Levy Laplacian AL becomes a scalar operator on E,,o,, U EoJ,, such that A L C=~ 0 f o r all nXZ 'p E Eu,o,n and ALP= 'p f o r all cp E %,A,".

-m

For N E N and X E R let D Y be the space of 'p E (L2)o,x which admits an expression 'p = Cr=l'pnr 'pn E EoJ,,, such that

~r=i

I I I ~ I I I K =, ~ , ~ ~k(n)/l~n1li,x < 00, where

2e

=

CL

(g).

By the Schwartz inequality we see that D Y is a subspace of (L2)0,xand becomes a Hilbert space equipped with the new norm 111. I I I N , o , x . Moreover, in view of the inclusion relations:

(L2)o,x3 DYlA3 . . . 3 DOn;' 3 DOn;tl 2 . . . , we define DLx = projlimN4,D:x - n N00C 1 D$x. Then A, becomes a continuous linear operator defined on D N + , into D Y satisfying I l l A ~ V l l l ~ , o ,5x ~ ~ ~ ' p ~ ~ ~ N 'p +E iDLx, , o , ~N , E N. Therefore A, is a continuous linear operator on DLx. Moreover the operator AL is a self-adjoint operator densely defined in DSx for each N E N and X E R. For each t 2 0 and X E R we consider an operator G? on DLx defined 00 by G?'p = e - - t n ~ 2 / 1(Pn, ~ I 'p = (Pn E DZx. We also define G! on (L2),,o as an identity operator by G!'p = 'p, 'p E (L2),,o.

cr=1

Theorem 2.2. [ll, 231 Let X E R. Then the family of operators {G?; t 2 0) o n DZx is a n equi-continuous semigroup of class (CO)of which the infinitesimal generator is A,.

3. Extensions of the LBvy Laplacian

Let &(A)

be a finite Bore1 measure on R satisfying

334

Fix N E N. Let D& be the space of (equivalent classes of) measurable vector functions cp = (p') with 'pA = Cr=lp ' ; E D%A for all X E R \ {0}, and 'po E ( L 2 ) o , such ~ , that

Then Dg becomes a Hilbert space with the norm given in (3.1). In view of the natural inclusion: 9g+l c 9%for N E N, which is obvious from construction, we define D& = proj limN,, 9%= 9%. The Levy Laplacian AL is defined on the space D& by ALP = (A,px), cp = ('pA) E D&. Then AL is a continuous linear operator from D& into itself. Similarly, for t 2 0 we define Gtcp = (G?cpX),cp = (pA)E D&. Then by Theorem 2.2 we have the following:

n= ;,

Theorem 3.1. The family {Gt;t 2 0) is an equi-continuous semigroup of class (CO) on D& whose generator as given by AL. Remark: Let Gt be an operator defined on U[D&] by Et = UGtU-' , t L 0. Then by the above theorem, { E t ; t 2 0) is an equi-continuous semigroup of class (CO) generated by the operator LL. 4. Associated infinite dimensional stochastic processes

For p E R let EF be the linear space of all functions X H & E Ep x E p , X E R, which are strongly measurable. An element of EF is denoted by 6 = (&)XER. Equipped with the metric given by d p ( E , ~=) It*- 7 A I dv(X), 5 = (&,), 7 = ( q ~ )the , space : E becomes a corn-

SR

1+IEX-v:lp

plete metric space. Similarly, let CR denote the linear space of all measurable function X H zx E C equipped with the metric defined by p ( z , u) = SR dv(X), z = ( z x ) , u = (.A). Then CR is also a complete metric space. In view of d p 5 dq for p 2 q, we introduce the projective limit space ER = proj limp-+m EF. The U-transform can be extended to a continuous linear operator on D& by Ucp(E) = (U(P~(&))AER, E = (
wr

335

e-tlZI, z E R, j = 1 , 2 , 3 , 4 .Take a smooth function v~ E E with QT o n T . Set %A=

{

( X i t m ,- X & ~ T ) ( X - V T , -X!

Xtr]~),

= 1/ITJ

if x 2 0, otherwise.

Define an infinite dimensional stochastic process { Y t ; t 2 0) starting at E ER by Yt = (Ex %')xER, t 2 0. Then this is an ERvalued stochastic process and we have the following

< = (EX)XR

+

Theorem 4.1. If F is the U-transform of an element an 92, we have

Ed'(<)= E [ F ( Y t ) l Y o=
t 2 0.

(4.1)

Proof. We first consider the case when F E U[9&] is given by

F(c)=

(F'(EX))XER,

Fo E u [ ( L 2 ) u , 0 ] ,

with f E L&(R)On.Then we have

Since Fo E IA[(L2),,o]and F," E U[DzX],there exist 'po E (L2),,0 and E DLXsuch that F o = U['po]and F," = U [ p k ]for v-almost all X and each n. By the Schwarz inequality, we see that

'pk

336

n=l

n=O

where rp~, = C(Jx)-’ei(.t€X) for v-almost all X E R and each N E N.

-

Therefore by t h e continuity of G?, X E R, we get t h a t

Thus we obtain the assertion.

0

Acknowledgments This work was written based on a talk at the 26th International Conference on Quantum Probability and Infinite Dimensional Analysis held in Levico, Italy, 2005. This work was partially supported by JSPS Grant-in-Aid Scientific Research 17540136. T h e author is grateful for the support.

References 1. L. Accardi and V. Bogachev: The Omstein-Uhlenbeclc process associated with the Livy Laplacian and its Dirichlet form, Prob. Math. Stat. 17 (1997), 95-114. 2. L. Accardi, P. Gibilisco and I. V. Volovich: Yang-Mills gauge fields as harmonic functions for the L i v y Laplacian, Russ. J . Math. Phys. 2 (1994), 235250. 3. L. Accardi and 0. G. Smolyanov: Trace formulae for Levy-Gaussian measures and their application, in “Mathematical Approach to Fluctuations Vol. I1 (T. Hida, Ed.),” pp. 31-47, World Scientific, 1995. 4. D. M. Chung, U. C. Ji and K. SaitB: Cauchy problems associated with the L i v y Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 5. M. N. Feller: Infinite-dimensional elliptic equations and operators of Livy type, Russ. Math. Surveys 41 (1986), 119-170. 6. T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, No. 13, Carleton University, Ottawa, 1975.

337 7. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise: An Infinite Dimensional Calculus,” Kluwer Academic, 1993. 8. I. Kubo and S. Takenaka: Calculus on Gaussian white noise I-ZV, Proc. Japan Acad. 56A (1980) 376-380; 56A (1980) 411-416; 57A (1981) 433436; 58A (1982) 186-189. 9. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 10. H.-H. Kuo, N. Obata and K. Sait6: Lkvy Laplacian of generalized functions on a nuclear space, J. f i n c t . Anal. 94 (1990), 74-92. 11. H.-H. Kuo, N. Obata and K. Sait6: Diagonalization of the LLvy Laplacian and related stable processes, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 5 (2002), 317-331. 12. P. LBvy: “Leqons d’Analyse Fonctionnelle,” Gauthier-Villars, Paris, 1922. 13. R. LBandre and I. A. Volovich: The stochastic L i v y Laplacian and Yang-Mills equation on manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 4 (2001) 161-172. 14. K. Nishi and K. Sait6: An infinite dimensional stochastic process and the L b y Laplacian acting on WND-valued functions, “Quantum Inofrmation and Complexity” pp. 376-390, World Scientific, 2004. 15. N. Obata: A characterization of the Le‘vy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 16. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. 17. N. Obata: Quadratic quantum white noises and LLvy Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 18. N. Obata and K. Sait6: Cauchy processes and the L b y Laplacian, Quantum Probability and White Noise Analysis 16 (2002), 360-373. 19. J . Potthoff and L. Streit: A characterization of Hida distributions, J. f i n c t . Anal. 101 (1991), 212-229. 20. K. Sait6: It6’s formula and L6vy’s Laplacian I, Nagoya Math. J. 108 (1987), 67-76; IZ, ibid. 123 (1991), 153-169. 21. K. Sait6: A (C0)-group generated by the L i v y Laplacian 11, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 22. K. Sait6: A stochastic process generated by the L i v y Laplacian, Acta Appl. Math. 63 (2000), 363-373. 23. K. Sait6: The Lkvy Laplacian and stable processes, Chaos, Solitons and Ractals 12 (2001), 2865-2872. 24. K. Sait6: A n infinite dimensional Laplacian acting on multiple Wiener integrals by some Lkvy processes, to appear in “Infinite Dimensional Harmonic Analysis” I11 (2005). 25. K. Sait6 and A. H. Tsoi: The Livy Laplacian as a self-adjoint operator, in “Quantum Information,’’ pp. 159-171, World Scientific, 1999. 26. K. Sait6 and A. H. Tsoi: The L6vy Laplacian acting on Poisson noise functionals, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 503-510.

ON THE GENERAL FORM OF THE INTEGRAL-SIGMA LEMMA IN SYMMETRIC FOCK SPACE

KATFUN SCHUBERT Brandenburgische Technische Universitat Cottbus, Institut f u r Mathematik, PF 101344, 03013 Cottbus, Germany, EMail: [email protected] An important tool for integral transformation in symmetric Fock space is the socalled integral-sigma or

SE

-lemma. It exists in several versions depending on the

description of the random point configurations [4, 17, 15, 201. Usually, one has to m u m e that the point configurations have no multiple points. The goal of this paper is to present a detailed proof of a very general form of it including configurations with multiple points.

