Quantum Information III

90 sum as follows: OC

GO

*> = £ '»(/») = £ < : •*":- /»>• / » e C'-

(i)

where / „ is the multiple Wiener integral of order n. : •®" : is the Wick tensor [6], and the sub-index c denotes the complexificatioii. In addition, the (L 2 )-norm of ip is given by 1/2

IMIo = ( X>!|/ n |* ' \n=0

Let C + i i / 2 be the set of positive continuous functions u on [0, oo) such that hm

log u(r) ' = oo.

We need to consider the following conditions on a function u € C + 1 / 2 : (Gl) (G2) (G3)

u is increasing and u(0) = 1. l i m s u p , . ^ ^ r " 1 logu(r) < oo. log«(x 2 ) is convex for .;; G [0,oc).

These G-conditions are stated as U-conditions in the papers [1] [2] [3]. The only difference is between conditions (G2) and (U2). Condition (U2) says that limr-Kx; r _ 1 log u(r) < oo. However, it can be replaced by the weaker condition (G2) in these papers. Let « £ C + ,i/2- Its Legendre transform tu and dual Legendre transform M* are defined as in [1] by f.u(n)

= inf

'-^-,

n = 0,1,2....,

(2)

«*(r) = s u p ^ — - , re[0,oo). (3) ,>o uys) We have the following facts about Legendre and dual Legendre transforms: Fact 1. If « e C+,1/2 satisfies (Gl) (G3), then l i m , , - ^ A,(n) 1 A ' = 0. (See Lemma 3.6 in [1] or Theorem 2.3 in [3].) By this fact, the following function £ „ is an entire function oo

Cu(r) = Y,Un)rn.

(4)

Fact 2. If u e C"+li/2 is increasing and satisfies (G3), then u and Cu are equivalent, namely, there exist constants K\, Ki, a.\. 0,2 > 0 such that AVu(air) < £„(»•) < K2u(a2r),

Vrg.[0,oo).

(5)

(See Theorem 3.13 in [1] or Theorem 2.6 in [3].) Fact 3 . If u e C+a/2 satisfies (G3). then l i m , , - ^ (fu(n)(n\)2) _ 1 / " = 0. (See Lemma 3.2 in [3].) By this fact, the following function £ * is an entire function

* "

=

|wr'

(6)

91 Fact 4. If u £ C + 1 / 2 - then u* belongs to C+.1/2 and is increasing and satisfies (G3). If in addition u satisfies ( G l ) , then u* also satisfies ( G l ) . (See Lemma 4.5 in [1] or Theorem 2.7 in [3].) Fact 5. If u e C + , 1 / 2 satisfies ( G l ) (G3), then («")* = ?t on [0. oo). (See Theorem 4.7 in [1] or Theorem 2.9 in [3].) Fact 6. If u £ C+,1/2- then M* and £ „ . are equivalent. If in addition u satisfies (G3), then £„• and Cf are equivalent. (See above Facts 2 and 4, and Theorem 4.10 in [1].) Now, let u e C+.1/2 t>e a fixed function satisfying conditions ( G l ) (G2) (G3). For each p > 0 and for

S^) 1 7 - 1 '] •

(7)

where ^„ is the Legendre transform of u defined by Equation (2). Let [£p}n = {

0 } . This space [£]u is the space of test functions on £' given by the growth function u. Its dual space [£\*L is the space of generalized functions on £'. By using conditions (a) and (G2) we can show that [£p]u C (L2) for all large p (see Section 3 in [3] for the proof.) Hence [£]„ C (L2). Moreover, Condition (b) implies that [£]„ is a nuclear space. The space (L2) can be identified with its dual space and so we get a Gel'fand triple [£}u c (L 2 ) C [£]'u. Note that this Gel'fand triple is exactly the CKS-space [£]„ C (L2) C [£]* in [5] given by the sequence

«,.(«) = - 7 7 W

n = 0,l,2,....

(8)

Let £ £ £c. The renormalized exponential function : e ^ ' ' ^ : is defined by OC

-

n=0

OO

'

^

»i=0

Use Equations (6) and (7) to get

ii-wMuu = ( E ^ 4 ^ i e ' ) 1 / 2 = ^(^)1/2-

m

Hence ||: e < , f ^: \\PtU < 00 for all p > 0 and so : e<'"^ : £ [£]„ for any £ £ £c. The S-transform of a generalized function $ in [£]* is the function defined by 5 * ( 0 = «*,:e(-'°:)),

C G &.

where ((•,•)) is the bilinear pairing of [£]* and [£]„. Let $ £ [£]*. By the continuity of $ there exist constants A",p > 0 such that

K(*^»l <-KIMU-

vP e[£]„.

f

Put 9? = :t'('- ): and use Equation (9) to get

|s$«)|
vce£,,

(10)

92 But from the above Fact 6 the function £ * is equivalent to u*. Hence the inequality in Equation (10) is equivalent to the existence of constants K.a.p > 0 such that

|5*(OI<«"«'H^) 1/2 ,

Vfefc.

This is the growth condition in the following theorem due to Asai-Kubo-Kuo (see Theorem 3.4 in [3].) T h e o r e m 2 . 1 . Assume that, u e C+A/2 satisfies (Gl) (G2) (GS). Then a function F: £c —> C is the S-transform of a generalized function in [£]*, if and only if it satisfies the conditions: (a) For any £. q £ £c, the function F(z£ + r;) is an entire function of z e C. (b) There exist constants K,a,p > 0 such that

|F(0|
^efc.

Next consider test functions. Let ip e [£\u be represented as in Equation (1). Then its S-transform is given by

"

n-0

n—0

''

"

Therefore, for any p > 0, .30

|s^)l < El/„ue P 71=0

Apply the Schwarz inequality and use Equations (4) and (7) to get /«,

\

1 / 2

\1/2

/~

= iMuMieP)1/2-

(ID

But by the above Fact 2 the function £ „ is equivalent to u. Hence the inequality in Equation (11) is equivalent to the statement: For any constants a.p > 0, there exists a constant K > 0 such that

|5$(0|

V{e£e.

This is the growth condition in the next theorem due to Asai-Kubo-Kuo from Theorem 3.6 in [3]. T h e o r e m 2.2. Assume that 11 £ CV.1/2 satisfies (Gl) (G2j (G3). Then a function F: £l: —>• C is the S-transform of a test function in [£]n if and only if it satisfies the conditions:

93 (a) For any £, r\ e £c. the function F(z£ + r/) is an entire function of z e C. (b) For any constants a.p > 0, there exists a constant K > 0 such that \F{()\
V(6fc

We give a well-known example. Let 0 < /3 < 1 and consider the function u(r) = exp [(1 + fi)r7h].

r e

[0, oc).

It is easy to check that u belongs to C + 1 / 2 and satisfies conditions ( G l ) (G2) (G3). Moreover, the Legeiidre and dual Legeiidre transforms of u are given by t.u(n) = ( - )

,

«*(r) = exp [(1 - /?)rT^T] ,

n = 0,1,2,...,

(12)

r 6 [0. oc).

(13)

where 0° = 1 by convention. We can use the Stirling formula to show that t h e sequence au(n) = (ri!£„(n)) given by Equations (8) and (12) is equivalent to the sequence a\n) = (n!)^. Thus the GeTfand triple [£]„ C (L2) C [£]*, is exactly the triple {S)p C (L 2 ) C (£)£ introduced by Kondratiev-Streit (see [6].) In this case Theorems 2.1 and 2.2 are due to Kondratiev-Streit (see Theorems 8.2 and 8.10 in [6].) Note that when B = 1, the function u(r) — e 2 v ^ does not belong to C+A/i and so its dual Legeiidre transform u' is not defined. This fact is also evident from Equation (13). 3. G R O W T H FUNCTIONS F O R GENERALIZED FUNCTIONS

Consider the Bell number spaces introduced in [5]. Let expj.(»') be the A;-th iterated exponential function, k > 1. It has the power series expansion exp J : (r) = 2

w

— — •'• •

n=0

Let bic{n) = Bfc(n)/expj t (0). The numbers in the sequence {fjfc(n)}^L0 are called Bell numbers of order k. Then we have Wk{r) =

^ l

y

b

jMr».

(i4)

The Bell number space is the CKS-space [£];,k C (L 2 ) C [£]£ given by the sequence {6/t(n)}^_ 0 . To view the Bell number space as a Gel'fand triple in Section 2, we need to find a growth function u 6 C + j l / 2 satisfying ( G l ) (G2) (G3). In order to find such a fimction, note that the growth condition (b) in the characterization theorem in [5] for generalized functions in [£] £ takes the form: There exist constants K,a.p>0 such that

\F(t)\
Vfefc.

where the function to*, is given by Equation (14). By comparing this growth condition with the one in Theorems 2.1, we see that we may take u such that u* = ttv If we can check that u (assuming its existence) belongs to C+_\/2 and satisfies ( G l ) (G2) (G3), then by the above Fact 5 we get « = («*)* = u>£. Thus the function u is given by u = ujjl, the dual Legeiidre transform of w^. However, it is impossible to find the explicit form of uijj.

94 From the above discussion we see that it is desirable to find conditions on u" so that [£]„ C (L 2 ) C [£],* is a Gel'tand triple as given in Section 2. For this purpose we will need to consider the condition: (G*2)

l i m i n f r - ^ r _ 1 logui(r) > 0.

The following three lemmas can be easily checked. L e m m a 3 . 1 . Let u 6 C + 1 / 2 - Then u satisfies condition (G2) if and only if there exist constants c\,c2 > 0 such that u(r) < C\e''lT for all r > 0. L e m m a 3 . 2 . Let w he a positive continuous function on [0, oo). If there exist constants Ci.c2 > 0 such that w(r) > Cie':'*r for all r > 0, then w G C + 1 / 2 L e m m a 3 . 3 . Let w be a positive continuous function on [0. oo). Then w satisfies (G*2) if and only if there exist constants c^.c2 > 0 such that w(r) > c\eC2r for all r > 0. T h e o r e m 3.4. / / u £ C+A/2 satisfies (Gl) (G2) (G3), then u* satisfies (Gl) (G*2) (G3). Conversely, if w is a positive continuous function on [0, oo) satisfying (Gl) (G-2) (G3), then w' belongs to C+A/2 and satisfies (Gl) (G'i) (G3). R e m a r k . Note that if w is a positive continuous function on [0. oc) satisfying (G*2), then by Lemmas 3.2 and 3.3 a belongs to C +>1 /2- Hence the dual Legendre transform v.* is defined. Proof. Assume that u £ C+.1/2 satisfies ( G l ) (G2) (G3). We can use the above Fact 4 to see that u* satisfies ( G l ) (G3). By Lemma 3.1 there exist constants C\, c2 > 0 such that u(r) < e1e"-T, Vr > 0. Therefore, by Equation (3), 11.'(r) = s u p — — »>o «(s) > -isupe2^-C2\

Vr > 0 .

Cl « > 0

But it is easy to check that s u p , > 0 (2^/rs — c2s) = r/c2. Thus we get " * ( ' • ) > —er/<:\

Vr>0.

Cl

Hence by Lemma 3.3 u* satisfies (G*2). Conversely, assume that u i s a positive continuous function on [0,00) satisfying ( G l ) (G*2) (G3). By Lemmas 3.2 and.3.3 the function w belongs to C + . i / 2 . Since to satisfies ( G l ) by assumption, we can use the above Fact 4 to see that w* satisfies (Gl) (G3). By Lemma 3.3 there exist constants Ci,c 2 > 0 such that w(r) > Clecrr. Vr > 0. Then we have «!*(?•)

=

<

SUp——»>0 « " ( « )

isupc2v^-<=»» Cl » > 0

= — er/,:K Cl

Vr > 0 .

95 Hence by Lemma 3.1 the function w' satisfies (G2).

•

As a simple example, consider the function u>k(r) = expj.(r)/exp f c (0) defined in Equation (14) for the Bell number spaces. Obviously. u;fc is a positive continuous function on [0. oc). Moreover, it is easy to check that to*, satisfies ( G l ) (G*2) (G3). Thus by Theorem 3.4 «>£ belongs to C+A/2 and satisfies ( G l ) (G2) (G3). Hence the function u/t = '"'it determines a Gel'fand triple [£}Uk C (L2) C [£],*k- This Gel'fand triple turns out to be exactly the Bell number space associated with the sequence {h{n)}'ZLo (See Example 4.3 in [7].) An interesting example of w is given in a recent paper by Asai-Kubo-Kuo [4] on Feynman integrals. Let v be a complex measure on R with total variation \u\. Assume that v satisfies the conditions: (1) | i / | ( R \ { 0 } ) > 0 . (2) /% e';|'X' d|v|(A) < oc for any constant c > 0. By condition (2) we can define a function w by w(r) = exp

f ( e ^ W - l ) d|„|(A)

re[0,oo).

(15)

Obviously, w is a positive continuous function on [0, oo). It is easy to see that w satisfies ( G l ) (G3). To check condition (G*2), note that ex - 1 > \x2 for x > 0 and so logto(r) > -r f A2 d\v\{\), 2 Jut. Hence we have the inequality iim.nflogHr)^i

r-+oc

r

r

Vr > 0.

2

2 Jm

It follows from condition (1) that J R A2 d|i/|(A) > 0. Hence the function w(r) satisfies condition (G"2). Thus the function w(r) defined in Equation (15) satisfies ( G l ) (G*2) (G3). Then by Theorem 3.4 the function u = w* belongs to C+A/2 a n d satisfies ( G l ) (G2) (G3). With this function u we have a Gel'fand triple [£}» C (L 2 ) C [£}l For such a complex measure v on R, it has been shown in [4] that the Feynman integrands associated with the following potentials V(x) = [ eiXx dv{\),

V{x) = f eXx dv(X)

are generalized functions in the space [5],* given by u = w* with w defined by Equation (15). Here <S is the Schwartz space replacing the countably-Hilbert space £ in Section 1. A c k n o w l e d g e m e n t s . This research was partially supported by the Academic Frontier in Science (AFS), Meijo University and the Centro Vito Volterra (CVV), Universita degli Studi di Roma "Tor Vergata." I would like to give my deepest appreciation to Professors T. Hida and K. Saito (AFS), L. Accardi and R. Monte (CVV) for their warm hospitality during my visits March 5-13. 2000 (AFS) and May 24 July 31, 2000 (CVV).

96 REFERENCES [1] Asai. N.. Kubo. I., and Kuo. H.-H.: Log-con cavity, log-convexity, and growth order in white noise analysis; Preprint (1999) [2] Asai, N.. Kubo. I., and Kuo. H.-H.: CKS-space in terms of growth functions: Preprint (1999) [3] Asai, N.. Kubo. I., and Kuo, H.-H.: General characterization theorems and intrinsic topologies in white noise analysis: Preprint (1999) [4] Asai, N.. Kubo, I., and Kuo. H.-H.: Feynman integrals associated with AlbeverioH0eghKrohn and Laplace transform potentials; Preprint (2000) [5] Cochran. W. G.. Kuo, H.-H.. and Sengnpta. A.: A new class of white noise generalized functions: Infinite Dimensional Analysis. Quantum Probability and Related Topics 1 (1998) 43-67 [6] Kuo. H.-H.: White Noise Distribution Theory. CRC Press, 1996 [7] Kuo. H.-H.: Growth functions in white noise theory: Preprint (2000) DEPARTMENT OK MATHEMATICS, LOUISIANA STATE UNIVERSITY. BATON ROUGE. LA

USA

70803.

Quantum Information III, pp. 97-103 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

97

SOME PROPERTIES OF A R A N D O M FIELD D E R I V E D B Y VARIATIONAL CALCULUS K.S. L E E Department

of Math., School of Science and Korea University Chochiwon, Korea E-mail: [email protected]

Technology

A famous formula called Levy's stochastic infinitesimal equation for a stochastic process X{t) is expressed by the form SX(t) = <S>(X(s),s < t,Yt,t,dt),

t £ R.

There is a higher dimensional parameter generalization of this equation, i.e. a random field X(C) indexed by a contour C

SX(C) = *(X(C'),C' C C,Ys,s eC,C,8C). In this note, we get some nice properties for a random field X(C) and it's probabilistic structure by using the classical theory of variational calculus and the modern theory of white noise analysis.

1

Introduction

The probabilistic structure of a stochastic process X(t),t £ R, is completely determined by the so-called stochastic infinitesimal equation, which was proposed by P. Levy in 1953. The equation can be expressed in the form 6X(t) = $(X(s)
(1)

where $ is a nonrandom function. One might think that the above equation has only formal significance, however it still has profound meaning and makes us to be able to investigate the structure of a stochastic process X(t). In the expression (1), the Yt is the innovation of X(t); namely Yt is an independent system such that each Y% contains exactly the same information as that obtained by the X(t) during the time interval [t,t + dt). Having been motivated by the study of actual phenomena in quantum dynamics and in molecular biology we are led to investigate random field X(C) indexed by a manifold C and discuss its probabilistic structure by observing the variations 6X(C) when C varies slightly within a certain class C. This is a generalization of the method of the innovation approach mentioned above for X(t) with one-dimensional parameter to the case of a random field X(C) with the parameter C that runs through C. The stochastic infinitesimal equation

98 for the r a n d o m field X{C) is introduced in order to characterize the probabilistic structure of X(C). T h e idea is to form the innovation of the given random field X{C) and express it as a functional of the obtained innovation. This approach is of course in line with the white noise analysis. A possible counterpart of (1) for a random field X(C) depending on manifold C may be an equation expressed in the form 6X(C)=$(X(C',C'

cC,Y(s),s<=C,C,6C),

(2)

where C C C means that C is inside of C. i.e. the domain (C ) enclosed by a contour C is a subset of ( C ) , and where <J> is, as before, a n o n r a n d o m function and the system Y={Y(s),seC,CeC}

(3)

is t h e innovation of the random field X(C). T h e formula (2) is proposed by T. Hida. See the references of Hida [2],[3], and Hida, Si Si [4] and references cited therein. It is i m p o r t a n t t h a t the parameter set C = {C} has to be taken rich as enough so t h a t the variation gives us sufficient information to get the innovation. First we investigate a method of establishing an innovation and form the given field as a functional of the innovation . T h e n we discuss the innovation which is taken to be a white noise, and slightly generalize. In any case such an innovation may be called a system of idealized elementary random variables^.e.r.v.) or random field(i.e.r.f.), because the system of those generalized random variables is most elementary and atomic. The basic concepts will be discussed in Section 3. Basic concepts and tools from infinite dimensional analysis are employed for our purpose. They are generalized white noise functionals operators t h a t will be prescribed in section 2. For the short historical view and development of variational calculus, see the same title reference of Si Si [10]. 2

Background

In this section, the theory of white noise analysis is quickly reviewed. For more details and generalizations, see the references of Hida et. al [1], H.-H Kuo [5] and N. O b a t a [9]. We s t a r t from the Hilbert space L2(Rd,dx) where dx is the Lebesgue measure. Consider the Gel'fand triple

EcL2(Rd)

= H

CE*,

(4)

99 where £ is a nuclear space and E* is its dual space. Then, by Bochner-Minlos theorem, there is a Gaussian measure ft on E*. The space (£"*,//) is called white noise space. The Gaussian measure fi and its characteristic functional C(£) are related by C(£) = /

exp(i < x,£

>)dn(x)

JE*

= exp[-||K||2Ue£,

(5)

where ||£|| is L 2 -norm. Then, we get the complex Hilbert space (L2) = L2(E*,vi) = { complex valued, fi — square integrable function on E* with fi.](6) From (L?) we can construct a new Gel'fand triple

(E)c(L*)c{Ey, where (E) and (E)* are the space of test functionals and the space of generalized functionals, respectively. The S-transform Sf of a generalized functional, if G (E)*, is defined to be the function (Sp)(0 = e x p H K | | 2 ] « p , e < " * > »

= / (p(x + t)dn(z),
(7)

JE'

where < < •, • > > is the pairing of (JE1)* and (E). The following theorem plays an important role in this note. Theorem 1 (Wiener-Ho-Segal) For each

,

(8)

n=0

where Fn G H®n = L 2 y m ((R d )») In this case, OO

IM|2 = £ " ! | | ^ H 2 = I|F||2. n=0

The expression in (8) is called the Wiener-Ito expansion of

100

3

Basic Concepts and Variational Calculus

In this section, we briefly explain the concepts of variational calculus. For simplicity, we consider the 2-dimensional parameter space R 2 of white noise x(u),u 6 B?,x G E*. Assume that the parameter C runs over the C containing smooth contours^.e. loop) that is C = {C : contour, ovaloid, and C°°]

(9)

Define 6C to be the set 6C = {Sn(s) : s is the arclength parameter of C, 8n(s) is the outward normal vector at point C(s)}. Put C + 6C as C + SC= {C{s) + Sn(s) : Sn(s) £ 6C} G C.

