Quantum Probability and Infinite Dimensional Analysis: Proceedings of the 26th Conference [illustrated edition] 9789812708519, 9812708510

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QP-PQ: Quantum Probability and White Noise Analysis

Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy

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QP-PQ: Quantum Probability and White Noise Analysis VOl. 20:

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

VOl. 19:

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

Vol. 18:

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

Vol. 17:

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

Vol. 16:

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

Vol. 15:

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

Vol. 14:

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

Vol. 13:

Foundations of Probability and Physics ed. A. Khrennikov

QP-PQ VOl.

11:

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

VOl. 10:

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

VOl. 9:

Quantum Probability and Related Topics ed. L. Accardi

Vol. 8:

Quantum Probability and Related Topics ed. L. Accardi

VOl. 7:

Quantum Probability and Related Topics ed. L. Accardi

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QP-PQ Quantum Probability and White Noise Analysis Volume XX

zyxw zyx zyxwv zyx

Proceedings of the 26th Conference 20 - 26 February 2005

Levico, Italy

Editors

L. Accardi

Universith di Roma Tor Vergata, Italy

W. Freudenberg

Brandenburgische Technische Univevsitat Cottbus, Germany

M. Schurmann

Greifswald University, Germany

v

World Scientific

N E W JERSEY * LONDON * SINGAPORE

BElJlNG

SHANGHAI

.

HONG K O N G * TAIPEI

CHENNAI

Published by

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

All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

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ISBN-I3 978-981-270-851-9 ISBN-I0 981-270-851-0

Printed in Singapore by World Scientific Printers (S)Pte Lid

FOREWORD

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

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Luigi Accardi Wolfgang F’reudenberg Michael Schurmann

V

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CONTENTS

Foreword

zyxwvut zyxwv zyxwvut V

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

1

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

13

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

26

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

33

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

45

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

53

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

64

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

79

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

90

vii

viii

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

zyx 102

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

128

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

134

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

145

Quantum Probabilistic Model for the Financial Market 0. Choustova

156

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

163

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

173

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

181

Markov States on Quasi-Local Algebras F. Fidaleo

196

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

205

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

212

zyxwv ix

Note on the Time Operator T. Hida

224

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

233

An Analytic Double Product Integral R. L. Hudson

241

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

251

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

258

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

267

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

2 74

Hyperbolic Quantization A . Khrennikov and G. Segre

282

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

288

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

295

A Theorem on Liftings of Statistical Operators J. Kupsch

302

X

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

308

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

319

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

323

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

. 330

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

338

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

348

A Characterization of Poisson Noise S i Si

356

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

365

Note on Quantum Mutual Type Entropies and Capacity N. Watanabe

373

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A COMBINATORIAL IDENTITY AND ITS APPLICATION TO GAUSSIAN MEASURES

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

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2000 Mathematics Subject Classifications: 05335, 60H40.

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

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2

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1. Introduction

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

For all i E {1,2,. . . ,d } , we denote by Xi the operator of multiplication by the variable xi. This operator is defined on the space V, which is dense in H, as ( X i f ) ( x l , x 2 , . . ,xd) := x i f ( x 1 , ~ 2 ., ..,Q).

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The following lemma leads us to the notion of creation, neutral (preservation), and annihilation operators. See [2] for details.

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

+

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

+

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

= D:(i)

..,d}.

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

z zyxw zyxwvu zyxwv zyx z zyxw 3

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

c c 00

a+(i)cp :=

D,+(i)fn,

n=O

zyxw

00

aO(i)cp:=

D:(i)fn,

(3)

n=O

and

c 60

a-(i)cp :=

D,(i)fn.

(4)

n=O

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

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

(5)

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

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

zyxwvu zyxwv zyxw zyxwv zyxw z zy zyx

4

random variables X I , X2, . . . , x d on Rd. Therefore, one can see that by working only on Rd we do not loose anything from the generality of this theory. The following theorem was proved in [ 2 ] :

Theorem 1.1. A probability measure p o n Rd, hawing finite moments of any order, is polynomially factorisable, if and only if for any j , k E {1,2,. . . , d } , such that j # k, any operator f r o m the set {a-(j), ao(j), a+(j)} commutes with any operator from the set {a-(k), ao(k), a+(k)}. 2. A Combinatorial Identity

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

1 Ij1< j z

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

1, and no choice (room) if jl = 1. Thus, we can say that in both cases, the number of ways to select a younger partner for person ajl is (jl - 1)+. Let us assume now that the partner ail of ajl has been chosen and fixed and let us proceed to select the partner ai, of aj, from team 2. Since aiz is younger than aj,, we must have a2 < j 2 . Since there are j 2 - 1 numbers (positions) less than j 2 , namely: 1, 2 , . . . , j 2 - 1, out of which two are already occupied by a1 and j1, we can see that there are j 2 - 3 choices left for 22, if j 2 > 3, and no choices if j 2 5 3. Thus the number of ways in which 22 can be selected is ( j 2 - 3)+. Similarly we can see that after Zl and 22 have been selected and fixed, the number of ways in which we can select a younger partner for the player aj, is ( j 3 - 5)+, + and so on. Finally, there are (jn - ( 2 n - 1)) ways to select a younger partner for the player aj,. Using the generalized principle of counting, we conclude that if the older players from each team: aj,, a j , , . . . , aj, have been fixed and ordered as jl < j 2 < < j,, then the number of ways in which we can pair each of them with a younger partner is: (jl - 1)+(j2 - 3)' . . . ( j n - ( 2 n - 1))'. Thus the total number of partitions

zyxwv zyx

is: Cl> . >

30

zyxw zyxw zyx I. zyxw zyx zyx zyxwvu zyxw Using the commutation relations (2.2) we find that

A=

[

( 2 n )!c2n-zp ( I )

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

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

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

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

and

1 d2 = 2~~("-~)p(1>~(n!)~(2n)!(cp(1) - 1) 2 0 ($ p ( I ) 2 .;

Thus the interval I cannot be arbitrarily small.

0

3. The q-Deformed Fock Case

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

- s)

and letting, as in the Boson case,

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

where

zyxw 31

=

{

;(n-A)(k-A)

z

zyxwvu zyxwvuts zyxwv

[k[klq! A],!

(&,A

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

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

z zyx zyxwvu

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

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

for a f i e d interval I

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

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

32

zyxwvutszzy

zyxwvu zyxwv zyxw

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

dl = p ( I ) c

~ [2&! ~ -

~

which is non-negative for all I and

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

~

zyxwvu zyxw

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

which cannot be true for arbitrarily small I.

0

References

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

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POWERS OF THE DELTA FUNCTION

LUIGI ACCARDI Centro Vito Voltem, Universitci di Roma Tor Vergata via Columbia, 2- 00133 Roma, Italy E-mail: accardiQ Volterra.mat.uniroma2.it

ANDREAS BOUKAS Department of Mathematics and Natural Sciences, American College of Greece Aghia Paraskevi, Athens 15342, Greece E-mail: [email protected]

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Our attempts t o establish a Fock representation for the renormalized higher powers of white noise, involve the assignement of a meaning to the powers of the Dirac delta function. In this paper we give meaning to the expression 6" = Ck dk) where n 2 2, dk)is the k-th derivative of the Dirac delta function and co, ...,cn-l E C are arbitrary.

c;:,'

1. Introduction: The Square of the Delta Function

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

zyxwvu z

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

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

(1.1)

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

34

zyxw

where we have used the substitution x =

y, and so

zyxwvu lim fn(x) = 6(x) n++m

in the sense of generalized functions. To give a meaning to a2(x) we notice that for E SO

+

=o thus, as a distribution on SO,

zyx zyxwv zyxw zy zyxw zy zy

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

+(x) = d.1

- 4(0)

+ 4(0) N z )

(1.7)

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

zy zyxw 35

F(4J)= +(O) F(dJ)

and so, by the arbitrariness of $, we may define

F = c ~ i.e

b2 = c b

(1.10)

zyxwv zyxwv z

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

(1.11)

where n 2 2 and c, E C is arbitrary, has not been very easy to use in order to prove the existence of a Fock space representation for the Lie algebra associated with the higher powers of the white noise functionals (cf. [2] ). 2. The Square of the Delta Function Revisited

In this section we describe a method for defining the square of the Dirac delta function that allows for a generalization to higher powers. Definition 2.1. For k E {0,1, ...} we define

b ( k ) ( 4 J )=

(-1)Q‘”‘O)

Theorem 2.1. On S,

b2 = c16

+ c2 6‘

36

zyxwvutsr zyxwvut zyxwv zyxwvu

where c1, c2 E C are arbitrary.

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

Then, for

zyxwv 4 E SI

zyxwvu zyx zyx =o

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

x

= a1 ( X I

+ $642 + ( X I

(2.6)

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

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

zyxwvu zyxw zyxwvu zyxw zyx zyxw zyxw

where c E C is arbitrary, and so

F = cb’

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

(2.10)

4 E S and write

+ 4 ( 0 ) $‘(z)

(2.11)

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

= 4 k ) - 4 ( 0 ) Nz)

(2.12)

Applying F to both sides of (2.11) we find

F = CI 6 + ~2 6‘

(2.15)

J2 = c16 + c2 6’

(2.16)

i.e.

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

z

38

zyxwv zyxwv zyxwv zyxwvutsrqpon zyx zyx zyxw zyxwvu

3. The Cube of the Delta Function

Theorem 3.1. On S,

Is3

where c1

c2 c3

E

= c16

+ c2 6' + c3 6l'

C are arbitrary.

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

zyxw f n ( z )=

{

n

if -1

< 2 5 -1

-

0 otherwise

n

Then, for q5 E SZ

nlym

f:(x) +(x) dx

= lim

= lim n-+m

x 3 $(x) dx

-L J' 8n

(3.3)

y3$(;y)dy 1

-l

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

x2 4(x) = a2(.)

+ 4(0) x2 $(x)

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

(3.4)

39

zyxwvut zyxwvut zyxw zyx zyxw zyx zy zyxwvu

where c E C is arbitrary, and so

F = cb“

(3.9)

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

2

4(x) = a1>.(

+ 4(0) 2 $(.)

(3.10)

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

Ql(Z)

=x

($(.I

- 4(0)

Applying F to both sides of (3.10) we find

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

(3.11)

40

zyxwvutsrq zyxw zyxw zyxw zyxwv zyxwv zyx F = ~ 1 6+ ’ ~2 6”

(3.14)

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

4(.)

=

+ 4(0) +(.I

(3.15)

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

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

(3.16)

Applying F to both sides of (3.15) we find

where cl, ..., cs E

C are arbitrary, and so (3.19)

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

4. The General Case

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

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

(3.20) 0

zyxwvu zyxw zyx 41

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

lilim

zyx

x k $(x) dx

f k ( x ) $ ( x ) d x = lim = lim

n++m

l 2kn

-

/

1

-l

1 yk$(;y)dy

zyx zyx zyxwv = lim

n++m

=o

/

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

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

4(Z) = ak-l(x)

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

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

where

(Yk-l(x)

is defined by

(4.4)

zyxw

zyxw zyxwvut zyxwv

42

+

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

+

$)

(4.8)

C1

(k - l)!

x=o

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

(k-l)(&l+) 8

4)

where c E C is arbitrary, and so

F =c

on

sk-2.

zyxwv zyxwvut p - l )

(4-9)

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

(+(.I

sk-3

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

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

Qk-2(2) = xk-2

+E

ak-2(z)

- +(O)

$(.))

E

sk-2

and write

(4.10)

is defined by

(4.11)

Applying F to both sides of (4.10) we find

zyxwv

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

(4.14)

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

zyxwvu zyxwvuts zyxw zyxwv zyx zyx zyx zyxwv 43

k- 1

F =

C cm drn)

(4.15)

m=l

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

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

(4.16)

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

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

zy (4.18)

1=1

k- 1

zyxw

k- 1

=

C

1 =o k-1

=

c

6, cl,...,&-I

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

21 S("(q5)

1=0

where

el

E

C are arbitrary, and so k-1

F

=

C

m=O

i.e.

~ , 6 ( ~ )

(4.19)

44

zyxwvuts

zyxwvut zyxwv zyxw z zyxwvu k-1

(4.20)

m=O

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

0

References

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

zyx zyxw

DISPERSION RELATIONS IN THE STOCHASTIC LIMIT OF QUANTUM THEORY *

zyxwv

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

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

4 7005,

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

1. Introduction

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

zyxwv

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

zyxw zy

46

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

zyxw zyx zyxw

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

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

HR :=

J

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

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

zyx zyxwv zy zyxw zz zyx

zyx 47

We will assume that the total Hamiltonian has the form

H(’) := Ho

+ XHI = H s + H R + XHI,

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

HI =

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

where Dj are system operators and

A * ( g j ) := / d k g j ( k ) a + ( k ) , A ( g j ) := / d k g * ( k ) a ( k ) ,

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

a

-U(’) at

= -iXHI(t)

U,“),

H I ( t )= eitHoHIe-itHo 7

and study the limits, in a topology to be specified, of the rescaled interaction Hamiltonian and of the rescaled propagator: 1 lim - H I

A-0

x

lim

A-0

(+)

=: ht,

~ $ =:1 ut. ~

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

x

*

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

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

(3)

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

48

z

zyxw zyxwvu zy zyxw (s)) z zyxwv zy

where

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

A2

1 l1 (!$) t+dt

dtl

dt2 (HI

HI

(4)

is known as the drift term. Among the usual assumptions to achieve this program we have the following: the cut-off functions g j are Schwartz functions; the dispersion function w ( k ) and the cut-off functions g j are related by the following analytical condition:

0 0

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

0

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

zyxwvu

3. The Dispersion Function

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

H

rl u r2u r3,

where: (i)

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

:= Rang(wlr,),

and assume that the boundary zero.

8ri of I'i

has Lebesgue measure

zyxwvu zyxwv zyxwv zyxwvu zyxwv 49

(ii)

raj

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

w ( k ) = aj,

(iii)

Q k E raj.

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

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

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

/

zyx

4, cp, f

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

and g,

k’)).

zyxwvu zy A-0

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

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

k)a:,,t (t’,k’))

= bu,,,2~b(t

-

t’)b(k

(b) Over each raj,

-

jrl

k’)b(w(k) - w)Xr:(w).

of I’i,

50

zy

zy zy z zyxwv zyxwv zyxwv zyxwv

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

A-0 lim x2

4’

(y )v A(%$)

= lim A-0

x

lQj

dk f ( k ) g ( k )

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

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

I’3

or when w E

Xi? For example, for dispersion

w ( k ) = klP7 CL

> 0,

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

-

if d

0,

-

p

SO

> 0,

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

< 0, our techniques do not give an answer.

5. The Drift

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

fA

and the inverse Fourier

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

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

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

zy zy zy zyxwv zyx zyx zyxw 51

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

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

G=

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

w

(Oj)

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

+ The part corresponding t o the singular region r31,

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

1PE,-wDjPEr,

Er

Fw := {

E E ~

E Fu

Spec H s : E~ - w

E

Spec H s } ,

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

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

52

Acknowledgements

F.G. Cubillo is grateful to L. Accardi and Centro Vito Volterra for support and kind hospitality. References

zyxwv

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

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

zyxw zyxw

W. AYED

Institut PrLparatoire a m Etudes d ’Inghieurs de Nabeul, 8000 Nabeul, Tunisia E-mail: wided. [email protected] H. OUERDIANE Department of Mathematics Faculty of Sciences of Tunis University of Tunis El-Manar 1060 Tunis, Tunisia. E-mail: habib. [email protected]

zyxwvutsr

In this paper, we give a new criterion for positivity of generalized functions and positive operators on test functions space of entire function on the dual of a nuclear space N’ and with 0 order growth condition, denoted by F’o(N’). This definitions are used to prove that every positive operator has integral representation given by positive Radon measure and these measures are characterized by integrability conditions. This new criterion of positivity can be easily applied t o several examples.

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

54

zyxwvu zyxwvuts zyxwzy vutsrq zyxwv zyx zyxwv

sures associated to a positive generalized function or positive operator are characterized by integrability conditions of Fernique type.

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

N = projlimp,JVp,

M = projlimq,ooMq

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

N’ = indlimp+ooN-p,

M’ = indlim,+,M-,

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

M

@ N = projlim,,,M,

@ Np

Similarly,

( M @ N)’ = M’ @ N’ = indlim,,,M-,

€3

N-,

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

Eql(N-,

@ M-p, (&Cp),

(m1, m2))

zyxwv zyxwvutsrq zyxw zyx 55

zyxw zy zyxwvut zyx

becomes a projective system of Banach spaces and we put:

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

@ M-p,

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

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

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

(m1,m2)LP

E

N,ml > 0,7732 > 01

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

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

(m1, m2)).

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

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

F(e,vp)((N@ MI’)

. We recall, from [ll],that

Fe(N’)GFv(M’)

(1)

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

+

+

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

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

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

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

zyxwvut zyxwvu zyxwv zyx zyxwv zyxwv zyx zyxwv zyxw

56

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

C(.Fo (N‘),Fp(M‘)*)

.Fe

(,I)*

6~~(M’)*

(2)

F[e,+,)(N’ CBM‘)*.

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

zyxw

zyxwvu

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

Theorem 1.2. A function 0 : N @ N + CC is the symbol of some Z E C(.Fe(N’), Fe(N’)*)if and only if 0 E G‘p ( N @ N ) .

In the next, we suppose that the two Young functions 8 and cp satisfy the additional conditions:

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

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

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

zyxwv zyx zyx zyxwvu zyxwv zyx zyx zyxwvu 57

2. Positive generalized function in two infinite dimensional variables

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

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

a),

z E N’,

w E M’.

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

(7)

f1

E

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

z

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

V(z,y) E X ’ X Y ’

we have

((CP7.f))

b 0.

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

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

zyxwv

(ff*)(a:

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

+ io @ y + i0)l2 2 0.

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

58

zyxwvutsr zyxwvu

has:

P

z zyxwvu

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

1

X’CBY’

“XPi((GE) + (Y,v))dPa(z@?4). ( 9)

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

z zyxwvutsr zyx zyxw zy z zyxwv zyxw 59

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

-

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

(10)

L,$y,

(11)

- fm>(z@ Y)I

I(fn

2dP a@@Y).

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

~

zyxwvu zyxwv =

J

@ @ y)dCLa(z @ Y)

/

f b @ Y)dPa(.

X’$Y‘

= XJ$Y‘

@ Y).

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

Fe,,+,(N’CBM’);, = Fe,,+,(N’CBM’);.

(12)

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

Fe,,+,(N’@ M’)

c L ~ ( x ’x Y’,71

72)

c F,,,+,(N’ @ M’)*

zyxwv zyxwv zyxwv z zyxw

60

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

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

zyx

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

Theorem 2.2. Let p a finite measure o n X' @ Y' equipped with the Bore1 u-algebra of B(X' @ Y ' ) . The measure p represent a positive generalized function @ E Fo,,+,(N'@ MI): if and only if it verifies the two following properties:

zy

(i) There exist q > 0 such that the measure p is supported by X - q @ Y - q . (ii) There exist m l , m2 > 0 such that

Jx-,

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

@

y ) < 00.

@Y-,

(13)

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

zy zyxwvu

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

Jx-,

$Y-,

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

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

121

( 2 W W e m2

*

~2,Cp;l

El7l:7l.:l

(14)

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

z zyxwv zyxw zyx zy zyxwvu zyx zyxw 61

3. Positive operator in L(Fe(N‘),F,(M’)*)

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

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

(16)

Definition 3.1. An operator E E L(F~(N‘),Fq(M’)*) is positive if its kernel EK is an element of Fo,,+,(N’@ M’):.

Theorem 3.1. For any positive operator E E L(Fe(N’),F,(M’)*) there exists a unique positive Radon measure p~ o n XI @ Y‘ such that for all f E Fe,,(N’ @ M I ) , one has: ( ( E K ,f)) =

1

X‘$Y’

+

f ( x 20 a3 y

+ iO)dp=(x@ y ) .

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

@

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

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

>=

~f

E

Fo,,+,(N’CBM I ) .

62

zyxwv zyxw zyx zyxw

T h e characterization of pz is a consequence of theorem 2.2. We deduce from theorem 3.1, t h e equation (16) and t h e corollary 2.2, the following corollary that can be needed to characterize unitary solution of

quantum stochastic differential equations. I n a future paper, we generalize this new criterion and we apply this results to several concrete examples in order to give regularity property for solution of some quantum differential equations.

References

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

zy zyxwvu 63

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

zyx zyxw

COMPARISON OF SOME METHODS OF QUANTUM STATE ESTIMATION

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

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

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

1. Introduction

zyx

The aim of quantum state estimation is to decide the actual state of a quantum system by measurements. Since the outcome of a measurement is stochastic, several measurements are to be done and statistical arguments lead to the reconstruction of the state. Due to some similarities with X-ray tomography, the state reconstruction is often called quantum tomography [2]. More precisely, in physics-related books, journals and papers, tomography refers to both the state and parameter estimation of quantum dynamical

zyxw zyxw

*Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology. tSupported by the Hungarian grant OTKA T032662. *Supported by the Hungarian grant OTKA T042710.

64

zyxwv zyxw 65

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

66

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

zyxw zyxwvu zy zy zyxw zyx zyxw zyxwvu

2. Preliminaries about two level systems

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

[; I

,

c2=

[p -;]

,

c3=

[o-l]

9

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

+

+

+

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

C:.,

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

z zyxw zyxwvu zyxwv zyxwvu zyxw 67

3. Measurements on qubits

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

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

Dl = { D l ( j ): j

(i = 1 , 2 , 3 ) .

= 1,e.a , n }

(3)

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

+

4. Quality of the estimates

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

J1

(5)

pTwpT

zyxwvu zyxwv

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

F(p,w)=l

* p=w,

0 I F(P,W) I 1

F(p,w)=O

*w l p .

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

=

A +A.

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

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

68

zyx zy zyxw zyxwv

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

if m estimates are available. Alternatively, the Halbert-Schmidt distance

d(p,w) := J T q j T p

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

m

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

zyx zyx

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

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

zy

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

zyx zy zy zyxw zyxwvu zyxw 69

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

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

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

zyxw zy

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

mi = and the variance is

+ +

l(i) 1 X n+lc+2

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

(l(2) 1 (n

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

K.

(12)

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

qi) +1+x -1

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

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

(14)

z

70

zyxw zyxw zyxw zy zyxwvu zyxw zyx z zyxw

where both integrals are over the domain and

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

+ ui + uz I 1)

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

Then the conditioned estimate of si will be 2(fiZ

- 1).

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

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

:= T i ( + )

~i(f is )the

relative frequency of

-Ti(-)

zyxw

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

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

j = llr1I2 ) ~

(rj - ~

+ 1 1 ~ 1 -1 ~2 r .

T

(15)

j=1

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

L(w) llrll 5 1

subject to

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

r=-.

I1

11n-11

z zyxwvu zyxwvutsrqp zyxwvu 71

7. Simulation experiments

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

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

zyxwvut zyxwv zyxwv

8.1. Number of measurements

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

72

zyxwvut zyxwvutsrqponm

zyxwvutsrqponmlkj .i I’..r I : : : : . ] .................... i

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

0.9,

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

w

#?

~

Bayesian regld. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC ..a d a n

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

0.94

WihD”1

:

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

p.g.k

zyxwvuts

.: .........

0.98

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

:

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

1...

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

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

..........

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

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

i

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

0.9,

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

~

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

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

ZOO

~

..>................. t

(4) X /

=

v v

c

X ; for o I s 0 we define the Hilbert space

where f n E N:qn and

zyx

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

{

zyxwvu

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

,LO

zyx zyx zyxw zyxwv zyx zyxw -

98

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

Theorem 3.1. $ ( N ) is a Fre‘chet reflexive nuclear space.

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

: F,q,@P4

F&(NP)

is of Hilbert-Schmidt type. For any n E

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

+

basis of

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

and put

an orthonormal basis

I,,’P,’ 9P in the following way

Then,

{ &},

becomes

estimate the norm of the

z zy

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

$

4. Duality Theorems

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

(@,)r=,,

an E N ! ; ~ , we put

Define

and

Gi(N’) :=

indlim

p - - r + m ; ,--roo

G:,,(N-p).

Then, we come t o the following duality relation.

zyx

z zyxwv zyxwv zyxw zy zyxwvu zyxw zyxw zyxw 99

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

n=O

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

(an, fn)

is the canonical bilinear dual pairing on

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

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

!?

=

(Tn),EN with Tn E N@qn

00

we have

n=O

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

=

with

an := q

w

((.Iq!)-l

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

T,.

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

N7qn.

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

100

zyx zyxw

zyx

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

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

$ l l i p ~ , p 0, the sequence F ( t / n ) n , n E N, converges to etc as n -+ 00 in the strong operator topology, and this convergence is uniform with respect to t E [0,TI. Let us also present the definition (introduced in [23]) of Chernoff equivalence for one-parameter families of operators { F ( t ) , t > 0 ) and {etc, t > 0 ) . These two families of operators are said to be Chernoff equivalent if IIF(t)g - etcglJ = o ( t ) as t -+ 0 for any g E D1 C Dom(C), where D1-is the essential domain of the operator C. 3. Functional integrals corresponding to the Cauchy-Dirichlet problem for the heat equation

zyxwvu

Let G - be a domain of a smooth m-dimensional compact Riemannian manifold K c RN with smooth boundary dG. Consider the Cauchy-Dirichlet boundary value problem in this domain for the heat equation with bounded continuous potential V : G -+ R (we can extend V to a continuous function on = G d G , we will denote this extension also as V ) .

c

U

g ( t , X )= ( - $ A K f ) ( t , x ) f(0,X) =fob) f ( t ,x ) = 0

+

V(X)f(t,X)

t 2 0,s E G xEG t 2 0,x E dG

(I)

Here A K stands for the Laplace-Beltrami operator -tr V2 on the manifold K . Assume that f and fo satisfy the conditions fo E C O ( ~f) ,: [O, m) x R, f ( t ,.) E CO(G),vt 2 0. Let an operator A in the space Co(c) be the generator of the semigroup resolving the problem (I). Then for any f E Dom(A):

c

150

zyxwvuts zyxwvu zyxwvu zyx

zyx

Note, that D3(G)c Dom(A) is the essential domain for the self-adjoint operator A, and for any f E D 3( G )the operator A acts as following: ,

1 (Af)(z) = (--&f)(.) 2

+ V ( z ) f ( x ) , z E (7.

Let E ( . ) : [O,+m) -+ [O,+m) be a smooth function tending to zero when t 4 0. Suppose that ( P ~ ( ~ ) ( .is) a set of functions in D 3 (G)which approximate the indicator of the domain G as t +. 0 with respect to the pointwise convergence topology. Consider the following operators acting in the Banach space CO(G):

zyxw zyxw

T t ( t ): ( T , E ( t ) f > ( x=) ( P E ( t ) ( 4 j- e t v ( z ) f( Z ) Q E ( t , x , z)volddz), G

T f ( t ):

j- ,tV(

( q V ) f ) ( z=) ( P E ( t ) (

z) t s c a l ( z )e-

6 r2 (z) f ( 4 P E ( t ,z, z)vok(dz)

G

T,'(t) : T,'(t)fI(.)

=(PE(t)(4

j- et"(")f(z)q1(t,2,z)voMdz), G

T,'(t) : (T,'(t)f)(z)= ( ~ ~ ( ~ )e t(vx( z)) e ~ S C afl (z>pI(t, (z) z, z)volddz) G

One can check (see [2]), that these families of operators are Chernoff equivalent to the semigroup etA. Therefore, by the Chernoff theorem we obtain the following statement.

zyxw

Theorem 2. Let etA be the semigroup of operators on Co(G), resolving the Cauchy-Dirichlet problem (I). Then 1) etA = s - lim (T,E(t/n)"), 2) etA = s - lim (T,E(t/n)"), n'w n+w 3) etA = s - lim (T,'(t/n)n), 4) etA = s - lim (T,'(t/n)n). n+w 71-00 where s-lim stands for the limit with respect to the strong operator topology on Co(G).

It can be shown (see [2]) that the limits of finite-dimensional integrals in the theorem (2) coincide with limits of finite-dimensional integrals which are of the same kind as those used in definitions of measures W z ' , W z E , S;' and S;". Hence, Feynman formulas obtained in the theorem (2) can be understood as functional integrals over surface measures:

Theorem 3. Let f ( t , x ) be the solution of the Cauchy-Dirichlet problem (I) with the initial condition fo E Co(G). Then the solution f (t,x ) can be

151

represented by a functional integral over the Wiener measure WE.

zy

by a functional integral over the internal Smolyanov surface measure S2':

and by a functional integral over the external Smolyanov surface measure S;E:

zyx zyxw zyx

4. Functional integrals representing solution of the Cauchy

Problem for the Schoedinger equation

Consider the Cauchy problem for the Schroedinger equation on the manifold

K:

i g ( t , X )= ( i A K f ) ( t , X ) 4- v ( X ) f ( t , X ) t 2 0, 5 E K (11) X E K f(0,z) = fO(5) Assume that f and fo satisfy the conditions fo E C(K), f : [ O , o o ) x K + C, f ( t , . )E C(K), W 2 0. Here C(K) for the Banach space of complex valued continuous functions on K equipped with the norm

152

zyxwvuts zyxwv zyxw

11 11, l l f l l *

= SUPzEK

If(x)l.

+

zyxwvu zy zyxw zyxw zy

Consider the set K = U x C ~ { x &(K - x)}. We say that a function g : K 4 C belongs to the class A, iff the following conditions are satisfied: 1) there is a domain 0, of some complex manifold such that the closure of 0, contains the set K; 2) there exists the unique analytical in 0, and continuous on 0, K function i j such that the restriction of B to K coincides with g. If in condition (2) the function i j is twice continuously differentiable on 0, U K , then we say that the function g belongs to the class Az. Let functions V and fo belong to the class A . Suppose that the Schroedinger equation with the potential V and the initial condition fo has in K the unique solution f ( t , z ) , which belongs to the class Az:

u

ig(t,z ) = ( i A & f ) (zt ), + V ( z ) f ( t z, ) t 2 0, t E K

{ f(0,

zyxw

z ) = fob)

z E K.