1. Introduction One possible representation of the symmetric Fock space is the one using the language of point process theory. This approach was developed in the middle of the 1980s in Jena [l,4,2, 31. Consecutively there were published a lot of articles by several authors using and expanding this approach 115, 5, 6, lo]. It allows to understand the Fock space as C2-space C2(M,!JX, F ) where M is the space of random point configurations and F is a a-finite measure on M . The integral-sum lemma then replaces multiple integrals by a single one and summation and makes kernel calculus more comfortable:

//

F(dcpl)F(~cp2)h(cpl,9 2 )

=

/

F(dv)

c

h(@,cp - @I

FGP

where @ cp means that @ is a subconfiguration of cp. The integral-sigma lemma exists not only in this language but in many in different versions depending on the description of the random point configurations. Many versions require the absence of multiple points. If the underlying configuration space G is well-ordered (for example G = R+) point configurations (without multiple points) can be described by sequences from G [13, 171. If there are no multiple points one can also use 338

339

subsets of G instead of sequences (see [18], Ex. 19.12). In the language of point process theory [14] there exist versions of the lemma for diffuse reference measures by Fichtner and Freudenberg [4] and measures with atoms by Liebscher [15]. If one describes point configurations using multisets, one gets the integralsigma lemma for measures given by von Waldenfels [20]. In the work at hand we want to give a detailed proof of a general version of the lemma for possibly atomic reference measure.

2. The Symmetric Fock Space

First we give the definition of the symmetric Fock space following mainly [4,6, 9, 81. Let G be an arbitrary complete separable metric space and 6 the associated a-algebra of BOREL sets from G. The ring of all bounded BOREL sets from 6 is denoted by 93. Moreover, let v be a locally finite measure on [G,631,i.e. v ( B ) < M for all B E 93. By No := N U (0) we denote the set of all natural numbers including zero. Let M be the set of all locally finite counting measures on [G,61, i.e.

M := {'p : 'p is a measure on [G, 6 ] , ' p ( B E) No for all B E 93). Each 'p E M is of the form 'p =

C SZi

with an at most countable index set

j€J

J, xj E G for all j E J and the sequence (zj)jEJ having no accumulation points. S, denotes the Dirac measure in x. Each element of M describes a locally finite point configuration in G. Multiple points are allowed. For two counting measures @, 'p E M we write @ 'p, if for all B E 93 there holds @ ( B 5 ) 'p(B).This means that @ is a subconfiguration of 'p. By supp 'p := {x E G : ' p ( { x } )> 0) we denote the support of 'p E M . Let for 'p E M J'pI:= 'p(G) be the number of points in the configuration 'p (including multiplicities). Mf denotes the set of all finite point configurations:

M f := {'p E M : IpI < M). The set

{'p E

M :

I'pI = n } of

all n-point counting measures from M for 00

n E N is denoted by M,, hence, M f = IJ M,. n=O

Here, MO = {o) is the

set containing only the zero measure o, i.e. the empty configuration in M (o(G) = 0).

340

The u-algebra generated by all sets of the type {'p E M : cp(B) = n } , B E 23 , n E No,is denoted by M. It is the smallest u-algebra making the map 'p H 'p(B)measurable for all B E 23.

Definition 2.1. A probability measure on [M,M] is called a POINT PROCESS. Such a measure can be interpreted as distribution of a random point configuration in G. Now we define the Fock space measure F on [ M ,MI.

Definition 2.2. For Y E M let

Here x y denotes the indicator function of a set Y E M. F is a a a-finite measure concentrated on Mf, i.e. F ( M \ Mf) = 0.

Definition 2.3. The space M := L2(M,3n,F) is called the (SYMMETRIC)FOCK SPACE over G according to the reference measure v.

M is again a separable Hilbert space. For all n E N let M" := M B n be the n-fold tensor product of the Hilbert space M . Obviously, M" can be identified with L 2 ( M n , M n ,F n ) . By (., .)Mnwe will denote the scalar product in L2(M",Mn, Fn). Remark 2.1. Usually, the symmetric Fock space over 'H is defined as follows. Let 'H be a separable Hilbert space. Then the symmetric Fock space over 'H is defined by

'7-lgrn

where 'HF$, := C and for n E N is the n-fold symmetric tensor product of If, i.e. the Hilbert subspace of 'HBn generated by vectors uBn, u E 7-l arbitrary. Consequently,

341

The space M given in Definition 2.3 is isomorphic to r ( L 2 ( G , v ) )under the isomorphism I determined by

(Iu@'n)(cp) := 6& ,1

n

u(z)V({"})

X€SUPPV

where 6i,j denotes the Kronecker delta symbol. For further details and proof see [ll]. 3. Generalized Binomial Coefficients In many publications like [7] and [lo] there are considered only diffuse measures v on [G,651,i.e. V ( Z ) = 0 for all singletons z E G. In this work, like in [16], also reference measures v with atoms are allowed. Hence, point configurations cp E M are represented by cp = C,,Jk,S,3with k, E N. Therefore, some formulae must be supplemented by additional factors. Definition 3.1. Let cp, $3 E M with $3 bY

C cp.

We define the number

(z)

EN

with (f) being the usual binomial coefficient and o the empty configuration from M . We give some properties of these generalized binomial coefficients. For B E 6 we denote by BC:= G \ B the complement of B. Lemma 3.1. Let n E N,cp, (PI, ..., pn E M . Furthermore, let f o r Bore1 sets B E 23 piB(.):= cp(B n (.)) be the restriction ofcp to B . Then there holds

(a)

342

4. The Integral-Sigma Lemma in Symmetric Fock Space Lemma 4.1. For functions f :

M

x

M

----+

@, m,n E

n

E

Mn,

=

C b x j , ~j

j=1

E G,

N, m 5 n and

there holds

n

Remark 4.1. By definition, cp E Mn is of the form

~ 5 Since ~ ~ in.

=

'p

j=1

this representation not all xj are necessarily distinct, we rewrite it: 1

1

c p = c k j b g j w i t h y l ,... yldistinct, 1 < n , k j > l b ' j = l , j=1

..., l , C k j = n . j=1

Using this, lemma 4.1 can be proved easily.

Corollary 4.1. For all functions h : N x holds

N

-

@ and all cp E

M f there

Particularly, for all a , b E @ we have

Proof. From lemma 4.1 we get also for h : N x N

-

@ and

'p

E

Mn

IGl=m

(6) By summing up over all m E (0,. . . , n } we get the first statement. The second statement follows immediately from the first one by applying the binomial theorem. 0

Lemma 4.2. Let n,m E No, h : M x M with respect to F 2 (or 2 0). Then

-

343

C be a function integrable

n+m

Proof. Let cp E Mm+n. Then cp =

C

dXj, with not all xj being necessar-

j=1

ily distinct. Using lemma 4.1 we get

E {I,...,n+m)-

Using lemma 4.2, we get by induction

Lemma 4.3. Let n 2 2, mi E N for i E n] and f : M n integrable with respect to F n (or 2 0). Then

-

C be a function

344

lvn-i I=w+ ...m,-1

The following proposition contains a general version of the integral-sum lemma in point process language. Since the integral-sum lemma is a powerful tool for many proofs, replacing integration over M n (with respect to F n ) by integration over M (with respect to F ) , we will give a complete proof for the general case of possibly atomic reference measure v and arbitrary n E N,n 2 2. For part (a) see

1151. Proposition 4.1. (a) Let h : M x M

(or 2 0 ) . Then

-

C be a function integrable with respect to F 2

(b) Let n 2 2 and f : M n to Fn (or 2 0). Then

-

C be a function integrable with respect

00

Proof. (a) According to definition there holds Mf =

F ( M \ M f )= 0. From this and lemma 4.2 we conclude

c J' 00

J ' F ( W p ( d y l ) h ( ' p l , ' p 2 )= M

M

n,m=O

M,

F(d'p2)

IJ Mn and n=O

J' F(d'pl)h('pl,'p2)

M,

345

(b) For n = 2 we have part (a). Now let for n

2 2 and f : Adn-'

-

C

(7)

Using (7) and (a) we obtain for f : M n

-

C

346

T h e second equation follows from this and part (c) of lemma 3.1.

0

References 1. K.-H. Fichtner and W. Freudenberg. On a probabilistic model of infinite quantum mechanical particle systems. Math. Nachr., 121:171-210, 1985. 2. K.-H. Fichtner and W. Freudenberg. Point Processes and Normal States of Boson Systems. preprint, NTZ Leipzig, 1986. 3. K.-H. Fichtner and W. Freudenberg. Point Processes and States of Infinite Boson Systems. preprint, NTZ Leipzig, 1986. 4. K.-H. Fichtner and W. Freudenberg. Point processes and the position distribution of infinite boson systems. J. Stat. Phys., 47:959-978, 1987. 5. K.-H. Fichtner and W. Freudenberg. Characterization of states of infinite boson systems i. on the construction of states of boson systems. Comm. Math. Phys., 137:315-357, 1991. 6. K.-H. Fichtner and W. Freudenberg. Remarks on stochasic calculus on the Fock space. In L. Accardi, editor, Quantum probabitity and Related Topics VII, pages 305-323. World Scientific Publishing Co. Singapore, New Jersey, London, Hong Kong, 1991. 7. K.-H. Fichtner, V. Liebscher, and W. Freudenberg. Time Evolution and Invariance of Boson Systems Given by Beam Splittings. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(4):511-531, 1998. 8. K.-H. Fichtner and G. Winkler. Generalized Brownian Motion, Point Processes and Stochastic Calculus for Random Fields. Math. Nachr., 161~291-307, 1993.