We assume that 6n(s) is continuous and C -f 6C converges to C if \\Sn\\ = sup |£n(s)| —• 0, where |<5n(s)| means the length of 6n(s). First we consider a nonrandom function G, G:C

—>R

such that G(C + 8C) - G{C) = 8G{C) + g(C, 6C), where (a) 8G(C) is continuous and linear of 6n(s) and (b) g(C,6C) is o(\\6n\\). According to (a) there is a

Jc

Define
#„(&). Then we have

6G(C) = f ^Q(s)6n(s)ds. Jc vn Now X be a generalized random field such that X

:C—*(E)*.

We shall consider variation 6X(C) of X(C). Our approach to random fields X(C) is again based on the innovation theory. This theory for random fields

101

is to be understood more precisely in the following sense. The system is independent of every X{C ) with C < C. Eq.(2) tells us the new information that the random field gains while C runs between C + 6C and should be completely determined by Eq.(2), if it exists, although it has only a formal significance. The following definitions are adopted from references of Hida, Si Si [4] and Si Si [11]. Definition 1 (Casuality) Random field X{C) is casual if for any supp £ C

(cy, (SX(C))(t)

= 0.

(10)

Remark. The meaning of casuality is that X(C) is a function only of x(u),u £ (C), (C) being the domain enclosed by C, x £ E*. Definition 2 (Consistence) Random field X{C) satisfies the consistent condition if

(sx(c'))(0 = (sx(c))(0

(ii)

holds for any £, where supp £ C (C ) C (C). Consider the following random field X(C); X:C^(L2), where X(C) is real valued. Then for the above random field X(C), there exists unique (F„(C))~ = 0 where Fn{C) = Fn(C;uu • • • ,un) e £ ^ m ( ( R 2 ) n ) , Ui £ R 2 , i — 1, • • • ,n, by Wiener-Ito-Segal isomorphism. Definition 3 (Canonical representation ) Random field X(C) has canonical representation property if E(X(C)\X(C ),C C Ci) has following Wiener-Ito-Segal isomorphism (Fn(C))%L0 = (l( C l )»F„(C))~_ 0 , for any C\ C C. The notation E means the weak conditional expectation in the sense of Doob. 4

Main Results

Theorem 2 Random field X(C) satisfies the condition of casuality and consistence condition, then the random field X(C) has the Wiener-Ito-Segal isomorphism: ( J J ' n ( C ; U l , - - - , U n ) ) ~ = 1 = (l( C )"jF'n(«l, • • - , « „ ) " = ! ,

where Fn(ui, • • • ,un) is independent from C.

102

Proof By (10), it is clear that F0 — 0 because F0 is constant. For n > 1, by (11), it is clear that F„(C) and Fn(C ) have to satisfy the equation Fn{Cuuir--,un)

= l ( C l ) „ . F „ ( O i , ••-,«„),

(12)

where («i, ••-,«„) G ( C ' ) n C (C) n C ( R 2 ) n . Hence the inductive limit of Fn(C; ,Ui, • • •, un),C G C, exists and we denote it as F„. It is clear Fn(uir--,un)£LlCtSym((R2)n). Then again by (12), we get Fn(C; ult • • •, un) = l ( C l ) „ F n ( u 1 , • • •, un). Corollary 1 Under the same assumptions of Theorem 2, the random field X(C) has canonical representation. Corollary 2 Under the same assumptions of Theorem 2, 6X(C) has the form 6X(C) = I Jc

d an

^^{s)x{s)6n{s)ds,

where —^—-(s) is given by dX(C) (C)

f (s) = \2®n

Fn(s,u2,---,un)

: x(u2),

• • •, x(un) :

du"'1.

Proof If we apply the variational calculus to the equation (12), then we have 6(l(C)"Fri(ui,

••-,«„)) = nlc

• l(c)"-i-F,n(*,t«2> ••-,««),

(13)

because Fn is symmetric. Applying S~x, taking into account the equation (13), we get the result. We have so far discussed only somewhat special random field, but we hope that the more general classes of random field which are indexed by a general manifold and the codomain is (E)*. Acknowledgment The author is grateful to the Professor T. Hida, Professor K. Saito and the Advanced Science Research Center of Meijo University at Nagoya, Japan. This research is partially supported by Korea University.

103

References 1. T. Hida, H.H Kuo, J. Potthott and L. Streit. White noise. An Infinite Dimensional Calculus, Kluwer Academic Pub., 1993. 2. T. Hida, White noise analysis and Gaussian random fields, Proc. the 24th Winter School of Theoretical Physics, Karpacz, Stochastic Methods in Math, and Phys. (ed. by Gielevak et. al.) 1989,277-289. 3. T. Hida, A note on stochastic variational equations. Exploring Stochastic Laws (Korolyuk volume), ed. by A.V. Skorohod and Yu. V. Borovskikh, 1995,147-152. 4. T. Hida, Si Si, Innovations for Random Fields, Inf. Dim. Anal., Quan. Prob. and Related Topics, Vol. 1, No. 4, 1999, 499-509. 5. H. -H. Kuo, White Noise Distribution Theory, CRC Press, 1996. 6. K.S. Lee, Construction of the innovation for random fields by means of conformal transformation 1, Journal of Science and Technology, Korea University, Vol 15, 1997, 91-96. 7. P. Levy, Problems Concrets d'Analysis Fonctionelle, Gauther-Villars, 1951. 8. H. P. Mckean, Jr., Brownian Motion with a several-dimensional time, Theory Prob. Appl., 8, 1963, 335-354. 9. N. Obata, White Noise Calculus and Fock Space, Springer-Verlag. LNM, 1577, 1994. 10. Si Si, Historical view and some development of variational calculus Applicable to Random Fields, centro Vito Volterra, University Degli Studi Di Roma. N 379, 1999. 11. Si Si, A Variational Formula for Some Random Fields, An Analovue of Ito's Formula, Inf. Dim. Anal. Quan . Prob. and Related Topics, Vol.2, No. 2, 1999, 305-313

Quantum Information III, pp. 105-117 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

105

A STOCHASTIC E X P R E S S I O N OF A S E M I - G R O U P G E N E R A T E D B Y T H E LEVY L A P L A C I A N KENJIRO NISHI, KIMIAKI SAITO Department of Information Sciences Meijo University Nagoya 468-8502, Japan ALLANUS H. TSOI Mathematics Department 202 Mathematical Sciences Building University of Missouri Columbia, MO 65211, USA In this paper, we introduce some domain of the Levy Laplacian and give a stochastic expression of an equi-continuous semigroup of class (Co) generated by the Laplacian on the domain. 1

Introduction

An infinite dimensional Laplacian, the Levy Laplacian, was introduced by P. Levy 18 . This Laplacian was discussed in the framework of white noise analysis initiated by T. Hida 4 . L. Accardi et al. l obtained an important relationship between this Laplacian and the Yang-Mills equations. It has been studied by many authors ( see Refs. 1, 2, 3, 5, 7, 8, 13, 15, 16, 19, 22, 23, 24, 25 and others ). In the previous papers 29 3 0 , we introduced a Hilbert space as a domain of the Levy Laplacian and extended the Laplacian to a self-adjoint operator. In Refs. 27 and 28, we obtained stochastic processes generated by the powers of an extended Levy Laplacian and also in Ref. 31 we obtained stochastic processes generated by some functions of the Laplacian. In this paper we introduce Hilbert spaces consisting of eigenfunctions of the Levy Laplacian acting on generalized white noise functionals and give a semigroup generated by the Levy Laplacian on the projective limit space of those Hilbert spaces. We introduce those Hilbert spaces as a generalization of our previous Hilbert spaces in Refs. 26-28. Moreover we give a stochastic process generated by the Levy Laplacian. This result implies that the stochastic process generated by the Levy Laplacian depends on the choice of eigenfunctions of the Laplacian. The paper is organized as follows. In Section 2, we give a brief background

106

in white noise analysis which is necessary for our paper. In Section 3, we introduce some Hilbert spaces E ^ p A r , N > 1, h £ J7, in (E)-p for each p > 1, to discuss the self-adjointness of the Levy Laplacian, and we give an equicontinuous semigroup of class (Co) generated by the Laplacian defined on w n E-P,OO = Pljv>i E-p.iv ^ projective limit topology. In Section 4, using a strictly stable process, we give an infinite dimensional stochastic process generated by the Levy Laplacian on 5 [ E ^ p o o ] . Moreover we give also an operator-valued stochastic process associated with the Levy Laplacian acting on ~E_ for each h £ J-. In particular it is an important part that the stochastic process generated by the Levy Laplacian depends on the choice of eigenfunctions of the Laplacian.

2

Preliminaries

In this section we prepare some concepts of white noise theory following Refs. 7, 12, 15 and 20. Let £ 2 ( R ) be the Hilbert space of real-valued square-integrable functions on R. We take the space E* = <S'(R) of tempered distributions with a probability measure /x which satisfies J ^exp{i(x,t)}

dfi(x) = exp f - ^ K I o ) >

£ € E = <S(R),

where (•, •) is the canonical bilinear form on E* x E and | • |o is the i 2 (R)-norm. The differential operator A = —(d/du)2 + u2 + 1 is a densely defined self-adjoint operator on L 2 (R). There exists an orthonormal basis {e„; v > 0} for L 2 (R) such that Aev = 2{y + l)e„. Let Ep be the completion of E with respect to the norm | • \p defined by | / | p = |A p /|o for / S E and p £ R. Then Ep is a real separable Hilbert space with the norm | • \p and the dual space E'p of Ep is the same as E~p (see Ref. 10). With the projective limit space E of {Ep;p > 0} and the inductive limit space E* of {E-P;p > 0}, we have a chain of Hilbert spaces: for 0 < p < q,

EcEgCEpd

L 2 (R) C E-p C E-q C E*.

We denote the complexifications of L 2 (R), E and Ep by Ec,p, respectively.

L Q ( R ) , EC

and

Let (L 2 ) = L2(E*,fi) be the Hilbert space of complex-valued squareintegrable functionals defined on E* with norm denoted by || • | (o- By the

107

Wiener-Ito theorem every ip in (L2) can be represented uniquely by oo

fuGL2c(B.fn,

¥> = £ l „ ( / n ) , n=0

where I„ denotes the multiple Wiener integrals of order n G N . Let Lc(R)®n denote the n-fold symmetric tensor product of . ^ ( R ) . Moreover, for the (L2)norm ||y||o of

\f\\o

where | • | 0 is the norm of L2z(R)<s>n. For p e R , let ||y|| p = ||r(v4) p ^||o, where T(A) is the second quantization operator densely defined on (L2) by oo

oo

T(A)

¥> = £l„(/„).

n=0

n=0 2

If p > 0, let (E)p = {ip G (L ); | M | P < oo}. If p < 0, let (E)p be the completion of (L 2 ) with respect to the norm || • || p . Then (E)p, p G R, is a Hilbert space with the norm || • || p . It is easy to see that for p > 0, the dual space (E)p of (E)p is given by {E)-p. Moreover, for any p € R, we have the decomposition oo

(I?)p = ®tf, n=0 p)

where # l is the completion of { I „ ( / ) ; / G E^n} with respect to || • || p . Here Ef.n is the n-fold symmetric tensor product of EQ- In fact we have HLP) = { I „ ( / ) ; / G E®Tp] for any p G R, where £ g ^ is also the n-fold symmetric tensor product of EC,P. The norm ||y>||p of

n!

\ 1/2

= (E l/«lp)

' fn&E®rp,

where the norm of E^n is denoted also by | • | p . The projective limit space (E) of spaces (E)p, p G R is a nuclear space. The inductive limit space (E)* of spaces [E)p,p G R is nothing but the

108

dual space of (E). The space (E)* is called the space of generalized white noise junctionals. The canonical bilinear form on (E)* x (E) is denoted by . Then we have oo

oo

«$,^»=^n!(F„,/n)

oo

$ = Y/In(Fn)€(E)*,

n=0

n=0

Y,In(fn)&(E),

ra=0

where the canonical bilinear form on (.£§")* x E^n is denoted also by (•,•). The Schwarz inequality takes the form: | < $ , ¥ > > |<||$||_P|M|P,

peR.

Since (j)^(-) = exp ((-,0 - 1 ( ^ , 0 ) € (-E)> the S-transform is defined on (£)* by

S[$](0 = « < M e » , 3

£e£c

A semigroup generated by the Levy Laplacian

Let F € S[(E)*]. Then, by the characterization theorem of the S-transform (see Refs. 15, 20, 22) we see that for any £, rj s Ec the function F(£ + zrj) is an entire function of z € C. Hence we have the series expansion: oo

n

n=0

where F^ (£) : Ec x • • • x 2?c —>• C is a continuous n-linear functional. Fix a finite interval T in R. Take an orthonormal basis {Cn}£Lo C E for L2(T) satisfying the equal density and uniform boundedness property ( see Refs. 7, 15, 16, 19, 25 etc). Let VL denote the set of all $ € (E)* such that the limit ~ 1 N~1 A L S [ $ ] ( 0 = ton - £ S [ < T O K n , C n ) n=0

exists for any £ £ Ec and is in S[(22)*]. The Levy Laplacian Ax, is defined by AL$

=

S^ALS®

for $ e P i . W e denote the set of all functionals $ £ % such that 5[$](r?) = 0 for all rj € £ with supp(??) C T c by X>£. Let {50 = 0, g i , . . . , g„,...} be the set of all of non-negative rational numbers and let T be the set of all of complex-valued continuous functions h satisfying the following conditions:

109

1) h(0) = 0, 2) there exists a stochastic process {Xt;t for all t > 0 and z £ R,

> 0} such that eth^

=

E[eizXt]

3) there exists a polynomial r(z) such that \h(z)\ < r(\z\) for z e R . 4) Jt = { ( a i , . . . , a n ) G C»; E " = i ^ - VWkn, for all n e N.

E"=i «' = m % » ) } ^ 0

A function /i 7 (z) = — |z| 7 , 1 < 7 < 2, is an element of .F. Therefore, T is not empty. Take a generalized white noise functional $= /

/(ui,...,u„)

: e «i*(«i)... e«»«(«») : d u ,

(3.1)

/ e ^ § n , a f c e C , A = l,2,...,n, where the notation : • : denotes the Wick ordering. This functional is in for all p > 1 and also in 2?£, and so its 5-transform 5[$] is given by 5[$](0=/

/(u)e 0 l «( U l >... e a "«(""W.

(E)-p

(3.2)

Set D£ = LS | ^

/ ( u ) : f [ ea»^„)

:du&VTL]f&

E®n, ( f l l , . . . , fl„) e ^

|

for each n e N, /i e T and set D§ = C, where LS means the linear span. Define a space D£ by the completion of D^ in (E)-p with respect to || • ||_ p . Then for each n € NU{0}, D£ becomes a Hilbert space with the inner product of (£)_„. L e m m a 3.1.(c.f. Ref. 29) The Levy Laplacian AL is a self-adjoint operator from D£ into itself such that A L $ = % „ ) $ for all $ £ D J .

(3.3)

L e m m a 3.2.(c.f. Ref. 29) Let $ = E^Lo^"" be a generalized white noise functional such that <J>„ is in D£ for each n € NU {0}. If $ = 0 in (E)*, then $ n = 0 in (E)* for all n G N U {0}.

110

Proof: For each n £ N, $„ is expressed in the form: $ „ = Jim

V

/

a [N] e _4h-

/

/.^(uJiTTe^'-^idu,

-<

„= 1

where X^al^le-4'1 m e a n s a s u m or" finitely many terms on a ^ l = ( a ^ 1 , . . . ,a[f ! ) G A%. Suppose that $ = 0 in (£)*. Then, it is obvious that $o = 0, and taking the S'-transform, we have oo

p

n

Y: iim Y, I /..-.(«) n^,c(u,')*»=o n=0

"=1

aWe.AJ

for any £ € Ec. Take £ T £ Ec such that £ T = y/\T\ on T. Put £ = a$T + TJ, where a £ C and 77 € Ec- Then we get OO

r.

yy-iim

E

n=0

Tl

/ /«w(u)TieOi'Arl,'(u,')du=0

aWe-AJ

»=1

for any o € C and rj £ Ec- Therefore we obtain lim

J2

I

/a["i(u)neaLJV1"(u-W = 0

for any n e N and 77 € i?c- This implies $ n = 0 in (E)* for each n £ N U {0}. D Fix a polynomial r in the condition 3) for h £ T. Let a%{n) = ^2e=0r(q„)2e. Since Lemma 3.2 says that ^^L 0 $n,3>n € Djj, is uniquely determined as an element of (E)*, we can define a space E^. p jv f° r a n y A T e N , p > l , by

{

00

00

^

X) *n e (£)*; 2 «5v(»)ll*n||ip
n=0

J

with the norm ||| • |||- P ,jv given by

($>fr(n)||*n|£J 00

n=0

Then the space E ^ N£N,p>l.

N

\

/

X

/2

oo

, $ = X;$«eEVn=0

is a Hilbert space with the norm ||| • |||-p,w for each

111

Put E?Lp>0o = ri;v>i E-P,JV with the projective limit topology. Then, for any N > 1, we have the following inclusion relations: -p,oo

c

^ - p . A T + i C i^_ p ,jv C ^ _ p , i C (,-Gj-p-

The space E^. P)00 includes D£ for any n £ N U {0}. By Lemma 3.1 the Laplacian A L can be defined on E ^ p 2 and is a continuous linear operator defined on E* P ) 2 into E^ p>1 , satisfying |||Ai$|||_ P ) jv < | | | $ | | | _ P , J V + I , $ € E?l JV_|_1, for each A f g N . Any restriction of A L is also denoted by the same notation A^. Let h £ T. For each < > 0 w e consider an operator G\ on E ^ p ^ defined by oo n=0

for $ = Y^=o ®n G E ^ p . Then we have the following: T h e o r e m 3.3. For each h € J- the family {Gj1; t > 0} is an equi-continuous semigroup of class (Co) generated by A/, as a continuous linear operator defined on E ^ p o o . Proof: Since eth^ is a characteristic function, we have Reh(z) < 0. For any t > Q,p ^_1 and N e N , the norm | | | G £ $ | | | _ P I J V for $ = £ ~ = 0 $ n G E - p ooi ®n € D£, n = 0,1, 2 , . . . is estimated as follows:

\G$n\2-p,N =

J2aN(n)\\ethi9n)*n\\2-p n=0 oo


= lll$ll|2_ p,N Hence the family {Gh;t > 0} is equi- continuous in t. It is easily checked that G§ = I, GhG% = Gh+S for each t, s > 0. We can also estimate that

|Gtfc* - G^\\\lPiN

= 5 > f r ( n ) eth{qn) - etoKqn) n=0 oo

<4£>k(n)||$n||2_p n=0

2

IPI * n. . l 1l1 *-

p

112 = 4|||$|

< OO

-p,N -\oo

for each t,*o > 0 , i V e N a n d $ = E^=o $ n G E -P,OO- Therefore, by the Lebesgue convergence theorem, we get that lim G^

= G&$ in E*

for each t0 > 0 and $ e E^ PiOC . Thus the family {£?£;< > 0} is an equicontinuous semigroup of class (Co). We next prove that the infinitesimal generator of the semigroup is given by A/,. For any i V e N and p > 1, we see that 2

*

oo

thi„

eth(qn)

-AL$

£

~P,N

n=

1

$n ~

t

o

2

_ i

h(qn)$n (3-4)

Since $ = £ ~ = 0 $n € E ^ p o o , we have OO

$>fc +1 (n)|| n || 2 _ P <°°.

(3.5)

n=0

Hence, by the mean value theorem, for any t > 0 there exists a constant 6 e (0,1) such that eth(qn)

_ j

\h{qn)\et°X*hb»)
=

t

Therefore we can estimate each term in (3.4) as follows. eth(qn)

a%(n)

t

eth(qn)

_ i

$n -

h(qn)$r

=

a

_ j

n

h(q„)

Ni )

1$,

<4afc+1(n)||n||lp. By (3.5), eth(qn)

lim

t

t->o

_ I

~ Kin)

0

and the Lebesgue convergence theorem, we obtain lim

GH-$

._

* t

-AL$

t->o

Thus the proof is completed.

2

= 0. -p,N

•

113

4

A stochastic process g e n e r a t e d by t h e Levy Laplacian

Let {Xj*; t > 0} be a stochastic process with the characteristic function of X£ given by E[eizXt]

th{z)

=

e

for any h G J-. Take a smooth function rjx £ Ec with TIT = pfr on T. Put G\ = SG^S'1

on 5[E|LPi00] with the topology induced from E^ P ] 0 0 by the S-

transform. Then by Theorem 3.3, {£?£; t > 0} is an equi-continuous semigroup of class (Co) generated by the operator A/,. Let {[X/ 1 ];* > 0} be an I?c-valued stochastic process given by \X^\ = £ + iXtr]T, £ G Ec- Then we have the following a stochastic expression of the operator G^. Theorem 4.1.