The operator A& for functions from the class Az is defined as follows: the value ( A & B ) ( z ) equals the value of the analytical continuation of the function A K g at the point z. For functions fo, V of the class A and functions f ( t , .) of the class A2 for any y E K and any fixed x E K we can define following functions: cp"(t,Y) = f(t,a: &(Y - x)), cpti(Y) = f o b + J;l(Y - X I ) , V"(Y) = 4 V ( x &(y - x)). Let y in the symbol Ak mean that the Laplace-Beltrami operator AK acts on the variable y. Then ( A k c p " ) ( t , y ) = ( i A & f ) ( t , z ) for z = x &(Y - x). Hence, for any fixed x E K the function cp" solves the Cauchy Problem for the heat equation:

+

+

+

%(t,Y) = (-$Ak'P")(t,y) v"(0, Y) = cptib) Using results of works [21, 26,

+ V(Y)cp"(t,Y) t 2 0,YE K YEK

(1)

18, 191, we can represent solutions

9" of the Cauchy problem for the family of heat equations (1) with the

zyxwv

help of functional integrals over Smolyanov surface measures and over the Wiener measure, generated by Brownian motion in the manifold. If in obtained formulas we come back to functions f, fo, V and notice that f ( t , x) = cp"(t,x) then we get the following:

Theorem 4. Let K be a smooth m-dimensional compact Riemannian manifold isometrically embedded into the space RN c CN.Let functions V and

zyxwvutsrqp zyxwvu z zyxw z 153

fo belong to the class A . Suppose that the Cauchy problem (11) for the Schroedinger equation with the potential V and the initial condition fo has in K the unique solution f ( t ,z ) , which belongs to the class Az. Then f ( t ,x) can be represented by a functional integral over the Wiener measure WE:

c(lo,tl, K ) by a functional integral over the external Smolyanov surface measure S 2 E :

and by a functional integral over the internal Smolyanov surface measure S2':

t

( c I ( t ,x ) ) - 1 =

.I

zyx zy

zyxw

Q JSC~(E(T))~T e o

S2'( d c ) .

Acknowledgments Author expresses her deep gratitude to Prof. O.G. Smolyanov for useful discussions. References 1. L. Andersson, B.K. Driver, Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds, J. Funct. Anal., 65 (1999), no. 2, 430-498.

154

zyxwvu zyxwvu zy

2. Butko Ya. A., Representations of the Solution of the Caushy-Dirichlet Problem for the Heat Equation in a Domain of a Compact Riemannian Manifold by Functional Integrals, Russian Journal of Mathematical Physics, 11 N 2 (2004), 1-9. 3. Butko Ya. A., Functional integrals for Schroedinger equation in a compact Riemannian manifold, Math. Zametki, 79 N 2 (2006), 194-200. 4. R. Chernoff, A Note on Product Formulas for Operator Semigroups, J . Funct. Anal., 2 (1968), 238-242. 5. R. Chernoff, Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators, Mem. Amer. Math. SOC.,140 (1974). 6. De Witt-Morette C., Elworthy K.D., Nelson B.L., Sammelman G.S., A stochastic scheme for constructing solutions of the Schroedinger equations, Ann. Ins. H. Poincare Sect. A (N. S.) 32 N 4 (1980), 327-341. 7. Doss H., Sur une Resolution Stochastique de 1'Equation de Schroedinger a Coefficients Analytiques, Communications in Math. Phys, V.73, N3, 1980, 247-264. 8. Eells J., Elworthy K.D., Wiener integration on certain manifolds, Problems in non-linear analysis (C.I.M.E., IV Ciclo, Varenna, 1970 ), Edizioni Cremonese, 1971. 9. R.P. Feynman, Space-time Approach to Nonrelativistic Quantum Mechanics, Rev. Mod. Phys., 20 (1948), 367-387. 10. R.P. Feynman, An Operation Calculus Having Application in Quantum Electrodynamics, Phys. Rev., 84 (1951), 108-128. 11. F'roese R., Herbst G., Realizing holonomic constrains in classical and quantum mechanics, Com. Math. Phys. 220 (2001), 489-535. 12. Nash J. F., The imbedding problem for Riemannian manifolds, Ann. Math., 63, (1956), 20-63. 13. 0.0.Obrezkov, The Proof of the Feynman-Kac Formula for Heat Equation on a Compact Riemannian Manifold, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6 (2003), no. 2, 311-320. 14. Sidorova N. A., The Smolyanov surface measure on trajectories in a Riemannian manifold, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 7, N3, September 2004, 461-472. 15. Sidorova N. A., Smolyanov O.G., Weizsaecker H. v., Wittich O., The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold, Journal of Functional Analysis, 206 (2004), 391-413. 16. Smolyanov O.G., Tokarev A.G., Truman A., Hamiltonian Feynman Path Integrals via the Chernoff Formula, J . Math. Phys., 43 (2002), no. 10, 51615171. 17. Smolyanov O.G., Truman A,, Hamiltonian Feynman Formulas for the Schrodinger Equation in Bounded Domains, Doklady Acad. Nauk, 70 (2004), 899-904. 18. 0. G. Smolyanov, H. von Weizsacker, 0. Wittich, and N. A. Sidorova, Surface Measures Generated by Diffusions on Paths in Riemannian Manifolds, Doklady Mathematics, 63 N 2 (2001), 203-208. 19. 0. G. Smolyanov, H. von Weizsacker, 0. Wittich, and N. A. Sidorova , Wiener

zyx zyxwvu zyxwvut 155

Surface Measures on Trajectories in Riemannian Manifolds, Doklady Mathematics, 65 N 2 (2002), 239-244. 20. 0. G. Smolyanov, Smooth measures on loop groups, Doklady Acad. Nauk, 345 N 4 (1995), 455-458. 21. Smolyanov O.G., Weizsiicker H. von, Wittich O., Brownian Motion on a Manifold as Limit of Stepwise Conditioned Standard Brownian Motions, Canadian Math. Society Conference Proceedings, 29 (2000), 589-602. 22. Smolyanov O.G., Weizsiicker H .v., Wittich 0. ”Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds” http : //arxiv.org/PS-cache/math/pdf /0409/0409155.pdf 23. Smolyanov O.G., Weizsiicker H. von, Wittich O., Chernoff’s Theorem and the Construction of Semigroups, Evolution Equations: Applications to Physics, Industry, Life sciences and Eqonomics - E V E Q 2000, M. Ianelli, G. Lumer, Birkhauser (2003), 355-364. 24. Smolyanov O.G., Weizsiicker H. von, Wittich O., The Feynman Formula for the Cauchy Problem in Domains with Boundary, Doklady Acad. Nauk, 69 N 2 (2004), 257-262. 25. Trotter H.F., On the Product of Semigroups of Operators, Proc. Amer. Math. SOC.,10 (1959), 545-551. 26. H. von Weizsacker, 0. G. Smolyanov, and 0. Wittich , Diffusion on Compact Riemannian Manifolds and Surface Measures, Doklady Mathematics, 61 N2 (2000), 230-235.

zyx zyxw

QUANTUM PROBABILISTIC MODEL FOR T H E FINANCIAL MARKET *

zyxw zyxw

OLGA CHOUSTOVA International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vaxjio, S-35195, Sweden E-mail: Olga. ChoustovaQm.se

zyxw z

We use the formalism of quantum mechanics in the framework of the Bohmian (pilot wave) model to describe stochastisity of the financial market. We interpret nonclassical contribution to stochasticity as a psycho-financial (information) field $ ( q ) describing expectations of agents of the financial market.

1. Introduction

In economics and financial theory, analysts use random walk techniques to model behavior of asset prices, in particular share prices on stock markets, currency exchange rates and commodity prices. This practice has its basis in the presumption that investors act rationally and without bias, and that at any moment they estimate the value of an asset based on future expectations. Under these conditions, all existing information affects the price, which changes only when new information comes out. By definition, new information appears randomly and influences the asset price randomly. Corresponding continuous time models are based on stochastic processes (this approach was initiated in the thesis of L. Bachelier' in 1890). However, empirical studies have demonstrated that prices do not completely follow random walk. Low serial correlations (around 0.05) exist in the short term; and slightly stronger correlations over the longer term. Their sign and the strength depend on a variety of factors, but transaction costs and bid-ask spreads generally make it impossible to earn excess re*This work is supported by Profile Mathematical Modelling of Vkixjo university and EU-network on Quantum Probability and Applications

156

157

zyxw zyxwvu

turns. Interestingly, researchers have found that some of the biggest prices deviations from random walk result from seasonal and temporal patterns. Therefore it would be natural to develop approaches which are not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. In particular, there are two well established (and closely related ) fields of research behavioral finance and behavioral economics which apply scientific research on human and social cognitive and emotional biasesa to better understand economic decisions and how they affect market prices, returns and the allocation of resources. The fields are primarily concerned with the rationality, or lack thereof, of economic agents. Behavioral models typically integrate insights from psychology with neo-classical economic theory. Behavioral analyses are mostly concerned with the effects of market decisions, but also those of public choice, another source of economic decisions with some similar biases. In physics, researchers are concerned with observer effects - which set clear limits on the process of observing as in Heisenberg's uncertainty principle. Quantum entanglement also introduces problems in some observational situations. These are all well-accepted foundations of 20th century philosophy of science and along with a few other such discoveries (like a universal maximum for the speed of light) form its core epistemology. Knowing these limits has helped develop a cognitive science by which humans might reasonably characterize the limits of their own perception. However, bias does not end with cognition. How to interpret the data on what humans 'can' observe becomes controversial when there are few individuals capable of reproducing experiments and compiling new models. Since the 1970s, the intensive exchange of information in the world of finances has become one of the main sources determining dynamics of prices. Electronic trading (that became the most important part of the environment of the major stock exchanges) induces huge information flows between traders (including foreign exchange market). Financial contracts Wognitive bias is any of a wide range of observer effects identified in cognitive science, including very basic statistical and memory errors that are common t o all human beings and drastically skew the reliability of anecdotal and legal evidence. They also significantly affect the scientific method which is deliberately designed to minimize such bias from any one observer. They were first identified by Amos Tversky and Daniel Kahneman as a foundation of behavioral economics. Bias arises from various life, loyalty and local risk and attention concerns that are difficult t o separate or codify. Tversky and Kahneman claim that they are at least partially the result of problem-solving using heuristics, including the availability heuristic and the representativeness.

158

are performed at a new time scale that differs essentially from the old ”hard” time scale that was determined by the development of the economic basis of the financial market. Prices at which traders are willing to buy (bid quotes) or sell (ask quotes) a financial asset are not more determined by the continuous development of industry, trade, services, situation at the market of natural resources and so on. Information (mental, marketpsychological) factors play very important (and in some situations crucial) role in price dynamics. Traders performing financial operations work as a huge collective cognitive system. Roughly speaking classical-like dynamics of prices (determined) by ”hard” economic factors is permanently perturbed by additional financial forces, mental (or market-psychological) forces, see the book of J. Soros [2]. In this paper we develop a new approach that is not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. Our approach can be considered as a special econophysical model in the domain of behavioral finance. In our approach information about financial market (including expectations of agents of the financial market) is described by an information field $(q) - financial wave. This field evolves deterministicallyb perturbing the dynamics of prices of stocks and options. Since psychology of agents of the financial market gives an important contribution into the financial wave $(q), our model can be considered as a special psycho-financial model. This paper can be also considered as a contribution into applications of quantum mechanics outside microworld, see also books [3,4]. The complete version of the present paper was submitted as the quant-preprint [5].

zyxwv zyxw zyx zy zy

2. Financial phase-space

Let us consider a mathematical model in that a huge number of agents of the financial market interact with one another and take into account external economic (as well as political, social and even meteorological) conditions in order to determine the price to buy or sell financial assets. We consider the trade with shares of some corporations (e.g., VOLVO, SAAB, IKEA, ...).‘ We consider a price system of coordinates. We enumerate corporations which did emissions of shares at the financial market under consideration:

zyxwv

bDynamics is given by Schrodinger’s equation on the space of prices of shares. ‘Similar models can be developed for trade with options, see E. Haven [S]for the Bohmian financial wave model for portfolio.

zyxwv zy z zyxwv 159

j = 1 , 2 , ...., n (e.g., VOLV0:j = 1, SAAB:j = 2, 1KEA:j = 3,...). There

z zyxwvu zyxw

can be introduced the n-dimensional configuration space Q = Rn of prices, q = (q1,. . . ,q n ) , where qj is the price of a share of the j t h corporation. Here R is the real line. Dynamics of prices is described by the trajectory q ( t ) = ( q I ( t ) , . . . ,qn(t)) in the configuration price space Q. Another variable under the consideration is the price change variable: vj ( t )= q j (t)= limAt+o q j ( t t AAt t ) - q J ( t ) , see, for example, the book [7] on the role of the price change description. In real models we consider the discrete time scale At, 2At,. . . . Here we should use discrete price change variable zj(t) = qj(t +At) - qj(t). We denote the space of price changes by the symbol V ( = Rn),v = (211,. . . ,vn). As in classical physics, it is useful to introduce the phase space Q x V = R2n,namely the price phase space. A pair (q,v) = (price, price change) is called a state of the financial market. (Later we shall consider quantum-like states of the financial market. A state (q,v) is a classical state.) We now introduce an analogue m of mass as the number of items (i.e., in our case shares) that trader emited to the market.d We call m the financial mass. Thus each trader has its own financial mass mj (the size of the emission of its shares). The total price of the emission performed by the j t h trader is equal to Tj = m j q j . Of course, it depends on time: Tj(t) = mjqj(t).To simplify considerations we consider a market at that any emission of shares is of the fixed size, so mj does not depend on time. In principle, our model can be generalized to describe a market with timedependent financial masses, mj = mj(t). We also introduce financial energy of the market as a function H : Qx V R. If we use the analogue with classical mechanics. (Why not? In principle, there is not so much difference between motions in ”physical space” and ”price space”.), then we could consider (at least for mathematical modeling) the financial energy of the form:

zyxwvu --f

zyxw zyxwv

4

Here K = C;==, mjv; is the kinetic financial energy and V(q1,.. . ,qn) is the potential financial energy, mj is the financial mass of j t h trader.e d‘Number’ is a natural number m = 0,1,. . . , - the price of share, e.g., in the US-dollars. However, in a mathematical model it can be convenient to consider real m. This can be useful for transitions from one currency to another. eThe kinetic financial energy represents efforts of agents of financial market to change

160

zyxw zyxwvu zyz

The potential financial energy V describes the interactions between traders j = 1,...., n (e.g., competition between NOKIA and EFUCSSON) as well as external economic conditions (e.g., the price of oil and gas) and meteorological conditions(e.g., the weather conditions in Louisiana and Florida). For example, we can consider the simplest interaction potential: V(q1,.. . ,qn) = z y = l ( q i - q j ) 2 . The difference Iq1 - q j l between prices is the most important condition for arbitrage. To describe dynamics of prices, it is natural to use the Hamiltonian dynamics on the price phase space. As in classical mechanics for material objects, it is useful to introduce a new variablep = mu, the price momentum variable. So, instead of the price change vector v = (211,. . . ,vn), we shall consider the price momentum vector p = ( P I , . . . ,p,), p j = mjvj. The space of price momentums is denoted by the symbol P. The space Q x P will be also called the price phase space. Hamiltonian equations of motion on the 8H price phase space have the form: q = =,@j = = 1,.. . , n .

zy z

zyxw -E,j

3. Financial Pilot W a v e

We now consider a model in that dynamics of prices of shares is driven by an information field (or psycho-financial wave) representing the psychology of agents of the financial market. We represent such a wave in the same way as in the Bohmian mechanics for quantum particles. In fact, we need not develop a new mathematical formalism. We will just apply the standard pilot wave formalism to traders of the financial market. The fundamental postulate of the pilot wave theory is that the pilot wave (field) $(q1,. . . ,qn) induces a new (quantum) potential U ( q 1 , .. .,qn) which perturbs the classical equations of motion. A modified Newton equation has the form:

-=

where f = 89 and g = financial mental force.f

P=f+g,

-m. We call the additional financial force g a 89

zyxw

prices: higher price changes induce higher kinetic financial energies. If the corporation jl has higher financial mass than the corporation j z , so mj, > mj,,then the same change of price, i.e., the same financial velocity vj, = vj2, is characterized by higher kinetic financial energy: K j , > Kj, . We also remark that high kinetic financial energy characterizes rapid changes of the financial situation at market. However, the kinetic financial energy does not give the attitude of these changes. It could be rapid economic growth as well as recession. fThis force g(q1,.. . ,qn) determines a kind of collective consciousness of the financial market. Of course, the g depends on economic and other ‘hard’ conditions given by the

zyxwvu zyxwv zyxw 161

By using the standard pilot wave formalism we obtain the following rule for computing the financial mental force. We represent the financial pilot wave $ ( q ) in the form: $(q) = R(q)eiS(q)

where

R(q) = I$(Q)I is the amplitude of $(q) and S(q) is the phase of $(q). Then the financial mental potential is computed as

zy zyxw

zyxwvuts zy

and the financial mental force as gj(q1,. . . , qn) = = aU ( q l , . . . ,qn). These formulas imply that strong financial effects are produced by financial waves having essential variations of amplitudes. Example 1. (Financial waves with small variation have no effect). Let R = const. Then the financial mental force g E 0. There are no nonlocal effects which can be induced by nontrivial financial force. Thus if R = const, then it is impossible to perturb the psychological state of the whole financial market by varying the price of shares qj of the fixed trader j. The constant information field does not induce psychological financial effects at all. As we have already remarked the absolute value of this constant does not play any role. Waves of constant amplitude R = 1, as well as R = lo1'', produce no financial effect. Let R(q) = cq,c > 0. This is a linear function; variation is not so large. As the result g = 0 here also. No financial mental effects. Example 2. (Successive speculations) Let R(q) = c(q2 d ) , c, d > 0. Here U ( q ) = (it does not depend on the amplitude c !) and g(q) = The quadratic function varies essentially more strongly than the linear function, and, as a result, such a financial pilot wave induces a nontrivial financial mental force. In particular, there are nonlocal financial effects.

+

-&

.a.

financial potential V ( q 1 , .. . , q,,). However, this is not a direct dependence. In principle, a nonzero financial mental force can be induced by the financial pilot wave cp in the case of zero financial potential, V 0. So V = 0 does not imply that U 0. Market psychology i s not a totally determined by economic factors. Financial (psychological) waves of information need not be generated by some changes in a real economic situation. They are mixtures of mental and economic waves. Even in the absence of economic waves, mental financial waves can have a large influence to the financial market.

=

=

162

zyxw zyx zyxwv zyxwv zyxw

The only problem which we have still to solve is the description of the time-dynamics of the financial pilot wave, $(t,q). We follow the standard pilot wave theory. Here $ ( t , q ) is found as the solution of Schrodinger's

equation.

with the initial condition $(O, q1,. . . ,qn) = $(q1,. . . ,qn).g We underline two important features of the financial pilot wave model: a) all traders are coupled on the information level; b) reactions of the financial market do not depend on the amplitude of the financial pilot wave: financial waves $, 2$, lOOOOO$ will produce the same reactionsh. References

1. L. Bachelier, Theorie de la speculation, Ann. Sc. 1'Ecole Normale Superiere 111-17,21-86 (1890). 2. J. Soros, The alchemy of finance. Reading of mind of the market (J. Wiley and Sons, Inc.: New-York, 1987). 3. L. Accardi, Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica (I1 Saggiatore, Rome, 1997) 4. A. Yu. Khrennikov, Information dynamics in cognitive, psychological and anomalous phenomena (Kluwer, Dordreht, 2004). 5. 0. Choustova, Pilot wave quantum model for the stock market, http://www.arxiv.org/abs/quant-ph/O109122. 6. E. Haven, Bohmian mechanics in a macroscopic quantum system. Foundations of Probability and Physics-3, ed. A. Yu. Khrennikov (Melville, New York: AIP Conference Proceedings, 2006). 7. R. N. Mantegna and H. E. Stanley, Introduction to econophysics (Cambridge, Cambridge Univ. Press, 2000).

gWe make a remark on the role of the constant h in Schrodinger's equation. In quantum mechanics (which deals with microscopic objects) h is the Planck constant. This constant is assumed to play the fundamental role in all quantum considerations. However, originally h appeared as just a scaling numerical parameter for processes of energy exchange. Therefore in our financial model we can consider h as a price scaling parameter, namely, the unit in which we would like to measure price change. hThe amplitude of an information signal does not play so large role in the information exchange. The most important is the context of such a signal. The context is given by the shape of the signal, the form of the financial pilot wave function.

zyxw zyxw

A NEW PROOF OF A QUANTUM CENTRAL LIMIT THEOREM FOR SYMMETRIC MEASURES

VITONOFRIO CRISMALE Dipartimento d i Matematica, Universitii d i Bari via E. Orabona 4, I-70125 Bari, I T A L Y e-mail: crisma1evQdm.uniba.it YUN GANG LU

Dipartimento di Matematica, Universitii d i Bari via E. Orabona 4, I-70125 Bari, I T A L Y e-mail: 1uQdm.uniba.it

zyxwvu

We present a new proof of the central limit theorem performed in [2] for symmetric measures based on a different approach.

1. Introduction In [2] a constructive quantum central limit is proved for any mean-zero real probability measure with moments of any order. The most important tool there used is interacting Fock space (IFS) (see and references therein for more details). More recently in the authors gave another proof of such result based on the realization that the convolution arising from addition of field operators in one-mode type IFS is the universal one of Accardi-Bozejko [l].In this note we give a new proof of such a theorem for symmetric measures, where, even in the framework of IFS, a new approach is privileged. Namely, after the introduction of a creation-annihilation process on a suitable IFS, we prove that the central limits of its even moments satisfy a system of equations whose unique solution is given by the (even) moments of the measure. It is worth to mention that, in [5] a similar central limit result is obtained. That result has been successively generalized in [8]. But between that result and our central limit theorem proved in the present paper, there

zyx

163

164

zyxwvu zyxw

are two differences: i) the random variables considered here satisfy only the singleton condition and the uniform boundedness of the mixed moments (see [3]) and do not satisfy the weak independence used in [5]; ii) we explicitly realize both the approximating random variables and their limits as sums of creation and annihilation operators in suitable interacting Fock spaces.

2. Interacting Fock spaces

zyxwvu zyx zy

In this section we define interacting Fock spaces and give some properties about them that will be used in the following results.

Definition 2.1. Let ( X , X , p ) be a measure space and let family of functions with the following properties:

be a

(i) for any n E N, A, : ( X n , X n ) -+ R+ is bounded, positive, measurable; (ii) for any measurable function F, : (Xn, Xn)4 C if

zyxw

then for any measurable function f : ( X ,X)-+ C,

J If (.>I2

IF,

2

(5%.

. .,z1)1

An+l(z,zn, . . . , 5 1 ) p ( d z ) p (dz,)

. . . p (dz1) = 0

We define, for each n E N, the (not necessarily finite) measure p, on X n by

and the associated L2-space:

H,

:= L2 (Rn,p,),V n 2

2

with pre-scalar product such that for any F,, G, E H ,

By taking the quotient and completing H , becomes an Hilbert space and with the convention that

Ho

:= C,

H := Hi

zyxwvu zyxwvut z zyxwvu z zyxwvu 165

The space

00

F(H,{A,},) : = @ H , ,

@::=1@0@0@*..

(1)

n=O

is called the (standard) interacting Fock space with weight functions {A,},='. 00 In particular if the are constant, then the corresponding space r ( H ,{A}), is called a 1-mode type free interacting Fock space (1MT-IFS in short).

zyxwv

Definition 2.2. On the Interacting Free Fock space r ( H ,{A,},), f E H , for any n E N and for any F, E H , the creation operator

(A+ (f)Fn) (xn+1,xn,.. *

,21):= f (xn+1). F n (2n.r. . .

for any

21)

is well defined as a linear operator A + ( f ) : H, -+ H,+l and has an adjoint A ( f ) : H, HH,-1 (on an appropriate domain) called the annihilation operator

A ( f ) := ( A + ( f ) ) * Remark. The condition ii) in Definition 2.1 guarantees that the creation operator is well defined. The @-statistics of the operator stochastic process { A (f),A+ (9) : f , g E H } is coded into the mixed moments

zyxwvu

(A"(")(f,) . . . where (a)

:= (@, .@)

1

(f2)

A"(') (f~))

(2)

( A (f)A+ (9))= ( f , 9 > H ,

Remark. The following simple results are easy consequences of the definition of IFS. For any n E N and E = ( E (1), . . . ,E ( n ) )E (-1, l}, 0

if among { A"(,) (f,) ,. .. ,A"(1)( f l ) } there are same number of annihilators and creators, then

A"(,) (f,) . * A"(') ( f 1 ) @ = C@

(3)

166

zyxwv zyxwv zyx zyx zyx

if among { A E ( l()f i ) , . . . , than creators, then

(fn)}

~ d n(fn) ) . . . A'(')

there are more annihilators

( f i ) Q, = O

if n is odd

(Aacn)(fn). .

2N

(f2)

(fl)) = 0

if C c ( k )# N k=l

more generally, if there is i = 1 , 2 , . . , 2 N such that among { AE(i-1) (fi-1) , * * , (fi)} there are more annihilators than creators, then

zyxw zyxwv zyxwv zyxw

By the above remarks, one knows that in order to calculate the mixed moments (2) it is sufficient to consider only the even mixed moments (f2N)*.

*

(f2)

(fl))

and only for those E E {-l,l}:N , i.e. E E {-l,l}2Nand such that the number of creators is equal to the number of annihilators, i.e. x k2=Nl & ( k ) = N and such that, counting from right to left, at each step the number of creators is larger than the number of annihilators, i.e. a

E

(k) 2 0, for all i = 1,2,. ,2N

k=l

The simplest class of standard IFS is that for which the functions (An)n in Definition 2.1 are constants. The condition (ii) of Definition 2.1 becomes in this case: c R+ and A, = 0 + A n + l = 0 Vn. For them the moments of the field operator A (f)+ A + (f)depend only on the L2-norm of the test function f , as the following result state.

Proposition 2.1. For any f , g E H the moments o f A ( f )+ A + ( f )and of A(g) + A+(g) are the same if and only zf llfll = 11g[[.

zyxwvut zyxwvu zy zyxwvu zyxw zyxwv zyxwvu zyx zyxw 167

Proof. See [2].

Remark. There are many interacting Fock spaces in which the distribution of A ( f ) A + ( f ) depends not only on llfll but also on f itself. For example, if we take H := L2 ([0,1]) and A, ( z n ,... ,z1) := 5 2 2 ; .z;-' for any n, then the distribution of A(x[o,q) A + ( x [ ~ ,is~ the I ) arcsine law but A ( f i ~ [ o , ~ / A~ +] )( f i ~ [ ~ ,has ~ / a~different ]) distribution. For more example see [4,71 references within.

+

1

+

+

3. Central limit theorem for symmetric measures

In this section we present our main result. The notations and definitions are the same as in [2], but in this case we deal only with symmetric measures. We consider the following operators on the 1-MT-IFS (L2 (R+) { A n I L ) : 1

Ak := A(X[k,k+l)); A: := A+(X[k,k+l)), k = 071,. . .

(4)

and denote

w1

:=A1

;

wn :=

A ;V n L 1 An- 1

We give the following technical lemmata, whose proof can be obtained similarly as in [2].

Lemma 3.1. O n 1-MT-IFS r (L2 (It+), { A n } r = 1 ) , for any N E M, for 2N any E E {-1,1}+ and for any { f l , . - . ,f 2 ~ )C H , with the convenience that A0 := 1

where, E

{ l k , rk}:=1

E {-1, l}"," .

as the unique non-crossing pair partition determined by

Remark For details on non-crossing pair partition, see '. Lemma 3.2. The family { A k , with respect to the state (@,.@).

satisfies the singleton condition

zyxwvu zyxwvu zyxwvut zyx zyxw

168

Remark For details on the singleton condition, see [3].