347 9. W. Freudenberg. Characterization of states of infinite boson systems ii. on the existence of the conditional reduced density matrix. Comm. Math. Phys., 137~461-472, 1991. 10. W. Freudenberg. Quantum Stochastic Integrals on a General Fock Space. Quantum Probability and Related Topics, VZII:189-210, 1993. 11. W. Freudenberg, K.-H. Fichtner and V. Liebscher. Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings. Math. Nachr., 218:25-47, 2000. 12. W. Freudenberg, K.-H. Fichtner and V. Liebscher. On Exchange Mechanisms for Bosons. Random Oper. Stochastic Equations, 12(4):331-348, 2004. 13. A. Guichardet. Symmetric Hilbert spaces and related topics, volume 261 of Lecture Notes in Mathematics. Springer-Verlag Berlin, Heidelberg, New York, 1972. 14. J. Kerstan, K. Matthes, and J. Mecke. Unbegrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin, 1974. 15. V. Liebscher. Bedingte Zustande und Charakterisierung lokalnormaler Zustande uon unendlichen Bosonensystemen. PhD thesis, FSU Jena, 1993. 16. V. Liebscher. Beam Splittings, Coherent States und Quantum Stochastic Differential Equations. Habilitation thesis, FSU Jena, 1998. 17. J. M. Lindsay and H. Maassen. An integral kernel approach to noise. In L. Accardi and W. von Waldenfels, editors, Quantum probabitity and Applications III, volume 1303 of Lecture Notes in Mathematics, pages 192-208. Springer-Verlag Berlin, Heidelberg, New York, 1988. 18. K. R. Parthasarathy. A n Introduction to Quantum Stochastic Calculus. Monographs in mathematics ; vol. 85. Birkhauser Verlag, Basel, Boston, Berlin, 1992. 19. K. Schubert. Quantum Markou Chains and Position Distributions for States of Boson Systems. PhD thesis, BTU Cottbus, 2005. 20. W. von Waldenfels. Symmetric differentiation and hamiltonian of a quantum stochastic process. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 8(1):73-116, 2005.

SPATIAL &-SEMIGROUPS ARE RESTRICTIONS OF INNER AUTOMORPHISM GROUPS

MICHAEL SKEIDE*

If 6 is a strict Eo-semigroup on some fBa(E),then Skeide [Ske02,Ske04] and Muhly, Skeide and Sole1 [MSS04] associate with 6 a product system of correspondences. We say the Eo-semigroup is spatial, if the associated product system is spatial in the sense of Skeide [SkeOlb]. The main goal of these notes is to establish the following theorem that just restates the title of these notes in a more specific form. Further terminology used in the theorem will be discussed after the notes and the example.

Main theorem. Suppose that E+ is a Hilbert module over a (unital) is a spatial strict Eo-semigroup on C*-algebra B and that 6 = (6t)tET Ba(E+). Then there exists a correspondence E- over B and a semigroup w = (wt)tETof unitaries wt o n E := E+ 0 E- such that the canonical homomorphism Ba(E+) t Ba(E+)0 idE- is a n isomorphism and such that f o r every t E T the restriction of at := w t 0 w: to Ba(E+) 0 idE- is 0 idE-. Notes. 1.) The result has an obvious variation for normal Bo-semigroups when E+ is a von Neumann (or W*-)module. Just replace correspondences by von Neumann (or W*-) correspondences and their tensor products. 2.) Following the lines of Skeide [SkeO3] it is easy to show that the unitary semigroup w reflects the continuity properties of the Eo-semigroup 6. But we do not have enough space to include the proof of such technicalities here. 3.) Unfortunately, for Hilbert modules the condition that 'B"(E+)--+ Ba(E+) 0idE- injective is not automatic, if the the left action of B on Efails to be faithful. *This work is supported by research fonds of the Department S.E.G.e S. of University of Molise.

348

349

Example. The usual time-shift endomorphism (CCR-flow) on the symmetric Fock space r(L2(R+,K ) ) may be understood as the restriction of of the unitarily implemented time-shift automorphism on r(L2(R,K ) ) = r(L2(R+, K ) ) @ r(L2(R-,K)) to B(r(L2@+,K))) idr(LP(R_,K)). The same is true for time-ordered Fock modules [BSOO],the module analogue of the symmetric Fock space. Now we explain in detail the terms used in the theorem. The semigroup with identity T is either R+ = [O,m) or No = {0,1,. . .}. A correspondence over B (or a Hilbert B-bimodule) is a Hilbert B-module with a nondegenerate left action of B as a representation by adjointable operators. B a ( E ) denotes the C*-algebra of all adjointable operators on a Hilbert module E . The tensor product is the internal tensor product over B . An Eo-semigroup is a semigroup of unital endomorphisms, and a unital endomorphism of B a ( E + )is strict if its restriction to the compacts 3C(E+) := m { z y * (z,y E E+)} acts nondegenerately on E+, where zy* denotes the rank-one operator z H x(y,z). (This is the simplest and most useful criterion for strictness; see, for instance, Lance [La11951 .) A product system is a family E@= (Et)tcTof correspondences over B with Eo = B and an associative family of isomorphisms (that is, bilinear unitaries) u,.: Es @ Et -+ Es+t (being the canonical ones when s = 0 or t = 0). Using the representation theory of [MSS04], [SkeO4] associates with every 6t (t > 0) the correspondence ET o tE+, where E; is the dual correspondence from B to B a ( E +)of E+ with inner product (z*, y*) = zy* and obvious B-Ba(E+)-bimodule operations, while otE+ is E+ viewed as correspondence from B a ( E + )to B with left action of B a ( E + )via 9t. (If E+ is f u l l , that is, if span(E+,E + ) = B,then this definition applies also for t = 0. Otherwise, we have to put Eo = B by hand.) The isomorphisms u,t I * I are determined by u,t((zQ*0,xi) 0 (y,* Ot yl)) = z: Os+t flt(z,yt )yt, where by at we indicate that the tensor product is that of E; @ etE+ = Et. Product system and Eo-semigroup are related by the family u = (ut)tET of unitaries ut: E 0 Et -+ E determined by ut(z 0 (y* Ot z ) = 6t(zy*)z which give back dt as 9t(u) = ut(u 0idEt)u,*. A unit for a product system Ea = (Et)tETis a family E@ = (Jt)tET of vectors Et E Et such that u,t(J, 0 &) = &+t and l o = 1 E EO = B. This implies necessarily that B is unital. (As observed in Bhat and Skeide [BSOO] the definition of the inner product in internal tensor products implies that the mappings Tt := (&, *&) form a CP-semigroup on B. Without the condition on t o , the mapping TOcould never be the identity. See also

350

the discussion of nonunital B in Skeide [SkeO4].) According to [SkeOlb] a product system is spatial if it admits a central unital unit w" = (wt)tET. Here central means that bwt = wtb for all t E T,b E B,and unital means that (wt, wt)= 1for all t E T. Spatiality of the product system of 6 implies that E+ is full. Of course, a central unital unit w@ generates the trivial CP-semigroup (wt, owt)= ida.

Remark. For B unital and E+ full one may show that spatiality of the Eo-semigroup 6 is equivalent to existence of a semigroup w = (wt)tET of intertwining isometries wt E Ba(E+)for 6 , that is, 6t(a)wt = wta. (This is closest to Powers' original definition of spatial Eo-semigroups [Pow871 in the case when E+ is just a Hilbert space.) In fact, if there is a central unital unit w",then vt := ut(idE+ @ d t ) defines such a semigroup of intertwining isometries. (Here idE+ Owt denotes the mapping x H x 0wt.)Conversely, given such a semigroup w by general abstract nonsense one may show the converse. (This involves Rieffel's fundamental results on Morita equivalence [Rie74a,Rie74b] together with such simple observations like E; (full!) is a Morita equivalence from B to X ( E + ) and B"(B,E+) = E+ (B unital!) thinking of x E E+ as the mapping b H xb.) We will also show a supplement to the main theorem regarding weak dilations in the sense of [BSOO]. The pair (E+,6) is a weak dilation of a (necessarily unital) CP-semigroup, if there exists a unit vector <+ E E+ (unit vector means that (t,<) = 1)such that the projection PO := <+<$ is increasing for 6, that is, St(p0) 2 po for all t E T. In this case Tt(b):= ([+, 6t(c+b<;)<+), indeed, defines a unital CP-semigroup T and 6 , clearly, is a dilation of T (under the embedding b H <+b<:). Also, it is not difficult to check that under these circumstances E@has a unital unit such that Tt= (&,.St); see [Ske02]. [@

Supplement. If (E+,6 ,<+) is a weak dilation of the unital CP-semigroup T , then the correspondence E- in the main theorem can be chosen such that E- contains a (central) unit vector w- and the semigroup a is a dilation of T with respect to the embedding b H [b<* 0 id& in the vector expectation (<+ 0w-,.[+ 0w-). We note that (E+ 0 E-,a,<+ @ w-) is a weak dilation of T , if and only if T is the trivial CP-semigroup and if the projection idE+ 0w-w: is invariant for a. It is, generally, a weak dilation with respect to the filtration c.ut(po0idE-) = 8t(po) 0idE-, if we apply the weaker hypothesis

351

of Bhat and Parthasarathy [BP94] (rephrased suitably in terms of Hilbert modules). 1. Units and inductive limits

As a motivation for the construction of E- we repeat an inductive limit construction from [BSOO] that reverses in some sense the construction of the product system E@from 6 in the case when (E+,6) is weak dilation. So let E@ be a product system and a unital unit for E @ . Then Q 0 idEt : xt H u,t(<, 0 x t ) defines an isometry in Ba(Et,E,+t). The family (Et)tETtogether with the family of embeddings ts0 idEt forms an inductive system of Hilbert B-modules (not of correspondences!) and the completion of the algebraic inductive limit is Hilbert B-module which we denote by Em. The factorization E, 0Et = Es+t (we surpress the mappings u S t )survives the inductive limit and gives rise to a factorization E, 0Et = E, fulfilling the associativity condition (Em0E,) 0 Et = E, 0 ( E , 0 Et). Therefore, gt(a) := a 0idEt defines a strict Eo-semigroup on 'BBa(E,). AS 0& = ts+t, the unit vector & E Et, when embedded into Es+t, coincides with &+t. Therefore, E, contains a distinguished unit vector Em, the inductive limit of all the & , and ( E m ,$, 5,) is a weak dilation of the CP-semigroup Tt = (&, o & ) . <@

<,

1.1 Remark. If (E+,6, [+) is a weak dilation and if E @and

are product system and unit associated with that dilation, then E , is identified naturally as the submodule UtET6t(po)E+ of E+. The dilation is primayl, if E+ = Em. In this case also 6 = 6 and E+ =

-

[@

c,.