Let h € T. Then it holds that G?F(0=ElF(lXth})\lX£}=Z)

for all F G S[Eh_p>00}. Proof: have

Put F(£) = / T „ f{u)ea^u^

• • • e°»«(u»)du in 5[E^ PiOC ]. Then we

E[F([X* ] ) | p t f ] = £] = E[F(£ +

X

iXtvr)}

f(u)ea^u^

• • • ea»«u»>E[e*«»**h]du

Let F = Y,™=0 Fn € S[E* p>0O ]. Then for any n G N U {0}, F„ is expressed in the following form:

Fn(0 = Jim N—tOO

X) *—*

/" / a i ^ u ) ^ ^ ' " 1 1 " ' ^ ^ ' ^ lrpn

where (/a[jv]);v is a sequence of functions in i?® n . Hence we have oo

j2mFn^+ix^T)\} n=0 oo

^ ^=0

E l i m ^ E a ^ e ^ / m /..-i (u) I E L I e ' ^ ^ e "

W ^ W

114

= Y 71=0

lim aWe-A*

oo

i=0

Since F„ G 5 [ E ^ p o o ] , there exists some $ „ G E ^ p o o such that .Fn = £[<&„ for any n. By the Schwarz inequality, we see that

E 1^(01 <En^ii-Pii
n=0

1/2

( oo

<<E«Af(n)~ (n=0

1/2

a

n

|E k( )ii*»ii-pf

iwip<°°.

for all £ G -E and some M > 1. Therefore by the continuity of G^ we get that

E[F(£ + iXthrjT)} = E

E[F„(£ + iXthvr)}

oo n=0

Thus we obtain the assertion.

• 1

Theorem 4.1 says that the infinite dimensional stochastic process {[X/ ];* > 0} is generated by Ajr, defined on E ^ p o o for each h G T. For any $ G (E)* and 77 G -Ec, the translation r,,* of $ by 77 is defined as a generalized white noise functional TV$ whose S-transform is given by •SK^IU) = S[$}(£ + rj), £ G Ec. (See Ref.14.) Then we can translate Theorem 4.1 to be in words of generalized white noise functionals. Corollary 4.2.

Let h G T. Then it holds that G$*{x)=E[TiXttn.${x)]

for all $ in E ^ p ,

115

Concluding Remark In this paper the domain of the Levy Laplacian is depending on non-negative rational numbers. Using the notion of direct integral spaces we can discuss the results in the paper more generally. Those discussions with recent developments on the Levy Laplacian will appear in Ref.17. Acknowledgments This work was written based on the second author's talk in the Workshop "White noise approach to classical and quantum stochastic calculus " held in Centra Vito Volterra, Universita di Roma Tor Vergata, June 21 - 30, 2000. He would like to express his deepest gratitude to Professors L. Accardi, H.-H. Kuo and R. Monte for their warm hospitality and supports during my visit to Rome, and also to Professors T. Hida and N. Obata for their advice. This work was also supported in part by the Joint Research Project " Quantum Information Theoretical Approach to Life Science " for the Academic Frontier in Science promoted by the Ministry of Education in Japan and JMESSC Grantin-Aid for Scientific Research (C)(2) 11640139. The authors are grateful for their supports.

References 1. Accardi, L., Gibilisco, P. and Volovich, I.V.: The Levy Laplacian and the Yang-Mills equations, Rendiconti delPAccademia dei Lincei, (1993). 2. Accardi,L., Smolyanov, O. G.: Trace formulae for Levy-Gaussian measures and their application, Proc. The HAS Workshop "Mathematical Approach to Fluctuations, Vol. I I World Scientific (1995) 31 -47. 3. Chung, D. M., Ji, U. C. and Saito, K.: Cauchy problems associated with the Levy Laplacian in white noise analysis, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol.2, No.l (1999), 131-153. 4. Hida, T.: "Analysis of Brownian Functionals", Carleton Math. Lecture Notes, No.13, Carleton University, Ottawa, 1975. 5. Hida, T.: A role of the Levy Laplacian in the causal calculus of generalized white noise functionals, " Stochastic Processes A Festschrift in Honour of G. Kallianpur " (S. Cambanis et al. Eds.) Springer-Verlag, 1992. 6. Hida, T., Kuo, H. - H. and Obata, N.: Transformations for white noise functionals, J. Fund. Anal. (1990)

116

7. Hida, T., Kuo, H. - H., Potthoff, J. and Streit, L.: "White Noise: An Infinite Dimensional Calculus", Kluwer Academic, 1993. 8. Hida, T. and Saito, K.: White noise analysis and the Levy Laplacian, "Stochastic Processes in Physics and Engineering" ( S. Albeverio et al. Eds. ), 177-184, 1988. 9. Hida, T., Obata, N. and Saito, K.: Infinite dimensional rotations and Laplacian in terms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93. 10. ltd, K.: Stochastic analysis in infinite dimensions, in " Proc. International conference on stochastic analysis ", Evanston, Academic Press, 187-197, 1978. 11. Kubo, I.: A direct setting of white noise calculus, in:Stochastic analysis on infinite dimensional spaces, Pitman Research Notes in Mathematics Series, 310 (1994) 152-166. 12. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I, II, III and rV, Proc. Japan Acad. 56A (1980) 376-380; 56A (1980) 411-416; 57A (1981) 433-436; 58A (1982) 186-189. 13. Kuo, H. - H.: On Laplacian operators of generalized Brownian functionals, in " Lecture Notes in Math." 1203, Springer-Verlag, 119-128, 1986. 14. Kuo, H. - H.: Lectures on white noise calculus, Soochow J. (1992), 229300. 15. Kuo, H. - H.: White noise distribution theory, CRC Press (1996). 16. Kuo, H. - H., Obata, N. and Saito, K.: Levy Laplacian of generalized functions on a nuclear space, J. Fund. Anal. 94 (1990), 74-92. 17. Kuo, H. - H., Obata, N. and Saito, K.: Diagonalization of the Levy Laplacian and Related Stable Processes, Preprint (2000). 18. Levy, P.: "Lecons d'analyse fonctionnelle", Gauthier-Villars, Paris 1922. 19. Obata, N.: A characterization of the Levy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 20. Obata, N.: "White Noise Calculus and Fock Space," Lecture Notes in Mathematics 1577, Springer-Verlag, 1994. 21. Obata, N.: Quadratic Quamtum White Noises and Levy Laplacian, Preprint (2000). 22. Potthoff, J. and Streit, L.: A characterization of Hida distributions, J. Fund. Anal. 101 (1991), 212-229. 23. Saito, K.: Ito's formula and Levy's Laplacian I and II, Nagoya Math. J. 108 (1987), 67-76, 123 (1991), 153-169. 24. Saito, K.: A (Co)-group generated by the Levy Laplacian, Journal of Stochastic Analysis and Applications 16, N o . 3 (1998) 567-584. 25. Saito, K.: A (Co)-group generated by the Levy Laplacian II, Infinite

117

26.

27. 28. 29. 30.

31.

32. 33.

Dimensional Analysis, Quantum Probability and Related Topics Vol. 1, N o . 3 (1998) 425-437. Saito, K.: A stochastic process generated by the Levy Laplacian, Volterra International School "White Noise Approach to Classical and Quantum Stochastic Calculi and Quantum Probability ", Trento, Italy, July 19-23, 1999; to appear in Acta Applicandae Mathematicae (2000). Saito, K.: The Levy Laplacian and stable processes, to appear in Proceedings of the Les Treilles International Meeting (1999). Saito, K.: Infinite dimensional stochastic processes generated by extensions of the Levy Laplacian, Publication in Centro Vito Volterra (2000). Saito, K., Tsoi, A.H.: The Levy Laplacian as a self-adjoint operator, Quantum Information, World Scientific (1999) 159-171. Saito, K., Tsoi, A.H.: The Levy Laplacian acting on Poisson Noise Functionals, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 2 (1999) 503-510. Saito, K., Tsoi, A.H.: Stochastic processes generated by functions of the Levy Laplacian, Quantum Information II, World Scientific (2000) 183194. Sato, K.-L: "Levy Processes and Infinitely Divisible Distributions", Cambridge, 1999. Yosida, K.: "Functional Analysis 3rd Edition", Springer-Verlag, 1971.

Quantum Information III, pp. 119-125 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

ON THE DESIGN OF EFFICIENT Q U A N T U M

119

ALGORITHMS

T E T S U R O NISHINO Department of Information and Communication Engineering The University of Electro Communications In this paper, we consider how to design efficient quantum algorithms by using the quantum database search algorithm designed by L. Grover, and show the following results: (1) we show how to apply an efficient quantum algorithm for finding the minimum designed by Diirr and H0yer to the shortest vector in lattice problem. The algorithm by Diirr and H0yer is designed based on Grover's algorithm, and (2) we estimate the time complexity of quantum algorithm for the collision problem, which are designed by using Grover's algorithm and various sorting algorithms.

1

Introduction

Current computers are implemented baed on Turing machine introduced by Alan Turing in 1936. Since Turing machine is a very simple and stable model of computation, it has been used as a standard model in recursive function theory and computational complexity theory. On the other hand, in 1985, David Deutsch introduced quantum Turing machines (QTMs for short) as Turing machines which can perform so called quantum parallel computations [3]. Then, in 1994, Peter Shor showed that QTM can factor integers with arbitrary small error probability in polynomial time [5]. Since it is widely believed that any deterministic Turing machines cannot factor integers in polynomial time, it is very likely that QTM is an essentially new model of computation. Many computer scientists are working hard in this area these days, and have obtained a lot of important results [1,4,5,6,7]. One tape cell of a Turing machine can contain a symbol 0 or 1, i.e. one bit of information. On the other hand, one tape cell of a QTM can be in an arbitrary superposition of the states 0 and 1, which is called one qiibit {quantum bit) of information. Here, a superposition of the states 0 and 1 is represented by a|0) +/?|1), where |0) and |1) are state vectors in some Hilbert space representing the states corresponding to 0 and 1, respectively, a and /? are complex numbers such that |a| 2 + |/?| 2 = 1, and a (/?) is called an amplitude of the state |0) (|1)). A computation of a QTM is a sequence of applications of unitary transformations to some qubits on its tape. After the computation, if we observe a tape cell in a superposition a|0) + /?|1), we will see 0 (1) with probability

120 l a | 2 (l/?| 2 )- T h u s , if we observe a t a p e cell in a superposition 4 s | 0 ) + -^= |1), we will see 0 or 1 with equal probability 1/2. Namely, this t a p e cell is an ideal r a n d o m bit. But, when we observe this t a p e cell, the superposition is completely destroyed. It is conjectured t h a t there exist bounded error polynomial time q u a n t u m algorithms only for the problems which belong to the class NP b u t which are not NP-complete. For example, this class of problems includes factoring, discrete log, graph isomorphism, and the shortest vector in lattice problems. In fact, P. Shor showed t h a t Q T M can solve factoring and discrete log problems with arbitrary small error probability in polynomial time [5]. In this paper, we consider how to design efficient q u a n t u m algorithms for these problems by using the q u a n t u m database search algorithm designed by L. Grover, and show the following results: (1) we show how to apply an efficient q u a n t u m algorithm for finding the minimum designed by Diirr and H0yer t o t h e shortest vector in lattice problem. T h e algorithm by Diirr and H0yer is designed based on Grover's algorithm, and (2) we estimate the time complexity of q u a n t u m algorithms for the collision problem, which are designed by using Grover's algorithm and various sorting algorithms.

2

Quantum Turing Machines

We first review the definition of Q T M s which was formulated in [1]. Like an ordinary Turing machine, a q u a n t u m Turing machine consists of a finite control, an infinite t a p e , and a tape head. D e f i n i t i o n 2.1 [1] A quantum Turing machine (QTM for short) is a 7-tuple M = (Q, E, T, 6, qo, B, F), where Q is a finite set of states, T is a tape alphabet, B G T is a blank symbol, S C T is an input alphabet, 6 is a state transition function and is a mapping from QxTxTxQx {L,R} to C (the set of complex numbers), qo G Q is an initial state, and F C Q is a set of final states. An expression 8(p, a, b, q,d) = c represents the following : if M in a s t a t e p reads a symbol a (let C\ be this configuration of M ) , M writes a symbol b on the square under the t a p e head, changes the state into q, and moves the head on the square in the direction denoted by d G {L, R} (let Ci be this configuration of M). Then, the complex number c is called an amplitude of this event, and the probability that M changes its configuration from C\ to Ci is defined t o be |c| 2 . This state transition function 6 defines a linear mapping in a linear space

121

of superpositions of M's configurations. This linear mapping is specified by the following matrix Ms- Each row and column of Ms corresponds to a configuration of M. Let C\ and C2 be two configurations of M, then the entry corresponding to C2 row and C\ column of Ms is 6 evaluated at the tuple which transforms Ci into Ci in a single step. If no such tuple exists, the corresponding entry is 0. We call this matrix Ms a time evolution matrix of M. Restriction: For any QTM M, the time evolution matrix Ms must be a unitary matrix. Namely, if Mj is the transpose conjugate of Ms and I is the identity matrix, then the relation MJMs = MgMJ = I must be satisfied by MsComputation of M is an evolution process of a physical system defined by the unitary matrix Ms- Let \ip(0)) be an initial state of M. If we denote the state of M at time s by \ip(s)), we have \rp(rt)) = Mj|V>(0)} where r is the time required by M to execute a single step. In quantum mechanics, observations from outside will change the state of the physical system. Thus, we can not observe the tape contents of a QTM from outside before the computation is terminated. When the computation is terminated, the tape contents will be observed as follows: if a QTM M in superposition ip = J2i aiCi is observed, configuration C,- is seen with probability |a,| 2 , and the superposition of M is updated to C,-. Especially, when a tape cell of a QTM is in a superposition a|0) -f /?|1), we will see 0 (1) with probability |a| 2 (|/?| 2 ). We may also perform a partial observation. Let us consider the case of a partial observation only on the first cell of the tape. Suppose the superposition was V = Yliai^i + Y^iPiC}, where C° (C/) are those configurations that have a 0 (1) in the first cell. In this case, if we observe the first cell, we will see 0 (1) with probability | J2i a «'| 2 (I J2i A| 2 )- Moreover, if a 0 is observed, the new superposition is given by

\2^iai\

i

and if a 1 is observed, the new superposition is given by

\2^iPt\

i

122 3

T h e Shortest Vector in Lattice P r o b l e m

L. Grover designed an efficient q u a n t u m algorithm which solves tue following problem [4]. I n p u t F States labelled by Slt S2, S3, • • •, SN,whereN

= 2n,

P r o b l e m F Find the only one s t a t e such t h a t C(S) = 1 (It is assumed t h a t the condition C(S) can be checked within constant time for an arbitrary s t a t e S.) Grover's algorithm runs Oi\/N)

steps when there exists only one s t a t e

such t h a t C(S) — 1, and O ( y / N / t 1 steps when there exist t states such t h a t C(S) = 1. By using Grover's algorithm, we design an efficient q u a n t u m algorithm for t h e following shortest vector in lattice problem : find the shortest non-zero vector in an n dimensional lattice L. Here, a lattice in R n is a set of the form :

L = L(b1,...,bn)=iJ2\ibi

\Xi e Z , » = l , .

where (&i,...,6 n ) is called the basis for L. And the length of a vector x = (xi,...,x„) G R n is defined by ||a;|| = {x\ + ... + x%)i. In this paper, we assume t h a t n — 2. Thus, the shortest vector is of the form v = Ai&i + \2b2(\i,\2

€ Z).

Boundaries of the rectangle is defined in the following fashion. Let &i = (bii,bi2),b2 = (621,^22) be the basis. First, we define

{

A = (&iicos0 + &2isin0) 2 + (b 12 cos e + b22 sine)2 B = (bn sin 9 - 6 2 i cos 9)2 + ( 6 i 2 sin 0 - b22 cos 9)2 C = b2n+b22

then, the boundaries of the rectangle is defined as follows : cos#| +

% I sin 01 +

J^\sin9\ ;9\

(1)

Let N be the number of the points of a lattice in the rectangle. Our q u a n t u m search algorithm for the shortest vector in lattice problem is as follows :

123 1. Pick randomly an index y (0 < y < N — 1). 2. Repeat the following steps 22.5v / N + 1.41g2 N times. (a) Initialize the registers to £ V -7= \j) \y). (b) Apply Grover's algorithm. (c) Observe t h e first register. If y' satisfies T[y'] < T[y] then let y = y'. 3. Observe the index y. It is easy to see t h a t the above algorithm runs 0(yN) the number of the points of a lattice in the rectangle. 4

steps, where N is

T h e Collision P r o b l e m

For a function F : X —• Y, a pair of two elements XQ, X\ £ X such t h a t F(xo) = F(x\) is called a collision. G.Brassard , P.H0yer and A.Tapp designed a q u a n t u m algorithm for the following collision problem[2]: I n p u t F A set X with N elements, and a function F of time complexity 0(T), P r o b l e m F Find a collision { x o , ^ i } which is assumed to exist. A q u a n t u m algorithm for the collision problem presented in [2] is as follows: 1. Pick an arbitrary subset K with k elements from a given set X with N elements. Then, construct a table L with k pairs (x, F(x)), x £ K . 2. Sort the elements in L by the second components

F(x).

3. Check whether there exists a collision in L. 4. C o m p u t e Xl - G r o v e r ( # , 1), where H : X -> {0, 1} and, H(x) (XQ, F(x)) £ L and there exists xo £ K such t h a t x ^ XQ. 5. Find (x0,

F(Xl))

= 1 iff

£ L.

6. O u t p u t the collision {zo,

x\}.

We will estimate the time complexity of the above q u a n t u m algorithm for the collision problem. When the quick sort is used in the step 2 of the above algorithm, we obtain the following :

124

1. 0(Tk + klogk) steps are needed for sorting, and O (y/N/k(T steps are needed for the execution of Grover's algorithm. 2. O ((T + log Jb)(* +

+ logfc)]

y/N/k))

steps are needed in total. 3. The algorithm runs fastest when jfe = ^ 7 7 with O ((T + log steps.

N)\YN)

On the other hand, in the case of two-to-one function F, if the radix sort is used in the step 2 of the above algorithm, we obtain the following : 1. 0(Tk + k) = 0(Tk) steps are needed for sorting, and o(y/N/k(T +log k)) steps are needed for the execution of Grover's algorithm. 2. o(Tk

+ Ty/Njk

+y/N/k

log k) steps are needed in total.

3. The algorithm runs fastest when k = \/N ( with 0(T\/N) steps ) if T > logjfc, or when k = k' ( with O(k'T) steps ) if T < logk, where k' =

References 1. Bernstein, E., and Vazirani, U. : "Quantum Complexity Theory", in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM, New York, 1993, pp.11-20. Also in Special issue of SIAM J. Comp., October, 1997. 2. Gilles Brassard, Peter Hoyer, and Alian Tapp, "Quantum Algorithm for the Collision Problem", Technical Report: quant-ph/9705002 (1997). 3. Deutsch, D. : "Quantum Theory,the Church-Turing Principle and the Universal Quantum Computer", Proc. R. Soc. Lond., Vol. A 400, pp.97117 (1985). 4. Grover, L. : "A Fast Quantum Mechanical Algorithm for Database Search, in Proc. 28th Annual ACM Symposium on Theory of Computing, ACM, New York, 1996, pp.212-219. 5. Shor, P. W. : "Algorithms for Quantum Computation : Discrete Log and Factoring", in Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp.124-134. Also in Special issue of SIAM J. Comp., October, 1997.

125

6. Simon, D. R. : "On the Power of Quantum Computation", in Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp.116-123. Also in Special issue of SIAM J. Comp., October, 1997. 7. Yao, A. : "Quantum Circuit Complexity", in Proc. 34th Symposium on Foundations of Computer Science, pp.352-361, IEEE Press (1993).

Quantum Information III, pp. 127-141 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

127

Phylogenetic relation of HIV-1 in the V3 region by information measure Taro Nishiokaf, Keiko Sato} and Masanori Ohyaf fDepartment of Information Sciences, Science University of Tokyo Noda City, Chiba 278-8510, Japan ^Department of Control and Computer Engineering, Numazu College of Technology 3600 Ooka Numazu 410-8501, Japan Abstract Sequences obtained from patients infected with human immunodeficiency virus (HIV-1) were analyzed by phylogenetic trees with a genetic measure called entropy evolution rate. In this analysis, we used the third variable (V3) regions of HIV-1 which are classified according to the clinical course of infection. Then we conclude that there exists some relations between the variation of V3 region and the clinical course of AIDS, so that the entropy evolution rate can be one of the measures characterizing the clinical course of AIDS as well as the CD4 counts.

1

Introduction

It is known that the V3 region of the HIV-1 envelope protein is highly variable and immunologically important because it is an epitope for neutralizing antibodies and cytotoxic cells. Moreover, it is known that this region has amino sites which decide biological nature of virus [1, 2, 3, 4]. The course of disease progression might be estimated by analyzing the V3 sequences of patients infected with HIV-1 from phylogenetic trees based

128

on information theoritical view. More details, in this paper, we use a genetic measure called the entropy evolution rate introduced in [5] as an Information theoretical treatment of genes based on the entropy and the mutual entropy of Shannon which are fundamental quantities in information theory, and we write phylogenetic trees of sequence variability in the V3 region, by which we can estimate the course of disease progression [6, 12].