Lem ma 3.3. (Uniform boundedness of the mixed moments). For any N E N , f o r any {kl,.. . ,k,} c N and f o r any E E {-1,1}+2N

I

I(

with the conventions:

A0

. . .AfL!”) 5 [A (N)]2N

:= 1

(7)

and, f o r any m E N :

Theorem 3.1. Let be given

(W,t3) with moments of any order and with the sequences of Jacobi coeficients given by {wn}, ; A : the *-algebra generated by {Ak, A:}:, where these operators are defined over the 1-MT-IFS space r (L2 (W+), {A,}). T h e relationships among the space, operators and coeficients {w,,A,}, are given as in [l] and in particular for any n w, = &.

p : a mean-zero symmetric probability measure o n

Then

zyx

zy zyxwvut zyxwvu

k = 0 , 1 , 2 . . . , the distribution of Ak + A: with respect t o the state (@, 4) is exactly the measure p; ii) for any k = 0 , 1 , 2 . . - , a) for any

where Qk is defined in (5). Proof. The point i) follows from Proposition 2.1 and [l].We turn to prove ii). We have to compute the limit for N + 0;) of E

(( 5 flk=l

for any m E N. In fact

Qk)

m, I

zyxwvut zyxwv z 169

If m is odd, then the vacuum state above is equal to zero. On the other hand, the measure p is symmetric, so all the odd moments vanish. Hence in this case, condition ii) is satisfied and we turn to the case in which m = 2n. (8) is equal to

zy zyxwv zyxw zyxw z

jF'rom Lemma 3.2 and Lemma 3.3 we know that the sequence{Aj, AT};, satisfies the singleton condition and the boundedness of the mixed moments. As a consequence, from 3 , Lemma 2.4, it follows that (8) does not vanish only if it is equal to 1 kz, *'* ki (9) N"

c

c

(

k : { 1 , 2,...,2n}-r{l, ...,N } ~ ~ { 0 , 1 } ~ "

I R a n d k ) I=P Ik-'(k(j))l=2 V j = l ,

)

+

...2n

where 1.1 denotes the cardinality. Each map k defined as in the above summation induces a pair partition of the set { 1,2, . . . ,2n} and we will use the following notation k-l(k(j))=:{lj,rj},

lj > r j

j = l ,..., n

We recall that in the interacting Fock space structure only the non crossing pair partitions may give a non-zero contribution in the computations of vacuum expectations (see [4]) and from [4], Lemma 6, Section 22, any fixed E E (0, l}? uniquely determines one non-crossing pair partition { l j , rj}y=l on { 1 , 2 , . . . ,2n} and viceversa. As a consequence (9) becomes

where N.C.P.P. {2n} denotes the set of all non-crossing pair partitions of { 1 , 2 , . . . ,2n} 2nd &k is the element of the set {0,1}? which is uniquely determined by the map k inducing the pair partition { l j , r j } y = l . We denote by

170

zyx zyxwz

and we prove the following relation by induction on n.

where c ( w ,m) is a function of w depending on m. In fact the case n = 1 is trivial. We suppose the assumption is true for any natural integer h 5 n - 1 and prove it for h = n . If we denote by ( { l j , ~ - j } y = ~ ) the subset of { Z j , ~ - j } y =such ~ that km is the position of the m

creator coupled with the first annihilator from the left (i. e. m = rn), then

zyxwv zyxwvuts c zyxwv

{ Zj,

E N.C. P. P. { m - 1 }

~ j

(

km-i

. . . A;:')))

2n

zyx zyxwvutsr c 2n

=

m=l

mEZN+l

where in the last equality we used the property up = 1. We now see that the limit for N -+ 00 of u? exists. In fact

uf = 1 NZpO 1.

zyxwv zyxw zyxw zyx zyxw zy zy 171

Let us suppose that for any m < n lim uz does exist. Then N-tW

2n

It follows that the limit on the left hand side exists. Let us denote V m := lim uz for any m = 0,1,2,. . . (notice that uf = 1 for any N ) . The N+W

o the system of discussion above implies that the sequence { v m } ~ =satisfies equations 2n

j R o m another hand, if we consider Um

= /x2mdp,

Vm = 0,1,2,. . .

from i) it follows that p is the distribution of namely urn=

1

(Ak

+ A;)

for any k E N,

(

z 2 m d p = (AkA;)2m)

zyxwvu

By using this fact, it is easy to prove that 2n

m=l

mE2N+l

The system (10) has a unique solution, then u,

References

= un

for any n, i.e.

z

1. L. Accardi, M. Bozejko: Interacting Fock Spaces and gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1,no. 4, pp. 663470 (1998). 2. L. Accardi, V. Crismale, Y.G. Lu: Constructive universal central limit theorems based o n interacting Fock spaces, to appear in Infin. Dimens. Anal. Quantum Probab. Relat. Top., preprint Volterra n. 591 (2005).

172

zyxwv zy zyxwvu

3. L. Accardi, Y. Hashimoto, N. Obata: Notions of independence related t o the free group, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, no. 2, pp. 201-220, (1998). 4. L. Accardi, Y.G. Lu, 1.Volovich: T h e &ED Hilbert module and interacting Fock spaces, International Institute for Advances Studies, Kyoto (1997). 5. T. Cabanal-Duvillard, V. Ionescu: U n the'ordme central limite pour des variables ale'atoires non-commutatives. ProbabilitBslProbability Theory, C.R.Acad. Sci. Paris, t. 325, SQrie 1, pp. 1117-1120 (1997) 6. A. Krystek, L. Wojakowski: Convolution and central limit theorem arising f r o m addition of field operators in one-mode type Interacting Fock spaces", Preprint (2005). 7. Y.G. Lu: O n the interacting free Fock space and the deformed Wigner law, J. Nagoya Math., 145,pp.1-28, (1997). 8. Mlotkowski W.: Fkee probability o n algebras with infinitely m a n y states, Probab. Theory Related Fields 115, no. 4,pp. 579-596, (1999).

ON THE MOST EFFICIENT UNITARY TRANSFORMATION FOR PROGRAMMING QUANTUM CHANNELS*

zyxwvut zyxw

GIACOMO MAURO D’ARIANO~ QUIT group, INFM-CNR, Dipartimento d i Fisica “A. Volta”, via Bassi 6, 27100 Pavia, Italy

PAOLO PERINOTTI~ QUIT group, INFM-CNR, Dipartimento di Fisica “A. Volta”, via Bassi 6, 271 00 Pavia, Italys

zyxwvuts zyxw

We address the problem of finding the optimal joint unitary transformation on system + ancilla which is the most efficient in programming any desired channel on the system by changing the state of the ancilla. We present a solution to the problem for dim(H) = 2 for both system and ancilla.

Keywords: Quantum information theory; channels; quantum computing; entanglement

1. Introduction

A fundamental problem in quantum computing and, more generally, in quantum information processing [l]is to experimentally achieve any theoretically designed quantum channel with a fixed device, being able to program the channel on the state of an ancilla. This problem is of relevance for example in proving the equivalence of cryptographic protocols, e. g. proving the equivalence between a multi-round and a single-round quantum *This work has been co-founded by the EC under the program ATESIT (contract no. ist-2000-29681), and the MIUR cofinanzzamento2003. tWork partially supported by the muri program administered by the U.S. Army Research Office under grant no. DAAD19-00-1-0177 Work partially supported by INFM under project PRA-2002-CLON. §Part of the work has been carried out at the Max Planck Institute for the Physics of Complex Systems in Dresden during the International School of Quantum Information, September 2005.

173

174

zyxwv

bit commitment [2]. What makes the problem of channel programmability non trivial is that exact universal programmability of channels is impossible, as a consequence of a no-go theorem for programmability of unitary transformations by Nielsen and Chuang [3]. A similar situation occurs for universal programmability of POVM's [4, 51. It is still possible to achieve programmability probabilistically [6], or even deterministically [7], though within some accuracy. Then, for the deterministic case, the problem is to determine the most efficient programmability, namely the optimal dimension of the program-ancilla for given accuracy. Recently, it has been shown [5] that a dimension increasing polynomially with precision is possible: however, even though this is a dramatical improvement compared to preliminary indications of an exponential grow [8], still it is not optimal. In establishing the theoretical limits to statcprogrammability of channels and POVM's the starting problem is to find the joint system-ancilla unitary which achieves the best accuracy for fixed dimension of the ancilla: this is exactly the problem that is addressed in the present paper. The problem turned out to be hard, even for low dimension, and here we will give a solution for the qubit case, for both system and ancilla. 2. Statement of the problem

zyx zy zyx zyxw z zyxw z

We want to program the channel by a fixed device as follows

Pv,u(p) = TT2[V(Pc3 .)V+],

(1)

with the system in the state p interacting with an ancilla in the state u via the unitary operator V of the programmable device (the state of the ancilla is the program). For fixed V the above map can be regarded as a linear map from the convex set of the ancilla states d to the convex set of channels for the system V. We will denote by 9v,& the image of the ancilla states B under such linear map: these are the programmable channels. According to the well known no-go theorem by Nielsen and Chuang it is impossible to program all unitary channels on the system with a single V and a finitedimensional ancilla, namely the image convex 9v,d c %?' is a proper subset of the whole convex '& of channels. This opens the following problem:

zyx

Problem: For given dimension of the ancilla, find the unitary operators V that are the most eflcient in programming channels, namely which minimize the largest distance E ( V )of each channel C E '& from the programmable set 9 v , d :

E ( V ) max min 6(C, P ) 3 max min 6(C, Pv,,,). (2) C€V P€Pv,, CEV U E d

z zyxwvu zyx zyxwvu zyxwvu zyxwvu zyxw zyx z 175

zyx

As a definition of distance it would be most appropriate to use the CBnorm distance IIC - P l l c ~ However, . this leads to a very hard problem. We will use instead the following distance

where F ( C , P ) denotes the Raginsky fidelity [9], which for unitary map C = U = U . Ut is equivalent to the channel fidelity [l]

xi

where C = Ci . C!. Such fidelity is also related to the input-output fidelity averaged over all pure states Fio(L4,P),by the formulaFio(U,P)= [l+dF(U, P)]/(d+ 1). Therefore, our optimal unitary V will maximize the fidelity

F ( V ) A UEU(H) min F(U,V ) , F ( U , V ) = maxF(U,Pv,u) U € d

(5)

3. Reducing the problem to an operator norm

In the following we will use the GNS representation I@)) = (Q €3 1)lI))of operators 9 E B(H), and denote by XT the transposed with respect to the cyclic vector II)), i. e. IQ)) = (Q €3 I ) l I ) ) = ( I €3 QT)lI)),and by X* the complex conjugated operator X* = (XT)t, and write lo*) for the vector such that (Iu)(uI€3 I) I I ) ) = Iv)lu*).Upon spectralizing the unitary V 51s follows

v=

c

eiekI Q k ) ) ( ( Q k 11

(6)

k

we obtain the Kraus operators for the map P V , ~ ( ~ ) k

nm

where Iun) denotes the eigenvector of (T corresponding to the eigenvalue A,. We then obtain

xI nm

ei(ek-eh) Tr[QiUtQk(TTQiUQh]

n[cAmU]12= kh

= Tr[aTS(U,V ) t S ( U V , )]

where

(8)

176

The fidelity (5) can then be rewritten as follows

zyxwvu zyxw zyx zyxwvu zyxwv zyxw z

4. Solution for the qubit case

The operator S(U,V) in Eq. (9) can be written as follows

S(U,V) = Tr,[(UT€3 I)V*].

(11)

Changing V by local unitary operators transforms S ( U , V) in the following fashion

S(u7 (Wl €3 W2)v(W3 €3 W4))= W,*s(W!uW$,v)W,*,

(12)

namely the local unitaries do not change the minimum fidelity, since the unitaries on the ancilla just imply a different program state, whereas the unitaries on the system just imply that the minimum fidelity is achieved for a different unitary-say WiUWJ instead of U . For system and ancilla both two-dimensional, one can parameterize all possible joint unitary operators as follows [lo]

V = (W1€3W2)exp [i(a1CTI€

3~1

+a2g2

@.a2

+a 3 0 3 €303 ')I

(W3€3 W4). (13)

A possible quantum circuit to achieve V in Eq. (13) can be designed using the identities [ga€3 ga7

go €3 go] = 0,

C(CZ €3 I ) C = CTZ €3 oz,

C ( I €3 gz)C= -c% (e-%cz

(14)

€3 g z ,

€3 e - -irr~ a z ) c(gZ€3 I>C( e % ~ z g % ~ = z )gg €3 gy,

where C denotes the controlled-NOT

c = lO)(Ol @ I + 11)(11 €3

gz.

(15)

This gives the quantum circuit in Fig. 1. The problem is now reduced to study only joint unitary operators of the form

+ (1202 €3 mzT + ~ 3 ~ €37 as')]. 3

V = exp[(i(alal@uiT This has eigenvectors

(16)

177

zyxwvu zyxw zyxw zyx zyx zy

Figure 1. Quantum circuit scheme for the general joint unitary operator V in Eq. (13). Here we use the notation Go = exp(i4uG) with G = X , Y,Z.

where oj,j = 0 , 1 , 2 , 3 denote the Pauli matrices oo = I , o1 = o,, o2 = oy, 0 3 = oz.This means that we can rewrite S(U,V ) in Eq. (9) as follows 1 2

S(U,V )= -

3

C

epiej aj

j=O

with

eo = a1+ a2+ a 3 ,

uoj ,

zyx

ei = 2ai - eo .

(19)

The unitary U belongs to SU(2), and can be written in the Bloch form

U = noI with

nk

E R and n8

+ in

.CT

+ 1nl2= 1. Using the identity

we can rewrite

S ( U , V )=fioI+fi.., where

+

t3, --e-ieo

,-is

to, 1 5 j ~ 3 ,

tj=Itjlei@j,0

~ j 5 3 ,

zyx

It is now easy to evaluate the operator S ( U ,V)tS(Ul V ) . One has

S(U,V ) + S ( UV, )= v o l +

2).

6,

+ n* x f i ] . (24) Now, the maximum eigenvalue of S(U,V ) t S ( U ,V ) is vo + Iwl, and one has vo

=lfiiOl2

+

lfL12,

3 1VI2

i,j=O

= i [2S(fiOfi*)

3

1fiiI2)fijl2 - fiir26,:

=

2)

=2

C lfii121fij12sin2(4ii,j=O

$j),

(25)

zyx

zyxwvu zyxw zyxw 4 zyxwvu zyxw zyxwv zy

178

whence the norm of S ( U ,V ) is given by 3

IIS(~,V)1I2=

C+jI2+

j=O

2

&

n:npIti121tj12 sin2(q5i - q5j) .

(26)

i,j=O

Notice that the unitary U which is programmed with minimum fidelity in general will not not be unique, since the expression for the fidelity depends on {n;}. Notice also that using the decomposition in Eq. (13) the minimum fidelity just depends on the phases { O j } , and the local unitaries will appear only in the definitions of the optimal program state and of the worstly approximated unitary. It is convenient to write Eq. (26) as follows

IlS(U,V)1I2= u . t + &.i-%. (27)

where u = (nz,n;,ng,ni), t = (l t 012,1t 112, 1t212, 1t312), and Tij = ltiI21tjl2sin2(q5i- q5j). One has the bounds u . t -t

JUTU 2 u . t 2 min I t j 12, 3

(28)

and the bound is achieved on one of the for extremal points u1 = Slj of the domain of u which is the convex set { u , uj 2 0, Cjuj = 1) (the positive octant of the unit four dimensional ball S:). Therefore, the fidelity minimized over all unitaries is given by

1 F ( V ) = - min l t j I 2 d2 3 The optimal unitary V is now obtained by maximizing F ( V ) . We need then to consider the decomposition Eq. (13), and then to maximize the minimum among the four eigenvalues of S(U,V)tS(U,V ) . Notice that t j = C , Hj,eiep, where H is the Hadamard matrix

H=f-"I 1 1

2

1

1

1-1 1 - 1 1-1-1 1

'

zy

zyx

which is unitary, and consequently Cj l t j I 2 = Cj leiej l2 = 4. This implies that minj l t j l 5 1. We now provide a choice of phases 8, such that l t j l = 1 for all j , achieving the maximum fidelity allowed. For instance, we can take 80 = 0,81 = 7r/2,82 = 7 r , 83 = 7r/2, corresponding to the eigenvalues i, 1,-2, 1 for V. Another solution is B0 = 0 , O1 = -7r/2, Q2 = 7r, e3 = -7r/2.

zyxwv zyxw z zyx 179

Also one can set Oi -+ -&. The eigenvalues of S(U,V)tS(U, V ) are then 1 , 1 , 1 , 1 ,while for the fidelity we have

and the corresponding optimal V has the form

(a, B a, f az 8 az)].

(32)

A possible circuit scheme for the optimal V is given in Fig. 2.

zyxw z zyxw I

I

Figure 2. Quantum circuit scheme for the optimal unitary operator V in Eq. (31). For the notation see Fig. 1. For the derivation of the circuit see Eqs. (14).

zyxwvuts

We now show that such fidelity cannot be achieved by any V of the controlled-unitary form 2

v = C v k B l$k)($kl, k=l

(+117/~2) = 0,

K , ~2 unitary on

H = c2.(33)

2 &) (k) (k) )(+j I the eigenvectors For spectral decomposition v k = &=l e k of v are I!ijjk)) = (k))I$k), and the corresponding operators are !ijjk =

\+:))($:I,

namely the operator S(U,V)is

C e+@ I $ ~ ) ( + ~ ) I U, ~ + ~ ) ) (34) (~~I with singular values xi=,e-iey)(q5y)lUl+~)) = Tr[ViU]. Then, the opS(U, V )=

j,k

timal program state is sponding fidelity is

l$h),

with h = argmaxk I Tr[ViU]I, and the corre-

F(U,V)= -ITr[V,tU]12,

1 4

(35)

F(V)= minF(U, V )= 0,

(36)

and one has U

zyxw zy zyxwvu zy zyxwvu

180

u

since for any couple of unitaries v k there always exists a unitary such that Tr[V,U] = 0 for k = 1,2. Indeed, writing the unitaries in the Bloch form (20), their Hilbert-Schmidt scalar is equal to the euclidean scalar product in R4 of their corresponding vectors, whence it is always possible to find a vector orthogonal to any given couple in R4.The corresponding U is then orthogonal to both v k , and the minimum fidelity for any controlled-unitary is zero.

References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University press, Cambridge, 2000). 2. G. M. D’Ariano, D. Kretschmann, D. Schlingeman, R. F. Werner, unpub-

lished 3. M. A. Nielsen and I. L. Chuang, Programmable Quantum Gate Arrays, Phys. Rev. Lett. 79,321 (1997). 4. 3. Fiurbek and M. DuSek, Probabilistic quantum multimeters, Phys. Rev. A 69 032302 (2004). 5. G.M. D’Ariano, and P. Perinotti, Eficient Universal Programmable Quantum Measurements, Phys. Rev. Lett. 94 090401 (2005). 6. M.Hillery, V. Buiek, and M. Ziman, Probabilistic implementation of universal quantum processors, Phys. Rev. A 65 022301 (2002). 7. G.Vidal and J. I. Cirac, Storage of quantum dynamics on quantum states: a

zyxwv

quasi-perfect programmable quantum gate, quant-ph/0012067. 8. J. FiurGek, M. DuSek, and R. Filip, Universal Measurement Apparatus Controlled by Quantum Software, Phys. Rev. Lett. 89 190401 (2002). 9. M. Raginsky, A fidelity measure for quantum channels , Phys. Lett. A 290 11 (2001). 10. B. Kraus and I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A63 062309 (2001).

zyxwv zyxwvu zyxw

STABILITY ANALYSIS OF QUANTUM MECHANICAL FEEDBACK CONTROL SYSTEM

P.K. DAS

Physics and Applied Mathematics Unit Indian Statistical Institute 203, B. T.Road, Kolkata-700108 e-mail:daspk@isical. ac. in

B.C. ROY The Institute of Radio Physics and Electronics Science College, Calcutta University 92, A . P. C. Road, Kolkata - 700009 e-mail:[email protected]

zyxwvut

In this paper we derive the state equation of the optical cavity in interacting Fock space as well as in boson Fock space. We design closed-loop feedback control system of a composite cavity QED in boson Fock space using beam splitter device and prove with the help of Nyquist stability criterion that the system is stable. The other physical characteristics, such as, phase margin and gain margin of the closed-loop feedback control system are also discussed.

1. Introduction The concepts and tools of control theory help us to understand the dynamics of complex networks. Extension of control theory to the quantum domain enables us to design complex quantum systems in a systematic way. A feedback control system is a system that maintains a relationship between the output and the input and their difference is a controller. The primary objective of the feedback control system is the elimination or reduction of error of the system output. The primary difference of quantum systems from the classical systems is that the input, output and state variables of the quantum system are operators rather than scalars acting on Hilbert spaces. The concept of interaction of single mode of quantized field in a cavity with a noisy external field has been utilized in this paper for finding the 181

182

state space model in interacting Fock space. The dynamics of the cavity in the interacting Fock space are obtained by utilizing the basic concepts of quantum stochastic process. The transfer function modelling of the closedloop system of a composite system of cavities utilizing a device of accepting two inputs and emitting two outputs of the beam splitter is used to design closed-loop feedback quantum control system. The state space modelling of quantum feedback control system in interacting Fock space is shown to be a generalization of description of quantum feedback control system in usual boson Fock space [l,21. The Nyquist stability analysis of the system in boson Fock space by using beam splitter has been discussed by constructing Nyquist plot of the transfer function of a composite system with a second cavity in the feedback path of the closed-loop control system. The Nyquist stability criterion along with the gain margin and phase margin of the composite cavity QED system is expressed in terms of the parameters of reflectivity and transmissivity of the beam splitter. The paper is organized as follows. In section 2, we discuss some basic facts which will be needed in the paper. In section 3, we model single QED system in interacting Fock space from which the state space modelling in bosonic mode can be derived easily as discussed there. In the remaining sections we are concerned in designing and analyzing the stability of feedback controlled cavity QED system in boson Fock space. In section 4,we design a single cavity QED feedback system with a second cavity in the feedback path using beam splitter. In section 5, we discuss in details the Nyquist stability analysis of quantum feedback system for composite QED system with a second cavity in the feedback loop. And finally in section 6 , we give a conclusion.

zyx z z

2. Preliminaries and Notations

In this section we discuss some basic preliminaries on interacting Fock space, interaction of optical cavity QED with the external field and the quantum stochastic process which will be needed throughout the paper.

2.1. Interacting Fock Space

As a vector space one mode interacting Fock space r(C) [3] is defined by

zyxwvu zyxwvut zyxwv z zyxwv 183

for any n E AT, where @In > is called the n-particle subspace. The norm of the vector In > is given by

< nln >= A,

(2)

where {A,} 2 0 and if for some n we have {A,} = 0, then A{}, = 0 for all m 2 n. The norm introduced in (2) makes r(@) a Hilbert space. We consider the following actions on I?(@) :

where A* is called the creation operator and its adjoint A is called the annihilation operator. In defining the annihilation operator we have taken the convention 010 = 0. The commutation relation takes the form [A,A*]= -- AN+1

AN

AN

AN-1

zy (4)

where N is the number operator defined by N J n>= n ) n >. In a recent paper [9]we have proved that the set { n = 0 , 1 , 2 , 3 , .. .} forms a complete orthonormal set and the solution of the following eigenvalue equation

5,

zyx zyxw Afa = afa

is given by

where

(5)

y.

= C,"==,

We call fa a coherent vector in I?(@).

2 . 2 . Interaction of Cavity and the External Field

We consider the interaction of an interacting single-mode of quantized field confined in an optical cavity with a noisy external field. Let 7 - i ~and X B be Hilbert spaces of the cavity and the external field respectively. The . composite system is expressed by the tensor product space 7 - i ~@ 7 - i ~ The total Hamiltonian is given by

Htotal = H A 8 IB

+ IA 8 H B + Hint

(7) where H A describes the Hamiltonian of the cavity mode. This Hamiltonian may be further decomposed into two parts

H A = Hca,

+H.

(8)

184

zyxwvut zyxwv zyx zyxwv zyx zyxwvu zyxw

Here H is the residual Hamiltonian determined by the optical medium in the cavity, referred to as a free Humiltoniun. HB is the Hamiltonian of the external field. The interaction Hamiltonian Hint consists of four terms. We drop the energy non conserving terms corresponding to the rotating-wave approximation and obtain the simplified Hamiltonian as

Hint(t) = i&[U(t)b+(t)

- U+(t)b(t)l

(9)

with

[b(t),b+(t’)] = q t - t’)

(10)

and y is a coupling constant. Here u is the annihilation operator of the cavity and b is the annihilation operator of the external field. The operator b ( t ) is a driving field at time t and we interpret the parameter t to mean the time at which the initial incoming field will interact with the system and not that b ( t ) is a time-dependent operator at time t.

2.3. Quantum Stochastic Process In order to describe quantum stochastic process we define first an operator

lo t

Bi,(t,tO) =

bin(s)ds

(11)

where bin@) satisfies the commutation relation (10).The operator bi,(t) represents the field immediately before it interacts with the system and we regard it as an input to the system. Now, from (11)we get

Then we write down the increments

dBin(t) = Bi,(t

+ d t ) - Bi,(t),

dBk(t)= B,?-,(t+ d t ) - B z ( t )

(13)

From (12) and (13) we get

[dBi,(t),dB&(t)]= dt.

(14)

This leads to the natural definition of quantum stochastic process as

+

dBi,(t)dBi+,(t) = ( N ’ 1)dt dBk(t)dBi,(t) = N’dt dBZ,(t)dBi,(t) = Mdt d B L ( t ) d B z ( t )= M*dt

(15)

zyx zyx zyxw zyxw 185

and all other products higher than the second order in dBi, are equal to zero. N’ and M are real and complex numbers satisfying

Nt(N’

+ 1 ) 2 [M12.

(16)

The evolution of an arbitrary operator X is given by

X ( t )= U+(t)XU(t)

(17)

in which the unitary operator U ( t ) is generated by the Hamiltonians ( 7 ) and (8). H,,, and HE drive the cavity and the external field respectively. We shall assume here H to be zero. The unitary operator of the system is then given by

U ( & ) = e&(adBL-a+dBin)

(18)

U+(dt) = eJ;/(a+dBin-adBk)

(19)

Also we have

The increment of an arbitrary operator r of the system driven by the stochastic input bin is given by

dr(t) = r(t

+ d t ) - r ( t )= U+(dt)r(t)U(dt)- r ( t )

Now

U+(dt)r(t)U(dt)

zyxw (20)

- e J ; i ( a + d B i n - - a d B ~ ) r ( t ) e ~ ( a d B -a+dBim) L =

r ( t )+ f i [ U + d B i , - a d B 2 , r(t)]+ +Z{(N’ + 1)(2a+ra - a+ar - ?-.+a) +”(2ara+ - aa+r - Tau+) +M(a+(a+r - ?-a+)- (a+r - ra+)a+) +M*(a(ar - ra) - (ar - ra)a)}dt

Hence we get

3. Modelling of Single QED System

To describe the state space model of open loop quantum system we must describe the state equation of the system along with the input-output relation of the system.

186

zyxwvut zyxwvut zyxw zyxw

zyxwv zyxwvu zyxw

3.1. State Equation of the Cavity

To describe the dynamics of the operator a ( t ) in the open quantum system we replace T in (22) by a to get da = ~

+

( td t ) - ~ ( t ) = f i [ a + d B i , - a d B 2 , a]+ +${(" 1)(2a+aa - a+aa - aa+a)+ +"(2aaa+ - aa+a - aaa+)+

+

+

+M[a+,[a+,a]] M*[a,[a,a]]}dt = +(hAN kAN-1 )bz,(t)dt :{-(" l)(* - &)a +"a( - &)}dt = { - 2 ( h - &)a - f i ( h- k ) b i n ( t ) } d t

*

2

AN

+

+

AN

AN-1

(23)

AN-1

This implies

The equation (24) defines the state equation of the single optical cavity QED in the interacting Fock space. The dynamics of the cavity in the bosonic mode can be obtained from (24) by using the operation Nln >= nln >. The state equation of the cavity then reduces to U(t)

Y

= --a(t) - &bzn(t)

2 which is the usual quantum Langevin equation.

(25)

z

3.2. Input-Output Relation of the Open loop System

Due to the interaction of the evolving incoming field with the cavity an outgoing field is produced. To describe this we need to define an operator

lo t

Bout(t,to) =

bout(s)ds

(26)

where

bout(t) = U+(dt)bi,(t)U(dt)

(27)

zyxw

The input-output relation after the interaction at time t is given simply by the following derivation. We have dBout( t )= (dt)dBi, ( t )U ( d t ) - , ~ ( a d ~ ~ - - a + d ~ ; , ) + d ~ ~ , ( t ) , ~ ( a d ~ ~ - a (28) + d ~

u+

+ f i U [ d B i , , dB&]

= dBin(t)

. , )

187

z

zyx zyx

zyxwvu zyx zyxw zyxwv zy G

Figure 1. The configuration of input-output relation of single cavity QED in bosonic mode.

Now using (14) the above relation gives us

+fiudt

(29)

fim + bin (t).

(30)

bout(t)dt = bi,(t)dt

and hence we have the required relation bout ( t )=

This gives the input-output relation of the single cavity QED system. The fig.1 shows the input-output configuration of a single cavity QED through the system operators which defines the open-loop single cavity QED system. 3.3. Transfer Function of the Open-Loop Quantum System

We have seen that the cavity dynamics may be thought of as a single input and a single output(SIS0) system. The equations (24) and (25) give the state equation of a single cavity in different modes. The operator bi,(t) is the input and the operator bOut(t)is the output of the cavity. The state equation of the cavity dynamics along with the output equation can be rewritten as

u(t)= A'a(t)+ B'bi,(t) bout(t) = C'a(t) D'bi,(t)

+

where

(31)

188

zyxwvu zyxwvu zyxw zyx zyxw z zyx

When A h = 1 we get the bosonic mode of the cavity system as given by (25). The dynamics of the cavity in interacting Fock space is a first order differential equation with operators as coefficients whereas in bosonic mode the equation is of first order with constant coefficient. Taking Laplace transform of equations (31) and assuming zero initial state the QED system can be represented by the gain G(s) of the cavity from the input to the output as bout(s) = G(s)bin(s)

(33)

where the gain G(s) of the cavity in the interacting Fock space is given by G ( s )= -YA;(SI

+ -A:)-' Y +1 2

(34)

In bosonic mode, the gain of the single cavity is simply described as

As the cavity dynamics in boson Fock space is linear with constant coefficient, one can successfully apply the Nyquist stability criterion for analyzing feedback controlled problems of the system in boson Fock space. Henceforth, we shall mainly concentrate our discussion in bosonic mode to the problem of modelling optical cavity QED feedback control system and shall study their stability. 4. Mathematical Model of the Feedback Control of the

Cavity QED Using Beam Splitter

A beam splitter is an optical device, a partially silvered piece of glass, which allows us to perform manipulations of two input signals and two output signals of a feedback control system. The input field bin(t) is sent to one port of the beam splitter which is chosen to have reflectivity a and transmissivity p, and the feedback operator with negative sign evolve from the feedback path is sent to the other port of the beam splitter. In this way the beam splitter may be used to split the input fields into two operators of which one may be taken to define the feedback controller of the feedback system. We now utilize the concept of beam splitter to design the quantum feedback control system. The input-output configuration of the composite QED system is shown in fig.2. The input signals bin and bz to the beam splitter are related to the

zyxwvut zyxwvu zyxwvu zyxwvu zyx zyx 189

outputs bo and

where cy and

b3

by

p are real and satisfy a2+ p2 = 1. Hence we get b3 = a b i n

+ pb2

bo = Pbin - ab?.