2. Proof of the main theorem and its supplement

Already in Skeide [SkeOla]we noted that in the case of a central unital unit w@ the preceding inductive limit can be performed into the other direction, that is, using embeddings idEt Ow, rather than w, 0 idEt. Indeed, in the identification Et = Ba(B,Et) (B is unital!), the mapping w t : b H wtb is actually bilinear and, therefore, can be amplificated as a right factor in a tensor product. The embeddings Et 4 Es+t we obtain in that way are, indeed, bilinear so that we obtain an inductive limit E- which is a bimodule, that is, a correpondence over B. We have now a family of bilinear unitaries u t : Et 0

352

E- -+ E-. Clearly, w- := lim indtETwt is a central unit vector, that is in particular, (w, bw) = b for all b E B. Now we can put together ut and ut to form a unitary wt on E+ 0E-. We define

wt = (ut 0idE-)(idE+ @u,*). What wt does is simply

or

Thinking of ut as identification E+ 0Et = E+, as we did in many papers, and of U; as identification Et 0E- = E-, then wt is just rebracketting

Clearly, the wt define a semigroup. And for every a E 'Ba(E+), as ( a 0 idE-)(idE+ O u t ) = (idE+ Ou,)(a 0 idEt 0idE-), we find at(aOidE-) = (ut OidE-)(idE+ @u,*)(aOidE_)(idE+ = (ut 0idE-)(a

OU;)(U:

@idE-)

0idEt 0idE-)(u: 0idE-) = & ( a ) 0idE- .

This proves the main theorem. The assertions of the supplement follow simply from the preceding calculation by observing that ((<+ 0 w-), (&(a) 0 idE-)(J+ 0 u - ) ) = (w-7 (E+, W a ) E + ) w - ) = (s+,st(.)r+). 2.1 Remark. For proving the note after the supplement, it is sufficient to observe, that (E+ 0 w-)(E+ 0w-)* is increasing for a , if and only if (ut,
We note further that the main theorem for the case where E+ = Em for the central unital unit w@ has been considered in [SkeOla]. Here we have the generalization where we assume neither that 6 is a dilation of the trivial semigroup (as in [SkeOla])nor of any other CP-semigroup.

353

3. A n open problem Eo-Semigroups (normal and strongly continuous in the case T = Ro) on B ( H ) ( H a separable infinite-dimensional Hilbert space) are classified by their product system up to cocycle conjugacy; see Arveson [ArvSga]. This means that under the restriction on the dimension of H , we have a complete characterization of Eo-semigroups up to cocycle conjugacy by product systems. (Also, under suitable technical conditions every product system of Hilbert spaces comes from an Eo-semigroup; see Arveson [ArvSgb]. The corresponding question for general product systems of Hilbert modules is completely open, so far, in the continuous case T = R+ and has been solved only recently in the discrete case T = No in [Ske04]. But this is not the open problem about which we wish t o speak.) The truth is that we are speaking about Eo-semigroups only on those Hilbert spaces that are in the same isomorphism class. Under this assumption we obtain the same statement also for Hilbert modules [SkeO2] (without any continuity assumption): If 6' and 6' are strict Eo-semigroups on B a ( E 1 )and B a ( E 2 )and El E E 2 , then d1 and f12 are cocycle conjugate, if and only if their associated product systems E'' and E2@are isomorphic, that is, if there exists a family w@ = (wt)tGTof unitaries wt E B B " ~ b i ' ( E ~ such , E ~that ) w,+t = u:+,(ws 0wt)u;& and wo = ida. (In [SkeO4] we have relaxed the condition El % E2 to 'Ba(E1)E B a (E 2 ). Under this condition we have cocycle conjugacy (in an obvious sense), if and only if the product systems are Morita equivalent. The notion of Morita equivalence of correspondences is borrowed from Muhly and Sole1 [MSOO].) In full generality, we do not know what we can say about the relation among El and E2 given the information that there exist Eo-semigroups 6' and g2 on them that have isomorphic (or Morita equivalent) product systems. We know by explicit examples that neither of the conclusions El E 2 or Ba(E1) % Ba(E2) needs to be true. But what can we say, if El and E2 are inductive limits in the sense of Section 1of the same product system E@ but with respect to possibly different units E" and t2@? To state the problem we wish to pose clearly: Are the inductive limits over a product sytem with respect to two different units always isomorphic Hilbert modules or not? Certainly the inductive limits will be isomorphic, if there exists an automorphism of E@that sends to t2'. In this case, necessarily the CP-semigroups generated by and 6'' coincide. But even in the case of Arvesons product systems of Hilbert spaces it is an open problem, whether

cl@

354

for every pair of (normalized) units there is an automorphism of the product system that sends one unit to the other, that is, whether the automorphisms of a product system act transitively on the set of units. A positive answer is known only for type I systems of Hilbert spaces, that is, for symmetric Fock spaces. For time ordered Fock modules we have the result provided the units generate the same CP-semigroup. (For more we cannot ask, so the statement is analogue to that for Hilbert spaces.) But while we know that the inductive limit over time-ordered Fock modules for a central unital unit (that plays the role of the vaccum) is a time-ordered Fock module (indepent of the choice of that unit), we do not know whether the same is true for a unit that generates a nontrivial CP-semigroup. This discussion includes the representation space of the minima1 weak dilation of an arbitrary nontrivial uniformly continuous unital CPsemigroup. The fact that dilations of such CP-semigroups may be obtained with help of quantum stochastic calculi lets us suspect that also the minimal weak dilation lives on a Fock module. Positive answers exist only in the case B = B ( H ) . The situation we met in the proof of the supplement when we start with a primary dilation and a spatial product system, so that there are two units arround, and w o , one of which generates a nontrivial CP-semigroup and the other unit generates the trivial one. We would be happy if we could show that the two inductive limits are isomorphic and, therefore, the minimal weak dilation a cocycle perturbation of a dilation of the trivial CP-semigroup. (This is exactly what quantum stochastic calculus usually does: Constructing a cocycle that transforms a dilation of the trivial CP-semigroup into a dilation of a nontrivial one.) We suspect that this might not be possible in general. But we have the feeling that chances might improve, when we try cocylce perturbations of CY instead of 19. References Arv89a. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC.,no. 409, American Mathematical Society, 1989. Arv89b. -, Continuous analogues of Fock space III: Singular states, J. Operator Theory 22 (1989), 165-205. BP94. B.V.R. Bhat and K.R. Parthasarathy, Kolmogorov's ezistence theorem f o r Markov processes in C*-algebras, Proc. Indian Acad. Sci. (Math. Sci.) 104 (1994), 253-262. B.V.R. Bhat and M. Skeide, Tensor product systems of Hilbert modules BSOO. and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519-575. Lan95. E.C. Lance, Hilbert C*-modules, Cambridge University Press, 1995.

355 MSOO. MSS04. Pow87. Rie74a. Rie74b. SkeOla.

P.S. Muhly and B. Solel, O n the Morita equivalence of tensor algebras, Proc. London Math. SOC.81 (2000), 113-168. P.S. Muhly, M. Skeide, and B. Solel, Representations of 'Ba(E), Preprint, ArXiv: math.OA/0410607, 2004. R.T. Powers, A non-spatial continuous semigroup of *-endomorphisms of 'B(fi), Publ. Res. Inst. Math. Sci. 23 (1987), 1053-1069. M.A. Rieffel, Induced representations of C* -algebras, Adv. Math. 13 (1974), 176-257. -, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. M. Skeide, Hilbert modules and applications in quantum probability, Habilitationsschrift, Cottbus, 2001, Available at http://~.math.tu-cottbus.de/INSTITUT/lswas/_skeide.html.

SkeOlb. SkeO2. SkeO3.

SkeO4.

-,

The index of white noises and their product systems, Preprint, Rome, 2001. -, Dilations, product systems and weak dilations, Math. Notes 71 (2002), 914-923. -, Dilation theory and continuous tensor product systems of Hilbert modules, QP-PQ: Quantum Probability and White Noise Analysis XV (W. Freudenberg, ed.), World Scientific, 2003. -, Unit vectors, Morita equivalence and endomorphisms, Preprint, ArXiv: math. OA/04 12231, 2004.

Address: Dipartimento S.E. G.e S., Universitci degli Studi del Molise, Via de Sanctis, 86100 Campobasso, Italy. E-mail: skeide(9math.tu-cottbus. de. Homepage: ht tp://www.math. tu-cott bus,de/INSTITUT/lswas/_skeide.html

A CHARACTERIZATION OF POISSON NOISE*

SI SI Faculty of Information Science Aichi Prefectural University Aichi-ken, Japan e-mail : sisiOist.aichi-pu. ac.jp

We shall compare Gaussian (white) and Poisson noises in order t o characterize the latter. First, their characteristics are discussed, then we show that approximations of the two noises indicate significant difference between them from the viewpoints of invariance under transformation group and optimality of information. There one can recongnize dissimilarity in addition to similarity. The dissimilarity is more significant. In fact, there are many different properties between the two noises in parallel, so that we can see the contrast which is most interesting for us. Then, we shall come to a characterization of the so-called fractional power distribution with the help of the significant properties of Poisson noise. It is our hope that special methods of approximation to the two noises would further reflect a duality between them. One is related to the rotation group which is continuous, while the other is to the symmetry group which is discrete.

1. Poisson noise and its probability distributions

We are interested in some latent optimality of a Poisson process. To discover optimality we introduce a partition of a probability measure space where the given Poisson process is defined. On each subset of the partition we will see that the Poisson process has maximum entropy. This trick can be compared to the method of approximation of a Gaussian white noise. To come to the main aim in Section 2 we shall recall some properties of a Poisson system, that are necessary to our approach. They are 1. Poisson distribution has the reproducing property. *AMS Mathematics Subject Classification 2000 : 60H40

356

357

2. Linear functionals of Poisson noise define processes with weakly linear property. 3. Poisson processes satisfy maximum entropy property. 4. Asymptotic behaviour of a distribution in the domain of attraction of a symmetric stable distribution over [0,co) with exponent a is

1- F ( x ) t k", as x + co, k 1 - F ( kz)

> 0.

(Tacitly used in Section 3.) Now we give a concrete set up of a Poisson noise.