2

Material and methods

The entropy evolution rate could be used both amino acid and base sequences. In this paper, we briefly explain it for the amino acid sequences. Let A and B be two aligned amino acid sequences, which are composed of 20 kinds of amino acids and the gap * [7]. The complete event system {A,p) of A is determined by the occurrence probability Pi (0 < i < 20) of each amino acid and the gap *; A

\

= ( * P ) \P0

A

C

'" W Y \ Pi P2 ••• Pl9 P20 J

In the same way, the complete event system (B, q) of B is ( B\ = ( * \Q J ~ \Qo

A C ••• W Y \ 9i Q2 ••• ?i9 ?2o J

The compound event system (A x B, r) for two sequences A and B is denoted by ( AxB\ \ T

=

J

( ** *A ••• YY \ \ r o o r 0 i ••• 7-2020 / '

where r^ represents the joint probability of the event i of A and the event jolB. These event systems define various entropies, among which the following two are important: (1) Shannon entropy 20

S{A) = - ^ P i l o g p i , t=0

129

which expresses the amount of information carried by (A, p). (2) The mutual entropy 20 20

J 04 *) = £ £ ' « log ^ , which expresses the amount of information transmitted from A to B (or B to A). Using the above information measures, a measure, called the entropy evolution rate, indicating the difference between two amino acid sequences was introduced in [5], which is defined as follows: Put r[B\A)-

S{A)

,

which is the rate how much information is transmitted from A to B, and it is symmetrized as r (A, B) = X- {r {A \ B) + r {B \ A)} The entropy evolution rate p (A, B) is defined by p(A,B)

=

l-r(A)B)

and another version p' (A, B) of the entropy evolution rate is defined by p'(A,B)

=

I(AB) S{A) +

S{B)-I{A,B)

We analyzed by phylogenetic trees with these two genetic measures. Consequently, the phylogenetic trees by means of two measures showed similar phylogenetic trees, so that the details of the result by means of p' (A, B) are not described here. In this paper, the variation among the V3 sequences was computed by the entropy evolution rate p (A, B). Note that the entropy evolution rate takes the value in [0,1], and p (A, B) = 0 if A and B are completely same and p {A, B) = 1 if they are completely different. Therefore the variation of virus becomes larger, the value of the entropy evolution rate is getting larger.

130

Data used in our analysis is the amino acid sequences and base sequences for virus clone stored in the International Nucleotide Sequence Database (DDBJ/EMBL/GenBank). Here, sixteen HIV-1-infected patients [1, 2, 3, 8, 9, 10] were designated as patient A to patient P. These facts are summarized in Table 1. It is reported that patients A and B were followed for a period of more than 5 years [1]. Patient A remained healthy although p24-antigenemia reappeared in 33 months after primary infection. It is known p24-antigenemia in blood reflects the viral road, and it has been used as a measure to estimate disease stage and AIDS development. Patient B was diagnosed as having AIDS in 55 months after primary infection. Zidovudine (AZT) treatment was started at the time of diagnosis. Patient C has been asymptomatic throughout study of 7 years [8]. He has never received antiviral therapy. The CD4+ T-cell count is reported for patient D, E, and F [9]. That has been used to estimate disease stage similarly as p24-antigenemia. The immunocyte for healthy people is around from 800 to 1,000 /fd. When the CD4 + T-cell count of patient decreases and it becomes less than 200, various infections are considered to appear in infected patient. Therefore according to the diagnosis standard of CDC (Center for Disease Control), when the count becomes less than 200, patient is recognized to have AIDS. For patient D, it fluctuates as 470, 826, 273 and 515 at every month of following. For patient E and F, it decreases as 1225, 756, 368 and 943, 575, 187, respectively. Patient G, H, I, J, K and L were categorized according to the rate of CD4 + T-cell count decline [10]. Patient G and H had a rapid, patient I and J had a moderate, and patient K and L had a relatively stable rate of the count. Patient H, I and J started Zidovudine treatment from 30, 48 and 64 months after primary infection respectively. Patient G and H died in 36 and 42 months. Patient M was diagnosed as having AIDS in 37 months after infection [2]. Patient N was diagnosed in 54 months, and died in 93 months [2, 3]. It is not reported when patient O and P acquired HIV-1 infection [3]. Patient 0 was followed from 10 months before AIDS diagnosis to 15 months after diagnosis, and died in 42 months after diagnosis. Patient P was followed from 7 months before diagnosis to 40 months after diagnosis, and died in the same month.

131 : Designation ol our analysis

Designation

patient D

patient A

patient B

patient C

1

496

82

si

Presumed trasnission mode

Homosexual contact

Homosexual contact

A single batch of factor VII

No information

Clinical status

Asymptomatic

AIDS in 1989

Asymptomatic

No information

CD4 counts

Decreasing

Decreasing

Decreasing

Fluctuating

Antiviral therapy

None

AZT from 1989

None

None

Term for the study

1985~after 59 mounths

1985-after 56 mounths

1984-1991

1985.111989.5

Tissue

Serum

Serum

Plasma

PBMC

Molecular type

Viral RNA

Viral RNA

Viral RNA

DNA

patient E

patient F

patient G

patient H

patient 1

patient J

s2 No information

s4 No information

PI No information

P2 No information

P3 No information

P4 No information

No information

No information

Died within

Died within

No information

No information

36 mounths

42 mounths

Rapid Progress

Rapid Progress

Moderate

Moderate

Decreasing

Decreasing

Progress

Progress

AZT frpm 48 mounths

AZT frpm 64 mounths

None

AZT frpm 30 mounths

1985-.1

1985~after

1985—after

1986~after

1986~after

1989.6

32 mounths

35 mounths

122 mounths

11 6 mounths

PBMC DNA

PBMC DNA

PBMC DNA

PBMC DNA

PBMC DNA

PBMC DNA

patient K

patient L

patient M

patient N

patient 0

patient P

P5

P6

ACH0039

ACH0208

PI 98

6052

No information

No information

Homosexual

Homosexual

No information

No information

No information

No information

AIDS

AIDS

None

None

1985.5 — 1987.1

contact

contact

AIDS in 1989.11

AIDS in 1980.6 And 1993.9 Died

Stability

Stability

Decreasing

Decreasing

Decreasing

Decreasing

None

None

None

None

None

None

1984-after

1985-after

1987—after

1985.12~after

52 mounths

47 mounths

47 mounths PBMC

93 mounths PBMC

37 mounths PBMC

93 mounths PBMC

PBMC

PBMC

DNA

DNA

DNA

DNA

DNA

DNA

Tablel. Data used in our analysis In order to examine the relation between sequence variability in V3 and disease progression, we constructed a phylogenetic tree using the neighborjoining method [11] with genetic matrix made by the entropy evolution

132

rate. The phylogenetic relationships were estimated using 213 amino acid sequences removed identical sequences from 772 sequences of 16 patients (Figurel). Here Ij — k denotes that / indicates the name of patient, j indicates the time from primary infection, and k indicates the number of identical sequences. However, the time j of patient C indicates not months but years. Filled symbol • represents syncytium-inducing (SI) sequences, and others represent non-syncytium-inducing (NSI) sequences. We grouped sequences from each patients in nine clusters and designated them " group i" (t = l , 2 , - . - , 9 ) . Moreover, the phylogenetic tree was made for patient M, 0 , and P. (Figure 2) is the phylogenetic tree made by using the entropy evolution rate for the amino acid sequences. (Figure 3) is the phylogenetic tree was made by using the entropy evolution rate for the base sequences, and (Figure 4) is one made by using the 2 parameter method for the base sequences. The phylogenetic relationships were estimated by 39 amino acid sequences and 56 base sequences removed identical sequences from 124 sequences. The markes in Figures are same as those mentoned above.

133

3

Results

®

®

134

ZJ ©

©

©

©

• p£s$a • Ml 7-1 9

®

Figure 1. Phylogenetic tree for amino acid sequences with entropy evolution rate

135 Group 3 showed a tendency to form population associated with sequences obtained at early stage of infection. Therefore, in that group, a lot of viruses of initial types seen at infected time have gathered in the amino acid sequences of this group. Furthermore, there exist several sequences obtained at different time for each patient, so that there are many viruses whose evolution speed is very slow. Table 2 shows the change of sequences in the group 3, namely, this table shows where the sequence stayed in the group 3 at early stage moves at later stage. Many sequences obtained at late stage were observed in one of the groups 1, 2, 6 and 7, few were observed in the groups 4, 8 and 9. Moreover, the amino acid sequences which belong to the group 3 shows the tendency to move the outside of the groups 1, 2, and 3 after they stagnate for a while in the group 3. Especially, the element in the group 3 has very slow speed of evolution, and it is formed with weak viruses causing disease, so that it may be considered that the evolution of these viruses to escape from the cellular and humoral immune responses has not begun yet. Early Patient C Patient D Patient E Patient J Patient K Patient L

| :
: (D I : (D I : (D

Late

1

-

©®

! 1 j

-

(D® (D®® (DdXD® 1

Table 2. Change for patients observed in group 3 Next, let us read the phylogenetic tree (Figure 1) by observing clinical symptom of AIDS. For the sequences gathered from nine asymptomatic, patients A,C,D,E,F,I,J,K,L, we observe how the sequences change among groups. The sequences obtained at early stage were observed in one of the groups 1,2,3 and the sequences obtained at late stage were observed in the group from 5 to 7, but few were observed in one of the groups 4, 8 and 9.

136

Early Patient A Patient C Patient D Patient E Patient F Patient I Patient J Patient K Patient L

© (D (D (D © ® (D (D ©

Late — — — —

© ©@ ©@ © © ©CD® ©CIXD©

Table 3. Change for asymptomatic patients On the other hand, for the sequences obtained from patient B, M, N, O who developed to AIDS, the sequences attached to each patient change the group as in Table 4. Before Patient B Patient M Patient N Patient 0 Patient P

® ® ©

After — -

©® ®@ ©®@ © ©©

Table 4. Change for AIDS patients Almost all sequences were observed in one of the groups 4, 8 and 9. In particular, the sequences in the group 8 or 9 were formed by viruses replicating SI genotypes, so that they strongly related to AIDS diagnosis.

137

Figure 2. Phylogenetic tree for amino acid sequences of patient M,0,P with entropy evolution rate

Figure 3. Phylogenetic tree for amino acid sequences of patient M,0,P with entropy evolution rate

138

Figure 4. Phylogenetic tree for base sequences of patient M,0,P with 2-parameter method Figure2 classifies the amino acid sequences into two types, namely, SI genotypes and NSI genotypes, due to the biological property of virus. On the other hand, the phylogenetic tree (Figure3) for the base sequences classifies the sequences according to not the biological property of virus but the grouping figure of patients. Comparing the entropy evolution rate method with the 2-parameter method, the phylogenetic tree with the entropy evolution rate more depends on the biological property of virus than that with the 2 parameter method. Therefore in order to see the phylogenetic relation for the biological property of virus, it is good to make the tree by means of amino acid sequences with the entropy evolution rate, which is one example of the usefulness of information measure.

4

Discussion

The V3 region of HIV-1 has been the focus of research to evaluate sequence variation during the course of infection. In our study, phylogenetic trees were constructed using the entropy evolution rate to quantify the sequence

139

variation in the V3 region. Phylogenetic analysis shows that the sequences obtained from each patient are classified into the clinical course. Therefore, we can infer that there exist some relations between the evolution in the V3 region and the course of disease progression. Sequences in V3 are classified into two categories of virus that appeared at early stage and late stage of disease progression. Shorter branch lengths were observed for the sequences emerged early in infection. Supporting cause for this is that viral sequences at early stage are not enough pathogen, because they have not occurred the necessary evolution yet to overcome host immune system. The sequences from virus at late stage are divided into 2 types, SI genotypes and NSI genotypes. Longer branch lengths were observed for SI genotypes relative to NSI genotypes. The emergence of SI phenotypes influences progression to AIDS due to rapid evolution. Viral sequences of NSI genotypes observed at late stage tend to evolve more slowly. We can infer that the evolution in the V3 region stagnates because of the declines of the immune defenses or the accumulated evolution to virus with strong pathogen. Therefore, if viral sequences within HIV-1 infected patients exist in this cluster of NSI genotypes, the disease course clearly progresses. According to the phylogenetic tree constructed by using standard distance, the substitution rate, sequences in V3 obtained at several points in time in the clinical course of each patient was divided into 2 types, SI genotypes and NSI genotypes. However, that could not be divided into different disease progression as measured by the entropy evolution rate. The weight substitution rate to evaluate the weight in the alignment indicated an interesting result. The phylogenetic relation of V3 was classified by the pattern of the amino acid appeared at position 11 and 25 that be supposed to control the biologic properties. The details of the above facts are not described here. It is difficult to estimate prognosis in patients infected with HIV-1 at the only V3 region, for it is supposed to progress to AIDS owing to various factors. However, we treat analyzing the evolution in V3 using information theoritic quantity called entropy evolution rate to predict progression to AIDS as important. Moreover, it is possible to estimate the course of disease progression according to phylogenetic analysis using the entropy evolution rate.

140

References [1] T.W.Wolfs, C.Zwart, M.Bakker, M.Valk, C.Kuiken, and J.Goudsmi: Naturally occurring mutations within HIV-1 V3 genomic RNA lead to antigenic variation dependent on a single amino acid substitution, Virology, Vol.185, pp.195-205 (1991). [2] A.B.van't Wout, H.Blaak, L.J.Ran, M.Brouwer, C.Kuiken. H.Schuitemaker: Evolution of Syncytium-Inducing and Non-Syncytium Biological Virus Type 1 Infection, J.Virol, Vol. 72, pp.5099-5107 (1998). [3] A.B.van't Wout, L.J.Ran, C.L.Kuiken, N.A.Kootstra, S.T.Pals, H.Schuitemaker: Analysis of the Temporal Relationship between Human Immunodeficiency Virus Type 1 Quasispecies in Sequential Blood Samples and Various Organs Obtained at Autopsy, J.Virol, Vol.72, pp.488-496 (1998). [4] T.Shioda, S.Oka. S.Ida, K.Nokihara, H.Toriyoshi, S.Mori, Y.Takebe, S.Kimura, K.Shimada, Y.Nagai: Naturally occurring single basic amino acid substitution in the V3 region of the HIV-1 env protein alters the cellular host range and antigenic structure of the virus, J.Virol., Vol.68, pp.7689-7696 (1994). [5] M.Ohya: Information theoretical treatment of genes, The Trans, of the IEICE, Vol.E 72, N0.5, pp.556-560 (1989). [6] K.Sato, S.Miyazaki and M.Ohya: Analysis of HIV by entropy evolution rate, Amino Acids vol.14, pp.343-352 (1998). [7] M.Ohya, S.Miyazaki, Y.Ohsima: A new method of alignment, Viva origino, Vol.17, N0.3, pp.139-151 (1989). [8] E.C.Holmes, L.Q.Zhang, P.Simmonds, C.A.Ludlam and A.J.L.Brown: Convergent and divergent sequence evolution in the surface envelope glycoprotein of human immu-nodeficiency virus type 1 within a single infected patient, Natl. Acad. Sci. U.S.A., Vol.89, pp.4835-4839 (1992). [9] T.McNearney, Z.Hornickova, R.Maxkham, A.Birdwell, M.Arens, A.Saah, and L. Ratner: Relationship of human immundeficiency virus type 1 sequence heterogeneity to stage of disase, Proc. Natl. Acad. Sci. U.S.A., Vol.89, pp.1024710251 (1992).

141

[10] S.M.Wolinsky, B.T.M.Korber, A.U.Neumann, M.Daniels K.J.Kunstman. A.J.Whetsell, M.R.Purtado. Y.Cao, D.D.Ho, J.T.Safrit. R.A.Koup: Adaptive Evolution of Human Immunodeficency Virus-Type 1 During the Natural Course of Infection. SCIENCE. Vol.272, pp.537-542 (1996) [11] N.Nei: Molecular Evolutionary Genetics, Columbia University Press, NewYork (1987). [12] K.Sato and M.Ohya: Analysis of the Disease Course of HIV-1 by Entropic Chaos Degree, to appear in Amino Acids.

Quantum Information III, pp. 143-156 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

143

Q U A N T U M LOGICAL GATE B A S E D O N E L E C T R O N S P I N RESONANCE MASANORI OHYA: IGOR V. VOLOVICH fAND NOBORU WATANABE * University of Tokyo, Noda City, Chiba 278-8510, Japan GSP-1, 117966, Moscow, Russia, Noda City, Chiba 278-8510, Japan Moscow, Russia,

In classical computer, there exist inevitable demerits for treating logical gates. One of the demerits is an irreversibility of logical gates, i.e., AND and OR gates. This property causes to the restriction 2 of computational speed for the classical computer. There are two kind of approaches for avoiding these demerits. One of two approaches is proposed by Feynman 1. He introduced two reversible logical gates, that is, a NOT gate and a Controlled NOT (CNOT) gate. He proved that every logical gates can be constructed by only combinations of these NOT and CNOT gates. State changes under these unitary gates can be formulated by using quantum channels in quantum communication theory. By the way, there are several approaches for realizing quantum logical gates. One of the approaches is the study by using a nuclear magnetic resonance (NMR) 5 . Quantum logical gate based on NMR is performed by controlling the nuclear spin under the additive magnetic fields from the environments. In these spin systems, two states expressing the spin-up and the spin-down are usually used as the basic computational states. The performance of the computational speed does depends on the processing time changed from the state of spin-up to the state of spin-down. Unfortunately, as the mass of nuclear is more heavy than that of electron, the computational speed of NMR is more slow than that of the electron spin resonance (ESR). In order to discuss the quantum logical gate based on ESR more precisely, we should consider the relativistic effect. As the interaction among the nuclear is very week, it might be difficult to make the logical gate being able to treat the lots of qubits. Since there are lots of electrons even in one atom, one of the authors •DEPARTMENT OF INFORMATION SCIENCES, SCIENCE UNIVERSITY OF TOKYO, NODA CITY, CHIBA 278-8510, JAPAN, \\E-MAIL: [email protected], [email protected] tSTEKLOV MATHEMATICAL INSTITUTE, GUBKIN ST.8, GSP-1, 117966, MOSCOW, RUSSIA, \\E-MAIL: [email protected]

144

proposed the atomic quantum computer 6 , which can treat the interactions among these electrons and nuclear. In this paper, we construct the quantum channels for the NOT and the CNOT gates under the nonrelativistic formulation (Bloch equation) of electron spin resonance based on the atomic quantum computer, which is connected with the investigation for realization of the effective quantum algorithm 3,4 of NP complete problem. In the future development of our study, we will construct these gates for the fine structure and for the superfine structure of electron spin resonance in order to make our discussion more rigorously. 1

Quantum channel for N O T gate based on ESR

In this section, we construct the quantum channel for the NOT gate based on ESR. Let 0,1 be propositions of false and truth, respectively. The truth table of NOT and Controlled NOT gates are denoted by IJVOT

OjvOT

0

IcNOT

CcNOT

OcNOT

0 1 0 1

0 0 1 1

0 1 1 0

Every quantum logical gates can be constructed by the combinations of NOT and Controlled NOT gates. Using quantum channel, we realize these gates based on ESR. First of all, let us consider one particle case. Let Hs be C 2 with its canonical basis u+ = \\) = I I

1 , U- = \i) =

1 , M(HS) be the set of all bounded operators on Hs and ~E{Hs)sa —

{A € B (Hs); A = A*} , where A* is the adjoint of A defined by (A*u, v) = (u, Av) B (Hs)sa

has the basis ax

for any u, v € Hs.

01 10

called Pauli spin matrices and I =

0-i i 0 10 01

' ^ ( J - l ) ' which are

is an identity matrix on 7-Ca. That

145

is, a = {ax, ay,az}

is an orthogonal basis of B CHs)sa with the scalar product (<7i, Uj) = -trcxiCTj, j e {x, y, z}

and satisfies the multiplication table as follows: <Ti<Jj

(Jy Oz

1=1

Ox

&x

-icrz iay

ay ioz

-iffy

I

iax

—iax

I

<*z

Let S = {Sx,Sy,Sz) be a spin (angular momentum) operator of electron, where Si = \<Ji is a component of spin operator of electron in the direction of i-axis (i = x7 y, z). We denote unit vectors of x, y, z axis by e j , e^, ez and S is the spin vector given by "J

=

(>->X) by, bz) = Oxex + £>y£y + bz -*Z&z•

Let us consider two magnetic fields Bo and B\. BQ is a static magnetic field given by Bo =

BQ

ez

in the z direction and B\ is a rotating magnetic field given by B\ (t) = B\ (ex coswi + e^sinwi) with frequency u> in the xyjAsAii, where Bo and B\ are certain constants due to the magnetic fields. If B (t) is a magnetic vector defined by

~3(t) = Bi(t) + B!>, then i

has 1 Q

—

= S x B (t) = Bi (Sx cosujt + Sy smut) + B0SZ.