From the input-output relation of each cavity, we get

bi = f

b2 = f

i a A

+ bo +

i a ~ bi

Each signal in the feedback loop can now be written as bo = &bin bl= A b i b2 = &bin b3 = bin

n

-e ( f i a A +f i a B ) +& f i a A -e f i a B + h ( f i a A +f i a B )

(39)

+& ( f i a A +f i a B )

Figure 2. The design of the composite system in which the second cavity in the feedback path by using beam splitter.

Assuming the non-interaction of the cavity A with the cavity B , the gain H ( s ) of the bath B in the feedback path is described, as in subsection 3.3, by equation (38) as

H(s)= -YB(s

+ -)2

Y B -1

+1

(40)

Then in the case of the composite system with a second cavity in the feedback path, the closed-loop transfer function is written as

M ( s )=

+

Ws)

1 aG(s)H(s)

zyxwvu

zyxwv zyxwv

190

The stability of the closed-loop system is characterized by the zeros of 1 aG(s)H(s),or equivalently, by the roots of

+

It then follows that R e s < 0 which implies that the closed-loop system is asymptotically stable. The stability of the composite system is characterized by computing the poles of the characteristic equation of the system. So there arises a difficulty in calculating the characteristic equation. However, the stability of a composite system can be easily analyzed by considering open-loop transfer function with the help of the Nyquist stability criterion. 5. Nyquist Stability Analysis of the Quantum Feedback Control System

So far, we have concerned with the construction of transfer functions of quantum feedback control systems which provide some information about the absolute stability of a composite system with a second cavity in the feedback path. This representation experiences system gain function that provide various valuable insights into the problems such as, stability, gain margin and phase margin of the QED system. This problem of the quantum mechanical system can be solved by simply analyzing the open-loop transfer function of the system using the Nyquist criterion. The Nyquist stability analysis is a graphical method that determines the stability of closed-loop system by analyzing the property of the frequency domain plot called Nyquist plot of the open-loop transfer function L ( s ) = a G ( s ) H ( s )of the system. Specifically, the Nyquist plot of L ( s ) is a plot of L ( j w ) in polar coordinates or in Cartesian coordinates of I r n L ( j w ) versus R e L ( j w ) as w varies from 00 to 0 and from 0 to -m. This gives also the information about the relative stability of an unstable system. It also gives the indication on how the system stability may be improved (if needed) for getting a desired output. A simple Nyquist contour rS in s-plane having no pole and zero on the imaginary axis is depicted in the adjoining figures 5 . The closed Nyquist contour consists of four parts: C1, C2,C3 and C4. The section Cl is defined by s = jw,O 5 w < 00; section C2 is defined by s = j w , -00 < w I 0; section C3 is defined by s = Rej', R -+ 00, -$ 5 9 5 0, and section C4 is defined by s = Rej', R -+ 00,0 5 B 5 f.

zyxwv zyx zyxw zyxwvu

zyxw zyxwvu zyxw 191

We now discuss in some details the Nyquist stability analysis of the cavity QED system. The closed-loop transfer function M ( s ) shown in fig.2 with a gain H ( s ) in the feedback path has been described in section 4 by equation (41) in the form

M(s)=

PG(s) 1 aG(s)H(s)'

+

The transfer function relating the feedback variable bz(s) to the input bo(s) is the open-loop transfer function L ( s ) = oG(s)H(s),shown in the fig.2. As the open-loop transfer function L ( s ) is in the factor form, the Nyquist plot in L(s)-plane can be easily obtained. We express L ( s ) as

zyx zyxwv zyx zy zyxw zyxw

The open-loop transfer function L ( s ) has two poles to the left half and two zeros to the right half of the s-plane. There are no zero or pole on the imaginary axis. So the Nyquist contour, as described in fig.5, may be used to describe the characteristic properties, such as, stability, phase margin, gain margin of the closed-loop feedback control system from the Nyquist plot in L(s)-plane. To describe the Nyquist plot in L(s)-plane, we put s = j w in L ( s ) for the portions Cl, C2 of the closed-loop contour rs. Then

L(jw)=

a ( j w - ?)(jw - ?f) ( j w ?)(jw y)

+

+

(44)

Hence IL(jw)l = Q for all values of w. Expressing the function L ( j w ) as L ( j w ) = u(w)+jw(w), the Nyquist plot of the portions Cl, C2 of the Nyquist contour rs can be easily programmed and the corresponding results are shown in fig.3 and fig.4. Again, to find the Nyquist plot in L(s)-plane of the curves C3,Cq, let us put s = Reje,-.rr/2 5 9 5 ~ / in 2 (44) we then get

192

zyxwvut zyxwvutsrqponm zyxwv z zy zyxwv

where E’ =a-+ 0 as R - t M. The Nyquist plot of the portions C3, C4 in L ( s ) are shown in fig.3 and fig.4 by the small circular arcs about the centre a j 0 . These circular arcs coincide at the point a j 0 when R -+ 00. We observe that the Nyquist plot in L(s)-plane of the closed contour rs encircles the origin twice, and hence the number of encirclement in the anticlockwise direction about the origin is N = Z - P where P = 0 and Z = 2, and hence N = 2. Thus, the Nyquist criterion of the number of encirclement about the origin satisfies N = Z - P and passes through the points E = a j 0 and F = -a j 0 , where 0 < a < 1. According to the Nyquist criterion the closed-loop feedback composite QED system designed by using a beam splitter is asymptotically stable. The phase margin(PM) is zero. The gain margin(GM) is given by

+

+

+

+

zyxwv zyxwv

In the special case of 50/50 beam splitter, a = it is expressed in decibel, then

GM

20Z0glo-

1

1

lOEl

5.And so, GM = a.If

= 3.01 > 3 d B

Note that the measurements of GM and PM of a closed-loop feedback control system indicate the information about the degree of stability of the control system.

6. Conclusion

The problem of controlling a composite cavity system with a second cavity in the feedback path can be utilized to study the experimental problem of coupling a single photon with an atom within an optical cavity. The Nyquist stability analysis of a system with a feedback gain evolved by using a second cavity in the feedback path described in this paper is a general study in phase plane. The process described in this paper can be easily extended to the case of cascaded system with a finite number of optical cavities.

zyxwv zyxwvu z 193

References

1. Yanagisawa, M. and Kimura, H.: Transfer Function Approach t o Quantum Control- Part I: Dynamics of Quantum Feedback Systems, IEEE Transactions

on Automatic Control, Vol. 48, no. 12, 2107,(2003). 2. Yanagisawa, M. and Kimura, H.: Transfer Function Approach t o Quantum Control- Part 11: Control Concepts and Applications, IEEE Transactions on Automatic Control, Vol. 48, no. 12, 2121,(2003). 3. Accardi, L. and Bozejko, M.: Interacting Fock Space and Gaussianization of Probability Measures, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(4), 663,(1998). 4. Wiseman, H. M., and Milburn, G. J.: Quantum Theory of Optical Feedback via Homodyne Detection, Phy. Rev. Letts, Vol. 70, no. 5, 548, (1993). 5. Doherty, A. C., Habib, S., Jacobs, K., Mabuchi, H. and Tan, S. M.: Quantum Feedback Control and Classical Control Theory, Phy. Rev. A, Vol. 62, 012105, (2000). 6. Ogata, K.: Modern Control Engineering, Prentice-Hall of India, (2004). 7. Kuo, Benjamin C.: Automatic Control Systems, Prentice-Hall of India, (2003). 8. Gopal, M.: Control Systems, Tata McGraw-Hill, (2003) 9. Das, P. K.: Coherent states and squeezed states in interacting Fock space. International Journal of Theoretical Physics. vol. 41, no. 06, 1099-1106, (2002), MR. No. 2003e: 81091 (2003).

zy

194

zyxwvutsr zyxwvuts zyxwv

TRlO Data

TRll Data

1.5

~

0

4.5

zyxw zy 1

-1.5 -15 -1

zyxwvu zyxwvut -0.5

0.89

- (1.00

0 0.5 1.11

1

15

- 0.00

Figure 3. Nyquist plot of G ( s ) H ( s )as s traverses C4 and C1

I

Figure 4. Nyquist plot of G ( s ) H ( s ) as s traverses CZ and C3

TR9 Data

jm

-IS'..... -15 -1

'

-0.5

'

0

0.88 -ow

Figure 5.

Nyquist contour rs which is di-

and an arc

&,

C4

of infinite radius.

'

0.5

'

1

'

15

195

z

zyxwvuts zyxwvutsrq zyxwvu

5 OPEN "O", #12, "tr9.out" 10 FOR I = 0 TO 1.57 STEP .01 15 X = 1 - .125 * COS(1) 20 Y = .125 * SIN(1) 25 PRINT #12, USING " # # # . # # " ; X; 30 PRINT 112, " . 35 PRINT #12, USING " # # # . # # " ; Y 40 NEXT I 45 FOR I = 8 TO 0 STEP -.01 50 X = (I 4 - .37 * I 2 t .004) / ((I 2 t .06) (I 2 - .06)) / ((I 2 t .06) 2) 55 Y = (I 60 PRINT #12, USING " # # # . # # " ; X; 65 PRINT #12, " . 70 PRINT #12, USING " # # # . # # " ; Y I5 NEXT I 80 FOR I = 0 TO -8 STEP -.01 85 X = (I 4 - .37 * I 2 t .004) / ((I 2 t .06) 90 Y = (I * (I 2 - .06)) / ((I 2 t .06) 2) 95 PRINT #12, USING " # # # . # # " ; X; 100 PRINT #12, " . 105 PRINT #12, USING " # # # . # # " ; Y 110 NEXT I 115 FOR I = 4.71 TO 6.28 STEP .01 120 X = 1 - .125 * COS(1) 125 Y = .125 SIN(1) X; 130 PRINT 812, USING " # # # . # # " ; . 135 PRINT #12, " 140 PRINT #12, USING " # # # . # # " ; Y 145 NEXT I 150 CLOSE #12 155 END IT

2)

PI

2)

A

*I

11

zyx

Figure 7. The basic programme to generate data to draw Nyquist plot in the G ( s ) H ( s ) plane.

z zyxw

MARKOV STATES O N QUASI-LOCAL ALGEBRAS *

FRANCESCO FIDALEO Dipartimento di Matematica Uniuersith d i Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy E-mail: fidaleoOmat.uniroma2.it

zyxw

We review the definition of Markov states on quasi-local algebras, their structure and the main properties on known models.

zyxw

1. Preliminaries

Since Refs. [l,51, the investigation of the Markov property had a inpetuous growth, in view also to natural applications to various fields such as quantum statistical mechanics and information theory. Recently, a sistematic investigation of Markov chains and states was extended to typically quantum models such as quasi-local algebras describing Fermions, e.g. the CAR algebra. Yet, there is not a satisfactory general theory of the Markov property in quantum setting. The present note is devoted to quote the known results about the structure, and the main properties on known models of the quantum Markov states. The reader is referreed to Refs. [2, 4,91 for the proofs and further details. A quasi-local algebra associated to the set I equipped with a Boolean structure, and an orthogonality relation Ibetween pairs of elements, is a C*-algebra U with an isotonic family {Ua},c~of local C*-algebras such that

zyx

I

zyxwv

(i) U {U. cy E I } is dense in U; (ii) the algebras U, have a common identity I; ~~

*The author is grateful to Italian CIRM and INDAM for hospitality and financial support.

196

zyxw zy zyx zyxwvut zyxwvu zy zyx zyxwv zyx 197

(iii) there exists an automorphism a of U with a2 = L and a(U,) such that

=

a,

(AB - €(A,B)BA) = 0 ,

w h e n e v e r A E U ~ U U , , B E U p + U U p , a I p ,a n d E ( A , B ) = - l i f AEU ,; B E U i , E(A,B)= 1 in the three remaining possibilities.

is the decomposition of A w.r.t. a in the even and odd part, see Sec. 2.6 of Ref. [8] for further details. Let p 4 a , and E : U, H Up be a completely positive identity preserving linear map. We call such a map a transition expectation. We say that

E

(i) is even if aE = Ea; (ii) is a quasi-conditional expectation if there exists y + ,B such that E ( X Y ) = X E ( Y ) whenever X E U,;” (iii) has the Markow property if there exists y 4 p such that E(U,\p) c UP\,.

Let cp E S(U), and J

c 1 such that

u

U, = U. The state cp is called a

aEJ

Markov state w.r.t. the filtration

J ,

if a,,B E J and

p 4 a implies

cpra,o~a,p = cpla,

for some transition expectation E,,p : U, H Up. We refer the reader to Sec. 2 of Ref. [4]for further details. The previous set up naturally applies to quasi-local algebras based on classes of subsets of a fixed set (e.g. spin systems living on standard lattices Zd). It is not sufficient in order to understand the fine structure of Markov states. Yet, it is explicit enough in order to establish natural connections with the KMS boundary condition, as well as phenomena of phase transitions and symmetry breaking for quantum Markov fields on Zd [3]. In the case of linearly ordered lattices, and quasi-local algebras on them arising from infinite tensor product or Canonical Anticommutation Relations (CAR for short [8]), we are able to exhibit the explicit structure of Markov states. say that E is a quasi-onditional

expectation w.r.t. the triplet

C 2lp

zy C K.

198

zy zyxw zyx zyxwvu zyxw zyx zyxwv

2. Markov states on linearly ordered sets

We specialize the situation to the linearly ordered countable sets I containing, possibly a smallest element j - and/or a greatest element j+. In other words, I is order-isomorphic to Z,Z-, Z+or to a finite interval [ j - - l,j+]c Z, the case 11)< +co being almost trivial. We consider the cases when C'

(i)

:= @ M d j

(c)

(non homogeneous infinite tensor product), in

j€l

this situation

(ii) U :=

v %ti)

(T

C'

= L;

, where U is the CAR algebra generated

by anni-

j€I

the mentioned d j annihilators and creators generate the local algebra %ii} (non homogeneous CAR algebra). In view of physical applications, we deal without further mention with locally faithful even states (e.g. locally faithful states cp such that y = p a ) , and with even transition expectations. The following definition specializes the matter in Sec. 1 to the present situation.

Definition 2.1. (Ref. [4], Definition 4.1) A state cp on U is called a Markov state if, for each n < j + , there exists a quasi-conditional expectation En w.r.t. the triplet U,-l~ c U,] c U,+11 satisfying

Let cp E S(U) be a Markov state, the ergodic limit

E,

1 k-l

:= lim -

k k

C(e,)h

h=O

of e, := E,[a[n,n+l, plays a crucial r61e. Indeed, it uniquely determines, and is determined by the conditional expectation &, : %,+I] H %,I, given for x E %-l], y E z'1[,+1] by

zyxw zy zyxwv zy zyx zyx zyxw zyxwvu zyxwv 199

see Sec. 4 of Ref. [4]. In addition, cp is uniquely determined, for every k by all the marginals

cp(Xk.. . X,)=cp(Ek(XkEk+l(Xk+l . . .cl-l(XZ-lXd. . )>) -cp(Ek(XkEk+l(Xk+l . . . & Z-l(Xl-lQ(Xd) ...I))

< 1,

(1)

*

7

where the Xk,. . . ,Xi linearly generate all of Up,,]. Let cp be a Markov state, together with the sequence {E,}~ 0 , is the same process as a’/’X(t). Then, by changing the time parameter t to e t , we are given a stationary N-ple Markov process. So the above trick can be applied.

5. Nonlinear case

A nonlinear version of the time operator theory can also be discussed in somewhat more detail. The basic Hilbert space is now taken to be the one involving all square integrable functions which are measurable with respect to the sigma-fields generated by the given stationary process X ( t ) . In this case, we do not need to assume existence of moment of any order of the stationary process X ( t ) .

Let B t ( X ) be the sigma-field generated by measurable subsets of R determined by the X ( s ) ,s 5 t. It is understood that B t ( X ) is the smallest sigma-field with respect to which all the X ( s ) , s 5 t , are measurable. Set B ( X ) = V B t ( X ) . Then, we have Hilbert spaces L : ( X ) = L’(R,Bt,P) and L 2 ( X ) = L2(w,B ( X ) ,P ) , respectively. There is naturally defined an orthogonal projection

E’(t) : P ( X ) -+ L ? ( X ) . Set E ( t ) = E’(t+). The collection { E ( t ) ;t E R} forms a resolution of the identity on the Hilbert space L 2 ( X ) . The oprtator T given by

T=

s

tdE(t)

zyxwvu zy zyx zyxw

zyx 229

is called the time operator on L 2 ( X ) . The same notations are used as in the case of M ( X ) , if no confusion occurs.

On the other hand, we can define a unitary group of shift operators {Vt, t E R}. Starting with a mapping V, :

V t X ( s )= X ( t

+ s),

implies a flow (one-parameter group of measure preserving transThen, formations) {Tt,t E R,} on the measure space ( w , B ( X ) , P )such that

zyxw Tt : B s ( X )

+

Bs+t(X).

More precisely, for a cylinder set w-set

A=

(W:

( X ( S l , w ) , . . . , X ( S n , w ) )E Bk),

with Bk being a Bore1 subset in Rn, we define TtA by

TtA= ( w : ( & X ( S ~ , W ) , .,V,X(Sn,W)) .. E Bk). Obviously, Tt is a measurable transformation and

P(TtA) = P(A) holds. Then, Tt extends to a measure preserving transformation on the algebra A ( X ) generated by all the cylinder sets of the form A above, where the operations union and intersection of cylinder sets commute with Tt. Since the sigma-field B ( X ) is generated by A ( X ) , the Tt can be extended to a measure preserving transformation on (0,B ( X ) ,P). It is easy to prove that

namely {Tt,t E R} forms a one-parameter group with TO= I . To proceed to the next step we assume that Assumption The measure space (Q, B ( X ) ,P ) is an abstract Lebesgue space. With this assumption the measure space (Q, B ( X ) ,P ) given above may be said to be isomorphic to the Lebesgue measure space (without atoms).

zyx

230

Examples. 1) If X(t) is a continuous stationary Gaussian process with canonical representation, then the space (R, B(X), P ) is an abstract Lebesgue space without atoms. 2) For a linear process expressed as a linear functional of a white noise and a Poisson noise the associated measure space is also an abstract Lebesgue space without atoms.

With this assumption, the family of set transformations {Ti}turns into a family of point transformations. In addition, the family forms a oneparameter group of measure preserving transformations on the measure space (R, B(X),P ) except a null set. Also, by assumption of the continuity of X ( t ) , it can be proved that

zyxw

Ttw is measurable in (t, w ) .

Hence, by the usual argument we can prove the following assertion. Proposition Set

(UtcP)(~)= cP(Ttz).

Then, {Ut, t E R} forms a continuous one-parameter group of unitary operators acting on the Hilbert space L 2 ( X ) :

Ut

4

I ast+O.

Having obtained the unitary group {Ut}, we can prove the commutation relation with the time operator T established before. Note that the commutation relation is exactly the same in expression as in the linear case where the entire Hilbert space is taken to be M(X). Theorem It holds that

T U t = U t T itUt. There, more profound results are included. Apply Stone’s theorem to Ut to have the infinitesimal generator which is self-adjoint and is denoted by H :

zyxw zyxw zyx zyx zyxwv 231

Then, we have the commutation relation

[ H , T ]= il

Thus, similar results are obtained as in M ( X ) , however, there is a short note to be added. Since M ( X ) c L 2 ( X )holds and two spaces have been introduced consistently so far as X ( t ) has second order moment, all the operators representing statements commute with the projection operator

P :P(X)

-+

M(X).

It is our hope that the above relationships between operators will give some help to the profound investigation of the evolutional phenomena described by X ( t ) .

z

6. Diffusion process

We now turn our eyes to a stationary Marlcov process X ( t ) with t 2 0. assume that the transition probability density p ( t , 5,y) exists and smooth in (t,u,w). Define an operator &, t 2 0, by

zyx

Then, we have a semi-group of operators & acting on C:

VtVs = &+s,

&

-+

I,(t+O).

We can therefore appeal to the Yosida-Hille theorem for one-parameter semi-group to obtain a generator A such that: -I w - t+O lim t f=Af

v,

for f in a dense subset of C.

To fix the idea, we consider the case where X ( t ) satisfies the Langevin requaton:

d X ( t ) = -AX(t)dt

+ dB(t).

Then, A has an explicit expression of the form 1 d2 A=---Au-. 2 du2

d du

232

zyxwv zyxwvu zyxw zyxw zyxwvu

With this operator A we establish an interesting commutation relation. Namely

Proposition Let A be as above and let S =

2. Then, we have

[A,S] = AS.

Thus, a transversal relation appears. This is a typical relationship in the theory of dynamical systems, which tell us some properties from the viewpoint of a dynamical system that is determined by the Langevin equation.

In addition, the operator S is the generator of the shift (the oneparameter group of translations of space variable), so that we can use harmonic analysis arising from the shift and the semi-group of the phase transition. Some more details on these topics will be seen in the forthcoming paper.

References

zyxwv

1. L. Accardi et al. (eds.), Selected Papers of Takeyuki Hida. World Scientific Pub. Co. Ltd. 2001. 2. T. Hida, Canionical representation of Gaussian processes and their applications. Memoires Coll. of Sci. Uniuv. of Kyoto, A33 (1960), 109-155. 3. T. Hida, Stationary stochastic processes. Mathematical Notes. Princeton University Press. 1970. 4. T. Hida, Complex white noise and infinite dimensional unitary group. Lecture Notes in Math. Mathematics, Nagoya Univ. 1971. 5. T. Hida, Brownian motion. Springer-Verlag. 1980. Japanese Original: Buraun Unidou. Iwanami Pub. Co. 1975. 6. T. Hida, Analysis of Brownian functionals. Carleton Math. Notes no.13, 1975. 7. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White noise. An infinite dimensional calculus. Kluwer Academic Pub. Co. 1993. 8. T. Hida and Si Si, Lectures on white noise functionals. World Scientific Pub. Co. 2005. to appear 9. H. -H. Kuo, White noise distribution theory. CRC Press Inc. 1996. 10. R. LBandre, Theory of distribution in the sense of Connes-Hida and Feynman path integral on a manifold. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 6 (2003) 505-517. 11. R. LBandre and H. Ouerdiane, Connes-Hida calculus and Bismut-Quillen superconnections. Stochastic analysis: Classical and Quantum, Perspectives of White noise theory, ed . T. HIda, World Scientific Publ. 2005, 72-85. 12. K . Yosida, Functional analysis. Springer-Verlag, 6th ed. 1980.

ON THE DYNAMICAL SYMMETRIC ALGEBRA OF AGEING: LIE STRUCTURE, REPRESENTATIONS AND APPELL SYSTEMS

zyxwvu zy :

MALTE HENKEL RENE SCHOTT~STOIMEN STOIMENOV JEREMIE UNTERBERGER

I:

The study of ageing phenomena leads to the investigation of a maximal parabolic subalgebra of conf3 which we call a1t We investigate its Lie structure, prove some results concerning its representations and characterize the related Appell systems.

1. Introduction

Ageing phenomena occur widely in physics: glasses, granular systems or phase-ordering kinetics are just a few examples. While it is well-accepted that they display some sort of dynamical scaling, the question has been raised whether their non-equilibrium dynamics might posses larger symmetries than merely scale-invariance. At first sight, the noisy terms in the Langevin equations usually employed to model these systems might appear to exclude any non-trivial answer, but it was understood recently that provided the deterministic part of a Langevin equation is Galilei-invariant, then all observables can be exactly expressed in terms of multipoint correlation functions calculable from the deterministic part only [8].It is therefore of interest to study the dynamical symmetries of non-linear partial differential equations which extend dynamical scaling. In this context, the so-called Schrodinger algebra 5cg has been shown to play an important r61e in phaseordering kinetics. In what follows we shall restrict to one space dimension and we recall in figure 1 through a root diagram the definition of 5 c I ~as a parabolic subalgebra of the conformal algebra confs [7].

zy zyxwv

* LPM, Universite Henri Poincar6, BP 239, 54506 Vandoeuvre-lb-Nancy, France t IECN and LORIA, Universite Henri Poincare, BP 239, 54506 Vandoeuvre-lb-Nancy, France LPM, Universit6 Henri Poincar6, BP 239, 54506 Vandoeuvre-16s-Nancy, France and Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria SIECN, Universite Henri Poincarb, BP 239, 54506 Vandoeuvre-lkNancy, France

233

234

zyxwvut 0

0

zyxw 0

Figure 1. (a) Root diagram of the complexified conformal Lie algebra (conf,)c and the labelling of its generators. The double circle in the center denotes the Cartan subalgebra. The generators of the maximal parabolic subalgebras (b) sc5 and (c) alt are indicated by the filled dots.

2. A brief perspective on the algebra alf

There is a classification of semi-linear partial differential equations with a parabolic subalgebra of conf3 as a symmetry [lo]. Here we shall study the abstract Lie algebra aKt, which is the other maximal parabolic subalgebra of conf3 (see figure 1) and its representations; we shall also see that, like the algebra 5 4 , it can be embedded naturally in an infinite-dimensional Lie algebra W which is an extension of the algebra Vect(S') of vector fields on the circle. Quite strikingly, we shall find on our way a 'no-go theorem' that proves the impossibility of a conventional extension of the embedding alt C ~ 0 n f 3 .

zyxwvut zyxwvut zyxwvu zyxwv zyxwv zyxwvu zyxwv

2.1. The abstract Lie algebra alt

Elementary computations make it clear that (see figure 1and D = 2x0-N) aKt=(V+,D,Y-g)K ( X I , Y + , M O ) : = ~ K $

(1)

is a semi-direct product of g N sK(2, R) by a three-dimensional commutative Lie algebra 5; the vector space $ is the irreducible spin-1 real representation of 5[(2,R), which can be identified with 51(2, R) itself with the adjoint action. So one has the following Proposition 2.1:

(1) aIt N 5[(2, R) @ R [ E ] / Ewhere ~ , E is a 'Grassmann' variable; (2) aKt N p3 where p3 z11 50(2,1) K R3 is the relativistic Poincare' algebra in (2+l)-dimensions.

zyxwv zyxw zyxw zy zyxw zyxw zyxw zyxw 235

Proof : The linear map CP : alt t d ( 2 , IR) €9 R [ E ] / Edefined ~ by

CP(V+)= Ll, @(D)= Lo, CP(Y-+) = L-1

1 qx,)= -Lq, 2

CP(Y+)= LZ, CP(M0) = L t 1

is easily checked to be a Lie isomorphism. In particular, the representations of a[t Wigner studied them in the 30’es.

N

p3

0 are well-known since

2.2. Central extensions: an introduction

Consider any Lie algebra g and an antisymmetric real two-form a on g. Suppose that its Lie bracket [ , ] can be ’deformed’ into a new Lie bracket

-

[ , ] on

5

-

x IRK, where [K,g] = 0, by putting [(X,O),(Y,O)] = ([X, Y], a ( X , Y)). Then 6 is called a central extension of g. The Jacobi identity is equivalent with the nullity of the totally antisymmetric threeform d a : A3(g) -+ IR defined by := g

da(X, Y, 2 ) = Q!([X,Y], 2 )

+ a([Y,21,X) + 412,XI, Y).

Now we say that two central extensions g1,gz of g defined by a1,a2 are equivalent if (12 can be gotten from g1 by substituting (X, c) H (X, c X(X)) (X E g) for a certain 1-form X E g*, that is, by changing the nonintrinsic embedding of g into 61. In other words, a1 and a2 are equivalent if a2 - a1 = dX, where dX(X, Y) = (A, [X,Y]). The operator d can be made into the differential of a complex (called Chevalley-Eilenbergcomplex), and the preceding considerations make it clear that the classes of equivalence of central extensions of g make up a vector space H 2 ( g ) = Z 2 ( g ) / B 2 ( g ) , where 2’ is the space of cocycles Q! E A2(g*) verifying d a = 0, and B2 is the space of coboundaries dX, X E g*. We have the well-known Proposition 2.2: The Lie algebra aIt has no non-trivial central extension: H2(aIt) = 0. All this becomes very different when one embeds ah into an infinitedimensional Lie algebra.

+

zyx

2.3. Infinite-dimensional eztension of alf

The Lie algebra Vect(S1) of vector fields on the circle has a long story in mathematical physics. It was discovered by Virasoro in the 70’es (see [Ill)

236

zyxwvu zyxw zyxw zyx zyxwvu zyx zyxw zyxwvuts zyxw

that Vect(S1) has a one-parameter family of central extensions which yield the so-called Virasoro algebra ~ i := r Vect(S1) B RK = ( ( L , ) , ~ Z , K )

with Lie brackets

[K,Lnl = 0,

[Ln,J5ml = .( - m)Ln+m + 6n+m,0 c n(n2 - 1)K ( c E R)

When c = 0, one retrieves Vect(S1) by identifying the (L,) with the usual Fourier basis (ei"edO),Ez of periodic vector fields on [0,27r],or with -zn+l a d t with z := eie. Note in particular that (L-1, Lo, L1) is isomorphic to sI(2, R), and that the Virasoro cocycle restricted to sI(2,R) is 0, as should be (since sI(2, R) has no non-trivial central extensions). It is tempting to embed aIt N d(2, R) @ R [ E ] / E into ~ the Lie algebra

w := vect(S1) @ rw[e]/e2= (L,),Ez

!x (Li)nEz,

with Lie brackets

[L,, Lml = .( - m)Ln+rn, [L,, L l l = .( - m)Li+,,

[G, -qnI = 0.