) Let El and E2 be suitably chosen Hilbert spaces (C Eo = L 2 ( R d ) that are topologized by Hilbertian norms 11 . 111 and 11 . 112, respectively. Assume that ((.I115 (1. ( ( 2and that there exist consistent injections Ti,i = 1,2, such that Ti from Ei to Ei-1 for i = 1 , 2 , is of Hilbert-Schmidt type. Then, we have inclusions of the form E2 C El C Eo C E;

C

E;.

(1.1)

The characteristic functional of Poisson noise with Rd parameter with intensity X is

We can prove that Cg(c) is continuous on El. Obviously it is positive definite and Cp(0)= 1. Thus by the Bochner-Minlos theorem, there exists a probability measure p p which is supported by E,* and then we have a Poisson noise measure space (E,*, pp). To fix the idea, consider one dimensional parameter case and let the time parameter space be a compact set, say I = [0,1]. In this case, Ch(<) is continuous in EO = L 2 ( I ) ,so that there is a Gel'fand triple of the form

El c L2(1) Eo

C E;.

(1.2)

A Poisson measure is now introduced on the space E;. Define P ( t , x ) = ( z , ~ [ ~ , ~5l )t, O 5 1, x E E;, by a stochastic bilinear form, where x is the indicator function. Then, P ( t ,z) is a Poisson process with parameter set [0,1].

358

Let A , be the event on which there are n jump points over the time interval I . That is

A , = {X

E E;; P(1,X) = n } ,

(1.3)

where n is any non-negative integer. Then, the collection { A n , n 2 0) is a partition of the entire space E;. Namely, up to measure 0, the following relations hold:

A,r)A,=$,

n f m ; UA,=ET.

(1.4)

Given A,, the conditional probability py is defined :

For C c Ak, the probability measure (Ak,B k , &), is such that

& on a probability measure space

where BI, is the sigma field generated by measurable subsets of Ak, determined by P(tl.).

For k = 0, the measure space is (Ao,Bo, p;) is trivial, where

Bo = { $ , A o ) mod 11; and &(A) = 1. Let z E A n r n 2 1, and let ~i statistics of jump points of P ( t ) :

0 = T o < 71

<

= ~i(x),i= < T, < T,+1

1 , 2 , . . . , n 1be the order

= 1.

(The T ~ ’ Sare strictly increasing almost surely.) Set

Xi(.)

= Ti(.)

so that n+l

- Ti-l(.),

359

Proposition 1. (Ref. [6]) On the space A,, the (conditional) probability distribution of the random vector ( X I ,X2, ..., X,+l) is uniform on the simplex n+l

EXj=1,

xj

20.

Cororally. (Ref. [ 6 ] ) The probability distribution function of each X j is 1 - (1 - U),,

0 5 u 5 1.

Proposition 2. (Ref. [6]) The conditional characteristic functional

CP,,(t)

= E [,i(p,C)

[A,]

(1.5)

is obtained as

Proposition 3. (Ref. [8]) The conditional probability measure p;, defined on the measure space (A,, B,, p;), is invariant under the symmetric group S(n 1) acting on ( X I ,X2, ..., X,+l).

+

2. Characterization of Poisson noise as compared with

Gaussian noise Some properties reviewed in the previous section are taken to be characteristics of Poisson noise. Among others, uniform probability distribution on a simplex and the invariance of the conditional Poisson distribution under the symmetric group are significant to characterize a Poisson noise. With this fact in mind, we are going to construct a Poisson noise in such a way that we decompose a measure space defining the noise and on each component of the decomposition we can see those characteristics in a visualized manner. Also our method can be compared to the approximation to Gaussian white noise by inductive limit of spheres, where one can see the characteristics; rotation invariant and maximum entropy. The comparison is successful to observe the two noises step by step. There latent properties appear explicitly in front of us.

360

We first observe the Gaussian case by reviewing the result by HidaNomoto [I]. G.0) Start with a measure space 1 -

27r.

(S1,do),

(s,,

where S1 is a circle and dB

=

sn

G.l) Take a measure space on), where is the n-dimensional sphere excluding north and south poles; and where do, is the uniform probability measure on Sn. G.2) Define the projection nn : sn+l 4 Sn in such a way that each longitude is projected to the intersection point of the longitude and the equator. Thus, Sn is identified with the equator of s n + l . Take the radius of Sn+1 be f i . G.3) Given the measure un on 3, which is the uniform probability measure, i.e. invariant under the rotation group SO(n l). Then the measure un+l is defined in such a way that

+

gn+1(ni1(E))= on(E), E

c Sn

and that on+l gives the maximum entropy under the above restriction. G.4) As a result on+l is proved to be the uniform distribution on $,+I and is invariant under the rotation group SO(n 2).

+

G.5) The inductive limit of the measure space is defined by the projections nn,and actually a measure space (Sm, o) is defined (&,o) = ind.lim (sn,u),

where "ind-lim" denotes the inductive limit. G.6) The limit can be identified with the white noise measure space ( E * , p )with the characteristic functional e-;11511', where E* is the space of generalized functions. In parallel with Gaussian case, we can form a Poisson noise by an approximation where optimality (maximum entropy) and invariance under the symmetric group are effectively used.

P.0) Start with a probability space (A1, p l ) , where A1 is defined as A1 = {(z1,z2);z1+z2 = 1) and

p1

is the Lebesgue measure,

361

P.l) Define a probability space ( A , , p , ) , where A, is an Euclidean n -simplex, n+l

A, = {(XI,...,zn+l),xi 2 0,

C X= 1)~ c Rn+'. 1

P.2) Define lr, to be the projection of A,+, down to A, which is a side simplex of A,,,, determined as follows. Given a side simplex A, of A,+,. Then, there is a vertex of A,+, which is outside of A,, let it be denoted by v,. The projection 7rn is a mapping defined by lr,

:

v,x -+ x.

where 21,2 is defined as a join connecting the v, and a point x in An * P.3) We introduce a probability measure pn+l under the requirements that pn+1(Tl1(B)) = pn(B),

B c An

and that p,+1 has maximum entropy. P.4) Since there is a freedom to choose a side simplex of A,+l, the rquirements on p,+1 in P.3), the measure space (A,+,, p,+l) should be invariant under symmetric group S(n 2) which acts on permutations of the coordinates xi.

+

P.5) We can form {(A,,p,)} successively by using the projection {lr,}, where p, is to be the uniform probability measure on A,. Set

Am = &An, B(A,) = a-field generated by U An, An E B(An), P(A) = C P n p n ( A n An)

A E am).

n

P.6) We have not yet specified p , in P.5). Thus we now set p , to be e-x X R X > 0, which has been determined in [8]. In fact, there ,!, we have used the reproducing property of a Poisson distribution, or equivalently Poisson variables can be imbedded in a Poisson process.

Remark 1. In P.6) above, we set p , = e-X$, X > 0, referring [8]. On the other hand, the natural reason of choosing p , in such a way can be also observed in the author's previous paper [6].

362

Theorem i) The measure space (An, pn) is isomorphic to the measure space (An,pn) defined in Section 1. ii)The weighted sum (Am,p ) of measure spaces (An,p n ) , n = 1,2,. . . is identified with the Poisson noise space. Proof. According to the facts P.l) to P.6), the theorem is proved. What have been investigated are explained as follows. Basic noises, that is Gaussian and Poisson noises, have 1) optimality in randomness, which is expressed in terms of entropy, and 2) invariance under the transformation group; each noise has its own characteristic group (one is continuous and another is discrete).

Remark 2. So far the parameter space is taken to be [0,1]. The characteristic properties that have been discussed can be generalized to the case of Rd-parameter with modest modifications. 3. Fractional power distribution in terms of Poisson noise

This section is devoted to a breif interpretation of fractional power distribution, since they are often used in information sociology application. Define Pu(t)= uP(t).Then the sample function of Pu(t)is of the form S

where s runs through a finite set depending on w . The characteristic functional is given by

where 00

~ ( =t A) /

- 1)dt 0

and +([) is called the 1c, function. Let Y ( t ) be a superposition of independent Poisson noises Pu(t)with which will be determined later. Then we have weight f(u),

363

Since Pu7s are independent, the $-functional of Y(<) can be expressed in the form

x SJ(exp[i~(t)u]- l ) f ( . u ) d u d t . where u extends over R - (0). Assumming self-similarity of Y ( t ) ,we come to the following result :

f(u)= And then we have to check the integrability as follows. i) For 0 < Q < 1, there is no problem of integrability. ii) For Q = 1 (Cauchy process) and 1 < Q < 2, it is necessary to compansate the integrand of the $-functional in such a way that & ( t b - 1 + &t)U - 1Note that the term i<(t)ucorresponds to a constant term. iii) For Q > 2, out of consideration. We have finally come to a “stable noise” with some exponent a.Thus, satisfying the it is now possible to analyse a random phenomena of Y([), self-similarity, by representing them as superpositions of various elementary Poisson noises. 4. Concluding remark

A stable distribution is often called a fractional power distribution in applications because of its tail behaviour. A study of a stable distribution is reduced to that of a stable process. If a statistics of actual data is subject to a fractional power distribution (see [5]), then one may think of characteristics that appear in the component Poisson noises.

References 1. T . Hida and H. Nomoto, Gaussian measure on the projective limit of spheres. Proc. Japan Acad. 40 (1964), 301-304. 2. T. Hida, Stationary stochastic processes, Princeton Univ. Press, 1970. 3. T. Hida and Si Si, Innovation Approach to Random Fields. An Aplication of White Noise Theory. World Sci.Pub. 2004. 4. J.R. Klauder and E.C.G. Sudarshan, Fundamentals of Quantum Optics. Benjamin. 1968.

364 5. S. Kumon, Introduction to Information Sociology. N T T Pub. 2004. (In Japanese) 6. Si Si, Effective determination of Poisson noise, Infinite Dimensional Analysis and Quantum Probability, Vol. 6 (2003), 609-617. 7. Si Si, Note on Poisson noise, Quantum Information and Complexity, World Sci. Pub. 2004, 411-425. 8. Si Si, A. Tsoi, Win Win Htay, Invariance of Poisson noise, Stochastic Analysis: Classical and Quantum. World Sci. Pub. 2005, 199-210.