Let It) = ( o ) and ||>

be spin vectors related to spin up and spin

down, respectively. Let us take an initial state ip (0) = ao |t) + bo \l) (ao, bo e C satisfying \a,o by

|6o| = 1), then state vector at time t is denoted

1>{t) = a(t)tt) + b{t)\l):

a(t) bit)

146 where a(t) ,b(t) are complex numbers satisfying |o (t)\2 + |6(t)\ 2 = 1. Theorem 1 / / we start from our spin |t), i-e., S = £~(?z to apply B\ (t) for time t = ii such that B\t\ = § (§ pulse) then our magnet will be in & = £ e y under the condition ui = B\, where £ is a certain constant. Proof. For the initial state vector ip (0), xp (t) can be obtain as the following time-dependent Schrodinger equation of electron spin in the static magnetic field and the rotating magnetic field:

dt

'

b

X B {t)

V {t} ~ { lBie™a (t) - \B0b (t)

Then we have ia(t) = \BQa{t) + ±Bie-iwtb(t) ib{t) = i f i j e ^ a ( t ) - \B0b(i)

(1)

We put the coefficients a (t) and b (t) into the form

U{t) = c{t)e-^\ i t \b(t)=d(t)e ^ , where OJQ is the spin frequency. Then we have (ia(t) = fa(t) + \ib(t)=id(t)ei^t-!fb(t).

ic(t)e-i^t, K

'

Comparing with the above equations under the condition UJQ = BQ, one can obtain ic(t) = iBid^e^0-^, id{t) = \Bic{t)e-i{-"°-^t. When WQ — u) holds, the above equation can be rewritten by c(t) + \Bfc(t)=Q, d(t) + \B2d(t) = 0. One can obtain (c(t) = asm (fyt + r) , \d(t) =iacos(^t + r) , where a is amplitude and r is phase. Thus we have the state vector at time t such as * (t) = a s m i^-tj

e • • * | t >+.acos ^ t j

e* • \±) = ( - a J

(

^

}

e

^« J

147

where we have taken r — 0. The expectation values of Sx, Sy and Sz are given by <5x> = - i c o s ( B i < ) (Sy) = — | sin (-Bit) sin (wot) (Sz) = | sin (Bit) cos (tvot) At a time t = 0, one can get

When Biii = f, the expectation value of Sz at time £i is obtained by (Sz) = 0 and the expectation value of Sy at time £i is

(sy) = -\ under the condition w = B\. It means that our magnet is in y direction at time h = •£; = £. (Q.E.D.) Theorem 2 If we have the eigenvalue equation i^Q-

= -~$ x # (i) ^ (t) = - [Bi (S x cos (wt) + 5 y sin (art)) + B 0 S Z ] V (t)

w/ien [SD 5j,] = iS^ is hold and Bo,B\,u) solution has the form

are arbitrary constants, then the

yj, (t) = e-**S.emu+Bo)S,+B1S.)ii> In particular, we see the resonance

(Q )

_

condition

UJ + B0 = 0,

that is, iP(t) =

eiBots*eitBlS*i>(0).

Based on the above results, we reconstruct the Not gate based on ESR using by quantum channel. Let us take u+ and u_ as u+ =

(o)'

u

-

=

(i)-

148 T h e n one can get t h e following relations: ( Sxu+ \SXU-

= +^u-, = +2«+,

Szu+ = T h e o r e m 3 For the

initial

+hu+,

SZU-

= — 2U--

state

vector

ip (0)

=

OQU+ + 6QU_

, (ao,bo € C ) , one can obtain by using the theorem 2

lM*) = a0 cos ( - — 1 + ib0 sin I

+ Proof.

—

iBQt exp

•iB0V exp | — - — ] u.

h I B l t \ J. • • fBlt b0 cos ( - — I + ia0 sm I — -

Since elB°tSz

and eltBlSx

u+

are expressed by

have ^{t)

eiB°ts*eiBlts*ip{0)

=

= | c o s | ' ^ 7 + 2isin^

N

]5

2

]x

x | cos I -~- j I + 2i sin I - y - j Sx ) (a0u+ + 6 0 u_) _ ( c o s ( f ) , +

2 i s i

„ ( ^ ) s ; ) x

x M a 0 cos f - y J + ib0 sin f - y J J u + + + ( «ao sin

^) + 6 o C O S (ir)) u -)

Bit\ ., . (Bit a0 cos [ - — I + ib0 sin I ——

+

6ocos

Bit

Bit iao sm

exp

iB0t

u+

-iB0V

exp | — - — | i t .

=

149

(Q.E.D.) For example, we take ao — l,6o = 0, that is, the state vectors at time 0 and t are V>(0)=u+, V> (t) = cos ( 4 p ) exp (i^i)

u+ + ism ( 4 p ) exp (=*§**•) i t -

If we take t = ( 0 ) = u_, \ V(t) = i s i n ( ^ ) e x p ( ^ ) u+ + cos ( ^ ) exp (=if<*) «_. If we take i = ij. such that %gi = ^ - = § (TT pulse) then V»(ti) = - u n it means that this gate is performed as the N O T gate based on ESR. Let Uxor (t) = eiBotszeiB1tsx ^ e a u n i t a r y operator expressing the NOT gate based on ESR. Quantum channel denoting the NOT gate based on ESR is defined by A

jvoT(ti) (•) = UNOT (*I) (•) U^OT

(*i).

For the initial state \ip (0)) {ip (0)| at time 0, the output state of ti*NOT(t ) *s obtained by

(IV(o)>ty(o)|) = |V(ti)MV'(ti)|. Thus we have the complete truth table of the N O T gate based on ESR as follows: \ l <—>u_ Ijvor 0 1

OjVOT

1 0

150

2

Quantum channel for C N O T gate based on ESR

In this section, we introduce the quantum channel for the CNOT gate based on ESR. Let us consider N particle systems to treat the Controlled Not gate. Let ei, S2, ez be unit vectors of x, y, z axis, respectively, and let S^-1', •••, S(N> be spin vectors of N electrons such as 5W = (
+ S^el

+

S^eg.

The spin operators satisfy the following commutation relations 3

7=1

where £Q/37 = <

and 5pq is a certain constant. Let us consider a Hamilto-

nian operator for AT particle systems given by

\i=l

/

\i=l

/

»,J = 1

where / (t) is a certain function, for example / (t) = cos uit and Jij is a coupling constant with respect to i-th. spin and j-th spin. Sfr is embedding Sk into i-th position of N tensor product. SP=I®---®Sk<8>---®I,

(fc = 1,2,3).

Let us take a Hamiltonian H(jy) as a Ising type interaction, that is

HW = B3 (£&) + E J^ ® &. If N = 2 then one can denote H ( 2 ) = B3 (S3 ® / + I ® 5 3 ) + J (5 3 ® S3) + flo (/ ® / ) , where Z?o> ^3 and J are determined by a certain phase parameter w. Then we have 5 -2ia rt (s 3 ®s,) =

y* (0n(-^)" ( 5 a ® 5 a ) n ^—'

n=0

n!

151 \2n

3 B-^^^c*®*) " 2n\

n=0

Nn(-2wt)

^E(-D"^V (53053) oo

/ -Uj \ In

n=0

/-u;t\2n+1

oo

'

v

n=0

'

= cos f y \ (7 ® 7) - 4. sin f y \ (S3 ® S3) • Indeed one can obtain

efc*(/®s.) = c o s

M\

(/ ® /) + 2i sin ( y )

(7 ® 53),

e ^t(W) = c o s

( ^

(/

( S 3 (8) 7 ) .

Let us take u+,U-,v+,V-

0

j)

+ 2f

sin (

^

as

/1\ 0 0

U+
u + i>_ =

w =

1 0

1

W /o\

/°\ U _ 1 > +

/ 0 \ 0

0 0

u _ <X> u _ =

W

Then one has

( S2u+ = + | u _ , 5 2 u _ = - | u + , \ 52V+ = +fu_, S2V- = -fU+, ( S3u+ = + i u + , 5 3 u _ = - 5 U - , \ S3u+ = + | v + , S3u_ = -;|v_, Let tj) (0) be an initial state vector given by

•0(0) = aou+®v++boU-V-.

=

,(aQ,bo,co,d0

G C).

152

For the initial state vector ip (0), if J = 2w, 7?3 = — u and 7?0 = \u are hold, then the state vector at time t is expressed by tf(i)=exp{-iiH(2)}V(0) = exp {-it (B3 (5 3 ® 7 + / ® S3) + J (5s ® S 3 ) + So (7 ® / ) ) } V (0) _

eiu>t(S3®I)eiut(I®S3)e-2iut(S3®S3)e-%iu>t(I®I)ij

/Q\

Theorem 4 For £/ie initial state vector %p (0), one can obtain ^U\

_

eiwt(53®/)etwt(/®Ss)e-2iwt(S3(853)e-^ia>t(/®/)^;/Q\

= (aou+ <8» v+ + boU- ® v+ + CQU+ v_) + do exp (—2iwt)u_ ® u_. Proof. Since e*""^® 7 ), e-2i«t{s3®s3)

ia,t 7 e

= cos

( ® 53 ) and e -2^t(s 3 ®s 3 ) M\

(/ 0 /}

e*-t(/®S,) = c o s

M A ( / g, / }

e *-t(5,®/) = c o s

/ ^ A (/

a r e expresse(i

_ 4-sin ^ A + 2i gin

0 7) +

ty

(53 0 5g)>

^

(/

g, 5 g ) ^

2isin f y ) (S 3 ® / ) ,

one can obtain ^LU\ _

e iwt(S 3 ®7) e iw((/®S 3 ) e -2iwt(S' 3 ®S'3) e -iiwt(i(gi/)

= (

c o s

./Q\

( y ) ( 7 ® 7 ) + 2 i s u W y J (S3®7)J x

x ( c o s f ^ - ) ( 7 ® 7 ) + 2 i s i n ( ^ - ) (7S3)) x x (cos ( y j (7 ® / ) - 4isin f y j (S 3 ® S 3 ) J exp

f-^

x (aou+ ® f+ + boii- ® i>+ + cou + v_ + d o u - ® v~) =

cos ( y ) (7® 7) + 2isin f y ) (S3 ® / ) ) x x (cos ( f ) (7 ® 7) + 2isin ( y ) (7 ® S3) ) x x (ao exp (—iujt) u+ ®v+ + 6ou_ ® v+ + +CQU+ <8> v- + do exp (—iwt) u_ «_)

cos Of)

(I®I)+

2i sin ( y ) (S 3 ® 7 ) ) x

153 x I a 0 exp I - — I u+ ® v+ + b0 exp I — I u_ i> + +

(

iwA

,

/— 3ia;A

—— ) u+ i>_ + d 0 exp I — - — l « _ ® v = (aou+ ®v+ + 6 o « - ® ^+ +COM + ® u_) + doexp(—2iujt)u- ® v-. (Q.E.D.) For example, if we take ao = 1 or bo = 1 or Co = 1, then we have rp(t) =ip (0). Let us take do = 1, that is ip(0) = u_ ®i>_, the state vector at time i is obtained by ip (t) = exp (—2iwi) u_ u_. If we take t = t\ such that 2wt = TT ( | pulse) then one has ^ (*i) =

—u

- ® v--

Therefore one can denote the matrix form E/$ (h) of e ^ ^ ^ e ^ M ' s ^ x xe -2iwti(S 3 ®S3) e -3««ti(/(»/)

^

/1000 \ 0100 tf* (*i) = 0010 \000-l/ Next we construct a unitary operator UJJ (t) related to a Hadamard transformation based on ESR. Let us define UH (t) by UH(t)=e-i"2tV®s*\ where u>i is a certain phase parameter. Then we have e-iu,2t(I®S2)

=

c o g

2 /

(/ ® I) - 2i sin

2 /

(7®52).

For the initial state vector ip (0), the state vector at time t is expressed by V> (t) = UH (t) i> (0) = e-*"a*(«9*«ty (0).

154

Theorem 5 For the initial state vector -0 (0) = a^u+ ® v+ + 6oit_ (g v+ + CQU+ «_ + d 0 u_ ® v_ , (aQ, bo,co,d0 € C) , one can obtain $ (t) = e - ^ 2 ' ^ ® ^ ) ^ ( 0 ) = aow+ 2t\ fu2t\ +doU- (g I — sin I —- I v+ + cos I -— I w_ Proof. Since e~iU2t^®s^ e -«, 2 t(/®5 2 )

is expressed by

= cos ^

V / o /) _

2i

sin ^ \ { I

® 5a),

one can obtain V> (i) = e-*"**^*3*)^ =

(0)

/®(cos(^)j-2*sin(^)s

2

)x

x (ao"+ ® t ) + + 6oM_ i>+ + CQU+ <8>v- + dou_
(w2t\

.

(u2t\

+b0u- (g I cos I — I v+ + sin I — 1 V+C 0 M+ I - sin I - | - J v+ + cos ( - | - J v_

(Q.E.D.)

+rfo«-
If we take t = t2 such that ^

=

f (§ P u ^ se ) > then one has

ip (h) = (aou+ + 6 0 u_) —= (-v+ + « _ ) .

155

Then one can denote the matrix form UH {t2) of e - ^ 2 * 2 ^® 5 2 ) by

^<«=(i:)^(lr1' Thus the unitary operator UCNOT (h + 2*2) related to the CNOT gate can be reconstructed by the combination of t/$ (*i) and UJJ (£2) as UCNOT (*I + 2t 2 ) = C/ir (t 2 )* ^ $ (*i) ^ff (*2) _ e Jw 2 t2(/®S 2 ) e ia;ti(S3ig>/) e iwt 1 (/(giS3)

x

x e-2iuti (S3®S3) e - 1 iwti (/»/) e -iw 2 t 2 (J®S2)

and the matrix form of UCNOT (*I + 2i 2 ) is obtained by rr

^

o^

UCNOT {H + 2*2) =

/1000\ 0 10 0 0

0 0 1

^0010/ It means that this unitary operator UCNOT (ti + 2*2) is performed as C N O T (Controlled N O T ) gate based on ESR. Quantum channel denoting the CNOT gate based on ESR is defined by A

cjvoT(t!+2t2) (•) = UCNOT (*I + 2t 2 ) (•) UCNOT (ti + 2t2).

For the initial state \ip (0)) (ip (0)| at time 0, the output state of ^CNOT(t +2t ) is obtained by A* IX

CNOT(ti+2t2)

(|V (0)) (V* (0)|) = |V (ti + 2t2)> (V (*i + 2t a )| •

Thus we have the complete truth table of the C N O T gate based on ESR as follows: ICNOT

1

OCNOTJ CCJVOT}

ICJVOT

CCNOT

OCNOT

0 1 0 1

0 0 1 1

0 1 1 0

In order to make our discussion more rigorously, we will discuss the quantum channels for the NOT and the CNOT gates under the Dirac formulation

156

(fine (Relativistic) formulation) of electron and under the hyperfine formulation including the interaction among electrons and nuclear. References 1. R.P. Feynman, " Quantum mechanical computers", Optics News, Vol. 11, 11-20 1985. 2. M. Ohya, "Mathematical Foundation of Quantum Computer", Maruzen Publ. Com., 1999. 3. M. Ohya and N. Masuda, "NP problem in quantum algorithm", Open Systems & Information Dynamics, 7, No.l, 33-39, 2000. 4. M. Ohya and I.V. Volovich, " Quantum Computing, NP-Complete Problems and Chaotic Dynamics", quant-ph/9912100. 5. C.P. Slichter, " Principles of Magnetic Resonance", Springer Ser. SolidState Sci., Vol.1, 3rd ed, Springer, Berlin, Heidelberg, 1992. 6. I.V. Volovich, "Atomic Quantum Computer", quant-ph/9911062; Volterra preprint, Roma Universita TorVergata.

Quantum Information III, pp. 157-176 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

157

C U R R E N T FLUCTUATIONS IN NONEQUILIBRIUM STEADY STATES FOR A O N E - D I M E N S I O N A L LATTICE C O N D U C T O R

SHUICHI TASAKI Advanced

Institute for Complex Systems and Department School of Science and Engineerings, Waseda 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, E-mail:[email protected]. ac.jp

of Applied University, JAPAN

Physics,

As a continuation of the previous work, currnet fluctuations of a one-dimensional conductor are investigated for nonequilibrium steady states ui±oo driven by temperature and/or chemical potential differences, which are obtained, respectively, as t —• ±oo limits. We show that (1) the current correlation functions for the states w±cx) are identical, (2) the variance of the current fluctuations at zero temperature and for small chemical potential difference is proportional to the averaged current as shown by Shimizu et al. and (3) the Kubo formula holds at equilibrium.

1

Introduction

The understanding of irreversible phenomena including nonequilibrium steady states is a longstanding problem of statistical mechanics. Several approaches were proposed 1,2 ' 3 and new formulations are still developed 4,5 ' 6,7 ' 8 . Since general features of irreversible phenomena are not well understood, rigorous approaches are important. So far, nonequilibrium steady states are analytically constructed for many open conservative systems. The reservoirs are modelled either by random forces or boundary conditions 9 ' 10,11,12,13 or by (infinitely extended) systems 14 ' 15,16,17 ' 18,19,20 ' 21 . Antal et al.22 used a Lagrange multiplier method instead. Examples of the purely dynamical approach include harmonic crystals 14,15,17 , a one-dimensional gas 16 , abstract multibaker maps 18 , a periodic Lorentz gas 19 , unharmonic chains 20 and an isotropic XY-chain 21 . In their study on nonequilibrium steady states for a classical infinite harmonic chain, Spohn and Lebowitz15 used semiinfinite left and right segments as reservoirs. They showed that any initial state, where the left and right reservoirs are in equilibrium with different temperatures, evolves towards a steady state with nonvanishing energy current. The current flows from the higher temperature to the lower temperature parts and is proportional to the temperature difference. They also obtained the corresponding full distribution function. Recently, following the same line of thoughts as Spohn and Lebowitz, and applying the method of C*-algebra, Ho and Araki 21 proved the approach to nonequilibrium steady states for an isotropic XY-chain.

158

As the works by Spohn-Lebowitz15 and Ho-Araki 21 , we studied nonequilibrium steady states for a one-dimensional conductor with the aid of the C*-algebra23. Left and right semiinfinite segments of the lattice are assigned for electron reservoirs. Initially the two reservoirs are set to be in equilibrium at different temperatures and/or different chemical potentials. The evolution of the initial states for t —> ±00 was investigated and two different quasi-free steady states u>±oo were obtained. We remark that, as the Xy-chain is equivalent to a free fermion gas on a chain, our construction of the steady states is essentially equivalent to the Ho-Araki construction 21 . The steady state w+0O carries nonvanishing electric and energy currents, which agree with the nonlinear generalization of the Landauer conductivity and which are consistent with the second law of thermodynamics 23 . Moreover, the state w+00 is equivalent to the nonequilibrium steady state proposed by MacLennan 24 and Zubarev 2 . The other steady state w_oo carries antithermodynamical currents and is the time-reversed state of w +tX) . Roughly speaking, in "a space of states", the state w+00 behaves as an "attractor" and w_oo as a "repeller". And initial states evolve unidirectionally from the "repeller" to the "attractor" in a way consistent with dynamical reversibility. As is well known, fluctuations of physical quantities characterize statistical states. Particularly, equilibrium fluctuations are related to averaged quantities at or near equilibrium 3 . For example, the transport coefficients in the linear response regime are given by the correlation functions of equilibrium fluctuations (the Kubo formula) 3,25 . The case of nonequilibrium steady states is more interesting since fluctuations are considered to be independent from averaged quantities (e.g., see Shimizu et al. 26 and references therein). In this article, as a continuation of the previous work23, we study fluctuations in the steady states u>±<x>- Sec. 2 is devoted to the summary of the previous results 23 . In Sec. 3, we show that the current-current correlation functions for the two states w±oo are identical. The explicit expressions of the correlation functions are obtained in Sec. 4. The leading term of the symmetrized correlation function is constant in time and is equal to that of the variance of the current fluctuation. Particularly, at zero temperature and for small chemical potential difference, the variance is proportional to the averaged current as was shown by Shimizu et al. 26 . In Sec. 5, we calculate the derivative at zero frequency of the Fourier-transformed antisymmetrized correlation function. At equilibrium, the derivative is proportional to the conductivity, or the Kubo formula holds. On the ohter # hand, for nonequilibrium states, there is no simple relation between the zero-frequency derivative and the averaged current. Sec. 6 is devoted to the summary and concluding remarks. In Appendix, we outline the derivation of the Kubo formula.