These brackets come out naturally putting W in the 2 x 2-matrix form

leading to straightforward generalizations. Note in particular that there exists a deformation of the so-called 'Schrodinger-Virasoro algebra', introduced in [6, 71 as an infinite-dimensional extension of the Schrodinger Lie algebra S C ~ that , can be represented as upper-triangular 3 x 3 Virasoro matrices (see [9]for more details). In terms of the standard representations of Vect(S') as modules of Qdensities Fa = {u(z)(dz)"} with the action d

f(4z(u(z)(dz)") = (fu' + (.f'4(4(dz)", we have Proposition 2.3: W N Vect(S1) !x F-1. There are two linearly independent central extensions of W : 1. the natural extension to W of the Virasoro cocycle on Vect(S1), namely [ , ] = [ , ] except for [L,, L-,I = n(n2-1)K+2nLo. In other words, Vect(S1) is centrally extended, but its action on F-1 remains unchanged; 2. the cocycle w which is zero on A2(Vect(S1)) and A2(3-1), and defined by w(L,, L L ) = 6 n + m , ~n(n2- l ) K on Vect(S1) x 3 - 1 .

-

-

zyxwv 237

zyxwvu zyxwv zyxwvu zyxw zyxwv zyx

A natural related question is: can one deform the extension of Vect(S1) by the Vect(S1)-module 3-1 ? The answer is: no, thanks to the triviality of the cohomology space H2(Vect(S1),3-1) (see [3], or [5]). Hence, any Lie algebra structure [ , ] on the vector space Vect(S1)$3-1 such that

-

I v

[(X,4), (Y, $)I

=

([X,YIVect(Sl),advect(sl)X.$ - advect(sl)Y.4+ B(X,Y))

is isomorphic to the Lie structure of W (where B is an antisymmetric twoform on Vect(S')). So one may say that W and its central extensions are natural objects to look at. 2.4. S o m e results on r e p r e s e n t a t i o n s of

W

We now state two results which may deserve deeper thoughts and will be developed in the future. Proposition 2.4 ('no-go theorem'): There is n o way t o extend the usual representation of aIt as conformal vector fields into a n embedding of W into the Lie algebra of vector fields on EX3. Proposition 2.5: The infinite-dimensional extension W of the algebra a h is a contraction of a pair of commuting Virasoro algebras bit $air ---f W . I n particular, we have the explicit diflerential operator representation

L, = -tn+lat + ( n+ l)t"ra, - ( n + l)ztn - n(n+ 1)ytn-'r L€ n = -tn+la, - ( n + 1)yt" where x and y are parameters and n E Z. 3. Appell systems

zyxw

zy

Definition 3.1. Appell polynomials {h,(x); n E N } on IR are usually characterized by the two conditions 0 0

h,(z) are polynomials of degree n, Dh,(x) = nh,-l(z), where D is the usual derivation opertor.

Interesting examples are furnished by the shifted moment sequences hn(x) =

s_(,.

00

+ Y)nP(dY)

where p is a probability measure on R with all moments finite. This definition generalizes to higher dimensions. On non-commutative algebraic structures, the shifting corresponds to left or right multiplication

238

zyxwvu zyx zyxwvu zy zyxw z z

and, in general, {h,} is not a family of polynomials. We shall call it AppeZZ systems (see [2]for details). Appell systems of the Schrodinger algebra scg have been investigated [I] but the algebra aIt requires a specific study. aIt has the following Cartan decomposition:

a

a[t = !$ CBI CB C = { Y1,X I } CB {Yo,X o } CB { Y-1, X - I }

(2)

and there is a one to one correspondence between the subalgebras J!3 and = aibi, where {bi, i = 1,.. . , 6 } is a basis of ak. The ai are called coordinates of the first kind. Here we use the basis bl = Y1, b2 = X I , b3 = Yo, bq = Xo, b5 = Y-1, bs = Y I . Let ALT be the simply connected Lie group corresponding to aKt. Group ellements in a neighborhood of the identity can be expressed as

2. Write X E aIt in the form X

,x

x!=l

= eAibi

. . . eA6b6

The Ai are called coordinates of the second kind. Referring to decomposition ( 2 ) , we specialize variables, writing V1,V2,B1, B2 for A l , A2, As, A6 respectively. Basic for our approach is to establish the partial group law: e B 1 Y - 1 + B 2 X - ~ e V ~ Y ~ +=?. V~X We ~ get

zyxwv

Proposition 3.1. In Coordinates of the second kind, we have the Leibnitz formula, 9(0,0,0 , 0, Bi, B2)g(Vi,V2,0,0,0,0) = g(Al,442,A37 A47 A59 A6) =

zyxwvuts zyxw zyxwv zyx zyx 239

Now we are ready to construct the representation space and basis-the canonical Appell system. To start, define a vacuum state R. The elements Yi, Xi of p can be used to form basis elements

Ijk) = Y,jXl"R,j, k 2 0

(6)

of a Fock space 5 = span{ I j k ) } on which Y1 ,Xi act as raising operators, Y-1, X-1 as lowering operator and Yo, XOas multiplication with the constants y, x (up to the sign) correspondingly. That is, Y1R = [lo),X1R = 101) Y-1R

= 0 ,X - l R =

0

(7)

YOR = -ylOO),XoR = -2100)

(8)

The goal is to find an abelian subalgebra spanned by some selfadjoint operators acting on representation space, just constructed. Such a twodimensional subalgebra can be obtained by an appropriate "turn" of the plane p in the Lie algebra, namely via the adjoint action of the group element formed by exponentiating X-1. The resulting plane, !Qp say, is abelian and is spanned by

zyxwvu y1

~

,PX-lyl,-BX-

= Yi

-

2PYo + P2Y-1

Xl = ePx-lxle-Px- = x1- 2 0 x 0 + p2x-1

(9)

Next we determine our canonical Appell systems. We apply the Leibniz formula ( 5 ) with B1 = 0 , B2 = P, V1 = 21, V2 = 22 and (7). This yields ,zlFleZzXl~ = ePX-le"lYlezzX1e-Px-l~ = e P X - l e z l Y l e z z x ' ~ =

+e ~p m(l

= e(1-P.z)

1-Pzz)e

1 - 0 ~ ~

-Pz2)-2ZZ;2

(10)

To get the generating function for the basis (jk)set in equation (10)

Substituting throughout, we have Proposition 3.2. The generating function for the canonical Appell system, Ijk) = Y,jX,kR is

+

(1 Pv2)-2Zf2

where we identify of Y11Y2.

(12)

= y1.l and X1R = y2.1 in the realization as function

240

zyxwvut

With v1 = 0, we recognize the generating function for the Laguerre polynomials, while 212 = 0 reduces to the generating function for Hermite polynomials. References

zyxw zy

1. P. Feinsilver, Y. Kocik and R. Schott, Representations of the Schrodinger Algebra and Appell Systems, Progress of Physics, 52 (2004) 343-359. 2. P. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus, Vol.3: Representations of Lie Groups. Kluwer Academic Publishers, Dordrecht, 1993. 3. D. B. Fuks, Cohohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. 4. M. Henkel, R. Schott, S. Stoimenov and J. Unterberger, O n the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems. Prepublication Institut Elie Cartan, 2005. 5. L. Guieu, C. Roger, preprint, available on http://www.math.univ-montp2.fr/~guieu/The~Virasoro~Project/Phasel/. 6. M. Henkel, J. Stat. Phys. 75, 1023 (1994). 7. M. Henkel, J. Unterberger, Schrodinger-invariance and space-time symmetries, Nuclear Physics B660 (2003) 407-435. 8. A. Picone and M. Henkel, Local scale-invariance and ageing in noisy systems, Nuclear Physics B688 (2004) 217-265. 9. C. Roger and J . Unterberger, in progress (2005). 10. S. Stoimenov and M. Henkel, Dynamical symmetries of semi-linear Schrodinger and diffusion equations, Nuclear Physics B 7 2 3 (2005) 205-233. 11. M. Virasoro, Phys. Rev. D1, 2933-2936 (1970).

zyxw zyxwvu zy zyxwv zyxw zyxw zy

AN ANALYTIC DOUBLE PRODUCT INTEGRAL

R L HUDSON School of Mathematical Sciences, University of Loughborough, Loughborough, Leicestershire LE11 3TU, Great Britain. -++

+

The double product integral fl (1 X(dAt C3 d A - d A @ d A t ) ) is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients.

1. Introduction

zyxw --re

n

+

A purely algebraic theory of double product integrals such as (1 X(dAt@dA-dA@dAt))[4, 91 has been developed based on formal power series in the formal parameter A, so that questions of convergence are avoided. This is not mere laziness; the resulting theory is useful in the construction of algebraic quantum groups [4,91. Moreover, if X is replaced by a complex parameter, it is known that the power series defining the matrix elements between nonvacuum exponential vectors of the double product in-

n

--tt

tegral (1+ XdA @ dA) have zero radius of convergence, thus precluding a meaningful analytic theory in this case. In the algebraic theory it is known [9] that, for an arbitrary formal power series dr[X]with coefficients in the tensor product Z @ Z with itself

n

-+t

of the algebra Z of It6 differentials,

n

t+

+

(1

+ Xdr[X])has a multiplicative

(1 Xdr'[X])) where the formal power series Xdr'[X] is inverse given by the quasiinverse of Xdr[h]defined by

+

+ X2dr[h]r"h]= Xdr"h] + Xdr[h]+ X2dr"h]r[h]= 0. (1) n (1+ X(dAt @ dA - dA @ dAt))it may be verified using

Xdr[h] Xdr"h]

--tt

In the case of the usual It6 multiplication table

dAdAt = dT, all other products vanish,

(2)

that the quasiinverse of X(dAt@dA-dA@dAt)is -X(dAt@dA-dA@dAt),

n

-it

and since the formal adjoint of

+

(1 X(dAt @ dA - dA @ dAt)) is 24 1

242

n

t+

zyxw zyxwvu zyx zy n

t+

+

( 1 ~ ( d@ ~d~ t- d~

n

+t

the double product

d ~ t ) t=)

( 1 - ~ ( d@ ~d~ t- d~ 8 d ~ t ) ) ,

( 1 + X(dAt @ d A - d A @ d A t ) ) is formally unitary.

n

4-

+

One might therefore expect that there is an analytic theory of (1 X(dAt @ d A - dA @ d A t ) ) for a real parameter X in which it consists of unitary operators in Fock space. This paper provides such a theory. 2. The double time orthogonal dilation [3]

Interpreting ( l + X ( d A t @ d A - d A @ d A t ) ) as a second quantised infinitesimal rotation and recalling the functoriality [ l o ]and continuity of the second

n

-++

( l + X ( d A t @ d A - d A @ d A t ) ) is quantisation map, one might expect that itself the second quantisation of a double product of infinitesimal rotations. Such a double product has been constructed [3].It consists of a family of operators w = w,":; in the Hilbert space L2(R+)@L2(IR+)

(

)la,bl,l~,tlc~+

enjoying the following properties.

zyxw

Each W,";,"is a unitary operator, acting non trivially on the subspace L2(]a,b ] )@ L 2 ( ] st,] )and as the identity on its orthogonal complement. 0

zy

For fixed Is, t ] ,W,";,"is a reverse evolution in ] a ,b] and, for fixed ] a ,b], W,";,"is a forward evolution in ] s , t ] that , is, for a < b < c and T < s < t , 0

w,";,"wbc,I& = w,;,",w;;gPWb+ a,t = Wb,' a,t . 0

(3)

W is covariant under shifts and time reversal; for arbitrary p, q E R+

(S, @s*)*w,~~~:;;(S,@s*) = w,"::,(R: @R:)*W:;,"(R: m

i ) = w,";;

where S, denotes the isometric shift through p and Ri the time reversal operator for the interval ] a ,b ] ;

The matrix operator W,";,"is given explicitly in two equivalent forms: (4)

(5)

243

where the notation is as follows.

zyxwvuts zy z zyx zy zyxwvz zyxwv

and K { K } denotes the integral operator on L2(R+) whose kernel is K,.: (z,y))= 1 if a < z < y < b (resp. b > z > y > a ) and 0 otherwise, xt is the indicator function of the interval Is,t ] ,and At, A: are the bounded mutually adjoint operators K { >;} and K { i)(f(k - 1)- f ( k ) ) ,

k

E N,

and the Laplacian

2 = -d:*d+

= d+

1 + -d-

P which generates a Markov process on N whose invariant measure is the geometric distribution 7r on N with parameter p E (0, l),i.e.

7 r ( { k } )= (1 - p ) p k , 267

k

E

N.

268

zyxw zyxw zyxw zyxw zy zyx zyxw zyxwvu

Denote by E, the expectation under defined as Ent,

T

and by Ent, the entropy under

T,

[fl = E7r [f1% fl - E,[fllog E7r [fl.

We recall the modified logarithmic Sobolev inequality proved in [2] for the geometric distribution 7r.

Theorem 1.1. Let 0 < c < - logp and let f : N + R such that Id+fl 5 c. W e have

In higher dimensions the multi-dimensional gradient is defined as d:f(k)

=f(k

+ ei) - f ( k ) ,

i = 1 , . . . ,n,

where f is a function on Nn, k = (kl,...,kn) E canonical basis of Rn, and the gradient norm is lld+f(k)l12 =

c

Id?f(k)I2 =

cIf(k +

N", ( e l , . . . , e n ) is the

ei) - f(k)I2.

(1.2)

i=l

i=l

From the tensorization property of entropy, (1.1) still holds with respect to 7rBn in any finite dimension n:

zyxwv

provided Idi f 1 5 c, i = 1,.. . ,n. As a consequence the following deviation inequality for functions of several variables under rBnhas been proved in [2] using (1.1) and the Herbst method.

Corollary 1.2. Let 0 < c < -1ogp and let f such that ld:fl 5 p, i = 1 , . . . ,n, and lld+f1I2 5 a2 for some a,P > 0. Then f o r all r > 0,

where ap,c

=

PeC (1 - P I P - @I

denotes the logarithmic Sobolev constant in (1.1). Our goal in the next section will be to extend these results to interacting spin systems under a geometric reference measure.

zyx zyxwv zy zy z zyxw 269

2. Logarithmic Sobolev inequality for an interacting spin

system

Given a bounded finite range interaction potential GJ = { ( a , i.e.

:

R c Zd},

zyx zyxwvutsr zyxw

let the Hamiltonian H A be defined as

c

R f l A#@

where 7~ denotes the restriction of 77 to N R , R c NZd. The Gibbs measure 7rX on N" associated to a N-valued spin system on a finite lattice A c Zd with boundary condition w E NZd\" is defined by its density with respect to 71" := "'r7 as:

where IT is the geometric reference distribution on N,2; is a normalization factor, and

H , w ( ~= I )H A ( v A w A ~ ) ,

rl E N'~,

where qw is defined as

+

(rlw)rc = r l k l ~ ( k ) ~ k l ~ ( k ) , k E Zd,

whenever 7 E NA, w E again

NB,and A , B c Zd are such that A n B = 0 . Let

zyxwvutsr zyxwvutsrq

270 7]k

zyx

> 0 , cf. [ l ]We . assume that there exists a constant C > 0 depending on

11@11

only, with 1

C < - c",k,v,+) For f : E 4 R we let:

5 C,

-

8 x ( e f )= kEh

/

v E NA,

A

c Z d , k E A.

(2.1)

c",k, a, +)ef(')Id~f(a)12d.lrX(a),

and

/

8 A ( e f )= kEh

zyxw

ef(u)Id~f(a)12d.lrh(a).

Next we consider the family of rectangles of the form

R = R ( k ,11, ...,Id) = k -k ([I,111

X

X

[I,I d ] ) I I Z d ,

zyxwvu zyxwv zyxw zyxwv zyxw

where k E Zd and

11,.

* * *

. . ,l d E N, with

size(R) = max l k . k = l , ...,d

Let 9~ denote the set of rectangles such that size(R) 2 L

and size(R) 5 10 min l k . k=l,...,d

Definition 2.1. We say that .lrx satisfies the mixing condition if there exists constants C1 and Cz, depending on d and \\@I1only, such that:

5

cl e - C z d ( A , B ) ,

for all L 2 1, A E 9~ and A , B

(2.2)

c A such that A , B

E92~ with

AnB

=

0.

We refer to [l]and [4]for conditions on Q, under which (2.2) holds under a geometric reference measure. Our goal is to prove the following logarithmic Sobolev inequality under the Gibbs measure .lrx.

Theorem 2.2. Assume that the mixing condition (2.2) holds, and let c < - logp. Then there exists a constant T~ > 0, independent of A and w,such that Ent,X [ e f ]5 r c 8 x ( e f ) ,

(2.3)

zy zyxwv zyxwv zyxwv zyxwv zy 271

for every f : E 4R such that lldff

Ilp(h)

I C , 7rA-a.e.

In particular we have

which implies, as in Corollary 1.2, a deviation inequality under Gibbs measures. Corollary 2.3. Assume that the mixing condition (2.2) holds, and let c - logp. Let f be such that [Id+f l l p ( ~I)P and

for some a l p> 0. Then for all r > 0, 7rx

(f - ET; [f]2 r ) 5 exp

c2r2 4yca2P2

rc

zyx zy

- -a2yc)) .

P


0 we get the Ruelle type bound: ~ { r El N~~ : 17121 2~IAII > >~ X (-(cr P - C~,)IAJ

7rx,

+ c ~ n [ 1 r ] n ,1 1 )r > ro,

for all finite subset A of Zd, under the mixing condition (2.2). Indeed, it suffices to apply the uniform bound (2.5) with f ( r ] ) = 1 ~ ~ 1a' , = CJAl, /3 = 1, and the compatibility condition

This shows in particular that II satisfies the (RPB)' condition in [3].

272

zyxwv zyxwvu zy zyxwv zyx zyx zyxwvu

3. Proof of Theorem 2.2

Recall that for c < - logp, by tensorization, Theorem 1.1 yields as in (1.3) the logarithmic Sobolev inequality ~nt,,[ef] I ScgA(ef),

(3.1)

R such that ~ ~ d + f ~ ~ I p (c,A.rrA-a.e., ) with an optimal which is independent of A c NZd.Let now S A , ~ , ,denote the optimal constant in the inequality

f : NZd constant s, I

for all

--f

Ent,;[ef]

I l d + f I l ~ (I ~ )c.

I SA,,,,g:(ef),

Lemma 3.1. For every A C Zd, there exists a constant A := Ce41Allloll > 0 depending only on IAl, c and independent of w E NZd, such that SC

A -< SA,W,C 5 A s , . Proof. We follow the proof of Proposition 3.1 in [l].From (2.1) we obtain: -

C-1e-21AIII*II

g A ( e f )< g x ( e f > I Ce21~lIl'll g A ( e f ) .

(3.2)

From the relation Ent,[f]

= %IE,[flogf

- flogt

-f

+ t]

and the bound

we have e-21A111*11Ent,,

[ef] 5 Ent,;

[ef]

zy

I e21AlllollEnt,,

from which the conclusion follows using (3.1) and (3.2). Let for L 2 1:

[ef] ,

0

which is finite by Lemma 3.1. Prop 3.1. Assume the mixing condition (2.2) is satisfied. Then there exists a constant K. depending on (I@((, such that S2L,C

for L large enough.

5 (1

-

5)

-l SL,C

(3.3)

zy z

zyxwv zyxw zyxw zyxwvu 273

Pro08 The proof of this proposition is identical to that of Proposition 4.1, pp. 1970-1972 and Proposition 5.1, p. 1975 in [l],replacing the Dirichlet form used in [l]with 8;. 0 Finally, Theorem 2.2 is proved by taking from Proposition 3.1.

"yc

=

S U P ~ Swhich L , ~ ,is finite

References

1. P. Dai Pra, A.M. Paganoni, and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab., 30(4):1959-1976, 2002. 2. A. Joulin and N. Privault. Functional inequalities for discrete gradients and application t o the geometric distribution. ESAIM Probab. Stat., 8:87-101 (electronic) , 2004. 3. Y. Kondratiev, T. Kuna, and 0.Kutoviy. On relations between a priori bounds for measures on configuration spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7(2):195-213, 2004. 4. F. Martinelli. Lectures on Glauber dynamics for discrete spin models. In Lectures on probability theory and statistics (Saint-Flour, 1997), volume 1717 of Lecture Notes in Math., pages 93-191. Springer, Berlin, 1999. 5. B. Zegarlinski. Analysis of classical and quantum interacting particle systems. In L. Accardi and F. Fagnola, editors, Quantum interacting particle systems (Trento, 2000), volume 14 of QP-PQ: Quantum Probab. White Noise Anal., pages 241-336. World Sci. Publishing, River Edge, NJ, 2002.

zy zyxw

TO QUANTUM MECHANICS THROUGH GAUSSIAN INTEGRATION AND THE TAYLOR EXPANSION OF FUNCTIONALS OF CLASSICAL FIELDS *

zyxwv

A.YU. KHRENNIKOV International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vaxjo, S-35195, Sweden E-mail:Andrei.KhrennikovOmsi.urn.se

zyxwvu

We propose a solution of the problem of hidden variables. In our approach QM can be reconstructed as an asymptotic projection of statistical mechanics of classical fields. Determinism can be reestablished in QM, but the price is the infinite dimension of the phase space. Our classical+ quantum projection is based on Gaussian integration on the Hilbert space and the Taylor expansion (up to the second order term) of functionals of classical fields. Our solution of the problem of hidden variables is given in the framework which differs essentially from the conventional one (cf. Einstein-Polosky-Rosen, von Neumann, Kochen-Specker,..., Bell). The crucial point is that quantum mechanics is just an asymptotic image of prequantum classical statistical field theory (PCSFT).

1. Introduction

The problem of reduction QM to classical statistical mechanics (problem of hidden variables) has been discussed in a huge number of articles, books, and conferences.a At the moment there is more of less general opinion that QM could not be reduced to classical statistical mechanics. The main reason for such a position are various ”NO-GO” theorems (e.g., von Neumann’s theorem or Bell’s theorem). However, a possible physical impact of ‘This work is supported by Profile Mathematical Modelling of V k j o university and EU-network on Quantum Probability and Applications are not able to present here a review on this problem. We just mention the book of von Neumann [l] and some recent publications [2, 31, see especially papers of L. Accardi, G. Adenier, L. Ballentine, W. De Muynck, W. De B a r e , Th. Nieuwenhuizen.

274

275

the "NO-GO"-activity was extremely overestimated.b Every "NO-GO" theorem is based on a list of physical and mathematical assumptions. Such assumptions are typically not justified [5, 61. One of such assumptions is the coincidence of classical (prequantum) and quantum statistical averages and the coincidence of ranges of values of classical physical variables and corresponding quantum observables. These conditions are violated in our approach. We consider a small parameter Q -+ 0 - dispersion of fluctuations of a (classical) background field $(z) :

zyxw zyxwv zyxw s s zyxwvu zyxwvu zyx zyxwv b2(4+ q2(z)1da:dP(Q,P) = Q,

Lz (R3) x Lz (R3)R3

+

z (1)

0.

Here $(z) is a vector field with two components $(z) = ( q ( z ) , p ( z ) ) and p is a Gaussian measure on the infinite-dimensional phase space R = L2(R3) x L2(R3).Quantum mechanics is obtained as the lim,,o of classical statistical mechanics with the infinite-dimensional phase-space. 2. Quantum mechanics as a projection of a classical model with the infinite-dimensional state space

A classical statistical model is described in the following way: a) physical states w are represented by points of some set R (state space); b) physical variables are represented by functions f : 52 + R belonging to some functional space V(R); c) statistical states are represented by probability measures on R belonging to some class S(R); d) the average of a physical variable (which is represented by a function f E V(R)) with respect to a statistical state (which is represented by a probability measure p E S(R)) is given by < f >,- Jaf(w)dp(w). A classical statistical model is a pair M = (S(R), V(R)). This is a measure-theoretic statistical model. We recall that classical statistical mechanics on the phase space R2n = R" x Rn gives an example of a classical statistical model. The quantum model (in the Dirac-von Neumann formalism [l]in the complex Hilbert space H,) is described in the following way: a) physical observables are represented by operators A : H , + H , belonging to the class of continuous self-adjoint operators L, (IT,);b) statistical states are represented particular, there was shown [4] that all distinguishing features of the quantum probabilistic model (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model.

zyxwv zyxwv zyxw

276

by density operators, see von Neumann [l](the class of such operators is denoted by D(H,)); d) the average of a physical observable (which is represented by the operator A E L,(H,)) with respect to a statistical state (which is represented by the density operator D E D(Hc))is given by von Neumann’s formula: < A >D= Tr DA. The quantum statistical model is the pair Nquant = (D(H,),Ls(Hc)).This is an algebraic statistical model. To simplify considerations, in this paper we shall consider the case of the real (separable) Hilbert space H . Thus in the definition of Nquant the complex Hilbert space H, should be changed to the real Hilbert space H . Everywhere below a Gaussian measure on H with the zero mean value and the covariation operator B is denoted by the symbol P B . Let us consider a classical statistical model in that the state space R = H (in physical applications H = L2(R3)is the space of classical fields on R3) and the space of statistical states consists of Gaussian measures with zero mean value and dispersion

zyxwvu zyxwvuz (2)

where a > 0 is a small real parameter. Denote such a class of Gaussian measures by the symbol S,%(R). We remark that scaling preserves the class of Gaussian measures. Let us make the scaling of the classical background field:

(we emphasize that this is a scaling not in the physical space R3, but in the space of fields on it). To find the covariation operator D of the image p~ of the Gaussian measure p ~ we, compute its Fourier transform: pD(Pe= JQf($)dp~($) with respect to the small parameter a. In this Gaussian integral we make the scaling (3):

where the covariation operator D is given by (4). We remark that JQ(f’(0),$)dp~($)= 0, because the mean value of p o is equal zero. Since p~ E Sg(R), we have Tr D = 1. The change of variables in (5) can be considered as scaling of the magnitude of statistical (Gaussian) fluctuations.c By (5) we have: < f > p = $ Ja(f”(0)y,y) d p o ( y ) o(a), a -+ 0, or

+

a

< f >p -- Tr D f”(0) + o ( a ) , 2

0.

4

(6)

We see that the classical average (computed in the model M a = (S,%(R),V(s2))by using the measure-theoretic approach) is coupled through (6) to the quantum average (computed in the model N q u a n t = (D(R), L,(O)) by the von Neumann trace-formula). The equality (6) can be used as the motivation for defining the following classical + quantum map T from the classical statistical model M a = (Sz,V) onto the quantum statistical model Nquant= (D,L,) :

B a

T : S,%(R)-+ D(R), D = T ( ~ B=)-

(7)

(the Gaussian measure p~ is represented by the density matrix D which is equal to the covariation operator B of this measure normalized by a ) ;

T : V(R) + Ls(R),

Aquant

zyxw 1 2

= T ( f )= -f”(O).

(8)

=Negligibly small random fluctuations a ( p ) = fi (where Q is a small parameter) are considered in the new scale as standard normal fluctuations. If we use the language of probability theory and consider a Gaussian random variables c(X), then the transformation (3) is nothing else than the standard normalization of this random variable: (in our case EE = 0).

278

zyxw zyxw zyxwv zyxwv zyxw z zyxwvu

Our previous considerations can be presented as Theorem 1. T h e m a p T : SE(R) -+ D(R) i s one-to-one; the m a p T : V(R) -+ L,(R) is linear and the classical and quantum averages are coupled by the asymptotic equality (6). We emphasize that the correspondence between physical variables f E V(R) and physical observables A E L,(R) is not one-to-one.d Example. Let fi($) = (A$,$) and f2($) = sin(A$,$), where A E L,(R). Both these functions belong to the space of variables V(R). In the classical statistical model these variables have different averages: J,(A$, $)@($) # sin(A$1 $)dP($). But J,[(A1CI,1CI) sin(A$,$)]dp(s) = o ( ~ ) , Q+ 0. Therefore by using QM we cannot distinguish these classical physical variables. Moreover, nontrivial classical observables can disappear without any trace in the process of transition from the prequantum classical statistical model to QM. For example, let f($) = cos(A$,$) - 1. This is nontrivial function on 0. But, for any p E S z ( R ) , we have < f >p= o(a),a + 0. Thus in quantum theory f is identified with g E 0. Physical conclusions. Our approach is based on considering the dispersion a of fluctuations of classical fields as a small parameter. For any classical physical variable f($) , there is produced its amplification fa($) = Then the quantum average is defined as

s,

if($).

zyxwv

< A >quantum= Or-0 lim < fa >classical,

where A = frf”(0). So QM is a statistical approximation of an amplification of the classical field model (for very small fluctuations of vacuum).