ON TWO CONJECTURES IN SEGAL-BARGMANN ANALYSIS

STEPHEN BRUCE SONTZ' Centro de Investigacibn en Matemciticas, A . C. (CIMAT) Guanajuato, Mexico Email: [email protected]

I present a review of two conjectures, each one about an inequality in the SegalBargmann space in the case when the dimension of the underlying phase space is finite. One concerns a reverse log-Sobolev inequality, and the other concerns a Hirschman inequality. If either conjecture is true, it allows us to prove the corresponding inequality in the case of infinite dimensional phase space.

1. Notations and Definitions

In this section I will present some of the basic notations and definitions in Segal-Bargmann analysis. For more details see Refs. [l]and [2]. We work inside of the space 7l(C") of all holomorphic f : C" + C. The integer n 2 1 is called the dimension. Bargmann [l]defined

B~ := L~ (c", p g ) n x(c"), known as the Segal-Bargmann space. (Also see Ref. [3].) I am identifying C" with the Euclidean space R2". I am also using the following notations for any Euclidean space R'. First, 1zI2= (C!=,zp) is the Euclidean norm of a vector z = (21,. . . ,5 ' ) in R'. Next, dpk(z) is Lebesgue measure on R ' and, finally, d&(z) := 7r-k/2e-1z12dpL(z) is the standard (normalized) Gaussian measure on R'. It turns out that ,132 is closed in the Hilbert space L2 (C",&?) and so is itself a Hilbert space. Next the Segal-Bargmann transform A : L2 (R",&) + ,132 is defined as an integral kernel operator, namely

*Research supported in part by CONACyT, Mexico, grants 32146-E and P-42227-F

365

366

+

+

L2 (R",pE) and z E C". Also: z2 := 21" ... z: and . . . Z,IC" for z = (21,. . . , z,) E C" (the phase space) and IC = ( X I ,. . . ,2,) E R" (the configuration space). This modifies Bargmann's definition [l],since he defined his transform on L2 (R", pE), which only differs from L2 (R", p z ) by a unitary change-of-measure transformation. For notational convenience, I also introduce G2 := L2 (R", p;). So, the Segal-Bargmann transform is a linear map A : Gz + B2. The fundamental theorem here is that A is a unitary transform from G2 onto B2. (See Ref. [l].) For every f E L2(R, p ) , where (R, p ) is a probability measure space, one defines its entropy by for all

z * IC

f

E

+ +

:= ~ 1 x 1

Here log denotes the base e logarithm and 0 log 0 := 0. Jensen's inequality then immediately shows that S(f) 2 0 for all f E L 2 ( R , p ) . However, S(f)= +oo can happen. This concept of entropy goes back to information theory as developed by Shannon [4]. There is a self-adjoint operator N (the energy or number operator) that can be defined in both of the spaces G 2 and B2 using n

j=1

where aj* (resp. a j ) is the creation (resp. annihilation) operator for the j t h degree of freedom, defined in both G 2 and 232. See Refs. [2] and [5]. The point here is that the Segal-Bargmann transform A intertwines all of these operators (a;, aj and hence also N ) . Since N = N* 2 0 , we also have quadratic forms (defined in G2 and B2) associated with N . So A preserves these forms as well. For more details, including domain considerations, see Ref. [l].Also, see Ref. [5]. 2. The Two Conjectures

The reverse log-Sobolev inequality in the Segal-Bargmann space is ($7

+

N+)

I CS(+> + wc>ll+ll;

(1)

for c > 1 and all E B2. Here K ( c ) is an explicitly given (although not necessarily optimal) finite constant that is proportional to the dimension n. (See Refs. [5, 61 and [7].) The expression on the left hand side of (1) is really the quadratic form associated to N . Note that this quadratic form is defined for all E 232, though it is sometimes equal to +oo. This quadratic

+

367

form is called the energy. The terminology “log-Sobolev inequality” comes from Gross [8]. The Hirschman inequality for the Segal-Bargmann transform is

S(f)I a S ( A f )+ 4a)Ilfll;

(2)

for all f E Q2. Here .(a) is an explicitly given (though not necessarily optimal) finite constant; moreover, it can be taken to be proportional to log (I IAl I P +, q ) which, in turn, is bounded by an expression proportional to the dimension n. Here p and q are any pair of Lebesgue indices satisfying 1 I p < 2, 1 I q < 2 and p > 1+ q / 2 . Also, ~ ~ A ~ is~ the P +operator q norm of A with respect to the L P norm in its domain and the Lq norm in its codomain. (See Refs. [5] an[9].) Hirschman’s original inequality in Ref. [lo] is for the Fourier transform. Either inequality immediately generalizes to the infinite dimensional case (roughly when the dimension n = co) if the optimal constant for the norm term does not depend on n. The simplest way for this to happen would be that the constant of the norm term is zero. 0 0

Conjecture 1: For some c the optimal constant in (1)is K ( c ) = 0. Conjecture 2: For some a the optimal constant in (2) is .(a) = 0.

These two conjectures are not independent. If Conjecture 1is true, then Conjecture 2 is true too. Here is the argument: 1

-W) I ( f , N f )= (Af,”) 2

54A.f)

for all f E Q2. The first inequality is the log-Sobolev inequality due to Gross [8]. The equality is the fact, alluded to above at the end of the previous section, that A preserves the quadratic forms associated to N . Finally, the last inequality is (l),given that Conjecture 1 is true. This proves (2) with a = 2c for some c > 1 and ~ ( u = ) 0, that is, it shows Conjecture 2. While I can find no way to prove the converse, it seems reasonable to study these two conjectures together. First, I want to give an argument that the norm term really is there in (I), namely that the optimal constant of the norm term there is strictly positive. The reader should beware that this argument has a gap in it! First, I take any 4 in Dom(N1l2), the domain of N1/2,with (4, l)a, = 0. Here 1 denotes the constant function. Then I form $J = 1 A 4 for any X E R and study the asymptotics in small X of the energy and the entropy for this choice of $J. So,

+

Il$Jl ;

+ A49 1+

= ($J,$J)= (1

+ X2Il4llE

=1

368

since (1,l)= 1 and

(4,l) = 0.

Next, using N * = N and N 1 = 0 , we have

($,wJ) = ( l + X 4 , N ( l + X 4 ) ) =X2(4,N4)=X211N1/24(1;, where N1/2is the positive square root of N . To expand S($), we first note that Il$ll;log

Ir$ll;

= (1

+ X211411;)

= (1 + X211411;)

1% (1 + X211411;)

(X211411$ + 0 (A4))

+ 0 (A4)

= ~211411;

by the Taylor expansion around X = 0 of the log term. Next,

1$12

+ = x ( 2 ~ e 4+ ) x2 (1412+ 2 ( ~ e 4 )+~o) +

log l$I2 = (1 Xqq2 log (1 X4I2

,

( ~ 3 )

again by a Taylor expansion around X = 0 of the function gZ : R defined by := 11

(3)

-, R

+ A4(z)l2 log 11+ X4(Z)l2,

(4)

where X E R is variable and where 4 and z are fixed. This implies that

S($) = x2

/ (!+I2

+2

+ 0 (A3)

-

(X211+l122

+ 0 (A4))

since J 2 ~ e 4= J ( + + f > = O + O = o using ($,I)= 0. Since we have the estimate

(5) 1412,

S ( $ ) I2X211411$ + 0 ( A 3 ) . Suppose Conjecture 1 is true, that is, that for some real number c we have ($3

for all $ E

B2.

N $ ) L q$J>

(6)

Taking $J = 1+ A 4 as above, we would then have X211N1/2$11;L 2X2clI+II;

+ 0 (A3)

(7)

for any 4 in Dom(N1/2) that is orthogonal to 1. Now dividing by X2 and then taking the limit X + 0, we obtain

llN1/2411;

L2cll4ll;,

(8)

which implies that N1/2is a bounded linear map. But it is known that N1/2is not bounded, since its spectrum is unbounded. So this contradicts the assumption that (6) holds. So this would seem to imply that there must be a nonzero norm term in the reverse log-Sobolev inequality (1). By the

369

way, note that this argument is an adaptation to the present situation of the argument of Rothaus [ll]and Simon [12] that shows that the existence of a log-Sobolev inequality implies the existence of a spectral gap. Nonetheless, as I noted above, this argument has a gap in it. And that gap is in the step that goes from (3) to (5). To see what is happening here, let's write (3) as 1+(z)l2 log 1+(.)12

+ X W I 2 1% I1+ X W I 2 (2Re4(z))+ x2 (14(2)1~ + 2 ( ~ e $ ( z ) )+~ )~

= I1 =

(9) ( zA),,

where z E C" and X E R. Of course, we can use this last equation to define the expression R ( z ,A), in which case the equality is trivial. The nontrivial content of Taylor's theorem is that R(z,X) is 0 (A3), that is, R(z,X)/X3 is bounded in a neighborhood of X = 0. But z E C" plays the role of a parameter, and so Taylor's theorem gives no a priori information on how this bound on R ( z ,X)/X3 depends on z. Now to pass to (5) I integrate both sides of (9) with respect to p$. Since the first two terms on the right hand side of (9) are integrable (this following from q5 E Bz),we see that R ( z ,A) is integrable if and only if l+(z)I2log 1 + ( ~ ) 1 ~ is integrable. The latter is a p g ) for some E > 0. rather weak condition; it is implied by $ E L2+' (Cn, (See Ref. [6].) Let's note that (9) comes from these elementary calculus results which follow from the definition (4)of the function g,(X) given above: g:(X)