159

2

Model and Nonequilibrium Steady States

The system in question consists of electrons on an infinitely extended chain interacting with a localized potential and is defined on a C*-algebra as follows. The basic dynamical variables are creation and annihilation operators, c*; a and Cji<7 respectively, of an electron at site j ( g Z) with spin a(= ± ) . They satisfy the canonical anticommutation realtions (CAR): [cji<7,ckiT}+ = [c*jia,c*kT}+ = 0 ,

[cj i<7 ,4 |T ]+ = 6jk6aTl

,

(1)

where [A, 5 ] + = AB + BA is the anticommutator, 0 the null element and 1 the unit. The C*-algebra A of dynamical variables is the CAR algebra 27 ' 28 , i.e., a Banach *-algebra with C* norm generated by + oo

B(f,g) =J2

Y, Vi.°ci,° + &><$,*> >

(2)


where the sequences {fj,a} and {gjl<7} are square summable. The physical states are defined as positive and normalized linear functionals u> over the algebra A, i.e., linear functionals satisfying (i) ui(B*B) > 0 for any B £ A and (ii) o>(l) = 1 with 1 the unit of A. The Hamiltonian H of the system is given by + oo H

h

L c

c

c

c

= ~ ~f J2 1Z { *j," 3+h^ + *i+i,a J,A + J2YlhejCJVCJ>'

(3)

where Ti is the Planck constant divided by 2ir, j(> 0) is the strength of the electron transfer and Cj stands for the localized potential. The corresponding "first quantized" Schrodinger operator is assumed to admit a complete set of outgoing scattering states and have no bound state. The outgoing state 4>q{j) (—7T < q < IT) is the solution of the eigenvalue equation corresponding to an eigenvalue Eq = —2hycosq: -hi

{i>q{j + 1) + 1>t(j - 1)} + hej^q(j)

= Eq^q(j)

,

(4)

with the outgoing boundary condition: 1>t(J) -+ - 4 = {eiqi + Rqe'igi

} ,

when j — -oo(+oo) for q > 0(< 0) , (5)

V27T

where Rq is the reflection amplitude. The time-evolution automorphism at : A —• A is generated via a truncated Hamiltonian in a standard way 27 . Initial states are prepared in the following way: Firstly, the chain is divided into three: (-oo, -M - 1], [—M,N] and [Ar + l,+oo) with M > 0

160

and N > L. The two semiinfinite segments serve as reservoirs and the finite one as an embedded system. Corresponding to this division, the algebra A is decomposed into a tensor product of the three subalgebras AL, AS and AR: A — AL ® As ® AR. NOW the Hamiltonian H is represented as a sum of a left-reservoir part HL, a right-reservoir part HR, an embedded-system part Hs and a reservoir-system interaction Vint: H = HL + HR + Hs + Vint • There is a similar decomposition of the number operator: N = NL + NR + Ns- Next we introduce an equilibrium state U>L over the algebra AL of the left reservoir variables with inverse temperature /3L and chemical potential HL corresponding to the Hamiltonian HL and the number operator NL . Similarly, let LOR be an equilibrium right-reservoir state over AR with inverse temperature J3R and chemical potential fiR corresponding to the Hamiltonian HR and the number operator NR. Then, for each embedded-system state u>s over As, an initial state uJin is given by a tensor product Wm = U i ® U>S U!R .

(6)

23

We showed that, for t —> ±00, the initial state u>i„ weakly evolves towards unique quasifree states to±^, i.e., for any B £ A, limj-^-too Win iat{B)) = ijj±oo(B), irrespective to the choice of the separating points M, N and the initial system state u s . As the state ui±oo are quasifree, they are fully characterized by the two-point functions. For example, W+00(c;acjla,)

= boa, fdq {FL(q)i>q(jyMJ') Jo

+ FR{q)^q{j)*^q{j')}

, (7)

where FL(q) = l/{ef}^E«~>i^ +1} and FR(q) = l / V * ( B » - ' " i ) + l } are Fermi distribution functions for the left and right reservoirs, respectively. Eq. (7) gives two-probe Landauer-type formula for the electric current 23 : w

+oo (Jj-w)

f'dqsinq\Tq\2{FR{q)-FL{q)}

= ^ "•

,

(8)

Jo

where —e is the electron charge, Tq the transmission amplitude and Jj-i\j stands for the electric current operator from the (j — l)th to the jth sites: J

i-i\i

3

= ~ie7 ^2 {c*j,*cj-i,v ~ c*j-\,ach")

•

(9)

Current Fluctuations and Their Symmetry

Current fluctuations can be characterized by correlation functions of arbitrary order and the simplest one is the current-current correlation functions. Thus,

161

we investigate correlation functions of the total current over N sites: (AT-l)/2

JN=

£

Jj-w

,

(10)

j=-(JV-l)/2

where N is an odd integer. Since current operators at different times do not commute, two correlation functions are possible, namely, the antisymmetrized correlation function (6JN(t);6JNy±>

= w±oo ( -[JN(t)

- W ± 0 O ( J J V ) , JJV -W±OO(^JV)]_

J , (11)

and the symmetrized correlation function {6JN(t);6JN)^'=uj±03l-[JN(t)-uj±cx,(JN),JN-Lj±00(JN)]+\

, (12)

where [^4,B]± = AB ± BA and ± are identical with each other. Proposition 1. (6JN(t);8JN)¥

= -(6JN(-ty,SJN)^

(13)

(6JN(t); 6JN)^

= (6JN(-t);

(14)

(6JN(t);6JN)W

= {6JN{i);6JN)W

(15)

(6JN(t); 6JN)W

= (6JN{t); SJN)^

(16)

SJN)^

Proof Eq. (13) follows from the invariance of the steady states w±oo: u±00(Q) w±oo(a_4(<5)): {6JN{t);6JN}^

- w±oo l-a-t

{[<M<) - u±00(JN),

JN - W ± 0 O ( J W ) ] _ }

=

J

162

= w±oo I -

[JN

-

^±OO(JN),

Jif(—t) — U±OO(JN)]_ J

= -(WivH);Wjv)La). Similarly, we have Eq. (14). Let / be the time reversal antiunitary involution 23 , then we have / jN(t) i = -jN(-t), Thus, since

W_ T O (JJV)

{6JN(t);6JNyy

«_«,(
.

= — OJ+OO{JN), one obtains Eq. (15):

=w_oo ( ~ [JN(t) + U+OO(JN),

JN + W + 0 O ( J ; V ) ] _

= W + 0 o [I-[JN(t)+U+oo(JN),JN

J

+W+0O(/AT)]_ /

= w+0O I —r- [JN(—t) - u)+OQ(J]\'), JN - w+0o(
= («Jjv(<);^JV>ia) •

In the last equality, we used Eq. (13). Eq. (16) is proved in the same way. 4

N —• oo Limit of the Correlation Functions

Since the average of JN is of order of N, correlation functions are expected to be of order of N: (6JN(t);6Jtf)+'/N = 0(1) (A = o or s). In this section, we study the limit: limN^^{SJN^y, 6J^y+'/N (A = a or s). Proposition 2. Suppose that the eigenfunctions ipp{j) are continuously differentiable with respect to p for each j £ Z. Then, we have

Jim ±:(6JN(t);6JN)P = 0

(17)

N—>oo jv

lim 1(WW(<);W^)W A T — • o o JV

2 7 F e 2 f^ ^

*-/

r/O

rfo^

T y_27

1e2

+~

t2"1

/

"" J-2-y

{|T ? | 4 +|/«:,| 4 }{^(9Ki-^(9))+^(?)(i-^(
aq dQ.

d f i ^ \Tq\2\Rq\2{Fn(q)(l-FL(q))+FL(q)(l-FR(qm

(18)

"9

with g = cos _1 ( -p- J and ^p = 27 sin q. Note that the limits don't depend on t.

163

Remark The t — 0 value of the symmetrized correlation function is the variance of the current fluctuation w+oo(<$Jjv) and, thus,

lim -j-w+0O(*J&) = lim N—*ool\

USJN(0);6JN)^

N—t-oo iV

d f i ^ {|T,| 4 +I^| 4 } { ^ ( g ) ( l - ^ ( 9 ) ) + ^ ( 9 ) ( l - ^ ( 9 ) ) }

= - / 2e2

'27

+ _- /

dfi— i T . P i ^ H ^ ^ l l - ^ ^ + f L ^ C l - ^ ^ ) ) } (19)

'-27

As easily seen, at zero temperature f3£ = 0^ = 0, the variance vanishes for the perfect conductor (i.e., for \Tq\ = 1), which is a particular feature of the one-dimensional system 26 . Moreover, at zero temperature /?£* = Z?^1 = 0 and for small chemical potential difference \HR — HL\ -C IMJJ + A'LI, the variance is proportional to the averaged current u+00(6JN) JV-^OO

_

TV

(dQ,

W+OO(JN)

2 U

q]

\dq'

m

+0(\m-LtL\2)

, 7

(20)

where qp is the Fermi wavenumber. This is consistent with the result by Shimizu et al. 26 . Proof of Proposition 2. With the aid of Eq. (7), we have (SJN(t);SJN)^

= f

{6JN(t);6JN)%>

= I

dqdp\KN(q,p)\2A^(q,p)

(21)

dqdp\KN(q,p)\2A^(q,p)

where the kernel KN(P, q) is given by (Af-l)/2

KN(q,p) = ~iej

]T j=

[iMj-lWi)-lMiW(j'-l)]

(22)

-(N-l)/2

and A(a

\i'P)

= Y^?;sin("« -

fi

p)*

(23)

164

In the above, £? = /^(/ifi, - HL) for q > 0 and £q = 0R(hQq — p,R) for q < 0. Because, for q > 0, ^ a )(g,«?) = A^(q, A^(q,

-q) = A(a\-q,

q) + A^(-q,

A^i-q,

q) = A^\-q,

-q) = 2 {FR(q)(l

q) + A<%, -q) = 2 {FR(q)(l

-q) = 0

- FR(q)) + FL(q)(l - FL(q))} - FL{q)) + FL{q){\ -

FR(q))}

the desired results (17) and (18) are derived from the following lemma. Lemma For a continuously differentiable function A(q,p) of q and p, we have lim

/

dpdq\^%^A(q,P)

e2 [*> ^dQ T7-27

+_/

1T,| 4 +1^| 4 fAI

°g

^

dn—irj^i^iM^Cg.-gJ + ^-g.g)}

f J-27

(24)

"g

Proof of the lemma The given integral reads as

+ /

dpdq \KN(q'-P^

{A(q,

-p) + A(-p, g)}

(25)

Now we observe that the kernel A'^v can be decomposed into a contribution from the sites j with / < \j\ < (N — l ) / 2 and the rest. For sufficiently large /, the former can be evaluated with the aid of the asymptotic form of tpP(j) (cf. Eq. (5)). For example, we have KN(q, -p) = (rN(p+q) +aN(q-p) where

gi(q,p) +
K2i-i(q,-p)

-q)

165

92(q,p) = %-. {T,iJ* p ( e -'« - e*) + T_pRq(e^ 2wi eiNp/2

crN(p) =

- c"'")}

_ei(2l-l)p/2

2J sin(p/2)

Substituting this and similar expressions of K^(q,p) and Kff(—q,—p) into Eq. (25), one can calculate the N —• +00 limit in a standard way. As an example, we consider a term of the form • 2 /

I

J[o,x]'

dqdp^«p^B(q,p)= N

f dq f dpB^p)^

Jo

Jo

M(q-p)\

> 2 *{

^sin

{"-f-)

,

where M = N — 21+1 and B(q,p) is continuously differentiable. Its integrand is sjn2

/•*

S ln sin

(M(q-p)\

II

N J0 ^"yH'y; + / dp Jo0

'-44

'" V

M(„-,)

}I

JV/

SM

4

9-P

f f 4^

y\4sin2V

.

2

, snTx

(P-?)2

N(q-p)

where, for N —> +00, the second and third terms converge to zero uniformly with respect to q and the first term converges to wB(q,q) at each q. As the absolute value of the first term is uniformly bounded by Trsup \B(q,q)\, Lebesgue's convergence therorem implies lim

I

dqdpl
P)?

B{q,p)

= TT f

dqB(q,q) = 7r P

dQ§-B(q,q)

Other terms can be calculated in the same way and we obtain the lemma. 5

Kubo Formula and Current Fluctuations

According to the Kubo formula 3,25,7 , the Fourier transform of the equilibrium antisymmetrized correlation function is related to the dc conductivity G via

G= „m llm!%^, iv—+00 w—0

iwN2

M

166

where Q\"q\LJ, N) is the odd part of the Fourier-transformed antisymmetrized correlation function: Q ^ ( " , N) = \ {Q ef (w, N) - Q e ,(-u>, N)}

,

(27)

with ge?(W,JV)=^y0°rfie-^/y[JjV(0-(JJv)o,^-(^)o]\

•

(28)

In the above, (• • -)o stands for the equilibrium average. At first sight, Proposition 2 seems to imply Qeq = O(N) and, thus, the limit (26) vanishes, but, as we shall see, it is not the case. When the limit is taken in an approriate order (i.e., N —> oo after LU —> 0), one obtains a finite value. Note that N in Eq. (26) is not the system size, but the range of the external field.0 A unique feature of the Kubo formula is the fact that it relates the current fluctuation (of an equilibrium state) to the averaged current (of a nonequilibrium state). Thus, it is interesting to compare the nonequilibrium current fluctuation with the averaged current. As we shall see, in nonequilibrium situation, there is no simple relation between the antisymmetrized current fluctuation and the averaged current. In this sense, the current fluctuation is independent from the averaged current. Of course, the Kubo formula (26) recovers at equilibrium. Indeed, we have Proposition 3. Suppose that the eigenfunctions ipp(j) are twice continuously differentiable with respect to p for each j £ Z. Then, we have ,2d{FL(q) w-o

+

iuN2

2e p

2irhJ_2y 'q 27 dU \FR{q)-FL{q)

M y - 2 7 ^ 3 P j 2Tsing

+ FR(q)}

on R

[

fd

d\

T ^ [dp-d-J

11

h

»^\l=p=q (29)

"In the conventional derivation of the Kubo formula, one starts from a finite size system and the infinite volume limit and u/ —* 0 limit are taken in this order after calculating the correlation function. On the other hand, in the present case where we deal with an infinitely extended system from the beginning, the formula (26) is derived via an external field localized on a finite-size interval [—(AT — l ) / 2 , (N — l)/2] C Z. Hence u - t O limit is taken before the infinite limit of the external-field range N. For details, see Appendix.

167

where Q^\u,N) = {Q+(u,N)-Q+(-u,N)}/2 is the odd part of the Fouriertransformed antisymmetrized correlation function Q+(u),N) defined by Q+{u,N)

=\

T n Jo

dte-iwt{6JN(t);6JN)^

,

(30)

q(Q) = cos - 1 ( — Y~J, Tq the transmission amplitude of the potential and FLiR\(q) the Fermi distribution function for the left (right) reservoir. The second term of Eq. (29) is finite (see Eq. (36) below) and vanishes for N —»• oo. Clearly, there is no simple relation between the right-hand side of Eq. (29) and the averaged current (8) unless FL{H) = FR{q). At equilibrium where Ft(g) = FR(q), the right-hand side of Eq. (29) reduces to the Landauer formula for the one-dimensional conductivity and the Kubo formula (26) does hold as expected. Proof Because of Eqs. (21) and (23), we have Q+{<*,N) _ 2TH [ + 2nv / + - p.v. /

dpdqI
,,KN(q,P)I
{6{w

_

_ ^(_w _ ^ 2(Qq - Qp) TTT p-^ -j ,

+

(31)

where p.v. stands for Cauchy's principle part, /"Ov(p,g) is the kernel defined by Eq. (22), £, = /3L(hQq - fiL) for q > 0 and £q = (3R{hSlq - nR) for q < 0. Obviously, the odd part (2+ (w, N) is the first line of Eq. (31). Note that, since Q+ is odd with respect tow, it is enough to restrict u to be positive for calculating the limit lirn^^o Q+ (w, N)/(iw). In terms of the integration variable f2 = — 27 cos g, we have iuN2 -

f2l*!-<* ~W , 0 /( \I
, 2TT / * - ^ | * j v ( p , - g ) | 2 - | t f j v ( « , - p ) l 2 F«(«)-FL(p) W2ft /i ._'2 7 w ^ ( 4 T 2 - n 2 ) ( 4 7 2 - ( f i + w) 2 ) '

168 where q = cos - 1 (—0/(27)) andp = cos _ 1 (-(ft+w)/(27)). Using the relations

KN(q,q) = -KN(-q,-q) I
= - ~ v V - ft2 \Tq\2 = ~ 9R

%/4 T 2 - ft2 Tq R*q /„.,

,/0

d\

.

.1

one finds that, in the limit of w —* 0, the integrands of the first and second terms of the right-hand side of Eq. (32) for each value of ft G (—27, 27) converge, respectively, to 2

e |.rrvdiMti + Mq)} r?l

2wh

(33)

on

and

M

^-ft

2

2 T sin,

^ f , f l » \TP ~ dq)

KNiP q)

'- \p-_q

(34)

The limit lirn^^o Q+ (<*>, N)/(iu>N2) is the integral of the sum of (33) and (34), but the proof is not straightforward due to the singularity of the integrands at ft = ±27. Firstly, we note that f2j-W ^ 7-2-y ,27-w

\KN(q,p)\2 + \KN(q,-p)\2 2 2 2 V/(47 -fi )(472-(n + W) )

FL(g)-FL(p) W

{l^^(g,P)| 2 + |g W (g,-p)| 2 }(Fz.(?)-fz-(p))

y-27

v/(4T

2

_ /^-^^^(sing-smp)^! J- 27 v ^ ^ ^ + w) 2

, JV 2 e 2 (4 7 2 -n 2 )|T 0 |' 9 F L ( , )

- f i 2 ) ( 4 T 2 - ( n + w )2) 2

^ ^ _ /^-" 7-27

^V|r,|2^(g) 47T2 3ft ' (35)

and that the numerators of the integrands of the first and second terms are bounded by C\\p— q\ with some positive constant C\. Then, because \q — p\ is uniformly bounded by C2\/u and 27_w

27 27

2-y w

dn „ , C47 , < C 3 log - ^ V ( 4 7 2 - « 2 ) (47 2 - ( 0 + " ) 2 ) ~
169

with positive constants Cj (j = 2, 3,4,5), the first and second terms of Eq. (35) vanish in the limit of w —• 0. The third term converges to the integral over the interval (—2j, 2j) as the integrand is bounded. In this way, one can calculate the u —+ 0 limit of the first integral of Eq. (32). Now we turn to the second integral of Eq. (32). With the aid of the eigenvalue equation for ipq, its integrand can be decomposed as \I
= ufi(q,p)

+ sin2 qf2(q,p) + smqsinpf3(q,p)

,

where fi(q,q)/sing is bounded and fv{q,q) = fs(q,q) = 0. Hence the second integral of Eq. (32) becomes 2 7

/

"^

27

Wg)-fl-(p)}/i(g.p) ^ ( 4 T 2 - f i 2 ) ( 4 T 2 - ( f t + u,)2)

1

I

f2y-^f2(q,p)-h(q,q)

+ 4j'

1

+ \~i'-

L

27

27

<m <m

f2y-W^M9,p)-f3(q,q)

{FR(q)-FL(p)}<

I

4 7 2 - fl2

\FR(q) - FL(p)}

each term of which is evaluated as before. By comparing the result with d{\Ks{p,-q)\2-\KN(q,-p)\^} dp

p= q

2JSinqf1(q,q)+S^q^i(i>P)±Uq>P» dp

=

p=q

one finally obtains the desired result (29). Now we show that the second term of Eq. (29) is finite and vanishes when N —• oo. The proof is similar to that of Proposition 2. We remind that the kernel A'JV can be decomposed into a contribution from the sites j with ' < Ul < ~f^~ a n d the rest, and that the former can be evaluated with the aid of the asymptotic form of i/>p(j). Then, we get Re

«l|-|)^,-*)

p= q

1-2

= -Zeysinq

£

(2j + 1) I m { T ? ^ V ? 0 > - ? 0 ' ) }

j=-l+i /V— 1

^ R e ( T , f l J c - ' « ) [2 Y^ R e ( r ? e 2 ! ' « ) + ( 2 / - l ) R e ( T ( ? e 2 ^ ' - 1 ) ) | (36)

170

Note that, because of the relation 23 TZqR-q = —TqR* and the periodicity with respect to q, He(TqR*e-^)/ sin q is finite for q —> 0 and q —• TT and, thus, the second integral of Eq. (29) is finite. Now, by changing the integration variable from fi to q = c o s - 1 ( - f i / ( 2 7 ) ) , the second term of Eq. (29) is found to be the real part of the integral of the form JV

—i

I, £dq{F{q)J2e^+G(q)} 1 / r

f

l + eiq

l-e'i

1 eiNq - e^ 2 ' - 1 )* 1i sin q

where F(q) and G(q) are continuously differentiable. The first and the last terms in the parentheses are finite, while the second term is of order of log TV. Thus, in the limit of TV —• +oo, the second term of Eq. (29) vanishes. This completes the proof of Proposition 3. 6

Conclusions

In this article, the current fluctuations for nonequilibrium steady states were investigated and we have shown: (1) The correlation functions for the antithermodynamical steady state are identical to those for the thermodynamic steady state. In other words, timereversal asymmetry of the steady states does not affect the correlation functions. (2) For the total current operator over TV sites, the antisymmetrized correlation function is o(TV) and the symmetrized one is O(TV) for large TV. The leading term of the symmetrized correlation function is independent from the time interval t and is equal to that of the variance of the current fluctuation. Particularly, at zero temperature and for small chemical potential difference, the variance is proportional to the product of the averaged current and the reflection coefficient |-R?|2. This is consistent with the known results for the mesoscopic conductor 26 . However, Eq. (19) is slightly different from the expression given by Shimizu et al. 26 since the former contains \Tq\A + \R,«i|4 instead of \Tq | 4 as a coefficient of F R ( 1 — FR) + FL(1 — F£). This is because the

171

waves propagating to the left and to the right do not interfere in the N -+ +00 limit. The expression similar to that of Shimizu et al. 26 is obtained when we consider the number of electrons passing between the (j — l)th and jth sites during time T:

nT(j) = — I dtJj.^it) . e Jo

(37)

Indeed, as the proof of Proposition 3, one finds ^ + 0 O (nT(j)) lim -ui+ao

T—-00 1

= - /

7


(38)

(<5«T0'))

= f1 — J-2-r w

\Tq\A{FR{
— \Tq\2\Rq\2{FR(q)(l-FL(p))+FL(p)(l-FR(q))} -27

where q = cos - 1 ( ^ ) of

TIT(J).