3. Pure quantum states as Gaussian statistical mixtures In QM a pure quantum state is given by a normalized vector $ E H : 11$11 = 1. The corresponding statistical state is represented by the density operator: D+ = $ 8 $. In particular, the von Neumann’s trace-formula for expectation has the form: Tr D+A = (A$,4). Let us consider the correspondence map T for statistical states for the classical statistical model M a = (S8,V). A pure quantum state $ (i.e., the state with the density operator D+)is the image of the Gaussian statistical mixture p+ of states

zyxwvuts

dA large class of physical variables is mapped into one physical observable. We can say that the quantum observational model Nquantdoes not distinguish physical variables of the classical statistical model M a . The space V(C2) is split into equivalence classes of physical variables: f N g c) f”(0) = g”(0).

z zyxwvutsr zyxw 279

zyx zyxw

$ E H . Here the measure p$ has the covariation operator B$ = ~ D Q . This implies that the Fourier transform of the measure p$ has the form: & ( y ) = e-T(Yi*)2, y E H . This means that the measure p+ is concentrated on the one-dimensional subspace H$ = {z E H : z = s$, s E R}. Conclusion. Quantumpure states $ E H , 11$11 = 1, represent Gaussian statistical mixtures of classical states 4 E H . Therefore, quantum randomness is ordinary Gaussian randomness (so it is reducible to the classical ensemble randomness, cf. with von Neumann’s idea 111 about so called “irreducible quantum randomness”).

zyx

4. Pure states as one-dimension projections of spatial

white-noise In section 3 we showed that so called pure states of QM have the natural classical statistical interpretation as Gaussian measures concentrated on one-dimensional subspaces of the Hilbert space H . On the other hand, it is well known that any Gaussian measure o n H is determined by its onedimensional projections. To determine a Gaussian random variable [ ( w ) € H , it is sufficient to determine all its one-dimensional projections: & ( w ) = ( 1 1 , , [ ( ~ ) ) , $ E H . The covariation operator B of E (having the zero mean value) is defined by ( B $ ,$) = E[$. We are interested in the following problem: Is it possible to construct a Gaussian distribution o n H such that its one-dimensional projections will give us all pure quantum states, $ E H , II$lI = 1” We recall that in our approach a pure quantum state $ is just the label for a Gaussian random variable &, such that E[$ = c~11$11~. Thus the answer to our question is positive and pure quantum states can be considered as one-dimensional projections of the &-scaling of the standard Gaussian distribution on H . The standard Gaussian distribution p on H (so the average of p is equal to zero and cov p = I , where I is the unit operator) is nothing else than the white noise on R3 (if one chooses H = L2(R3)). Thus pure quantum states are simply one-dimensional projections of the spatial white noise. It is well known that the p is not a-additive o n the (Tfield of Bore1 subsets of H . To escape mathematical difficulties, we consider the finite-dimensional case. We consider the family of Gaussian random variables [$, 11, E R”, E& = 0, E[$ = all$111~. This family can be realized as & ( w ) = ($, [ ( w ) ) where [ ( w ) = &q(w) and ~ ( w E) R” is standard Gaussian random variable (so Eq = 0,cov q = I ) . For any $ E R”, we define the projection P$ to this

zy z

280

zyxwvu zyxwv zyxwvut zyxwv zyx

zyx z zyx

vector: P$(k)= ($, k)$. Let f : R” + R be a real analytic function of the exponential growth. Then we have:

W(f’l(O)$,$) +

Thus Ef(P+ = c(n)c(n)= (CZ(.N2

- (C?4(.N2

(14)

In the same way it follows that it there exist an hyperbolic number d ( n ) = d,(n) j d y ( n ) E G such that:

+

+

atin > = d(n)ln 1 > and hence:

< nla

=

< n + 11J(n)

(15)

zy zyxwv zyxw

zyx zyx zyx 285

n + 1 = < n + l I N J n +1 > = < n + lJataln+1 > = c ( n + l)E(n + 1) < 72 + lln 1 > = c(n + l)E(n + 1) =

+

(cz(n

Let us observe also that:

+ 1)12 - (cy(72. +

(17)

n = < nlNln > = < nlatuln >= < nlaat - 11n >

+ 1IZ(n)d(n)ln+ 1 > - < nln > = Z(n)d(n) < n + lln + 1 > - < > = Z ( n ) d ( n )- 1 =n= nln >)

i+

( n = < nlNln > = < nlataln > 2 0)

(19) (20)

Thus one cannot guarantee that the spectrum consists of natural numbers. In principle, it may contain a number of series of the form n k,k = 0, f l ,f 2 , ..., where n is in general a real number - the generator of the series. This problem needs more studies.

+

5. Hyperbolic-quantization of the electromagnetic field

Let us consider the Lagrangian density of electromagnetism: L := := a,A, - &A,. Introduced the mo- ~ F , w F ~-wJ,AP, where: F,, mentum conjugated to A,: IF := = -FPO one has the primary a& first-class constraint lo :

and the secondary first-class constraint:

zyxwv

din' = Jo (22) Canonical quantization in Hyperbolic Quantum Mechanics (that we will call canonical hyperbolic-quantization from here and beyond) is given by the ansatz: {a,

')Poisson

+

j [ * *I ,

(23)

zyxw zyxwvu zyxw zyxw zyx z zy

286

In particular the equation:

{Ap(z,t ) ,~

~ t)}Poisson ( 2 ,

=

dLJ(2 - 2)

(24)

gives, under canonical hyperbolic-quantization, the relation:

[Ap(2, t ) ,~ ~ t )(] = 3 j6;6(2 , -2)

(25)

The existence of the constraints (21) and (22) complicates significantly the situation. In particular (25) is not consistent with the imposition of the Coulomb gauge condition . A’ = 0, since it implies that:

e [e X(2,t ) ) Ti@’, , t ) ] = j 6 W 6 ( 2 - 2’) # 0 *

(26)

Instead of embarking ourselves in the complex machinery of quantization in presence of constraints, let us follow the trick l 1 of replacing 6 i j with a suitable tensor Aij such that:

[e. A(?,t ) ,~ ~ (t2) ] ’=, j A i j @ 6 ( 2- 2’) = 0

(27)

A trivial computation implies that:

One arrives, consequentially, to the commutation relations:

[Ai(Z,t),Aj(Z’,t)] = [ ~ i ( 2 , t ) , d ( z ’ , t )= ] 0

(30)

As in the complex case, it is more convenient to work in the momentum represent ation: [uT(z),a!(V)]

=

6r,S6z,zl

[ a , ( i ) , a , ( V > ] = [a:(z),a!(R)] = 0

(31) (32)

One has furthermore that: (33)

P

= -pV(lc,r)

(34)

zyxwvuts zyxwv zyxw zyx zyxw zyxwv 287

where:

N ( & r ) := a;(i)a,(i)

(35)

The hyperbolic-quantum radiation field may be seen as an infinite collection of uncoupled hyperbolic-quantum harmonic oscillators. B u t at the moment we are not able t o provide a n analogue of the conventional corpuscular interpretation of the quantized electromagnetic field through introducing a hyperbolic analogue of photon (see our study of the hyperbolic harmonic oscillator).

References 1. B. Jancewicz, The extended Grassmann algebra of R3, in: Clifford (geometric) algebras with applications in physics, mathematics and engineering, W.E. Baylis, editor, p.389-421 (Birkhauser, Boston, 1996). 2. G. Sobczyk, Introduction to geometric Algebras, in: Clifford (geometric) algebras with applications in physics, mathematics and engineering, W.E. Baylis, editor, p. 37-43 (Birkhauser, Boston, 1996). 3. A. Yu. Khrennikov and G. Segre, An introduction to hyperbolic analysis, math-ph/0507053 (2005). 4. A. Yu. Khrennikov, Contextual approach to quantum mechanics and the theory of the fundamental prespace, J. Math. Phys., 45, 902-921 (2004). 5. A. Yu. Khrennikov, Interference of probabilities and number field structure of quantum models, Annalen der Physik, 12,575-585 (2003). 6. A. Yu. Khrennikov, Hyperbolic quantum mechanics, Advances in Applied Clifford Algebras, 13,1-9 (2003). 7. A. Yu. Khrennikov, Supernalysis (Nauka, Fizmatlit, Moscow, 1997, second edition - 2005, in Russian; English translation: Kluwer, Dordreht, 1999). 8. V. S. Vladimirov and I. V. Volovich, Superanalysis, 1. Differential Calculus, Theor. and Mathem. Physics, 59, 3-27 (1984). 9. V. S. Vladimirov and I. V. Volovich , Superanalysis, 2. Integral Calculus, Theor. and Mathem. Physics, 60, 16%198 (1984). 10. M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992). 11. L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1996).

zyxw zyxwv

zyx zyxw zyxwv zyxwv

DISCRETE E N E R G Y SPECTRUM I N DISCRETE TIME DYNAMICS*

A. KHRENNIKOV International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vixjo, S-35195, Sweden E-mail: Andrei. KhrennikovQmsi.mu.se

Ya. VOLOVICH International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Vixjo, S-35195, Sweden Yaroslav. VolovichQmsi.m u s e

zyx

We present the classical-type dynamical model which poses some distinguishing properties of quantum mechanics. The model is based on a conjecture of existence of fundamental indivisible time quantum. The resulting formalism - discrete time dynamics - is constructed and used to study the energy spectrum for motion in central potential. We show that the spectrum is discrete and present a method of reconstruction of the corresponding potential for which the discrete time dynamics leads to the experimentally observed spectra.

1. Introduction

In this paper we try to obtain a classical-type dynamical model which resembles some distinguishing properties of quantum mechanics. We show how starting from modifying classical physics with the following natural postulate - time is discrete - one obtains discrete energy spectrum. In our previous investigations we have shown how the resulting formalism leads to quantum-like interference of particles (see [lo, 111 for more details and motivations, for discussions of distinguished properties of quantum mechanics, its probabilistic structure, and motivations for construction of classical-type models see [l,2, 3, 4,5 , 61).

zyxw zyxw

'This work is supported by Profile for Mathematical Modeling of VriXjo University and EU-network on QP and Applications.

288

zyxwvu zyxwvu zyxw z zyxwv 289

Let us start from considering the well known dynamical equation for an observable A = A(p,q )

DtA = { A , H }

where H = H(p,q ) is a Hamiltonian of the system and in the right hand side is a Poisson bracket. The left hand side of (1) is the same in both classical and quantum dynamics, it contains a continuous time derivative DtA = The construction of descrete dynamics is done with the help of discrete derivative which is postulated to be D,(')A = $ [ A ( t T ) - A ( t ) ] ,where T is the discreteness parameter. This parameter is finite and is treated in the same way as Plank constant in quantum mechanical formalism. The discrete time dynamics could be solved in the sense of the following relation

g.

+

A(t

+

T)

= A(t)

+ T { A ,H }

(2)

which provides the evolution of any dynamical function A = A(p,q ) . Note that in our model the coordinate space is continuous. As we will see discreteness of time enriches classical mechanics with some new properties which are usually thought as having quantum nature. In particular as it will be shown below in discrete time mechanics stationary orbits (i.e. finite motion) have discrete energy spectrum. We point that the phase space is assumed here to be a continuous real manifold.

zyx

2. Motion in central field with constant radius

Let us study the motion in central field U = U ( r ) with constant radius. Following the general approach described in the previous section we start from the classical Hamiltonian and then write the dynamical equations. In polar coordinates (r,'p) the Hamiltonian of the system in central potential U ( r ) is given by P,2

H =2m

P; ++ U(T), 2mr2

(3)

where p , and p , denote momenta corresponding to r and 'p - radial and angular coordinates respectively. Using (2) let us write the dynamical equa-

290

tions. We obtain

zyxwvu zyxwz zyxwvut zyxw T(t

+

T)

=T(t)

zyx

+ T Pr m

(4)

+ 7 P, 2

(6)

d t + 7) = cp(4

P,(t + 7) = P,(t) (7) We are interested in the periodic trajectories with a period which is multiple of our time quant

Tn

N

(8)

nT

As it was shown in [ll]equations (4)-(8) lead to discrete energy spectrum. Our task here is to answer the following question: which potential U ( r ) leads to a given energy spectra of the system. The concept is that the energy spectrum is observable and the internal potentials are to be determined from the experimental data. Thus our main aim is to find potentials such that classical discrete time dynamics would produce experimental energy spectrum, in particular, those predicted by quantum mechanics. Let us study the case of circular orbits, i.e. the case when pr = 0. jFrom (5) we find that

where the constant rameter T )

+ 27r which

zyxwv

Now, the condition for the period (8) gives us being combined with (6) and (9) gives

(p(nT)

= cp(0)

c is given by (note that < depends on discreteness pa-

The relation (10) contains the radius of the n-th orbit, we underline the dependence on n writing r = r(n). Finding p , from (9) and substituting it to (3) we find the expression for energy spectrum

E(n) =

1 r(n)2 -c+ U(r(n)), 2 n2

zyxwvu zyxw zy zyx zyxw 291

where the subscript n = 1 , 2 , . . . denotes that the quantities correspond to the n-th orbit. jF’rom (10) and (12) we want to find U as a function of T in terms of a given energy spectrum E(n). We proceed as follows (see also [ l l ] ) , assuming the continuous parameter n we take derivative of both parts of (12) in n,

where prime denotes the derivative in n. The use of (10) allows us to get a differential equation for r(n)in terms of only known quantities. We have 1 n2 2r(n)r’(n)- -T(n)2= --E’(n) I n

(14)

Introducing a new variable p = r2 we obtain a linear differential equation which could be rewritten as

which can be integrated to obtain r(n)= { i n (nE(n)-

ln

E(k)dk)

Equation (15) expresses n-th radius in terms of n, i.e. it has the form T = f ( n ) now if we invert it we relate n in terms of T, n = f-’(r), which if substituted to (12) gives an equation for U in terms of T only (actually in terms of r(n),but we perform interpolation effectively ignoring the fact that the relation strictly holds only for orbit radii). 3. Energy spectrum for various models 3.1. Physical energy spectra and corresponding potentials

The procedure described in the previous section allows us to find potentials which lead to known spectra. Here we show the reconstructed potentials for some interesting physical spectra, the computational details could be found in[ll]. a). Spectrum of unperturbed hydrogen atom

En=--, I n2

n=1,2,...,

zyx zyxw zy zyz

292

corresponds to the following potential U(T) = -

1

r2t

+ 2y

Note that this is not Coulomb potential, although it is interesting that unlike Coulomb one it is nonsingular for all T . b). Spectrum of harmonic oscillator (in 2 0 with frequency w and mass m)

EA = h ( n + l), n corresponds to the potential U(T) = -3T 2 / 3 2

..

=O,l,.

zyxwvu (+tiwlrJm

2/3

Here one may note the Ti constant in the potential, it appears due to its presence in the original energy levels.

zyxwvu

3.2. Energy spectrum for given potentials

For a given potential it is straightforward to compute corresponding energy levels. Indeed, from (10) we find T = rn and upon substitution to (12) we get En.Let us consider several common potentials which result in rather simple expressions for energy spectrum. a). Polynomial potential n (y>l U(T) = m u , Tn = -

, E

n - 21

- -a(2+u)

r23* -

The case u = -1 corresponds to Coulomb potential, for which we have

b). Logarithmic potential

zyxwv

U ( T )= a l n r , rn = n

8 ,

En = a

[i

+ln ( n n ]

This potential is interesting since it gives the linear dependence of n-th radius on n.

z

zyxwvut z zyxwvu zyx 293

4. Discussion and conclusions

There is an open problem of the quantitative value of the discreteness parameter T . One might speculate its relation with Plank time constant [7] the smallest measurable time interval in ordinary QM and gravity - which is given by x 5.3910-44 (sec.)

tpl =

Although currently there is no direct relation with this quantity. It is interesting to note that the fact that presented model requires potentials which are different from classical ones is not surprising. Here there is a similarity with the Bohmian mechanics[l] in which potentials are also very different from original classical ones. In the presented approach we start from the measurable quantities - energy spectrum - and construct the internals of the model - the potentials. A more interesting question would be to study the “classical” limit in order to see whether the effective classical potentials are obtained in this limit. As we already noted in[ll] there might be a deep interrelation between the energy-time uncertainty relations [8] and Bohr-Somerfeld quantization rules [9] in quantum mechanics and our discrete time model. Indeed, if one writes the Bohr-Somerfeld quantization rules for energy and time as canonical variables, then for a system with conserved energy one might get

zyxwvu z zy EnTn nti N

This relation is similar to period quantization condition (8) which we use in our model. In fact one may argue that if we make the r in equations of motion depend on the energy of the system as 7-

E = To-,

E where 7-0 is the “fundamental” time quantum and E a “fuqdamental” energy quantum, we get precisely the semiclassical quantization rules. The question arise how to treat the energy E here and what will happen with the dynamics. Acknowledgments The authors would like to thank B. Hiley, A. Plotnitsky, G. ‘t Hook, H. Gustafson, and K. Gustafson for discussions on quantum-like models with discrete time.

294

zyxwvuts zyxwvu zyxw zyxw zyxwvu zyxwv zyx

References

1. D. Bohm, B.J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory, London: Routledge & Kegan Paul. 2. L. E. Ballentine, Rev. Mod. Phys., 42, 358-381 (1970). 3. R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGrawHill, New-York, 1965. 4. G. ’t Hooft, Determinism beneath Quantum Mechanics, Proceedings of “QUO Vadis Quantum Mechanics”, Philadelphia, 2002, quant-ph/0212095 -, Quantum Mechanics and Determinism, hep-th/0105105 5. L. Accardi, “The probabilistic roots of the quantum mechanical paradoxes” in The wave-particle dualism. A tribute to Louis de Broglie on his 90th Birthday, edited by S. Diner, D. Fargue, G. Lochak and F. Selleri, D. Reidel Publ. Company, Dordrecht, 1984, pp. 297-330. 6. A. Yu. Khrennikov, J. Phys.A: Math. Gen., 34, 9965-9981 (2001) -, J . Math. Phys., 44, 2471- 2478 (2003) -, Phys. Lett. A , 316, 279-296 (2003) -, Annalen der Physik, 12,575-585 (2003) 7. C. Callender, N. Huggett (edditors), Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Grauity, Cambridge Univ. Press, 2001. C. Rovelli, Quantum Gravity, Cambridge Univ. Press, 2004. 8. L. Mandelstam, I.E. Tamm, J. Phys. (Moscow) 9, 249, (1945). D. Ruelle, Nuovo Cimento A 61, 655, (1969). P. Pfeifer, J. Frohlich, Rev. Mod. Phys., 67 759, (1995). 9. V.P. Maslov, M.V Fedoriuk, Semi-classical Approximations in Quantum Mechanics, Boston: Reidel, 1981 10. A.Yu. Khrennikov, Ya.1. Volovich, Discrete Time Leads to Quantum-Like Interference of Deterministic Particles, in: Quantum Theory: Reconsiderations of Foundations, ed. A. Khrennikov, Viixjo University Press, pp. 455 -, Interference as a statistical consequence of conjecture on time quant, quant-ph/0309012 -, Discrete Time Dynamical Models and Their Quantum Context Dependant Properties, J.Mod.Opt. Vo1.51, No.6-7, pp. 1113, (2004). 11. A, Khrennikov, Ya. Volovich, Energy Levels of “Hydrogen Atom” in Discrete Time Dynamics, in: Quantum Theory: Reconsiderations of Foundations-3, ed. A. Khrennikov, AIP Conference Proceedings, 2005 (in press).

zyxw zyxw zyxw

CONVOLUTION ASSOCIATED WITH THE FREE cash-LAW

ANNA DOROTA KRYSTEK *

Mathematical Institute University of Wroctaw pl.Grunwaldzki 2/4 SO-384 Wroctaw, Poland Anna. [email protected]

LUKASZ JAN WOJAKOWSKI*

Mathematical Institute University of Wroctaw pl.Grunwaldzki 2/4 SO-384 Wroctaw, Poland [email protected]

In this paper we consider the deformation of conditionally free convolution, connected with the free cosh-law. Because that measure is freely infinitely divisible, we can define a new, associative convolution using the theory from [5]. We calculate the central and Poisson measure for that convolution and show that the coefficients of the continued fraction form of the Cauchy transform for the central and Poisson limit measures of that convolution are equal to the respective coefficients of the underlying measure starting from the third level.

1. Definitions In this paper we continue the investigations on the convolutions arising from the conditionally free convolution of Bozejko, Leinert and Speicher [2] through deformations of the second measure. Those investigations started with the tdeformation studied in the papers of Bozejko and Wysoczaiiski [3, 41 and Wojakowski [lo]. This was generalized by Krystek and Yoshida [6]. Further examples were provided by Oravecz [7, 81 and Krystek and Wojakowski [ 5 ] . In the latter paper, the authors introduced a family of deformations depending on a selected compactly supported freely infinitely divisible measure, and proved limit theorems in terms of properties of the R-transforms of the limiting measures. In this work we calculate the central and Poisson measures for the convolutiondriven

zyxw zy

"Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-(JT-2002-00279.

295

296

zyxwvu z zyxw zy zy zyxw

by the free cosh-law, that is by the measure with density

The measure C is infinitely divisible with respect to the free convolution and is a particular case of the central measures for conditionally free convolution [2]. Those measures were also considered by [I] and called free Meixner laws. It can be calculated that Voiculescu’s RB-transform and Cauchy transform of the measure C are equal to

Gc ( z ) =

dC(x) 3 = Z-X

z - d m -

2(22

+ 1)

’

where w is the Wigner law. Let us start with the definition of the infinitely divisible deformation and of the respective convolution connected with the free cosh-law, see [5] for details, definitions and theorems in more general cases:

Definition 1.1. Let C be the free cosh-law and p any measure with compact support. Let us consider the following map p H ’Dip:

which depends on the second free cumulant of the measure p and on the nonnegative parameter t. For s 2 0 we understand by Cs the s-fold free convolution power of C having the following RB-transform:

RE ( z ) = s . R F ( z ) .

Definition 1.2. For compactly supported probability measures p, v we define their L o n v o l u t i o n by p m v = (p,Dofp)rn(v,’Dofv).

zyxwvu

By a result of [ 5 ] we have that the convolution

is associative.

2. Central limit theorem

Theorem 2.1. Let p be a compactly supported probability measure on the real line with mean Zero and variance equal to I . Let C be the free cosh-law. Then the

z zyxwvu 297

sequence

zyx z zy

q/fim. . O B l / f i P *

= (D1/fip7 v ! D l / f i p ) "' '

is * - weakly convergent as N

+ 00

'

(D1/mp7 v!D1/fip)

to the measure &, such that & = V!& and

qt,Ct)(4

= z.

zyx

The measure & is absolutely continuous with density f t t ( x )

t J(4

1

+ 4 t - x2)

for x E [ - 2 & T i 7 2 d T i 7 .

f d X=)G x 4 + ( t 2 - t - 2 ) x 2 + t + 1

Proof. Using the general central limit theorem from [5] we obtain that the limiting measure Et satisfies the relation

R g , c t ) ( 4= z. Because of

2 t Get ( z )

- = z1 Get

1+

(z)

Jm7

we have the following expression for the Cauchy transform of the measure Ct

GEt

=

(2

+ t )z - J t 2

+

(-4 - 4 t 2 2 ) 2 (t2 9 ) Using the relation between conditional transform and Cauchy transforms we obtain

Get ( z ) =

+

zyxwv +2) + Jt2

2 (t2 ( 2 t 2 - t - 2 ) z + 223

( 9- 4 t - 4 ) ( 2 t 2 - t - 2 ) + 2 z3 - J.t 2 (.2 2 - 4 t - 4 ) - , 2 (z4+ ( t 2 - t - 2 ) 2 2 + 1 + t )

The appropriate choice of the branch of a square root gives in limit for real z = x

Jt2

(z2 - 4 t - 4 ) =

tJx2-4t-4 -tJx2 - 4 t - 4 itJ4t+4-x2

forx>2&Ti7 forx 5 - 2 & T i , for - 2 2 < x < 2 2 .

298

zyxw zy zyx zyxw zyxw zyxwvuts

We would like to find the atoms of the measure &, that is the zeros of the denominator of the Cauchy transform of the measure & . There is no atoms if either

(t2- t - 2 ) - 4 (1 + t ) < 0, 2

or

(t2 - t - 2 ) 2 - 4 (1+ t ) 2 0, that is when

and

that is for

t

E

(0,3)

(t2- t - 2 ) > 0 and (1+ t) > 0,

t 2 3.

Thus we have atoms only for t = 0. By Proposition in chapter X111.6 in [9] one can find that there is no singular parts of the measure &. It follows from regularity properties of Gtt (z). The density fct (x),x E R of the absolute continuous part of the measure & can now be calculated by the Stieltjes formula 1

fcL(x) = -- lim SGc, ( x + i ~ ) . 7r

€+o+

Thus we obtain

zyx

Moreover we can calculate that the measure & has the following continued fraction expansion 1

GEt ( z ) =

1

z-

L

z-

l+t l+t

z2--

z

-

..

A diagram of this measure for t = 1is presented on the following figure.

3. Poisson limit theorem Theorem 3.1. For X > 0 define for all N

0

z zyx 299

Figure 1 . Density of the central limit measure for the cosh-convolution

where

zyxw zyxw zyxw zyx zyxw

The absolutely continuous part of the measure px is given by

and where

= 2 ~ - - 5 3 (4+4x+2tx)

+ z2 (2 + 4X + 2 t X + 2X2 + 2 t X 2 + 2 t 2 A')

- 4 t 2 X 3z

+ 2t2X4.

The measure px may have at most four atoms.

Proof. Using the fact that the limiting measure px is determined by the relation [51

and because of

22

Gcxt ( z ) =

+ t z x - t Ad22 - 4 -4 t x 2 2 2 + 2 t 2 A2

7

300

zyxwvu zyxw zyx zy zyxwvut

we obtain

2X

1

--

GPX ( z ) Hence

G,,

- 222 - z (2

(2+tZX2)

+ t A) + t X

2 2 - z (2+tX)+tX ( (2)=

where

223 - 2 t 2 X3 - z2 (2

Z

zy

+ dz2- 4 - 4 t X )

(2tX

t

X

+ (2 + t ) A) + t z X

+

(2tX

d

'

W

+ dz2- 4 - 4 t A)

zyxw zyxwvut zyxw

Q ( ~= )2 z 4 - z 3 ( 4 + 4 X + 2 t X )

+ z 2 (2 + 4 X + 2 t X + 2 X2 + 2 t X2 + 2 t2A')

- 4 t 2 X3 z + 2 t 2 X4.

The denominator of the Cauchy transform of the measure p~ may have four simple real roots x j , thus the measure p~ has at most four atoms dZj with respective weights. The actual dependence of atoms and weights on X and t is rather complicated. The density f p , (x), x E R of the absolutely continuous part of the measure p~ can now be calculated by the Stieltjes formula 1

f p , ( x ) = -- lim SG,, ( x + i ~ ) 7r €+Of

- X JtW -

(4

+ 4 t X - x2)

7r Q ( x )

for x E [ - 2 4 X i i , 2 J l T I 4 . Moreover, we obtain the following continued fraction form of the Cauchy transform of the measure p x : 1

GP, (2) =

X

z-A-

t X

z-1-

1+tX

z-

l+tX 2-2

-

-.

zyxwvu zyxw zyx zyx zyxwv 30 1

References

1. M. Bozejko and W. Bryc, On a class of free Le'vy laws related to a regression problem, accepted for publication in JFA, 2005 2. M. Bozejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditinallyfree random variables, Pac. J . Math., 175no.2, (1996). 357-388 3. M. Boiejko and J. Wysoczaiski, New examples of convolutions and noncommutative central limit theorems, Banach Cent. Publ., 43, (1998), 95-103 4. M. Bozejko and J. Wysoczaiski, Remarks on t-transformations of measures and convolutions, Ann. Inst. Henri Poincare Probab. Stat., 37 (6), (2001), 737-761 5. A. D. Krystek and t.J. Wojakowski, Associative convolutions arising from conditionallyfree convolution, IDAQP, 8 no.3,(2005), 515-545 6. A. D. Krystek, and H. Yoshida, Generalized t-transformatonsof probability measures and deformed convolution, Probability and Mathematical Statistics, 24 no. l, (2004), 97-1 19 7. E Oravecz, The number of pure convolutions arising from conditionallyfree convolution, IDAQP, 8 no.3,(2005), 327-355 8. F. Oravecz, Pure convolutions arising from conditionally free convolution, preprint, 2004 9. M. Reed and B. Simon, Methods of Modem Mathematical Physics I y Analysis of Operators,AcademicPress, New York, 1978 10. t.J. Wojakowski, Probability Interpolating between Free and Boolean, Ph.D. Thesis, University of Wrodaw, 2004

zyxw zyx

A THEOREM ON LIFTINGS OF STATISTICAL OPERATORS

J. KUPSCH Fachbereich Physik, T U Kaiserslautern 0-67653 Kaiserslautern, Germany

In this note we investigate the problem of calculating the state of a system if the state of a subsystem is given.

1. Introduction

In quantum mechanics the states of a physical system are given by the statistical operators or density matrices in the Hilbert space associated to this system. It is well known that the state of a subsystem is uniquely calculated as the reduced statistical operator by the partial trace. But the inverse problem: to define an affine linear mapping from the set of states of a subsystem into the set of states of an enlarged system such that the reduced state coincides with the original state, has been studied in the literature only recently. These notes present the main results on this lifting problem on the basis of Refs. [l,21 and [3]. We use the following mathematical notations. Let 3-1 be a separable complex Hilbert space, then C(3-1) is the real Banach space of bounded selfadjoint operators, and Cl(1-I) is the real Banach space of selfadjoint nuclear operators. The respective norms are the operator norm and the trace norm. The quantum mechanical state space is the convex set D ( 7 f )of positive trace one operators on 3-1. The subset of rank one projectors on 7 f is denoted by p (3-1).

zyx zy zyxw zyxwvut

2. The problem of lifting

Let 7f and G be two complex Hilbert spaces, then F : D ( 7 f ) afine linear mapping if for W1, W2 E D(3-1)

F(X1Wl + X2W2) = XlF(W1) 302

--f

+ X2F(W2) E D(G)

D(G) is an

zyxwvu zyxwvut zyxwvuts zyxwv zyxwvu zyx zy 303

is true for all XI, XZ 2 0 , XI

+ Xz = 1.

Remark 2.1. For the importance of affine linearity in quantum mechanics see Ref. 4. Any affine linear mapping F : D(X)4 D ( E ) can be extended to a contraction mapping Ll(7-l)+. Ll(C7). See e.g. Ref. [5]. Now we consider a total system with Hilbert space 'H = 'Hs '8 ' H E . The system we are interested in has the Hilbert space 'Hs, the environment of this system has the Hilbert space ' H E . The main statement of this note is

-

Theorem 2.1. Let F : D('Hs) D(7-l~@ ' H E ) be a n a f i n e linear mapping such that trwEF(WS) = Ws f o r all Ws E D ( 7 - t ~ ) .Then there exists a n element WE E D ( X E ) such that F(Ws) = ws '8 WE.

(1)

zyx

The product ansatz (1) has been used as an obvious solution of the problem since a long time, see e. g. Refs. [6] or 7. The new result is that there are no other solutions. The Theorem has first been derived by Pechukas [l]for the two dimensional Hilbert space 3-1 = U?. A general proof was given in Ref. [2] (without knowing the publication of Pechukas) and then in Ref. [3]. The proof presented here borrows some ideas from Refs. [l]and [3]. The proof of Theorem 2.1 is given in three steps. The first step is Lemma 2.1. Let W E D ( 7 - l~ @'HE) with trEW = Ps E P('Hs) then there exists a statistical operator WE E V ( 7 - t ~ such ) that W = Ps @WE.