= (2ReW

+ 2Xl4(Z)l2) log I1 + X W I 2 + ( 2 R e W + 2XI+(z)l2)

and

+

so that g,(O) = 0 and gL(0) = 2Re+(z) and g:(O) = 214(2)l2 4(Req5(~))~. However, even knowing that R(z,X)/X2 -+ 0 as X -+ 0 pointwise for each z in Cn is not sufficient to conclude that

as X -+ 0. And the term called 0 ( X 3 ) in (5) is really just J d p ( z ) R ( z , X ) and the only property we subsequently use of this term is that X-' Jdp(z)R(z, A) -+ 0 as X -+ 0 (to go from (7) to (a)), where p means p g . But Taylor's theorem also allows us to write a formula for the remainder term R ( z , A). Will this help us? Well, Taylor's theorem gives us

370

a formula provided that a certain hypothesis is met. And that hypothesis is that gz be of class C3 in some open interval containing 0 and A. (See Ref. [13].) Now it turns out that gt is C1 on all of R, even in the “singular” case when 1 X+(z) = 0 (i.e., when $ ( z ) = -1/X for some real A) since the singularity of the log term in g: is wiped out by its coefficient being zero. Note also that if z is a zero of 4 (i.e., $ ( z ) = 0), then g,(X) = 0 is Coo. However, the singularity of the log term in :g is not controlled by its coefficient, provided that +(z) # 0. And so, gz is not C2 at -1/4(z), provided this expression makes sense and is a real number, namely that 4 ( z ) E R\{O). In the contrary case (namely that 4 ( z ) E C \ R or 4 ( z ) = 0) we have that gz(X) is C” in X E R. The point here is that

+

provided that gz is of class C3 in some neighborhood containing 0 and A. (See Ref. [13].) But if the singularity -1/4(z) falls between 0 and A, then gz is not even C2 between 0 and X and the remainder term R ( z ,A) (though well-defined and even integrable) can not be analyzed using the formula (lo), since that formula need not apply when the hypothesis of Taylor’s theorem does not hold. But (10) is a valid formula when + ( z ) E C \ R or 4 ( z )= 0. So how do we analyze R(z,X) in (9) and (lo)? If we want to use the dominated convergence theorem (this being the most obvious tool to apply) we have to show that the family XT2R(z,Xi) is uniformly bounded (in i) by an integrable function (in z ) for some sequence X i E R \ (0) that satisfies X i --t 0 as i -+ 00. Now the behavior on the measure zero set of z E C” such that 4 ( z ) E R \ (0) is not the only problem. The set of z such that 4 ( z ) is near the real axis is also a problem, as we are going to see now. Substituting the explicit formula (more elementary calculus),

for gL3’(X) into (lo), where z satisfies $ ( z ) $! R, we find the integrand to be a rational function in the variable t of integration, namely, a degree 5 polynomial in t divided by a degree 4 polynomial in t. Such integrals can be explicitly and exactly evaluated (more elementary calculus) - though the details can be hair raising! But the point is that the denominator of that rational function is

371

which is the square of a quadratic polynomial in t. The discriminant of that quadratic polynomial is

A := ( 2 X R e 4 ( ~ )-> ~4 . (X21$(z)I2) . 1= - 4X2 ( r m $ ( ~I) 0. )~ Now the explicit formula for the integral in (10) involves dividing terms with factors of + ( z )and +(z)* (and possible log factors) by powers of lAl. (Check a table of integrals, or do it yourself!) So even when we restrict ourselves to $ ( z ) E C \ R, there is no way to control (10) in terms of integral conditions on to get a function integrable in z , since the denominator goes to zero (as I m $ ( z ) + 0) in a way that the numerator can not cancel in general. So we are stuck with a gap; there is no justification here of the step from (3) to (5). Having tried to plug the gap (and thinking it would be trivial) and finding that some very standard tools of analysis are of no avail, I have come to believe it is because Conjecture 1 is true! Of course, this is not at all a proof of Conjecture 1. And what about Conjecture 2? From Ref. [9] we have 1 I IIAIIP+q < 00 for 1 I p < 2, 1 5 q < 2 and p > 1+ q / 2 . If there is just one such pair p , q with IIAIIp+q = 1, then we would have a proof of Conjecture 2. Is this plausible? One bit of bad news here is that there do exist pairs p0,qo satisfying 1 5 po < 2, 1 I qo < 2 and po > 1 q 0 / 2 such that IIAIIPo+qo2 Cn > 1, where C > 1 does not depend on n (though it may depend on p 0 , q o ) . This is shown by studying the action of A on trial functions of the form fa(z) := exp(-az2), for 2 E R”. For example, po = 1.6 and qo = 1.1is one such pair. (See Ref. [9].) But there are other pairs p l , q1 satisfying 1 5 p l < 2, 1 I q1 < 2 and p l > 1 q1/2 such that the “norm” of A when restricted to the functions fa is 1, i.e.,

+

+

+

where a is restricted so that fa is in Lpl(Rn, &). If these functions fa are the “measure” of the norm of the transform A (i.e., Cpl,ql= IIAllpl-+ql), then such pairs p 1 , q l have the property that IIAllpl+ql = 1. While it is not known if the functions fa have this property, this would not be so different from the results of Lieb [14] on integral kernel transforms with a Gaussian kernel function. Even though the division of the region 1 I p < 2, 1 5 q < 2 and p > 1+ q / 2 into the two subregions where Cp,q= 1 and Cp,q> 1, respectively, is explicit, it is algebraically quite complicated and is not very enlightening (at least not to this author as he is writing this). But this could also be the division between the regions where llA\lp+q = 1and ~ ~ A ~ ~>p1,+respectively. q If this is so, then Conjecture 2 is true. Also, as

372

noted before, Conjecture 2 is true (independent of the nature of IIAllp+q) provided that Conjecture 1 is true. 3. Concluding Remarks

Despite the fact that these two conjectures remain open problems, the above analysis leads me to believe that they are true. What is needed here apparently is a fresh idea or a new tool to resolve the issue.

References 1. V. Bargmann, “On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part I,” Commun. Pure Appl. Math. 14, 187-214 (1961). 2. B.C. Hall, Holomorphic methods in analysis and mathematical physics, In: First Summer School in Analysis and Mathematical Physics, Eds. S . PBrezEsteva and C. Villegas-Blas, Contemp. Math., Vol. 260, pp. 1-59, Am. Math. SOC.,Providence, 2000. 3. I.E. Segal, Mathematical problems of relativistic physics, In: Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. 11, Ed. M. Kac, Lectures in Applied Mathematics, Providence, Am. Math. SOC.,1963. 4. C.E. Shannon, The Mathematical Theory of Communication, University of Illinois Press, Urbana, 1949. 5. S.B. Sontz, Recent results and open problems in Segal-Bargmann analysis, In: Finite and Infinite Dimensional Analysis in Honor of Leonard Gross, Eds. H.-H. Kuo and A.N. Sengupta, Contemp. Math., Vol. 317, pp. 203-213, Am. Math. SOC.,Providence, 2003. 6. S.B. Sontz, “A reverse log-Sobolev inequality in the Segal-Bargmann space,” J. Math. Phys. 40, 1677-1695 (1999). 7. F. Gala-Fontes, L. Gross, S.B. Sontz, “Reverse hypercontractivity over manifolds,” Ark. Mat. 39,283-309 (2001). 8. L. Gross, “Logarithmic Sobolev Inequalities,” A m . J. Math. 97, 1061-1083 (1975). 9. S.B. Sontz, “Entropy and the Segal-Bargmann transform,” J. Math. Phys. 39, 2402-2417 (1998). 10. 1.1. Hirschman, Jr., “A note on entropy,’’ Am. J. Math. 79,152-156 (1957). 11. O.S. Rothaus, “Diffusions on compact manifolds and logarithmic Sobolev inequalities,” J. Fbnct. Anal. 42, 102-109 (1981). 12. B. Simon, “A remark on Nelson’s best hypercontractive estimates,” Proc. A m . Math. SOC.55, 376-378 (1976). 13. Y. Choquet-Bruhat, et al., Analysis, Manifolds and Physics, Rev. Ed., North Holland, Amsterdam, 1982. 14. E.H. Lieb, “Gaussian kernels have only Gaussian maximizers,” Invent. Math. 102, 17s208 (1990).

NOTE ON QUANTUM MUTUAL TYPE ENTROPIES AND CAPACITY

NOBORU WATANABE Department of Information Sciences, Tokyo University of Science Noda City, Chiba, 278-8510, Japan E-mail: [email protected]. tus. a c j p The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983. In this paper, we compare with mutual entropy-type measures and show some resuls for quantum capacity.

1. Introduction The development of communication theory is closely connected with study of entropy theory. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers [5, 61. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya [8] invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. Recently Shor [20] and Bennet et a1 [2, 3, 18, 191 took the coherent entropy and defined the mutual type entropy to discuss a sort of coding 373

374

theorem for communication processes. In this paper, we compare with mutual entropy-type measures and show some resuls for quantum capacity for the attenuation channel. 7-1 2. Quantum Channels The concept of channel has been carried out an important role in the progress of the quantum communication theory. In particular, an attenuation channel introduced in [8] is one of the most inportaint model for discussing the information transmission in quantum optical communication. Here we review the definition of the quantum channels. Let 7-11, 'Fl2 be the complex separable Hilbert spaces of an input and an output systems, respectively, and let B ( ' H k ) be the set of all bounded linear operators on x k . We denote the set of all density operators on 'Flk (k = 1,2) by

{ p E B ( a k ) ; p 2 0,trp = I}.