(39)

"•

and

= nT(j) — w+0O (nT(j))

6TIT(J)

is the fluctuation

This is exactly the same as that discussed by Shimizu et al. 26 .

(3) The odd part Q+'(N,u) of the Fourier-transformed antisymmetrized correlation function was obtained and the limit limjv-Kx> limu,_).o Q+ (N,u))/(N2u>) was explicitly calculated. The limit is not simply related to the averaged current for nonequilibrium states, but it reduces to the Kubo formula at equilibrium. It should be noted that the order of the limits cannot be changed. Indeed, because lirn/vwoo Q+'(N,u>)/N2 = 0 for fixed u> (cf. Proposition 2), the limit in the opposite order leads to the vanishing result. Before closing the article, we give a remark on a formal aspect. We showed23 that the state w + 0 0 is equivalent to the steady state proposed by MacLennan 24 and Zubarev 2 by comparing the two point functions. This correspondence can be seen more directly as follows. For simplicity, we assume that the system consists of two parts: Z = (—00, N] U [N + 1, +00). Formally our initial states are expressed by a density operator pin Pin = \ exp {-pL{HL

-

»LNL}

- pR{HR

- pRNR})

(40)

172

where Hi and NL (HR and NR) are the Hamiltonian and number operator for the left (right) reservoir, which are discussed in Sec. 2, and Z is the normalization constant. Then, the state at time t corresponds to p(t) = e-im/npineim'n

= ^exp(-r(f))

where r(t) = PL{HiX-t)

- »iNL{-t)}

+ pR{HR(-t)

-

tiRNR(-t)}

and, for example, HL(S) = exp(iHs/K)HL exp(—iHs/K). Let J^ and j£f be the energy and mass currents from the left reservoir defined by

4.

= J£(,)

-

-

" S ^ =-*<•>

then we have

L

HL(-t)

= HL+J

NL(-t)

= NL + J

dsj£{s) dsjM(s).

Note that the Hamiltonian H is the sum of Hi, HR and the interaction between the left and right parts VLR. Since N — NL + NR and H = Hi + HR + VLR are conserved, HR(-t)

= H-HL-

J

dsjZ(s)

-

VLR(-t)

NR(-t) = N - NL - /J; < dsJ^(s). Thus, because the local nature of VLR formally implies lim*_oo VLR(—t) = 0, we obtain a formal result: T+ =

lim r(t)

+ PRUH-HL-J

dsjE(s)^-m^N-NL-£

dsJ™(«))} (41)

and lirm^oo p(t) = exp(—F+)/Z, which is nothing but the MacLennanZubarev steady state 2 3 . Or the MacLennan-Zubarev expression can be derived directly from the time evolution of the initial state (40).

173

Acknowledgments The author is grateful to Professors T. Hida and K. Saito for their kind invitation and hospitality at The Third International Conference on Quantum Information, Meijo University, 7-11 March, 2000. He thanks Professors T. Hida, K. Saito, L. Accardi, M. Yamanoi, H. Araki, A. Shimizu and M.Suzuki for fruiteful discussions and valuable comments. Particularly, he acknowledges Professor H. Araki for sending the preprint of their work on nonequihbrium steady states of the XY-chain 21 and Professor A. Shimizu for sending reprints of their works including Ref. 26. This work is partially supported by Grantin-Aid for Scientific Research (C) from the Japan Society of the Promotion of Science and by Waseda University Grant for Special Research Projects (Individual Research) from Waseda University. Appendix Contrary to the conventional derivation of the Kubo formula where one starts from a finite system and takes an appropriate infinite volume limit, we are dealing with the infinite system from the beginning. Thus, it is worthwhile to outline the derivation of the Kubo formula Eq. (26). Note that all the calculations of this appendix are formal. We introduce an applied field via the vector potential. Then, the perturbed Hamiltonian is given by +oo

L

<7 = ± J = — OO

(7 = ± j = l

(42) where Aj is the vector potential, c the speed of light and (h.c.) stands for the Hermitian conjugate of the previous expression. For a monochromatic applied field of strength E abd frequency w extended on an interval [— ^ y ^ , ^f^], we have

Ai3

= {&*>* m<^) [0

(otherwise)

v

(43) '

Up to the first order with respect to E, the perturbed Hamiltonian H(p) and current operator J\_UJ are given by (JV-l)/2 j=

-(N-l)/2

174 2 ^'-l|j

=

^3-l\i

ft^T

Z^

3 \CioCj-l,o

+ Cj-ltacja\

,

a=±

where Jj-\u is the unperturbed current operator. To derive the Kubo formula, the Liouville-von Neumann equation for the density operator is solved perturbatively with respect to the external field E and the averaged current operator (Jj2ui)»t is calculated 3 ' 25 ' 29 . Then, we have

where (•••)() stands for the equilibrium averge, and rjj = 1 for \j\ < ^yJand rjj = 0 otherwise. At steady states, the charge is constant in time and {Jj-Uj)st does not depend on j . Thus, the averaged current (J)st can be calculated via (JU=N

£

^

-

=

l^Xm

dt([JN,JN(t)])0e^ K

j = -(JV-l)/2

J

~co

<7 = ± j = - ( J V - l ) / 2

J

The first term of the expression in the bracket is the Fourier-transformed antisymmetrized correlation function at equilibrium: Qeq(ui,N). Also the charge conservation implies (AT-l)/2

2

X

5Z

£

< ) ° = -<3e ? (0, AT) .

<7 = ± j = - ( J V - l ) / 2

Hence, we have (J)st=

—T7- Q e ? ( ^ , 7 V ) - Q e ? ( 0 , 7 V ) L

(44)

As V = .EJV is a voltage across the left and right ends, the dc conductivity GN = m r w o ^ U A W ' " " ) is given by = lim w^O

Q^,N)-Q

{,,N) ICON2

V

'

175 Noting t h a t only the odd part Qiy(w,N) contributes to the limit (45), one finally obtains the TV-independent conductivity G = limjv-*oo GN-

G=lim

,,m<%a

or the Kubo formula (26) is derived. We again emphasize t h a t N in Eqs. (26) and (45) is not the system size, but the range of the applied field.

References 1. I. Prigogine, Nonequilibrium Statistical Mechanics, (Wiley, New York, 1962). 2. D.N. Zubarev, Nonequilibrium Statistical Thermodynamics, (Consultants, New York, 1974). 3. M. Toda, R. K u b o and N. Saito, Statistical Physics I (Springer, New York, 1992); R. Kubo, M. T o d a and N. Hashitsume, Statistical Physics / / ( S p r i n g e r , New York, 1991). 4. I. Prigogine and T . Petrosky, in Generalized Functions, Operator Theory and Dynamical Systems, p. 153, ed. I. Antoniou and G. Lumer, (Chapman & H a l l / C R C , London, 1999); T . Petrosky and I. Prigogine, Adv. in Chem. Phys. 9 9 , 1 (1997) and references therein. 5. D. Driebe, Fully Chaotic Maps and Broken Time Symmetry, (Kluwer Academic, Dordrecht, 1999) and references therein. 6. P. Gaspard, Chaos, Scattering and Statistical Mechanics, (Cambridge Univ. Press, Cambridge, 1998) and references therein. 7. J.R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical Mechanics, (Cambridge Univ. Press, Cambridge, 1999) and references therein. 8. D.J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic, London, 1990); W . G . Hoover, Computational Statistical Mechanics (Elsevier, A m s t e r d a m , 1991). 9. A.J. O'Connor and J.L. Lebowitz, J. Math. Phys. 1 5 , 692 (1974) and references therein. 10. J.L. Lebowitz and H. Spohn, J. Stat. Phys. 19, 633 (1978). 11. S. Goldstein, C. Kipnis and N. Ianiro, J. Stat. Phys. 4 1 , 915 (1985) and references therein. 12. U. Zurcher and P. Talkner, Phys. Rev. A 4 2 , 3278 (1990) and references therein. 13. W.R. Frensley, Rev. Mod. Phys. 6 2 , 745 (1990) and references therein.

176 14. G. Klein and I. Prigogine, Physica 19, 74, 89, 1053 (1953); R. Brout and I. Prigogine, Physica 22, 621 (1956). 15. H. Spohn and J.L. Lebowitz, Commun. math. Phys. 54, 97 (1977) and references therein. 16. J. Farmer, S. Goldstein and E.R. Speer, J. Stat. Phys. 34, 263 (1984). 17. J. Bafaluy and J.M. Rubi, Physica A153, 129 (1988); ibid. 153, 147 (1988). 18. S. Tasaki and P. Gaspard, J. Stat. Phys. 8 1 , 935 (1995); Theor. Chem. Ace. 102, 385 (1999) and references therein. 19. P. Gaspard, Phys. Rev. E53, 4379 (1996). 20. J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Commun. math. Phys. 201, 657 (1999); J. Stat. Phys. 95, 305 (1999). 21. T.G. Ho and H. Araki, Asymptotic Time Evolution of a Partitioned Infinite Two-sided Isotropic XY-chain, Proc. Steklov Math. Institute 228, 191 (2000). 22. T. Antal, Z. Racz and L. Sasvari, Phys. Rev. Lett. 78, 167 (1997); T. Antal, Z. Racz, A. Rakos and G.M. Schiitz, Phys. Rev. E57, 5184 (1998); ibid. 59, 4912 (1999). 23. S. Tasaki, Nonequilihrium Stationary States of Noninteracting Electrons in a One-dimensional Lattice submitted to Chaos, Solitons and Fractals (1999); Statistical Physics, the Proceedings of the 3rd Tohwa University Conference on Statistical Physics M. Tokuyama and H. E. Stanley eds., pp.356-358 (AIP Press, New York, 2000). 24. J.A. MacLennan, Jr., Adv. Chem. Phys. 5, 261 (1963). 25. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). 26. A. Shimizu and M. Ueda, Phys. Rev.Lett.69 1403 (1992); A. Shimizu and H. Sakaki, Phys. Rev. B44, 13136 (1991); A. Shimizu, Solid State Physics (in Japanese) 28 771 (1993). 27. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1 (Springer, New York, 1987); Operator Algebras and Quantum Statistical Mechanics 2, (Springer, New York, 1997). 28. H. Araki, Publ. RIMS, Kyoto Univ. 20, 277 (1984). 29. E.N. Economou and C M . Soukoulis, Phys. Rev. Lett. 46, 618 (1981); D.S. Fisher and P.A. Lee, Phys. Rev. B23, 6851 (1981).

Quantum Information III, pp. 177-184 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

177

REPRESENTATIONS A N D T R A N S F O R M A T I O N S OF G A U S S I A N R A N D O M FIELDS SI SI Faculty of Information Science and Technology Aichi Prefectural University, Aichi-ken 480-1198, Japan We are interested in representations and transformations of Gaussian random fields X(C) with parameter C, running through the class C = {C; C £ C 2 , diffeomorphic to S 1 } . The canonical criterion for Gaussian random fields is established in this paper. Some detailed results of multiple Markov properties are obtained by taking C to be a class of concentric circles.

1

Introduction

We are interested in random fields with the class of the parameter C which is taken to be a class of manifolds such that C = {C; diffeomorphic to Sd~ *, convex}. Consider a random field X(C), where C runs through the class C, in particular, we concentrate on a Gaussian random field X(C)\C G C, with a representation in terms of R — parameter white noise. The canonical representation theory of Gaussian processes, developed by T. Hida in 1960, has been generalized to Gaussian random fields in [5], [6] and [7]. It is an interesting problem to find out the relation between the kernels of canonical and noncanonical representations. For that purpose we first deal with the multiple Gaussian processes and the observation on which can be generalized to the case of Gaussian random fields X(C),C E C. 2

Background

In this section we recall our previous work on Gauusian random fields. We consider the Gaussian random field {X(C);C £ C} assuming that X(C) can be expressed by a representation X(C)=

f J(C)

F(C,u)x(u)du,

(1)

178

in terms of i? 2 -parameter white noise x(u) and L 2 (/2 2 )-kernel F(C,u) every C g C . where

for

C = {C;C G C2,diffeomorphic to S1 ,(C) is convex}, in which (C) denotes the domain enclosed by C. In addition assume that X(C) £ 0 for every C, and E[X(C)} = 0. E[X(C)2] ^ 0 for every C. E[X(C)X(C')}

= r(C,C"); C > C, admits variation in the variable C.

Then we give the definition of canonical representation for a Gaussian random field as follows. Definition Let Bc'(X) be the sigma field generated by {X(C),C The representation (2.1) is called a canonical representation if E[X(C)\Bc>(X)]

= f(cl)

< C'}.

F(C, u)x(u)du,

(2)

holds for any C < C. Theorem 1 The canonical representation is unique if it exists. Remark. We now give a rather precise proof then that given in [7]. Proof. Take the variance of the conditional expectation, given in (2.2). E{E[X(C)\BC,{X)]2}

= / ( c # ) F(C, ufdu,

C < C.

(3)

We should note that the variance depends only on the probability distribution of {X(C)} and is independent of the choice of representation. If the representation is not unique, there are two canonical kernels F and F* and then /

F{C,ufdu=

J(C)

[

F*{C,ufdu

J(C>)

holds for any C < C. Hence we have F(C,u)=e(C,u)F*(C,u); where e is a measurable function of «.

|e(C,u)| = 1,

(4)

179

According to the two kernels F and F*, the covariance of (2.2) is obtained as the covariance E[E[X(C)\Bc»(X)]E[X(C')\Bc»(X)]]=

f Jc"

F(C,u)F{C',u)du.

On the other hand, we obtain E[E[X(C)\Bc»(X)]E[X(C')\Bc»(X)]\=

f

F*(C,u)F*(C

,u)du

JC"

= [

F(C,u)F(C',u)e(C,u)e(C',u)du

by (2.5). Similarly for any C" G C; C" < C". we have /

F{C,u)F{C',u)du=

JC"

I

F(C,u)F(C',u)e(C,u)e(C',u)du.

JC"

Thus the equality F(C, u)F(C',u)

= F(C, u)F{C, «)e(C, u)s(C, u),

holds almost everywhere. We can see that e(C,u)e(C',u)=

1, on C.

Fix C = Co, and determine e(Co, «)(= ±1) as a function of w. Thus u),

VC".

It means that £(C, u) is independent of C. Thus it is proved that F(C, u) is unique up to ± 1 . 3

Kernel criterion for canonical representation of Gaussian random field

Folowing the kernel criterion for canonical representation of Gaussian processes, we can give that of Gaussian random field. Assume that (i) X(C) has a causal representation (ii) there is no open set G such that fG F(C, u)tp{u)du = 0 for any if with supp{ip} C G.

180

Theorem 2 A random field X(C), satisfying the above assumption, canonical representation if and only if\/C C C\\ C\ : fixed, I

F(C,u)ip(u)du

has

= Oimplies
J(C)

Proof Since we are concerned with Gaussian, that E[X(C)\Bc] is the projection of X(C) down to the closed linear space spanned by {X(C")\ C" < C'}. By assumption the closed linear space Mc'(X), spanned by {(X(C"), C" < C")} is the same as the closed linear subspace Mc"(#)spanned by {*(«);« EC"}. The reason is that Mc0(X) C Mc0{x), in general, since X(C") is a (linear) funcion of x{u); u G (C"). If Mc0(X) ^ Mc0(x) then there exist p / 0 such that fc
€C, given in (3.1), is a canonical repre-

Example 2. Consider a random field X(C);C circles, with the representation X(C) = X0 [

G Co, where C 0 is a family of

e-kpi-c^d*uv(u)du,

J(C)

where p denotes the distance, k is a constant, ip and v are given continuous functions. We can prove that it is a canonical representation. Indeed it is the solution of Langevin equation, 6X{C) = -X(C)

[ k6n(s)ds + X0 [

Jc

Jc

v(s)d*6n(s)ds,

where C £ CoExample 3. Let {CR, R G R.} be a family of concentric circles with radius R and with center at the origin. Then X(C)=

i (3R JcR

4\u\)x(u)du

181

is a canonical representation of X(C), since there is a function
(3iJ-4|ti|)y>(|u|)du = 0.

JCn >C„

4

Representations and transformations of Gaussian random fields

In this section we shall deal with the multiple Markov Gaussian random fields. Thus we recall the definition of Multiple Gaussian random field, given in [7]. Definition For any choice of Ci 's such that if

CQ

< C\ < • • • <

CN

(i) E[X(Ci)\Bc0(X)],

i = 1, 2, • • •, N are linearly independent and

(ii) E[X(d)\Bc0(X)],iI

= 1,2, • • •, N + 1 are linearly dependent

<

CAT+I,

then X(C) is called N-ple Markov Gaussian random field.

Theorem 3 ([7]) If X[C) is N-ple Markov and if it has a canonical representation, then it is of the form N

X(C)=

Y)/i(C)flr,-(«)a:(«)d«, J(C)

(5)

!

where the kernel 53/j(C)fl(,-(w) is a Goursat kernel, namely {fi(C)},i 1, • • •, N satisfies det(/,-(Cj)) ^ 0,/or any ,/Vdifferent Cj

— (6)

2

and {gi(u)}, i = 1, • • •, N are linearly independent in L -space. With a particular choice of C = Cr, a circle with radius r and center at origin we have the following theorem. Theorem 4 The representation of N-p\e Markov Gaussian random field X{Cr) can be expressed in the form X(Cr) = ^2fi(r) j

/ J0

gi(v)x(v)dv.

(7)

182

in terms of white noise x(v),v £ R1, as a stochastic integral, where 521 fi(r)9i(v) ls a Goursat kernel. Proof. The representation (4.1) is rephrased as

X(Cr)=

fWy2fi(r)gi(s,e)x(s,0)sdSd0,

f Jo

Jo

j

(8)

by using the polar coordinate. We note that E[x{s, 9)x{s', 0')] = ±6(s -s',00'). This is guaranteed by computing the covariance function of white noise after changing the parameter u to (r, 9). Set /•27T

Zi(s)=

/ gi(s,0)sx(s,0)d0. Jo Then it is a generalized function with the characteristic functional /»oo

C-(0 = c a r p l - 2 y

/»2?r

^

g?(s,0)t2(8)8d8d9

(9)

£EC°°[0,oo),

which shows that Z{(s) is Gaussian and has independent values at every point s. Hence we can define an additive Gaussian process Yi(s) = [ Jo fS

Zi(v)dv, flit

=

9i{v,0)v x{v,9)d0dv Jo Jo which can be expressed as a white noise integral based on a white noise x(u), u £ R1, in the form

I I Jo Jo

gi(v,0)vx(v,0)d0dv

=

gi(v)x(u)du, Jo

where f fi{u)du= j I Jo Jo Jo and (ji is uniquely determined up to sign. Thus we obtain (4.3).

g?(v,9)vd0dv

(10)

183

We can compare this formula with the representation of a strictly TV-pie Gaussian processe X(t), ft

N

JO

j

for which there exists an N—th order ordinary differential operator Lt such that {fi(t)} is a fundamental system of solutions of Lt, and so is the system {gi(u)} for the differential operator L*, the adjoint operator for LtIn addition, if the kernel R(t,s) = J ^ fi(t)gi(u) is a Riemanian function, then LtX(t) = x(t). We then consider a Markov process with a non canonical representation of the form ft

X(t)=

N

/ Vh,-(i)ffi(*)*0»)
(12)

j

where gi(s)'s are required to be linearly independent L2— functions over any interval [0,t]. It is an interesting question to establish an analytic relations between A,-'s and fi's under the condition that <^'s ars fixed. There are two typical cases, worth to be mentioned, where a differential operator M< exists and satisfies Mtfi = h{,

i=l,---,N.

Namely in the first case, the kernel function is a polynomial in t and s, and in the second one, X(t) is stationary assuming t runs through (—00,00). The operator Mt shows how the given X(t) comes from the canonical representation. Returning to the case (4.1), we may apply the above observation to the representation (4.3). Proposition If /,-(r)'s and g^s in (4.7) satisfy the condition that ft and the Wronskian W(gi,...,g = i) never vanish for every i = 1,..., JV, then we can find a linear operator Mr such that X{Cr) =

MrY(Cr)

where Y(Cr) is strictly N-ple Markov process. Proof is given by a slight modification of the result in [1], Setion II. Remark This result suggests us to discuss more general cases.