Proof. Let W E D ( ' H s ' ~ ' H Ebe ) the statistical operator of the total system with the reduced state Ws = trEW = PS E ?(as) c D('Hs). Then tr?-ls@%~(PS '8 I E ) w ( P s '8 I E ) = t r X s (trE(Ps '8 IE)w) = t r ~ , P=~1

(2)

follows. The identity (2) implies that the range of the projector Ps '8 I E includes the range of the statistical operator W, and we have (PS @ IE)w(pS '8 I E ) =

w.

(3)

Since Ps has rank one, the space Ps'Hs is spanned by a single normalized eigenvector f s = Ps fs E 'Hs. All vectors in the range of Ps '8 IE have the structure $ = fs '8 g with some vector g E 'HE. As a consequence of (3) all eigenvectors of W with a non-vanishing (i.e. positive) eigenvalue must

zyxwv zyxwvu zyxwvuts z zyxw zyx zyx zyxw

304

have this form fs €3 g. The selfadjoint nuclear operator W has therefore the structure W = Ps @WE with WE = t r s W E ~ ( X E ) . 0

Remark 2.2. Generalizations of Lemma 2.1 can be found in Ref. [8].

Let F be the affine linear mapping of Theorem 2.1, then Lemma 2.1 implies that F ( P s ) = Ps C3 WE'

(4)

for all Ps E P('Hs),where WE' E D ( X s ) may depend on Ps. In the next step we exclude this dependence.

Lemma 2.2. I f F : D(lHs) D ( X s €3 ' H E ) is a n a f t n e linear mapping with the property F ( P s ) = Ps €3 WE' f o r all PS E P ( X s ) , t h e n WE' = WE does n o t depend o n P s .

Proof. Take two projection operators PI and P2 of P ( X s ) , PI # P2, then there exist two normalized linearly independent vectors fj E Xs,11 = 1, j = 1 , 2 , such that Pj fj = fj. We can choose the phases such that the imaginary part of the inner product (f1 1 fz) vanishes. Defining the normalized orthogonal vectors f3 = c3( f~ f2) and f 4 = C4(f1 - f2) with constants c3,4 > 0 we have

fjII

+

fi

= af3

zyx

+ Pf4, f2 = of3 - Pf4 with a,P 2 0, 'a + P2 = 1.

The projection operators P k onto f k , k = 1,..., 4, satisfy the identity PI

+ P 2 = a2P3 + @P4.

(5) By assumption the mapping F is affine linear and can be extended to a linear mapping Cl(7-l~)-+ C1('Hs €3 ' H E ) , which we also denote with F, see Remark 2.1. Hence we have F(P1 + Pz) = F(P1) + F(P2) and F(a2P3+ P2P4) = a2F(P3) + P2F(P4). Then the assumption of Lemma 2.2 and the identity (5) imply PI

C3 WI

+ P 2 €3 w, = a 2 P 3 8 w3 +PZP4€3 w4

with some statistical operators

w k

(6)

E D(XE).

Let P(6) be the projection operator onto the unit vector af3 + Pei* f4, 6 E R. As a consequence of (6) the partial traces trs(P(6)gIE) ( P I €3 WI ~2 ~ 2 =) \a2+ p2ei* WI+la2 - p2ei.9 wZ and trs(P(6) C3 IE) (a2&C3 W3 P2P4 €3 W4) = a4W3 P4W4 define the

+

+

l2

+

l2

zyxw zyx z zyxwv zyx z - zyxwv zyxwvu 305

same operator. But that leads to the identity (W1 - Wz)cos.9 = 0 for 19 E R, and W1 = Wz follows. Hence we have derived

F(PS) = PS €3 WE

(7)

with a statistical operator W E ,which does not depend on Ps.

In the last step of the proof of Theorem 2.1 the identity (7) is extended to (1) by affine linearity.

Remark 2.3. To derive the theorem only af€ine linearity is used; continuity and complete positivity are consequences. 3. Restriction mapping for observables

The Theorem 2.1 is equivalent to a statement for observables. To formulate this statement we first extend F to a linear mapping on C,('Hs), see Remark 2.1. Let F be a continuous linear mapping Ll('Hs) +Ll('Hs@"HE), then there exists a unique linear mapping F* : L('Hs €3 ' H E ) -+ L('Hs), continuous in the uniform and in the ultraweak topology, such that I W)N, = (A I ~(W)),,@N,for all A E C(7-f~€3 ' H E ) and all W E C l ( H s ) . Thereby ( B I W ) , = tr,BW is the standard bilinear pairing C ( H ) x C I ( H ) -+R.

P*W

Lemma 3.1. Let F : Ll('Hs) Ll(7-l~ €3 XE)be a continuous linear mapping and let F* : C('Hs @ ' H E ) + C('Hs) be its adjoint mapping, then F * ( B €3 I E ) = B f o r all B E L('Hs) i f and only i f tr,,F(W) = W f o r all

w E .Cl('HS).

Proof. The proof is straightforward, see Ref. [2].

0

Theorem 2.1 and Lemma 3.1 imply the following theorem for the restriction of an observable [2].

Theorem 3.1. If R : C('Hs €3 ' H E ) -+ L ( X s ) is a linear mapping, continuous in the ultraweak topology, and i f R ( B @ I E ) = B is true f o r all B E L ( X s ) then there exists a n element WE E D ( 7 - l ~such ) that

zyxwvut R(A) = trx,A(Is €3 W E )

f o r all A

E

L ( X s €3 'HE).

Remark 3.1. The restriction R is completely positive.

(8)

zyxwvut zyxwvu zyxwvu zyxwvu zyxw zyxw zyxwv z

306

4. Remarks about Zwanzig projectors

There is a more general notion of defining a subsystem based on the projection operators of Nakajima 101 and Zwanzig [ll]. These projection operators P are defined on the Banach space of trace class operators, P : Lx(3-1) -+ L1(3-1),such that P(D(3-1))c D(3-1). But one can also define adjoint Zwanzig projectors Q on the Banach space of observables, Q : L(3-1)4 L(3-1),which satisfy, see Sect. 7.2 of Ref. [12],

Q is a continuous linear operator onC(x), Q2=Q, QA>OifA>O, Q I = I .

(9)

Since QL(3-1)= B is a closed subspace of the Banach space L(3-1),the definition of an operator Q solves a restriction problem, which is more general than the problem of Theorem 3.1, as can be seen from the following example.

Example 4.1. The Hilbert space of a composite system is 3-1 = 3-1s@ 3-1~. Let WE E D(3-1~) be a reference state of the environment, then

A

E

L(3-1~ @%HE)-+ QA := (trEA(Is @ W E ) @ ) IE E L(3-1~ @ 3-13) (10)

is an operator on L(3-1),which has the properties (9). The non-trivial factor of the projection operator (10) A -+ trnA(I1 @W2) is exactly the restriction mapping (8). The restriction mapping of Theorem 3.1 is always completely positive. But there are Zwanzig projectors (9) which are not completely positive, see Sect. 7.7.2 of Ref. [12]. Hence using Zwanzig projectors the dynamics of subsystems might not be completely positive.

References 1. P. Pechukas. Reduced dynamics need not be completely positive. Phys. Rev. Lett., 73:1060-1062, 1994. 2. J. Kupsch, 0. G. Smolyanov, and N. A. Sidorova. States of quantum systems and their liftings. J . Math. Phys., 42:1026-1037, 2001. 3. T. F. Jordan, A. Shaji, and E. C. G. Sudarshan. Dynamics of initially entangled open quantum systems. Phys. Rev. A , 70:052110-1-14, 2004. 4. G. W. Mackey. The Mathematical Foundations of Quantum Mechanics. Benjamin, New York, 1963. 5. W. Guz. On quantum dynamical semigroups. Rep. Math. Phys., 6:455-464, 1974. 6. C. Favre and P. A. Martin. Dynamique quantique des systhmes amortis >. Helv. Phys. Acta, 41:333-361, 1968.

zyxwvut zyxw zyxwvu zy zyxwvut 307

7. E. B. Davies. Quantum Theory of Open Systems. Academic Press, London, 1976. 8. D. Giulini. Elementary properties of composite systems in quantum mechanics. In Ref. [9], pages 407-414. 9. E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I. 0. Stamatescu. Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin, 2nd edition, 2003. Corr. 2nd printing 2005. 10. S. Nakajima. On quantum theory of transport phenomena. Prog. Theor. Phys., 20~948-959, 1958. 11. R. Zwanzig. Statistical mechanics of irreversibility. In W. E. Brittin, B. W. Downs, and J. Downs, editors, Boulder Lectures in Theoretical Physics, Vol. 3 (1960), pages 106-141, New York, 1961. Interscience. 12. J. Kupsch. Open quantum systems. In Ref. [9], pages 317-356.

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POSITIVE MAPS BETWEEN M z ( C ) AND M n ( C ) . ON DECOMPOSABILITY OF POSITIVE MAPS BETWEEN M z ( C ) AND M,(C)*

WlADYSlAW A. MAJEWSKI Institute of Theoretical Physics and Astrophysics, Gdarisk University, W i t a Stwosza 57, 80-952 Gdarisk, Poland [email protected]

zyxw zyxw zyxwvuts z z

MARCIN MARCINIAK Institute of Theoretical Physics and Astrophysics, Gdalisk University, W i t a Stwosza 57, 80-952 Gdarisk, Poland [email protected]

+

A map 'p : Mm(@)+ Mn(C) is decomposable if it is of the form 'p = 'pi 9 2 where 'pi is a C P map while 'p2 is a cc-CP map. A partial characterization of decomposability for maps 'p : M z ( @ )4 M3(@)is given.

1. Introduction

z zyxwv

zyx

Let cp : M,(C) -+ Mn(C) be a linear map. We say that cp is positive if p(A) is a positive element in Mn((c) for every positive matrix from Mm(C). If k E W, then cp is said to be Ic-positive (respectively k-copositive) whenever [cp(Aij)]$=l (respectively [cp(Aji)]!,j=I) is positive in Mk(M,(C)) for every positive element [Aij]f,j=l of Mk(M,(C)). If cp is k-positive (respectively Ic-copositive) for every k E N then we say that cp is completely positive (respectively completely copositive). Finally, we say that the map cp is decomposable if it has the form cp = ( ~ 1 9 2 where cp1 is a completely positive map while cp2 is a completely copositive one. By P(m,n) we denote the set of all positive maps acting between Mm(C) and M,((c) and by PI(,, n) - the subset of P(m,n) composed of all positive

+

*Supported by KBN grant PB/1490/P03/2003/25. The authors would like also thank the support of EU RTN HPRN-CT-2002-00729.

308

zyxwv zyx zyxw 309

unital maps (i.e. such that 'p(1) = I). Recall that P (m,n) has the structure of a convex cone while PI(,, n) is its convex subset. In the sequel we will use the notion of a face of a convex cone.

Definition 1.1. Let C be a convex cone. We say that a convex subcone F c C is a face of C if for every c1, c2 E C the condition c1+ c2 E F implies C I , C ~E F . A face F is said to be a maximal face if F is a proper subcone of C and for every face G such that F C G we have G = F or G = C.

zyxwvu z zyxw zyx

The following theorem of Kye gives a nice characterization of maximal faces in P ( m ,n).

Theorem 1.1. [6] A convex subset F c P ( m , n ) is a maximal face of P(m,n) if and only zf there are vectors E E Cm and r] E Cn such that F = F C , where ~

and Pc denotes the one-dimensional orthogonal projection in Mm((C) onto the subspace generated by the vector 2 or n > 2 the problem of finding decomposition is still unsolved. In this paper we consider the next step for solving this problem, namely for the case m = 2 and n = 3 . Our approach will be based on the method of the so called Choi matrix. Recall (see [3, l o ] ) for details) that if cp : Mm -+ Mn is a linear map and {Eij}$j.=l is a system of matrix units in Mm(C),then the matrix

+

Hq =

[~(Eij)]$=E l Mm(Mn(c)),

(1.2)

is called the Choi matrix of cp with respect to the system {Eij}. Complete positivity of 'p is equivalent to positivity of H, while positivity of 'p is equivalent to block-positivity of H, (see [3, lo]). Recall (see Lemma 2.3 in [lo]) that in the case m = n = 2 the general form of the Choi matrix of a

310

zyxwvu zyxwv zyxwvut zyxwvu zyxwvut

positive map cp is the following

where a, b, u 2 0 , c, y , z , t E CC and the following inequalities are satisfied:

(1) Icl2 I ab, (2)

PI2 I bU,

(3) IYI

+

IZI

I(aUy2.

It will turn out that in the case n = 3 the Choi matrix has the form which similar to (1.3) but some of the coefficients have to be matrices. The main result of our paper is the generalization of Lemma 2.3 from [lo] in the language of some matrix inequalities. It is worth to pointing out that technical lemmas leading to this generalization are formulated and proved for more general case, i.e. for cp : M2(C) -+ Mn+l(@)where n 2 2. 2. Main results

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In this section we will make one step further in the analysis of positive maps and we will examine maps cp in Pl(2,n+ 1) where n 2 1. Let { e l , e2} and { f l ,f 2 , . . . ,f n + l } denote the standard orthonormal bases of the spaces CC2 and CCn respectively, and let {Eij}& and {Fkl};:Il be systems of matrix matrix units in Adz(@)and Mn+l(C) associated with these bases. We assume that cp E F C ,for ~ some E E CC2 and 7 E By taking the map A ++ V*cp(WAW*)Vfor suitable W E U ( 2 ) and V E U ( n 1) we can assume without loss of generality that [ = e2 and 7 = f l . Then the Choi matrix of cp has the form

+

. . .. ..

. ..

..

.. .

H=

. . ..

zyxw

zyxw zyx zyx 311

We introduce the following notations:

c = [ C l c 2 . .. c n ] ,

Y=

[y1

y2

.. . y n ] ,

2 = [z1 z2

. .. z n ] ,

.Uln

u11 u12

*.

U n l un2

. . . unn

The matrix (2.1) can be rewritten in the following form

The symbol 0 in the right-bottom block has three different meanings. It

roi

zyxwv zyxwvut

Proposition 2.1. Let cp : Adz(@) -+Mn+l(@) be a positive map with the Choi matrix of the form (2.2). Then the following relations hold: (1) a 2 0 and B , U are positive matrices, (2) if a = 0 then C = 0 , and if a > 0 then C*C 5 aB, (3) 2 = 0,

(4) the matrix

[TI T* U

E

M z ( M n ( @ ) )is block-positive.

[:]:

Proof. It follows from positivity of cp that blocks on main diagonal, i.e. a C cp(E11)= c* and cp(E22) = , must be positive matrices. This

[

zyxwvzyuts

zyxwv I;:[ zyxw zyxw z zyxwv

312

immediately implies (1) and ( 2 ) (cf. [14]). From block-positivity of H we conclude that the matrix =

[ (fly

1

cp(E1l)fl) ( f l l cp(E12)fl)

(fll cp(E21)fd (fll cp(E22)fd

is a positive element of Mz(C). So IC = 0, and ( 3 ) is proved. The statement of point ( 4 ) is an obvious consequence of block-positivity of H. 0

For X =

. . . 5,]

[ X I5 2

E M I , ~ ( Cwe ) define llXll =

(CZ, lzi12)1/2.

By 1 x1 we denote the square ( nx n)-matrix (X*X)1/2.Let us observe that for any X E Ml,,(C) we have

1x1 = IlXll%

where = IIXII-lX* and Pc denotes the orthogonal projection onto the one-dimensional subspace in Cn generated by a vector 5 E C" (we identify elements of Mn,1(C)with vectors from en). We have the following

Lemma 2.1. A map cp with the Choi matrix of the form

zyxw

is positive if and only i f the inequality

+ (z*,rT)+ n(nrqI2

I(y*,rT) < [Qu

[

+ n( K B )+ 2 x ( c * , r ~ n ) ](AW)

(2.4)

holds for every cr E C, matrices r = [ y1 79 . . . m ] and A11 A12 . . . A1" A = A21 A22 . . * A2n ] , ~ ~ ~ C , A i ~ € C f o r 2 ,,..., ~ =n,l 1suchthat 2

zyxwvu ~

An1 (1)

(2)

Q

An2

.

* *

Ann

2 0 and A 2 0 ,

r*r5

The superscript r denotes the transposition of matrices. Proof. Obviously, the map cp is positive if and only if w o cp is a positive functional on M2(C) for every positive functional w on Mn+l(C).

zyxwvut z zyxwvu zyx zyxwvu 313

Let w be a linear functional on Mn+l(C). Recall that positivity of w is equivalent to its complete positivity. Hence, the Choi matrix H, = [ ~ ( F i j ) ] : ; : ~of w is a positive element of Mn+l(C). Let us denote a = w(F11), yj = w ( F l j + l ) for j = 1,2,. . . ,n,X i j = w(Fi+l,j+l) for i , j = 1,2,. . . ,n and and A are defined as in the statement of the lemma. Then, we came to the conclusion that w is a positive functional if and only if the matrix

zyxwv

r

*; [

I;]

is a positive element of Mn+l(C). This is equivalent to the conditions (1) and (2) from the statement of the lemma. Similarly, if w’ is a linear functional on M z ( C ) , then its positivity is equivalent to positivity of the matrix [ ~ ’ ( E i j ) ] & =Consequently, ~. w’ is positive if and only if w’(Eii) 2 0 for i = 1,2, w’(E21) = w’(E12) and IW’(E12)I2

I W’(Ell)W’(E22)

(2.6)

(cf. Corollary 8.4 in [ l l ] ) . Now, assume that a, r and A are given and the conditions (1) and (2) are fulfilled. In described above way it corresponds to some positive functional w on Mn+l(C). Let w’ = w o p. Then n

zyxw zyx i,j=l

= aa

+ Tr(A‘B) + 2%(C*,r T )

c n

w’(E22) =

XijUij

= Tr(ATU)

i,j=l n

=

n

n

(Y*, r T+ ) ( Z * ,rT)+ Tr(A7T).

(2.9)

It follows from the above equalities that (2.6) is equivalent to the inequality (2.4). Hence, the statement of the lemma follows from the remark contained in the first paragraph of the proof. 0

Proposition 2.2. If the assumptions of Propositions 2.1 are fulfilled, then

I Y+I 1215 a1/2U1/2.

(2.10)

zyxwvut zyxwv zy

314

zyxw zy

Proof. Firstly, let us observe that the inequality (2.4) can be written in the form

+

I ( Y * , P ) (z',rr)12+/Tr(A~T)12+2X[ ( ( Y * J T ) + ( Z * , P ) )Tr ( A T ) ]

5 [aa+ Tr (A'B)

-r

+ 2X(C*,r')]Tr ( A v . l )

r

Putting instead of we preserve the positivity of the matrix (2.5) and the above inequality takes the form

12+

I ( Y * r') , + (Z*, P)

[(

+

ITr (ATT)12-2% (Y*,rT) (Z*, r.,) Tr ( A T ) ]

5 [a0+ Tr (ATB) - 2X(C*,r')]Tr ( A V )

If we add both above inequalities and divide the result by 2, then we get

and consequently

+ +

+

I(Y*,r y 2 I(z*,r7)12 2 X ( Y * ,T')(Z*, F') 5 aaTr(A7U) Tr(A'B)Tr(A'U) - p?r(A'T)12

(2.11)

Now, let rl be an arbitrary unit vector from C". Then

hence

zyxw zyxw ( Y *F , ) ( Z * ,r') = I(Y*,r~)lI(z*, 171.

Put also A = Er*r.Then from (2.11) and (2.12) we have

ll(lYl+1z1)q112I a T r ( ( r * r ) ' U ) + E 2 (Tr(A'B)Tr(ATU) - /Tr(ATT)12) Since E is arbitrary then we have

Lemma 2.2. Let cp : Mz(C) 4 Mn+l(C) be a linear m a p with the Choi matrix of the form (2.3). Then

z zyxwvu

zyxwvu [;*,.':I 315

(1) the map 'p is completely positive i f and only if the following conditions hold: ( A l ) Z = 0,

(A2) the matrix

C* B T

is a positive element of Mzn+l(@).

I n particular, the condition (A2) implies:

(A3) i f B is a n invertible matrix, then T*B-lT 5 U , (Ad) C ' C I a B , (AS) Y*Y 5 aU, (2) the map cp is completely copositive i f and only if ( B l ) Y = 0,

:*[ :I

zyxwv zyxwv

(B2) the matrix

C* B T* is a positive element of M2n+l(C).

I n particular, the condition (B2) implies:

(B3) if B is invertible, then TB-lT* 5 U , (B4) C * C I a B , (B5) Z*Z 5 a U ,

z zy

Proof. It is rather obvious that the conditions (Al) and (A2) imply positivity of the matrix (2.3), and consequently the complete positivity of the map cp. On the other hand it is easy to see that (A2) is a necessary condition for positivity of the matrix (2.3). In order to finish the proof of the first part of point (1)one should show that positivity of (2.3) implies Z = 0. Let L1 be a linear subspace generated by the vector f 1 and let L2 be a subspace spanned by f 2 , f 3 r . . . ,f n + l , so Cn+' = L1 @ L2. Any vector wE can be uniquely decomposed onto the sum w = dl) d 2 )where , dz) E Li, i = 1 , 2 . Blocks of the matrix (2.3) are interpreted as operators. Namely: B,T, U : L2 4 L2, C,Y,Z : L2 + L1, and a : L1 -+ L1. Recall (cf. [13]), that the positivity of the matrix (2.3) is equivalent to the following inequality

+

en+'

+ (w2,

[;;I

.,)

2 0

316

zyxwvu zyxwvuts zyxwvutsrq zyxwvut zyxwvu zyxw zyxwv z

for any vl,212 E (Cn+l. This is equivalent to

( u p7 a v y )

+ (up B v p ) + ( v p

U?p)

+ 2~(vj'),C~1~))+2X(v~~),Y~~))+2X(v~), Zviz))+2X (v$2)lTvp)) 2 0

(2.13)

+

vjl) vj2) for j = 1 , 2 , and ~ i ' ) ~ v ; ) E L1 and vr),v p ) E L2. Assume that v;') = 0, v p ) = 0, vi2) is an arbitrary element of L2, and = -rZv?) for some T > 0. Then (2.13) reduces to

where

ZIP)

vj =

( v p 7 B v j 2 ) ) - T I j Z v q 2 2 0.

(2.14)

This inequality holds for any vi2) E L2 and T > 0. It is possible only for = 0. To show the second part of the point (1) one needs to notice that posi-

z

implies the positivity of the matrices

tivity of the matrix

B T [T* U ] *' : [

i] [Y"*i]* and

Recall that the map is completely copositive if and only if the partial transposition of the matrix (2.3) is positive. The partial transposition (cf. [lo)) of (2.3) is equal to

p;$p Z*TO U

zyxwvutsr

So, to prove the point (2) we can use the arguments as in proof of the point (1).

Now, assume that cp : M2(@)+ A&(@) is a unital positive map, and cp E F e z , f l Hence . its Choi matrix has the form

(2.15)

zyxwvuts zyx zyxw zyxw zyxw zyxwv 317

+

where B and U are positive matrices such that B U = 1 and conditions listed in Propositions 2.1 and 2.2 are satisfied. From the theorem of Woronowicz (cf. [14]) it follows that there are maps cp1,cpz : Mz(C) --+ M3(Qj)such that cp = + cpz, and cp1 is a completely positive map while cp2 is a completely copositive one. From Definition 1.1 we conclude that both and cp2 are contained in the face F e , , f l . So, from Proposition 2.2 it follows that their Choi matrices H1 and H2 are of the form

+

where ai, Bi, Ui 2 0 for i = 1 , 2 , and the following equalities hold: a1 a2 = a , TI T2 = T , B1 B2 = B and U1 U2 = U . In [lo] we proved that if cp : Adz(@.)+ M2(C) is from a large class of extremal positive unital maps, then the maps cp1 and cp2 are uniquely determined (cf. Theorem 2.7 in [lo]). Motivated by this type of decomposition and the results given in this section (we ‘quantized’ the relations (1)-(3) given at the end of Section 1) we wish to formulate the following conjecture: Assume that cp : M2(C) + M3(C) is a positive unital map such that U # 0 , Y # 0, 2 # 0 and (Y( (21= U1/’. Then the decomposition cp = cp1 cp2 onto completely positive and completely copositive parts is uniquely determined.

+

+

+

References

+

+

z zyxwv

1. E. M. Alfsen and F. W. Shultz, State spaces of operator algebras, Birkhauser, Boston, 2001 2. 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer Verlag, New York-Heidelberg-Berlin, vol. I (1979), vol. I1 (1980) 3. M.-D., Choi, Completely Positive Maps on Complex Matrices, Linear Algebra and its Applications 10 (1975), 285-290. 4. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras, Illinois J. Math. 18(4) (1974), 565-574. 5. M.-D. Choi, Some assorted inequalities for positive maps on C*-algebras, Journal of Operator Theory 4 (1980), 271-285. 6. S-H. Kye, Facial structures for the positive linearmaps between matrix algebras, Canad. Math. Bull. 39 (1996), 74-82. 7. L. E. Labuschagne, W. A. Majewski and M. Marciniak, O n kdecomposability of positive maps, t o appear in Expositiones Math, e-print: math-phyd0306017.

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8. W. A. Majewski and M. Marciniak, On a characterization of positive maps, J . Phys. A : Math. Gen. 34 (2001), 5863-5874. 9. W. A. Majewski and M. Marciniak, k-Decomposability of positive maps, in: Quantum probability and Infinite Dimensional Analysis, Eds.: M. Schurmann and U. Franz, QP-PQ, vol. XVIII, World Scientific 2005, pp. 362-374; e-print: quant-ph/0411035. 10. W. A. Majewski and M. Marciniak, Decomposability of extremal positive maps on M 2 ( @ ) , t o appear in Banach Center Publications; e-print: math.FA/0510005 11. E. Stormer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. 12. E. Stormer, Decomposable positive maps on C*-algebras, Proc. Amer. Math. SOC.86 (1980), 402-404. 13. M. Takesaki, Theory of operator algebra I, Springer-Verlag, Berlin 2002. 14. S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), 165-183.

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THERMODYNAMICAL FORMALISM FOR QUASI-LOCAL C*-SYSTEMS AND FERMION GRADING SYMMETRY

HAJIME MORIYA Department of Mathematics, Graduate School of Science, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan.

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This paper is a summary of my talk on 26th conference of QP and IDA in Levico (Trento), February, 2005 with some supplemental explanation. Thermodynamical formulations such as characterizations of equilibrium states and spontaneous symmetry breaking are given for general quasi-local systems. We show that the unbroken symmetry of fermion grading (the univalence super-selection rule) follows essentially only from the local structure that satisfies the graded commutation relations.

zyxw zyx zyxwvut zyxwvu zyxwvu zyxwv

In [l,21 a characterization of local thermal stability for temperature states of quantum lattice systems is given. (The former is for quantum spin lattice systems, and the latter is for fermion lattice systems.) Let 5lOc be a set of all finite subsets of the lattice Z”, v being an arbitrary positive integer. Assume that there is a finite number of degrees of freedom on each site of the lattice. A typical example of the models under consideration is the following: The algebra on a site i, d { i } ,is generated by fermion operators ai, a;, and bosonic objects that are given by Pauli matrices u?, a:, af commuting with all fermion operators and all spin operators with different indexes. The interaction among sites is determined by the potential @, a map from 3lOc to A satisfying the following conditions: (@-a) @(I)E AI, @(0) = 0. (@-b) @(I)*= @(I). (@-c)@(@(I)) = @(I). (@-d)EJ (@(I)) = 0 if J c I and J # I. (@-e) For each fixed I E slot, the net (Hj(1))j with HJ(I) := C,{ @(K); K n I # 0, K c J} is a Cauchy net for J E 5lOcin the norm topology converging to a local Hamiltonian H(1) E A.

319

320

zyx zy zyxwvut

The above EJ denotes the conditional expectation of the tracial state (or any product state such as the Fock state) from A onto AJ for J E Slot. Let P denote the real vector space of all @ satisfying the above set of conditions. The set of all *-derivations with their domain Aloe commuting with 0 is denoted D(dloc).There exists a bijective real linear map from @ E P to 6 E D(dloc). The connection between 6 E D(AI,,) and @ E P is given by

6(A)= Z[H(I),A], A E A1 for every I E &,, where the local Hamiltonian H(1) is determined by (@-e) for this @. Let at (t E R) be a one-parameter group of *-automorphisms of A. A state ‘p is called an (at,P)-KMS state if it satisfies

zyxwv zyxwvu zyxw zyxwv

‘p(AQiL4B)) = ‘p(BA) for every A E A and B E dent, where dentdenotes the set of all B E A for which at ( B ) has an analytic extension to A-valued entire function az( B ) as a function of z E C. Our dynamics at is assumed to be even, namely at 0 = 0 at. We also assume the following in order to relate at with some 6 E D(dloc). (I) The domain of the generator 6, of at includes Aloe. (11) Aloe is a core of 6,. Let I, denotes the complementary of I, i.e. IUI, = Z”. Let w be a state of (A, { A I } I E Z l o c ) . For I E 51,,, the conditional entropy of w is defined in terms of the relative entropy by

3I(w) := -S(trI

0 WIAI,,

w ) = -s(w.

E I ~ ,w )

I 0,

where EI, is the conditional expectation onto AI, and w . EI=(A):= w(EI,( A ) )for A E A. Let @ E P. The conditional free energy of w for I E &, is given by

F;p,,(w) := 31(4- P w ( H ( I ) ) ,

where H(1) is the local Hamiltonian for I with respect to @.