6( x k )

(1)

A map A* from the quantum input system to the quantum output system is called a (fully) quantum channel. (1) A* is called a linear channel if it satisfies the affine property, i.e.,

2. A* : e(7-11) 4 e('7-l~) is called a completely positive (CP) channel if its dual map A satisfies n

Bj*A(A;&) Bk 2 0

(2)

j,k=l

for any n E N,any Bj E B('FI1) and any Ak E B(7-12),where the dual map A : B (X2) -+ B (7-11) of A* : 6 (7-11) -+ 6 ( 7 - l ~ )satisfies trpA ( A ) = trA* ( p ) A for any p E 6 (7-11) and any A E B (7-12). 2 .I. A t t e n u a t i o n channel

Let us consider the communication processes including noise and loss systems. Let I c 1 , Ic2 be the complex separable Hilbert spaces for the noise

375

and the loss systems, respectively. The quantun communication channel

b = 10) (01 and .rrG (.) = VO (.) vO*

AC(p) = trlc2nG ( p @ t o ) ,

(3)

is called the attenuation channel, where 10) (01 is vacuum state in 7-ll and VOis a linear mapping from 7-l1 @ K1 to 7 - l ~@ K2 given by

for any In) in 7-11 and a,P are complex numbers satisfying Ia12t ]PI2 = 1. 2 r] = la1 is the transmission rate of the channel. nt;is called a beam splittings, which means that one beam comes and two beams appear after passing throughrt;. This attenuation channel is generalized by Ohya and Watanabe such as noisy optical channel [14, 151. After that, Accardi and Ohya [l]reformulated it by using liftings, which is the dual map of the transition expectation by mean of Accardi. It contains the concept of beam splittings, which is extended by Fichtner, Freudenberg and Libsher [4] concerning the mappings on generalized Fock spaces. For the attenuation channel A t , one can obtain the following theorem: Theorem 2.1. The attenuation channel At; is described by 00

(P) =

C OiVOQpQ*&*Oi*, i=O

(5)

where Q = CEO (IN) €9 lo)) (yil, Oi = C,"=,Izk) ((a1€9 (il), {Ivi)) is a CONS inX1, { I z k ) } is a CONS in 7 - l ~and { li)} is the set of number states in K2. 3. Ohya Mutual Entropy and Capacity The quantum entropy was introduced by von Neumann around 1932 [7], which is defined by

s(p) E -trp

log p

for any density operators p in S (XI) . It denotes the amount of information of the quantum state p. In order to define such a quantum mutual entropy, we need the quantum relative entropy and the joint state, which is called a compound state, describing the correlation between an input state p and the output state A*p through a channel A*. For a state p E 6(7-l1), p = EkAkEk,

(6)

376

is called a Schatten decomposition [17] of p, where Ek is the one-dimensional projection associated with &. This Schatten decomposition is not unique unless every eigenvalue is non-degenerated. For p E B('H1) and A* : B('H1) + B(IFln), the compound states are define by (TE =

C X,E,

8 A*E,,

a0 = cp

(7)

~*cp.

n

The first compound state, which is called a Ohya compund state associating the Schatten decomposition p = &&&, generalizes the joint probability in classical dynamical system and it exhibits the correlation between the initial state p and the final state A*p. Ohya mutual entropy with respect to p and A* is defined by

where S ( ( T E ,(TO) is Umegaki's relative entropy [21]. I ( p ;A*) satisfies the Shannon's type inequality : 0 5 I ( p , A*) 5 min { S ( p ) , S ( A * p ) } . 3.1. Q u a n t u m capacity

The capacity means the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of the fully quantum mutual entropy with respect to a certain subset of the initial state space. The capacity of purely quantum channel was studied in Let S be the set of all input states satisfying some physical conditions. Let us consider the ability of information transmission for the quantum channelA*. The answer of this question is the capacity of quantum channel A* for a certain set S c S('H1) defined by 11112113314.

C t (A*) = sup { I ( p ; A*) ;p

E S} .

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When S = S ('HI), the capacity of quantum channel A* is denoted by C, ( A * ) . Then the following theorem for the attenuation channel was proved in [16].

Theorem 3.1. For a subset Sn E { p E S ('HI) ;dims ( p ) = n } , the capacity of the attenuation channel A: satisfies

where s ( p ) is the support projection of p.

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When the mean energy of the input state vectors {Id,) can } be taken 2 infinite, i.e., limT+m 1 d k l = 00, the above theorem tells that the quantum capacity for the attenuation channel A: with respect to S, becomes logn. It is a natural result, however it is impossible to take the mean energy of input state vector infinite. 3.2. Semi-classical mutual entropy

When the input system is classical, the state cp is a probability distribution and the Schatten-von Neumann decomposition is unique with delta measures 6, such that cp = C X ,S ,., In this case we need to code the classical state cp by a quantum state $, whose process is a quantum coding described by a channel I?* such that r*d, = $, (quantum state) and$ G r*cp= C , An$,. Then Ohya mutual entropy I (cp; A* o r*)becomes Holevo’s one, that is,

I (cp; A* o r*)= S (A*$) -

C XS,

(A*$,)

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n

when

C,, XS,

(A*$,) is finite.

4. Q u a n t u m M u t u a l Type Entropies

Recently Shor [20] and Bennet et a1 [2, 31 took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for quantum communication. In this section, we compare these mutual types entropy. Let us discuss the entropy exchange [18]. For a statep, a channel A* is denoted by using an operator valued measure { A j } such as

A* (.)

3

,

3

A; . Aj ,

which is called a Stinespring-Sudarshan-Kraus form. Then one can define a matrix W = (Wij)i,jwith

by which the entropy exchange is defined by Se(p,A*) = -trW log W.

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By using the entropy exchange, two mutual type entropies are defined as follows:

IC (P; A*)

S (A*p) - S e ( P , A*) 7

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378 It ( p ; A * ) E S ( P )

+

(A*p) - S e ( P , A*) *

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The first one is called the coherent entropy IC ( p ; A*) [19] and the second one is called the Lindblad entropy 11,(p; A*) [3]. By comparing these mutual entropies for quantum information communication processes, we have the following theorem [16]: Theorem 4.1. Let { A j } be a projection valued measure with dimAj = 1. For arbitrary state p and the quantum channel A* (.) E C jAj .AT, one has (1) 0 I I (p; A*) 5 min { S ( p ) ,S (A*p)}(Ohya mutual entropy), (2) IC ( p ;A*) = 0 (coherent entropy), (3) I t (p; A*) = S ( p ) (Lindblad entropy). For the attenuation channel AT;, one can obatain the following theorems [16]: Theorem 4.2. For any state p = C, A, In) (nI and the attenuation chan2 2 nel AT; with la1 = [PI = one has (1) 0 5 I ( p ; A:) 5 min { S (p) ,S (AT;p)} (Ohya mutual entropy), ( 2 ) I c ( p ; AT;) = 0 (coherent entropy), (3) 11,(p; AT;) = S ( p ) (Lindblad entropy).

a,

Theorem 4.3. For the attenuation channel AT; and the input statep = A ( 0 )(01 (1 - A) 10) (01, we have (1) 0 5 I ( p ; AT;) 5 min { S ( p ) , S (AT;p)}(Ohyamutual entropy), (2) -S ( p ) 5 I c ( p ; AT;) 5 S ( p ) (coherent entropy), (3) 0 5 II,(p; AT;) 5 2s ( p ) (Lindblad entropy).

+

Therem 4.3 shows that the coherent entropy IC ( p ; A:) takes a minus value for IaI2 < IPI2 and the Lindblad entropy 11,( p ; AT;) is grater than the von Neumann entropy of the input state p for la12 > \PI2. From these theorems, Ohya mutual entropy I (p; A*) only satisfies the inequality held in classical systems, so that Ohya mutual entropy can be a most suitable candidate as quantum extension of the classical mutual entropy. References 1. Accardi, L., and Ohya, M., Compond channnels, transition expectation and liftings, Appl. Math, Optim., 39, 33-59 (1999). 2. Barnum, H., Nielsen, M.A., and Schumacher, B.W., Information transmission through a noisy quantum channel, Physical Review A, 57, No.6, 4153-4175 (1998).

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3. Bennett, C.H., Shor, P.W., Smolin, J.A., and Thapliyalz, A.V., Entanglement-Assisted Capacity of a Quantum Channel and the Reverse Shannon Theorem, quant-ph/0106052. 4. Fichtner, K.H., Freudenberg, W., and Liebscher, V., Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematik und Informatik, Math/ Inf/96/ 39, Jena, 105 (1996). 5. Holevo, A.S., Some estimates for the amount of information transmittable by a quantum communication channel (in Russian)}, Problemy Peredachi Informacii, 9, 3-11 (1973). 6. Ingarden, R.S., Kossakowski, A., and Ohya, M., Information Dynamics and Open Systems, Kluwer, 1997. 7. von Neumann, J., Die Mathematischen Grundlagen der Quantenmechanik, Springer-Berlin, 1932. 8. Ohya, M., On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770-774 (1983). 9. Ohya, M., Some aspects of quantum information theory and their applications t o irreversible processes, Rep. Math. Phys., 27, 19-47 (1989). 10. Ohya, M., and Petz, D., Quantum Entropy and its Use, Springer, Berlin, 1993. 11. Ohya, M., Petz, D., and Watanabe, N., On capacity of quantum channels, Probability and Mathematical Statistics, 17, 179-196 (1997). 12. Ohya, M., Petz, D., and Watanabe, N., Numerical computation of quantum capacity, International Journal of Theoretical Physics, 37, No.1, 507-510 (1998). 13. Ohya, M., and Watanabe, N., Quantum capacity of noisy quantum channel, Quantum Communication and Measurement, 3, 213-220 (1997). 14. Ohya, M., and Watanabe, N., Foundatin of Quantum Communication Theory (in Japanese), Makino Pub. Co., 1998. 15. Ohya, M., and Watanabe, N., Construction and analysis of a mathematical model in quantum communication processes, Electronics and Communications in Japan, Part 1, 68, No.2, 29-34 (1985). 16. Ohya, M., and Watanabe, N., Comparison of mutual entropy - type measures, TUS preprint. 17. Schatten, R., Norm Ideals of Completely Continuous Operators, SpringerVerlag, 1970. 18. Schumacher, B.W., Sending entanglement through noisy quantum channels, Physical Review A, 54, 2614 (1996). 19. Schumacher, B.W., and Nielsen, M.A., Quantum data processing and error correction, Physical Review A, 54, 2629 (1996). 20. Shor, P., The quantum channel capacity and coherent information, Lecture Notes, MSRI Workshop on Quantum Computation, 2002. 21. Umegaki, H., Conditional expectations in an operator algebra IV (entropy and information), Kodai Math. Sem. Rep., 14, 59-85 (1962).