184 Acknowledgments This work due to the support of academic frontier " Q u a n t u m Information Theoretical Approach to Life Science" . References 1. T . Hida, Cannonical representation of Gaussian processes and their applications,Mem. Coll. Sci.Univ. Kyoto, 33 (1960),258-351. 2. T . Hida, Brownian motion, Springer-Verlag. 1980. 3. T . Hida and M. Hitsuda. Gaussian Processes. American M a t h . Soc.Translations of Mathematical Monographs vol.12. 1993. 4. T . Hida a n d Si Si, Innovation for r a n d o m fields, Infinite Dimensional Analysis, Q u a n t u n Probability and Related Topics, Vol 1, 409-509. 5. Si Si, A variation formula for some r a n d o m fields; an analogy of Ito's formula Infinite Dimensional Analysis, Q u a n t u n Probability and Related Topics, Vol 2, 1999. 6. Si Si, W h i t e noise approach to r a n d o m fields, t o appear in the proceeding of International summer school , " W h i t e noise approach to Classical and Q u a n t u m Stochastic Calculus" ,Trento, 1999. 7. Si Si, Gaussian processes and Gaussian r a n d o m fields, Q u a n t u m Information II, eds T.Hida k et. 2000 World Scientific, p 195-204.

Quantum Information III, pp. 185-196 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

185

T I M E R E V E R S A L S Y M M E T R Y OF F L U C T U A T I O N IN EQUILIBRIUM A N D NONEQUILIBRIUM STATES

M. Y A M A N O I Department

of Electrical Engineering Meijo University Tempaku-Ku, Nagoya 468-8502, Japan F. OOSAWA Aichi Institute of Technology Yakusa-cho, Toyota 470-0392, Japan The condition of time reversibility for stochastic process is reviewed for the informational analysis of the voltage fluctuation in biological system. The condition is expressed by the symmetry of the joint probability density function. Based on this condition, comparison is made on the difference between fluctuations in the states of equilibrium and nonequilibrium. It is shown that the joint probability is symmetric and the energy consumed is zero for the fluctuation in equilibrium, for the model represented by the Langevin equation. On the other hand, for the fluctuation in the state of nonequilibrium, calculation of characteristics function is made for the shot noise to show that the joint probability is asymmetric.

1

Introduction

Every physical quantity makes fluctuation with different characteristics depending on system and state. In the state of thermal equilibrium, it makes the thermal fluctuation. If the system is pumped by any means to leave the equilibrium state, it goes into the state of nonequilibrium and has a different aspect of the fluctuation compared with the thermal fluctuation. The basic unit of living biological systems is the cell, which is an open system and in the state of nonequilibrium. It is the open system because it takes energy into the body, and excess energy is taken out to reduce the entropy. It is nonequilibrium because substances are distributed unhomogeneously to generate biological functions. For example, the ion pumps on the cell membrane produce the unhomogeneous distribution of ions between the inside and outside of the cell. This unhomogeneity gives the motive force to the ions, carrier of electricity, in the ion channels. The conductivity and thus the current is controlled by the gate in the ion-channel, and then a special signal called as the active voltage is generated. In the single-cell animal Paramecium, the active voltage is generated spontaneously even in the absence of an external excitation. Since this sponta-

186

neous fluctuation has various important functions, this may be called as the functional fluctuation. Although the generation of the active voltage requires the state of nonequilibrium in open system, it may be different in someway from fluctuations in nonbiological system. The difference may be due to the life. Living biological systems must survive in changing environments, and for this adaptive control they may utilize the fluctuation. Therefore, not the average property, but the wave form itself of the fluctuation has significance for them, since it carries information to control themselves. Then it is interesting to study the characteristics of the fluctuation from the informational point of view. To study the characteristics of the waveform of fluctuation, the symmetry property of the fluctuation under the time reversal, and the energy consumed to generate the fluctuation have been studied by one of the authors [1]. In the present paper, we investigate further the mathematical aspects of these points. In chapter 2, the review is made on the time reversibility both in classical mechanics and in stochastic process. In chapter 3, the equilibrium fluctuation is considered first as a reference to compare with the fluctuation in nonequilibrium. It is shown that the Langevin equation gives the joint probability density with symmetry. The energy consumed for the fluctuation is shown to be zero. In chapter 4, the shot noise is considered as a nonequilibrium fluctuation. The characteristic function is calculated to show that the fluctuation is asymmetric. 2 2.1

Definition of Time-Reversibility Time reversibility in classical mechanics

Time reversibility in classical and quantum mechanics has been well known. Particularly, the trajectory of motion in classical mechanics can be represented pictorially, then its time reversibility can be easily understandable. Consider a simple harmonic oscillator with the hamiltonian, which is invariant under the time inversion;

H=^+^

(2-1)

Differential equation generated by eq.(2-l) is given by • * ( < ) •

At).

=

ro i i m

-k

0

At).

-A

At).

187

This gives the time evolution in the form '*(*)! -

PAt

.P(0J

ar(0) p(0)

_

coswtf ^sinut — muismuit coswt

x(0) p(0)

(2-3)

We see that under the time inversion t —+ — t;p —* — p the position is even, x(-t)

~ x(0) cosut - p ( 0 ) ^ s i n u ( - t ) = x(t)

(2 - 3a)

but on the contrary, the momentum is odd; p(—t) = x(0)mu sin tut — p(0) cos tot = —p{t)

(2 — 36)

Eq.(2-3a) shows that the trajectory of position is the same for the forward and the backward progressions of time, representing the time-reversible motion. 2.2

Time reversibility in stochastic process

On the contrary to the classical mechanics, the pictorial trajectory can not been drawn for the stochastic process. The trajectory or the wave form may be replaced by the joint probability density function. The following definition has been given by G. Weiss [2], and applied to various problems [3-5]. Stochastic process is defined as time-reversible if it is invariant under the reversal of the time scale. More specifically, stochastic stationary process X(t) is time-reversible if for every n, and every t\,... ,tn \X(ti),..., X(tn)} and {X(—ti),..., X(—tn)} have the same joint probability distribution; P{X{h),...,

X(tn))

= P ( A - ( - t i ) , . . .,X(-tn))

= P(X(tn),...,

X(h)) (2-4)

Theorem: Let X(t) be a stationary Gaussian process, i.e., all finitedimensional distribution of X(t) are multivariate normal. Then X(t) is timereversible. We notes that the first equality in eq.(2-4) is the stochastic version of eq.(2.3a). In this definition, the variants are taken on the discrete times, corresponding to the piece-wise approximation of wave form. If the discrete times are replaced by the continuous one, the precise definition of the reversibility may be obtained. This formulation has been done by Prof. Hida in the white noise analysis which deals with the infinite number of variants on the continuous time [6].

188

2.3

Two-variant normal distribution

For the two variants X(t\) takes the form

P(xi, x2) = N * exp

= xx and X(t2)

^(i — pz)

o\

= x2, the normal distribution


(Ti
(2-5)

The corresponding characteristic function defined by *0bi,Jb2) = (expfi^Xi + ik2X2}) =

/1eiklXl+ik^P(x1,x2)dxldx2

is given in the form

$(ki,k2)

= exp

{o-{k{ + a2k2 + 2a1
(2-7)

If the stochastic process is steady o\ =
= P(x2,x1),

(2-8)

and correspondingly the characteristic function (2-7) is also symmetric; (2-9)

*0bi,* 2 ) = *(*:,,*!). 3 3.1

Fluctuation in equilibrium state Onsagers microscopic reversibility

The principle of microscopic reversibility demands that a displacement ct\ = a[ of energy in the Xi direction, followed r seconds later by a displacement a2 = a 2 in the x2 direction, must occur just as often as a2 = a", followed r seconds later by a\ — a[. Consequently (ai(t)a2(t

+ T)) = (a2(t)ai(t

+ T))

(3-1)

The above statement (3-1) is given by Onsager [7]. More precisely, however, it should be expressed by the joint probability density in the form

189 P(a'1,t;a%,t + r) = P(a'i,t;a'lft

+ r)

(3 - 2)

Then, for the correlation functions in terms of the joint probability

(ai(t)a 2 (t + r)) = / 7 a l ^ a l . t j a j . i + T)da\da'{

(3 - 3)

(a2(0ai(* + i-)> = / / a ' X ^ K ^ ^ i ^ + ^^W^'

(3-4)

eq. (3-2) gives the equality

(ai(0«2(< + r)> = (a2(t)ai(< + r)) = {ai(t)a2(t

- r))

(3 - 5)

where the last equality follows from the stationarity and the commutativity for classical numbers. Here it is important to notice that ai(t) and a 2 M a r e different physical quantities. For a scalar quantity a(t), the equality (a(t)a(t + r)) = (a(t)a(t — T)) describes just the condition of stationarity, which has no connection to the time reversal symmetry. 3.2

Joint Probability for the Langevin equation

Let us consider the Langevin equation for a stochastic process v(t),

v(t) = -a v(t) + p F(t) (F(t)F(t')) =

(3 - 6)

DS(t-t')

We shall calculate the joint probability density for the process v(t) given by eq. (3-6). Under the assumption of steady and Markov process, the joint probability density f(q',t;q,t') may be decomposed into the probability density /(?) and the conditional probability f(q',t\q,t') :

f(q',t;q,t')

= f(q' ,t\q,t')f(q)

(3-7)

The conditional probability in eq.(3-7) is obtained from the Fokker-Planck equation

190 _____(

a t

,

/ ) +

_ _ _ _

(

3-8)

The solution of eq.(3-8) which reduces to the 6-function 6(q' — q) gives the conditional probability in the form [8];

f(q',t\q,t')

= Mr)]-1/2exp[-(«,-«e-'")2/a(r)]

where a ( r ) = f (1 - e~ 2 a T ), T = t-t',

(3-9)

and Q = /32D.

On the other hand, the probability density in eq.(3-7) is given by /(g) = {a/TcQ)1'2 exp[-ag 2 /Q]

(3 - 10)

The joint probability density is then obtained in the form;

f(q',t;q,t')

=f(q',t\q,t')f(q)

= N(r) exp[-(g 2 + q'2 -

2qq'e-aT)/a(r)]

Eq. (3-11) corresponds to the normal distribution in eq.(2-5) with the covariances;
p - exp(-ar)

(3-12)

After the similar calculation for the reverse joint probability

/(«,*;«',*') =f{q,t\q',t')f(q')

(3-13)

it follows that the forward and reverse joint probability is same;

f(q,t;q',t')

=f(q',t;q,t')

(3-14)

We have shown that the fluctuation given by the Langevian equation (3-6) has the symmetric joint probability density. In other words, (3-14) expresses the detailed balance between the fluctuation amplitudes q and q', representing the reversibility in the sense of Onsager as denoted in the section 3-1.

191

3.3

Energy consumed in the fluctuation

For the aim to study concretely the energy consumed in the fluctuation, let us consider the Langevin equation (3-6) as to describe the velocity v(t) of a Brownian particle in water. The right hand side in eq.(3-6) represents the total force to the Brownian particle exerted by the huge number of particles consisting a reservoir. The Langevin equation (3-6) gives the so-called Ornstein-Uhlenbeck process *

v(t) = J e-a(*-'')/?F(<')*'

(3 - 15)

— oo

We calculate the work per unit time {v(t)Ftot(t)) exerted on the Brownian particle;

done by the total force

Ftot(t) = - a v(t) + (3 F(t)

(3 - 16)

By making use of the stochastic property of F(t) given in eq.(3-6), we have

<«2W> = ^

{v(t)F{t)) = ^

(3-17)

Then it follows that (v(t)Ftot(t)}

=0

(3-18)

showing that there is no net flow of energy from the reservoir to the Brownian particle. The reservoir gives the energy to a Brownian particle through the work done by the pushing force represented by the second term in eq. (3-16). But, at the same time, the energy is returned to the reservoir by the resistive force represented by the first term. This is the balance between the dissipation and the fluctuation. The particle energy proportional to (v2(t)) is also constant. Therefore, the energy is not consumed to generate the fluctuation v(t). In conclusion, results of the sections 3-2 and 3-3 show that the Langevin equation (3-6) represents the fluctuation in the state of thermal equilibrium, with the properties of the time reversibility and the zero-consumption of energy.

192 4

Fluctuation in nonequilibrium state

Let us consider a situation in which a Brownian particle has an electric charge, and the work on the particle is done by an external electric field. Total force on the particle consists of a resistive force given from a reservoir and the electric force by an external source, as expressed by the right hand side in eq. (4-1) below. The electric force gives the energy to the particle, and the energy of the particle is delivered to the reservoir, which is different from the electric source. There is then the net flow of energy from the source to the reservoir, and the particle becomes an open system. Energy of particle may become time-dependent. Fluctuation in this condition may consume the energy. In such a nonequilibrium state, the detailed balance (3-14) can not hold. Therefore, fluctuations may become asymmetrical. W i t h the theorem given by Weiss described in section 2-2, the probability density of t h e fluctuation may become non-Gaussian. 4-1

Shot

noise

Statistical study of the shot noise is a classical work [9]. In this section we calculate the characteristic function to see if the shot noise has the asymmetric joint probability density, and if it is Gaussian. We consider the equation of the form

*& = -bx(t) + J2«i6{t-tj)

(4-1)

Solution of eq.(4-l) is given by n

x(t) = J2aMt~tj)

(4-2)

J= I

representing a train of wavelets, in which each single wavelet is given by u(t -tj)-

exp[-6(* - tj)] us(t - tj)

(4 - 3)

The characteristics of the wavelet (4-3) is the rapid rise and the slow decay; therefore, this shape may give the asymmetry under the time reversal. T h e similar shape of impulse occurs in the active voltage in biological systems as the action of ion channels.

193

Let us calculate the characteristic function $(*!, k2) = (expihX, where Xi(t) = x(ti) and X2(t) expressed as

+ k2X2))

(4 - 4)

kix(ti) + k2x(t2) = Ylai

= x(t2)- Since the exponent in eq.(4-4) is

lkl u(^i ~ti) + ^2 u ( < 2 - < J ' ) ]

(4-5)

the characteristic function can be rewritten in the form

$(fci, fc2) = ( exp{i ^2 aj [fci u (*i ~ *j) + ^2 «(<2 - tj)]} \

(4-6)

The average in eq.(4-6) can be performed in the following three steps [10]. The first average over {aj}. It is assumed that each wavelet is independent and amplitudes are distributed with the exponential probability density;

p(a)=iexp'(-a/0)

(4-7)

Then, the first average gives n

n a

r

w

(exp{i Y2 J [*i (*i - * ; ) + *2 u{t2 - tj)]}) = [ J /

= n ww=n

w

eiaA ]

* P(aj)daj

^ «(*i - *J)+*2 «(<2 - *,-)],

(4 - 8)

where dj[»] = aj [ki u(t\ -tj) and the function V7[o;(s)] is defined by

+ k2 u(t2 - tj)}

(4 - 9)

194

The second average over {tj}. It is assumed that for the given number of n wavelets, each wavelet has the uniform distribution of the occurrence in the interval [0,T]. Then, the second average gives

= ±[Jwd,r=(l)n

f w f . J w f . . . * J w f

(4-11)

o where 7 = / Jo /o

W[u(s)] ds

(4 - 12)

and w(s) = ibi u(ti -s) + k2 u(t2 - s)

(4 - 13)

The third average over n. It is assumed that the number n of the wavelets is distributed by the Poisson probability density. Then, the third average gives

£

(£)" ^ T

e

" " = eM-»(T

- 7)]

(4 - 14)

n=0 rp

In this stage, we need to evaluate the integration 7 = / 0 W[u(s)]ds defined in eq.(4-12). Since u(tj — s) = exp[—b(tj — s)]u,(tj —s), u(s) defined in eq.(4-13) is given by LJI(S) = (*i e~btl + k2 e-bt') ebs =

eb!

0 < s < tx

u2(s) = (k2 e-*' 2 ) eb° = T}2 ebs

tt < s < t2

W3 (s)

m

= 0 t2<s
(4 - 15) (4 - 16) (4-17)

By making use of the integral formula;

Ids. J

.

1

(h={x2-Xl)

l — ir)exp(bs)

We then obtain

+ \\og)-ir]eXJ2 b

I — iTjeDX2

(4-18)

195

1 =

T

«1

f W[u(s)] ds=

I

t2

W[UJX{S)}

J

ds + f W[u2(s)] ds+

f W[ui3(s)] ds «2 -bT

= T+\

{ - log[l - i9(h + k2e~»)] + log *

i

*k£~* }

(4-19)

where r = t2 - h. Substituting this result into eq.(4-14), we obtain the characteristic function in the form

This is asymmetric, (&!,fc2) ^ ^(^2,^1), and has not the Gaussian form as given in eq.(2-7). Correspondingly, the joint probability density is also asymmetrical, P(x\,X2) / P{x2,x\). We see therefore that the shot noise of the form (4-2) does not satisfy the detailed balance as mentioned in the beginning of the present chapter. Acknowledgment This work is supported by the Joint Research Project Quantum Information Theoretical Approach to Life Science at Meijo University. This project is assigned for the Academic Frontier in Science promoted by the Ministry of Education in Japan. References 1. F. Oosawa and J. Masai: Asymmetry of Fluctuation with respect to Time Reversal in Steady States of Biological Systems, Biophysical Chemistry 16, pp. 33-40 (1982). 2. G. Weiss: Time-Reversibility of Linear Stochastic Process, J. Appl. Prob. 12, 831-836 (1975). 3. M. Hallin, C. Lefevre and M. L. Puri: On time-reversibility and the uniqueness of moving average representations for non-Gaussian stationary series, Biometrika 75, pp. 170-171 (1988). 4. G. B. Giannakis and Michail K. Tsatsanis: Time-Domain Tests for Gaussianity and Time-Reversibility, IEEE transactions on signal processing, vol.42, No. 12 (1994).

196

5. D. R. Cox: Statistical Analysis of Time Series: Some Recent Developments, Scand J Statist 8: pp.93-115 (1981). 6. T. Hida: Complexity in White Noise Analysis, and references therein, in: Quantum Information II, ed. by T. Hida and K. Saito, World Scientific, (2000). 7. L. Onsager: Reciprocal Relations in Irreversible Processes. 1, Phys. Rev. vol. 37, pp. 405-426 (1931). 8. H. Haken: Synergetics (Second Enlarged Edition), Springer-Verlag. 9. S. O. Rice: Mathematical Analysis of Random Noise, in Selected papers on Noise and Stochastic Process, Dover. 10. R. P. Feynman and A. R. Hibbs: Quantum Mechanics and Path Integrals, Chap. 12, McGraw-Hill.

Quantum Information III, pp. 197-217 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company

197

S T O C H A S T I C C O N V E R G E N C E OF S U P E R D I F F U S I O N IN A SUPERDIFFUSIVE MEDIUM *

ISAMU D O K U Department

of Mathematics, E-mail:

Saitama University, Urawa 338-8570 [email protected]

Japan

We consider the Cauchy problem for nonlinear reaction-diffusion equation in a superdiffusive random medium. The degeneracy and non-degeneracy of L 1 -norm of the positive solutions can be established as a result of longtime asymptotic behaviors. We construct locally finite non-negative measure valued stochastic processes associated with the nonlinear equations in question and show stochastic convergence of the processes in a superdiffusive medium as a probabiblistic counterphenomenon.

1

Introduction and Main Results

The system that we treat here is related, in most cases, to stochastic models which are introduced, for instance, based upon the following two distinct viewpoints in random chemical systems or in random biological systems. The first one is a microscopic view in the chemical reaction, where a molecule reveals a certain chemical reaction only in the places where exists the specific reactant. The second one is just the case where, in the macroscopic view, the chemical reaction is described by reaction-diffusion equations and the effecting of reactor enters as a spatially heterogeneous rate function. In some cases there are reactants present only in the localized regions such as networks of filaments or the surfaces of pellets. Mathematically, such systems are modelled by the following nonlinear reaction-diffusion equations in R - ~

= -Au + Ps-R(u),

o<s
(1)

with terminal condition u\s=t = ps £ M(R ). Let p(r,b) denote the transition density of a standard Brownian motion in R d . Then the above (1) can be formulated •RESEARCH 10304006.

SUPPORTED

IN PART BY

JMESC

GRANT-IN-AID

CR(A)(1)

198 rigorously by the following integral equation [8]: u(s,t,a)

=

p(t — s,b — a)
a)R(u(r,t,b))pr{db).

(2) Our main concerns are firstly to formulate the equation (2) meaningfully for measure-valued paths p as generalized as possible, and secondly to investigate long-time asymptotic behaviors of the solutions, whereby we aim at studying the asymptotic behaviors of the associated stochastic processes. In this paper we will treat simply the typical case R(u) = u2. Next we introduce our main results in this paper. In connection with (1), let us consider the following one-dimensional random nonlinear parabolic equation

{

dv ~ds~

,

r

= Lv PsV

~

V \szzt =

•

0 < s

^

t

(*)

f-

Here L is a second order general elliptic operator (see the next section for details). Then naturally there corresponds some measure-valued process X to this problem (*), which we call a superdiffusion in a random medium. Then we can establish the degeneracy and non-degeneracy of L1 -norm of the positive solutions v of (*) in terms of the associated stochastic process X. Theorem A. If superdiffusion X has a finite time interference property with L -diffusion, then the positive solution v of (*) with