Definition 1. A state ‘p of A is said to satisfy the local thermal stability condition for @ E P at inverse temperature /3 or (@,P)-LTS condition if for each I E Slot

F;pp(‘p)2 F;pP(w) for every State w such that wIdI, = ‘p(dI,.

zy z zyx 321

There is another definition of local thermal stability [2] that has the same variational formula as above but takes the commutant algebra di as the complementary outside system of a local region I instead of dI,.We shall call this alternative version LTS’ condition, where the superscript ‘1’ may stand for the commutant. Also by ‘I’ we indicate that this formalism is not natural compared to Definition 1. The formalism of LTS’ using commutants makes it possible to exploit the known arguments for quantum spin lattice systems [l]where local commutativity holds. In fact we easily obtained several similar results for fermion lattice systems [2] to those known for quantum spin lattice systems. Among them, the equivalence of KMS and LTS’ conditions for the fermion lattice systems holds without assuming the evenness on states. For even states, LTS’ and LTS are shown to be equivalent. Hence we also obtained the equivalence of KMS and LTS conditions under the assumption that the states are even. We, however, think that LTS’ is not so natural, even though it is mathematically useful, because there is no physically good reason in twisting the given CAR local structure into the tensor-product one by Jordan-Wigner transformations. On the contrary, our LTS obviously respects the given quasi-local structure and seems to be natural. We have shown that LTS condition does not permit the broken symmetry of fermion grading [3]. If there would exist a KMS state that breaks fermion grading symmetry for some even dynamics, then there are noneven states (made of it by perturbation of a local Hamiltonian multiplied by the inverse temperature) that inevitably satisfy the KMS condition, however, violate the LTS. In other words, if there would be breaking of the univalence superselection rule for temperature states, then the equivalence of KMS and LTS conditions should be violated. We conjecture that such breaking will never happen for temperature states and our LTS and LTS’ turn to be equivalent. We note that non-factor quasi-free states, which are non-physical examples of breaking of the fermion grading [6], have all type-I representations. From now on, we consider general graded quasi-local systems that encompass lattice and continuous, also fermion and fermion-boson systems. For such systems, a criterion named macroscopic spontaneously symmetry breaking, MSSB, is defined. It is formulated based on the idea that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every complementary outside system of a local region specified by the given quasi-local structure.

zy

322

For bosonic (tensor product) systems, this MSSB reduces to the usual one, i.e. non-triviality of centers solely for the total system. But the former is in general stonger than the latter. We show the absence of MSSB for fermion grading symmetry in a model independent setting. For its proof, we make use of some observations noted in [4,51. The point is that no extension is possible for a pair of prepared states on disjoint regions if they are both noneven.

zyxwvu zyxw

Acknowledgements.

The author is a COE post-doctoral fellow of Department of Mathematics, Graduate School of Science, Hokkaido University and acknowledges this fellowship. The author is grateful to the organizer of this conference.

References 1. Araki, H., Sewell, G.L.: KMS conditions and local thermodynamical stability of quantum lattice systems. Commun. Math. Phys. 52, 103-109 (1977). Sewell, G.L.: KMS conditions and local thermodynamical stability of quantum lattice systems 11. Commun. Math. Phys. 55, 53-61 (1977). 2. Araki, H., Moriya, H.: Local thermodynamical stability of fermion lattice systems. Lett. Math. Phys. 60, 109-121 (2002). 3. Moriya, H.: Macroscopic spontaneous symmetry breaking and its absence for Fermion grading symmetry, preprint. 4. Moriya, H.: Separability condition for the states of fermion lattice systems and its characterization, preprint. 5. Moriya, H.: On a state having pure-state restrictions for a pair of regions. Interdisciplinary Information Sciences. 10, 31-40 (2004). 6. Manuceau, J., Verbeure, A,: Non-factor quasi-free states of the CAR-algebra. Commun. Math. Phys. 18, 319-326 (1970).

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MICRO-MACRO DUALITY AND AN ATTEMPT TOWARDS MEASUREMENT SCHEME OF QUANTUM FIELDS

IZUMI OJIMA RIMS, Kyoto University, Kyoto 606-8502, Japan

1. Micro-quantum systems vs. macro-classical systems

Here I consider the problem of consolidating the well-known heuristic idea of “quantum-classical correspondence”, according to which macroscopic classical objects are to be regarded as condensates of infinite quanta emerging from a microscopic quantum system. This attempt is for the purpose of establishing effective mathematical methods or guiding principles for controlling bi-directional transitions between microscopic quantum systems and macroscopic classical levels, whose essence can be boiled down into such an expression as “Micro-Macro duality”, mathematically formulated as categorical adjunctions found at many different levels in physical theories. This will be seen just a mathematically polished version of the above old heuristics, quantum-classical correspondence (q-c correspondence, for short), in physics. As exemplified by such interesting phenomena as “macroscopic quantum effects”, the contrasts between [Micro vs. Macro] (according to length scales) and between [Quantum vs. Classical] (due to the essential differences of structures) are, precisely speaking, independent of each other to certain extent, owing to the absence of an intrinsic length scale to separate quantum and classical domains. Since this kind of mixtures can be taken as exceptional, however, I restrict my consideration here to such generic situations that processes taking place at microscopic levels are of quantum nature and that the macroscopic levels are described in the standard framework of classical physics, unless the considerations on the last point become crucial. On this premise of the parallelisms among micro/quantum/noncommutative and macro/classical/commutative, respectively, the essential contents of q-c correspondence involve the following two levels: 323

324

zyxw

1) Superselection sectors and intersectorial structures detected by a centre: the major gap between the microscopic levels described by noncommutative algebras of physical variables and the macroscopic ones by commutative algebras can be clearly formulated and understood in terms of the notion of a (superselection) sector structure consisting of a family of sectors (or pure phases) described mathematically by factor states and representations, the totality of which describes physically mixed phase situations involving both classical and quantum aspects. Sectors or pure phases are faithfully parametrized by the spectrum of the centre of a relevant representation of the C*-algebra of microscopic quantum observables describing a physical system under consideration. Physically speaking, elements of the centre are mutually commutative classical observables which can be interpreted as macroscopic order parameters. 2 ) Intrasectorial structures detected by a MASA: while the above intersectorial structure describes and controls the coexistence of and the gap between quantum( =intrasectorial) and classical(=intersectorial) aspects, we need to detect the intrinsic quantum structures within a given sector, not only theoretically but also operationally (up to the resolution limits imposed by quantum theory itself). In the usual discussions in quantum theory with finite degrees of freedom, a maximal abelian subalgebra (MASA, for short) plays canonical roles in specifying a quantum state according to measured data, in place of the centre trivialized by Stone-von Neumann uniqueness theorem. As seen below, this notion of MASA need be reformulated in such a quantum system as quantum fields with infinite degrees of freedom, whose algebras of observables may have non type I representations. Under certain conditions, this formulation will be seen also to determine the precise form of the coupling between the object system and the apparatus required for the implementing a measurement process. In the attempt to utilize the obtained results for attaining a satisfactory measurement scheme of quantum fields, we encounter some subtle points caused by non-type I algebras like local subalgebras of type III in QFT, in close connection with the absence of maximal partition of unity consisting of minimal projections. This problem necessitates meaningful approximation schemes for bridging the gaps between continuous spectra inherent to type 111 structures at the mathematical level and the discrete spectra unavoidable in the actual measurements at the operational level. Once this point is resolved, it would suggest, at the same time, the possibility of reconstruction scheme of a microscopic quantum algebra from macroscopic classical observables combined with their data structure in duality.

zyx zyxwvuts zyxwvu zyxwvu zyxwvut zyxwv zyx 325

2. Sectors and order parameters as q-c correspondence

To attain a clear-cut separation between quantum and classical aspects in terms of sectors and order parameters, we first recall the standard notion of quasi-equivalence [I] 7r1 M 7r2 of representations 7 r 1 , 7 ~ 2of an abstract C*-algebra U describing the observables of a given microscopic quantum system: it is defined as unitary equivalence up to multiplicity by 7r1

M 7r2

4% 7r1(U)”

N

7r2(U)”

In the universal representation [l],(7r,

* c(7r1) = c(7r2). :=

@ 7r,,Ej,

:=

WEEN

El,),

@

WEEX

U** =: U”, of U consisting of all t h e GNS representations (T,, fj,, R,) for states w E En, the central support c(7r) of a representation (7r,fi, = P,fj,) with support projection P, E 7ru(U)‘ is the smallest projection in the centre := IU”n7r,(U)’ to pick up all the representations quasi-equivalent to 7r: c(7r) =projection onto 7r(U)‘fi, c 4,. On this basis, we introduce a basic scheme for q-c correspondence in terms of sectors and order parameters: the Gel’fand spectrum Spec(S(U”)) of the centre 7r,(U)”

21

s(%”)

zy

h

h

can be identified with the factor spectrum U of U: Spec(3(U”)) N U := Fa/ M, defined by all quasi-equivalenceclasses of factor states w E Fa with trivial centres 3(7r,(U)”) = 7r,(U)” n 7r,(U)’ = (cI~j,.

Definition 2.1. A sector (or physically, pure phase) of an observable algebra is defined by a quasi-equivalence class of factor states. In view of the commutativity of 3(U”) and of the role of its spectrum, we can regard [2] h

0

Spec(%”)) N U as the classifying space of sectors to distinguish

0

3(U”) as the algebra of macroscopic order parameters to spec-

among different sectors, and ify sectors. h

Then the dual of embedding map 3(U”) N Lbo(U)~f U”, h

Micro:

U* 3 Ea

--H

zyx

Prob(U) C L”(G)* : Macro,

can be interpreted as a universal q(uantum)+c(lassical) channel, transforming microscopic quantum states E En to macroscopic classical states h

E

-

Prob(U) identified with probabilities [2]. This basic q + c channel,

En 3 4

h

p+ = 4” l 3 ( n ) t E ) ~3 ( ~ p=) M 1 ( S p e c ( 3 ( U ” ) ) )= Prob(U) ,

326

zyxw zy zyxwvu zyxwvu zyxwv zyxw zyx

gives the probability distribution p4 of sectors contained in a mixed-phase state 4 of U describing a quantum-classical composite system:

,-.

B3A

~ " ( x A )= p4(A) = Prob(sector E

A 1 4),

where 4" denotes the normal extension of 4 E Ea to a". While it tells us as to which sectors appear in 4, it cannot specify precisely which representative factor state appears within each sector component of 4. 3. Intrasectorial structure & MASA as q-c correspondence

To detect operationally the intrasectorial structures inside of a sector w given by a factor representation (T,, 4,, Q,), we need to choose a maximal abelian subalgebra (MASA) A of a factor algebra M := 7r,(Iu)", by the condition A' n M = A L"(Spec(A)) [3]. Note that, if we adopt such a definition of MASA as A = A' found in the usual discussions of quantum systems with finite degrees of freedom, the relation A' = A c M implies M' c A' = A c M , and hence, M' = M ' n M = 3 ( M )is of type I, which does not fit to the general context of infinite systems involving non-type I algebras. Since a tensor product M 8 d (acting on the Hilbert-space tensor product fi, @I L2(Spec(A))) has a centre given by

3 ( M 8 A) = 3 ( M )8 A = 1 8 Lm(Spec(A)), we see that the spectrum Spec(A) of a MASA A can be understood as parametrizing a conditional sector structure of the coupled system of the object system M and A, the latter of which can be identified with the measuring apparatus d in my simplified version [2] of Ozawa's measurement scheme4. This picture of conditional sector structure is consistent with the physical essence of a measurement process as "classicalization" of some restricted aspects A(c M ) of a quantum system, conditional on the coupling M 8 A of M with the apparatus identified with A. In addition to the choice of relevant algebras of observables, what is important in the mathematical description of a measurement process is to specify coupling terms between algebras of observables, M and d,of the object system and of the apparatus so that a microscopic quantum state of M can be uniquely determined from the macroscopic data of the pointer position Spec(A) of the measuring apparatus. To solve this problem we note that the algebra A is generated by its unitary elements which constitute an abelian unitary group U ( A )=: U. Since this group is infinite-dimensional in general, the existence of an invariant Haar measure is not guaranteed.

zyx

zyxwvu zyxwv zyxwv zyxwv zy zyxw zyxw zyxwv 327

Just for simplicity, however, we assume here the existence of an invariant Haar measure du on U ,which requires U to be locally compact. Rewriting the condition A = A' n M for A to be a MASA of M into such a form ) , see that A is the fixed-point as A = M n A' = M n U(A)' = M A d ( U we subalgebra of the adjoint action Ad(u)X = uXu* ( X E M ) of u E U on M [5]. From this viewpoint, the relevance of the group duality and of the Galois extension can naturally be expected. Through a formulation in terms of a multiplicative unitary [6], the universal essence of the problem can be exhibited as follows. In the context of a Hopf-von Neumann algebra M ( c B ( f i ) ) with a coproduct r : M --t M @ M and a Haar weight, a multiplicative unitary V E U ( ( M@ M,)-) c U(4@ fi) is so defined as to implement r, r(x) = V * ( l@ x ) V , and is characterized by the pentagonal relation, V12V13V23 = v23v12,on rj @ Ej @ 4,expressing the coassociativity of r. Here subscripts i, j of &j indicate the places in Ej@fi@fi on which the operator V acts. It plays fundamental roles as an intertwiner, V(X '8 L ) = (X@X)V,showing the quasi-equivalenceamong tensor powers of the regular X(w) := (i @ w ) ( V ) E h;r, defined by a representation X : M , 3 w generalized Fourier transform, X(w1 * w2) = X ( w l ) X ( w 2 ) ,of the convolution algebra M,, w1 * w2 := w1 @ w2 o I?. On these bases, a generalization of group duality can be formulated for Kac algebras 161. In the case of M = L"(G,dg) with a locally compact group G equipped with a Haar measure d g , the multiplicative unitary V is explicitly given on L2(G x G ) by

-

for t E L2(G x G ) ,s,t E G,

( V t ) ( s t, ) := t ( s , s-lt)

or symbolically, V ( s t, ) = Is,s t ) , in the Dirac-type notation. Identifying the above M with L"(u^) = U(A)" = A for G := u^, the character group of our abelian group U = U(d),we adapt this machinery to the context of the MASA A. In terms of the group homomorphism E :U M associated with A M , the spectral decomposition of U is given via SNAG theorem by E ( u ) = &,ay(u)dE(y) (u E U ) , with dE an

-

-

zyx

M-valued spectral measure E ( A ) = E ( x A ) on u^ (for Bore1 sets A c @. Then V is represented as E,(V) = &a d E ( y )'8 A, on L 2 ( M )@ L2(u^)by

E*(V)(CA@ Ix)) =

1

dE(y)tA'8 Irx),

-YEA

for Y , X E 6,

[A

E E(A)L2(M),

(1)

satisfying the modified pentagonal relation, E, (V)12E*(v)13v23 =

328

zyxwvutsr zyxw

zyxwv zyxwv zyxwv zyxwv zyx -

V23E*(V)12. (Here the Hilbert space L 2 ( M ) provides M with its standard form.) By the equality A = L m ( S p e c ( A ) ) = Loo(fi),the same algebra A of observables to be measured maximally consistently can be viewed in two ways, respectively, as a function algebra on S p e c ( d ) consisting of characters on the commtutative algebra d , and as the corresponding one on the commutative group with product only: in contrast to the absence of an intrinsic “base point” in S p e c ( d ) (regarded as a Hausdorff space), the identity character L E 6, L ( U ) 3 1 (Vu E U ) present in the context of group duality U can be distinguished physically by its important role as the neutral position of measuring pointer. In the usual approaches, the importance of neutral positions remarked earlier by Ozawa has been overlooked for lack of the suitable place to accommodate it in an intrinsic way. It can also be related to the breakdown, AR # L 2 ( M ) ,of A-cyclicity of the cyclic and separating vector R of M in L 2 ( M )= P[dn]l E A’\(MUM’) = ( AV M’)\(MUM’), when A = A ’ n M # A’ [7]. The important operational meaning of the equality (1) and the role of the neutral position L can clearly be seen, especially when is a discrete group which is equivalent to the compactness of the group U (or the almost periodicity of functions on it): choosing x = L, we have the equality, E*(V)(&@ ( L ) ) = Er @ Iy) (Vy E which gives the required correlation (“perfect correlation” due to Ozawa8) between the states & of microscopic system M to be observed and those Iy) of the measuring probe system A coupled to the former. With a generic state [ = C,,a c7& of M , an initial uncomlated state E @ I L) is transformed by E, ( V )to a correlated one:

fi

fi

m,

zyxw zyxw fi

fi),

E*(V)(J@ 14) =

c

C7‘5-Y @

17).

7€ii The created perfect correlation establishes a one-to-one correspondence between the state & of the system M and the measured data y on the pointer, which would not hold without the maximality of d as an abelian subalgebra of M . On these bases, we can define the notion of an instrument 3 unifying all the ingredients relevant to a measurement as follows:

J(Alwc)(B):=(wC@ 1 =

L)(

LI)(E*(V)*(B @ XA)E*(V))

(( tI @ ( LI)E*(V)*(B @ xA)E*(v)(lC) @I

L)).

In the situation with a state wc = ( tI ( - ) E ) of M as an initial state of the system, the instrument describes simultaneously the probability p(A Iwc) =

zyx zyxwvu zyxw zyxwv zyxw zyxwvu 329

3 ( A l q ) ( l ) for measured values of observables in A to be found in a Bore1 set A and the final state J(Alwt)/p(Alwt)realized through the detection of measured values [4]. In this way, we have attained two important improvements, to remove the restriction inherent to the usual characterization of MASA and to specify the coupling term necessary for the measurement scheme due to Ozawa originally formulated in systemd of finite degrees of freedom with type I algebras; these results constitute the crucial step in extending his scheme to such general quantum systems with infinite degrees of freedom as QFT. To be fair, however, we note that there remain some unclear points, such as the possibility for the Haar measure du to be absent and the non-uniqueness of MASA A = A' n M , both of which are the difficulties caused by the infinite dimensionality. As remarked at the beginning, the consistency problem should be taken serious between the mathematically universal structures of type 111 and the finite discrete spectra inevitable at the operational level, which is closely related with such type of criteria as the nuclearity condition in algebraic QFT to select the most relevant states and observables. By starting from a suitable choice of an operationally accessible subalgebra, a scheme is examined to recover a full local subalgebra of observables from the former, within which the standard determination of a semisiple Lie algebra by its root system can be regarded as a special case of the scheme [7].

zy

References 1. Dixmier, J., C*-Algebras, North-Holland, 1977; Pedersen, G., C*-Algebras and Their Automorphism Groups, Academic Press, 1979. 2. Ojima, I., A unified scheme for generalized sectors based on selection criteria, Open Systems and Information Dynamics, 10 (2003), 235-279; Temparature as order parameter of broken scale invariance, Publ. RIMS 40,731-756 (2004). 3. Dixmier, J., Von Neumann Algebras, North-Holland, 1981. 4. Ozawa, M., J. Math. Phys. 2 5 , 79-87 (1984); Publ. RIMS, Kyoto Univ. 21, 279-295 (1985); Ann. Phys. (N.Y.) 259, 121-137 (1997). 5. Ojima, I., Micro-macro duality in quantum physics, t o appear in Proc. Intern. Conf. on Stochastic Analysis, Classical ?nd Quantum; math-ph/0502038. 6. Baaj,S. and G. Skandalis, Ann. Scient. Ecole Norm. Sup. 26 (1993), 425-488; Enock, M. and Schwartz, J.-M., Kac Algebras and Duality of Locally Compact Groups, Springer, 1992. 7. Ojima, I. and Takeori, M., work in progress. 8. Ozawa, M., Perfect correlations between noncommuting observables, Phys. Lett. A, 335, 11-19 (2005).

zyxw zyxwv zyxw zyxw

THE LEVY LAPLACIAN ACTING ON SOME CLASS OF LEVY FUNCTIONALS

KIMIAKI SAITO Department of Mathematics, Meijo University Nagoya 468-8502, Japan E-mail: [email protected]

ALLANUS H. TSOI Department of Mathematics, University of Missouri Columbia, M O 65211, U.S.A. E-mai1:tsoiQmath.missouri. edu

In this paper we shall discuss the Ldvy Laplacian as an operator acting on some class of the Ldvy functionals. The Laplacian acts on Gaussian and Poisson functionals in the class as a scalar operator. Based on this result, we introduce some domain of the Laplacian on which is a self-adjoint operator. We also discuss associated semigroups and associated stochastic processes.

Introduction An infinite dimensional Laplacian was introduced by P. LBvy in his famous book [12]. Since then this exotic Lapiacian has been studied by many authors from various aspects see [l-6,9,10,13,15,16,17,20] and references cited therein. In this paper, we study recent results on stochastic process associated with the LBvy Laplacian generalizing the methods developed in the former works [11,14,18,22-261. We construct a new domain of the LBvy Laplacian consisting of some LBvy functionals and associated infinite dimensional stochastic processes. This paper is organized as follows. In Section 1 we summarize basic elements of white noise theory based on a stochastic process given as a difference of two independent LBvy processes. In Section 2, following the recent works Kuo-Obata-Sait6 [ll],Obata-SaitB [18], SaitB [23, 241 and SaitbTsoi [25], we formulate the LBvy Laplacian acting on a Hilbert space consisting of some LBvy functionals and give an equi-continuous semigroup 330

z zy zyxwvuts zyxwvu zyx zyxw zyx zyxw zyxwv 331

of class (Co) generated by the Laplacian. In Section 3, we generalize this situation by means of a direct integral of Hilbert spaces. In Section 4, based on infinitely many Cauchy processes, we give an infinite dimensional stochastic process associated with the LBvy Laplacian.

1. Preliminaries

Let E = S(R) be the Schwartz space of rapidly decreasing R-valued functions on R. There exists an orthonormal basis {ev},20 of L2(R) contained d2 in E such that Ae, = 2(v+l)e,, Y = 0,1,2,.. ., with A = -=+u2+1. For p E R define a norm I . , 1 by lfl, = ( A p f l t 2 ( ~for ) f E E and let E, be the completion of E with respect to the norm I . Ip. Then E, becomes a real separable Hilbert space with the norm I ., 1 and the dual space EL is identified with E-, by extending the inner product (., .) of L2(R) to a bilinear form on E-, x E p . It is known that E = proj limp-+, Ep, and E* = indlim,,, E-,. The canonical bilinear form on E* x E is also denoted by .). We denote the complexifications of L2(R), E and Ep by Lg(R), E c and Ec,,, respectively. - be independent LBvy processes of Let {L;,,(t)}t>o - and {L;,,(t)}t>o which the characteristic functions are given by (a,

02

+ (ezxz- I ) , 2 where m E R , c 2 0 and X E R. Set h,,x(t) = Li,,(t) - 15:,~(t) for all t 2 0. Then we have E[eizLL.x(t)] = eth('), t

2 0,

j = 1 , 2 , h ( z )= i m z - - z 2

+

E[eiz"~,A(t)] = e t ( h ( z ) + h ( - z ) ) = exp { --ta2z2 t(eixz + e-ixz - 2 ) ) 7 t 2 0.

s

Set C(S) = ""P{JR(h(&(4) + h(- 0, which has been determined in [8]. In fact, there ,!, we have used the reproducing property of a Poisson distribution, or equivalently Poisson variables can be imbedded in a Poisson process.

Remark 1. In P.6) above, we set p , = e-X$, X > 0, referring [8]. On the other hand, the natural reason of choosing p , in such a way can be also observed in the author's previous paper [6].

zyxwv zyxw zyx zy zyx

362

Theorem i) The measure space (An, pn) is isomorphic to the measure space (An,pn) defined in Section 1. ii)The weighted sum (Am,p ) of measure spaces (An,p n ) , n = 1,2,. . . is identified with the Poisson noise space. Proof. According to the facts P.l) to P.6), the theorem is proved.

What have been investigated are explained as follows. Basic noises, that is Gaussian and Poisson noises, have

1) optimality in randomness, which is expressed in terms of entropy, and 2) invariance under the transformation group; each noise has its own characteristic group (one is continuous and another is discrete).

Remark 2. So far the parameter space is taken to be [0,1]. The characteristic properties that have been discussed can be generalized to the case of Rd-parameter with modest modifications.

zyxw zyx zyx zyxwv

3. Fractional power distribution in terms of Poisson noise

This section is devoted to a breif interpretation of fractional power distribution, since they are often used in information sociology application. Define Pu(t)= uP(t).Then the sample function of Pu(t)is of the form S

where s runs through a finite set depending on w . The characteristic functional is given by

where

00

~ ( =t A) /

- 1)dt

0

and +([) is called the 1c, function. Let Y ( t ) be a superposition of independent Poisson noises Pu(t)with which will be determined later. Then we have weight f(u),

zyxwvut zyx zyxwvu zy zyxwvu 363

Since Pu7s are independent, the $-functional of Y( + wc>ll+ll;

zy (1)

for c > 1 and all E B2. Here K ( c ) is an explicitly given (although not necessarily optimal) finite constant that is proportional to the dimension n. (See Refs. [5, 61 and [7].) The expression on the left hand side of (1) is really the quadratic form associated to N . Note that this quadratic form is defined for all E 232, though it is sometimes equal to +oo. This quadratic

+

zyx zyxw zyxw zyxwv zyxwvu 367

form is called the energy. The terminology “log-Sobolev inequality” comes from Gross [8]. The Hirschman inequality for the Segal-Bargmann transform is

S(f)I a S ( A f )+ 4a)Ilfll;

(2)

for all f E Q2. Here .(a) is an explicitly given (though not necessarily optimal) finite constant; moreover, it can be taken to be proportional to log (I IAl I P +, q ) which, in turn, is bounded by an expression proportional to the dimension n. Here p and q are any pair of Lebesgue indices satisfying 1 I p < 2, 1 I q < 2 and p > 1+ q / 2 . Also, ~ ~ A ~ is~ the P +operator q norm of A with respect to the L P norm in its domain and the Lq norm in its codomain. (See Refs. [5] an[9].) Hirschman’s original inequality in Ref. [lo] is for the Fourier transform. Either inequality immediately generalizes to the infinite dimensional case (roughly when the dimension n = co) if the optimal constant for the norm term does not depend on n. The simplest way for this to happen would be that the constant of the norm term is zero. 0 0

zyxwv zyxw zyx

Conjecture 1: For some c the optimal constant in (1)is K ( c ) = 0. Conjecture 2: For some a the optimal constant in (2) is .(a) = 0.

These two conjectures are not independent. If Conjecture 1is true, then Conjecture 2 is true too. Here is the argument: 1

-W) I ( f , N f )= (Af,”) 2

54A.f)

for all f E Q2. The first inequality is the log-Sobolev inequality due to Gross [8]. The equality is the fact, alluded to above at the end of the previous section, that A preserves the quadratic forms associated to N . Finally, the last inequality is (l),given that Conjecture 1 is true. This proves (2) with a = 2c for some c > 1 and ~ ( u = ) 0, that is, it shows Conjecture 2. While I can find no way to prove the converse, it seems reasonable to study these two conjectures together. First, I want to give an argument that the norm term really is there in (I), namely that the optimal constant of the norm term there is strictly positive. The reader should beware that this argument has a gap in it! First, I take any 4 in Dom(N1l2), the domain of N1/2,with (4, l)a, = 0. Here 1 denotes the constant function. Then I form $J = 1 A 4 for any X E R and study the asymptotics in small X of the energy and the entropy for this choice of $J. So,

zy zyxw +

Il$Jl ;

+ A49 1+

= ($J,$J)= (1

+ X2Il4llE

=1

zyxw zyxw zyxw z zyxw zy zy

368

since (1,l)= 1 and

(4,l) = 0.

Next, using N * = N and N 1 = 0 , we have

($,wJ) = ( l + X 4 , N ( l + X 4 ) ) =X2(4,N4)=X211N1/24(1;,

where N1/2is the positive square root of N . To expand S($), we first note that Il$ll;log

Ir$ll;

= (1

+ X211411;)

= (1 + X211411;)

1% (1 + X211411;)

zy

(X211411$ + 0 (A4))

+ 0 (A4)

= ~211411;

by the Taylor expansion around X = 0 of the log term. Next,

1$12

+ = x ( 2 ~ e 4+ ) x2 (1412+ 2 ( ~ e 4 )+~o) +

log l$I2 = (1 Xqq2 log (1 X4I2

,

( ~ 3 )

again by a Taylor expansion around X = 0 of the function gZ : R defined by := 11

(3)

-, R

+ A4(z)l2 log 11+ X4(Z)l2,

(4)

where X E R is variable and where 4 and z are fixed. This implies that

S($) = x2

/ (!+I2

+2

+ 0 (A3)

-

(X211+l122

+ 0 (A4))

since J 2 ~ e 4= J ( + + f > = O + O = o using ($,I)= 0. Since we have the estimate

(5) 1412,

zyxwv

S ( $ ) I2X211411$ + 0 ( A 3 ) .

Suppose Conjecture 1 is true, that is, that for some real number c we have ($3

for all $ E

B2.

N $ ) L q$J>

(6)

Taking $J = 1+ A 4 as above, we would then have X211N1/2$11;L 2X2clI+II;

+ 0 (A3)

(7)

for any 4 in Dom(N1/2) that is orthogonal to 1. Now dividing by X2 and then taking the limit X + 0, we obtain

llN1/2411;

L2cll4ll;,

(8)

which implies that N1/2is a bounded linear map. But it is known that N1/2is not bounded, since its spectrum is unbounded. So this contradicts the assumption that (6) holds. So this would seem to imply that there must be a nonzero norm term in the reverse log-Sobolev inequality (1). By the

369

zyxwv zzyxwvuts yzyxwvu xwvutsrqponzyx mlkjihgfedcbaZ zyxw zyxw

zy

way, note that this argument is an adaptation to the present situation of the argument of Rothaus [ll]and Simon [12] that shows that the existence of a log-Sobolev inequality implies the existence of a spectral gap. Nonetheless, as I noted above, this argument has a gap in it. And that gap is in the step that goes from (3) to (5). To see what is happening here, let's write (3) as 1+(z)l2 log 1+(.)12

+ X W I 2 1% I1+ X W I 2 (2Re4(z))+ x2 (14(2)1~ + 2 ( ~ e $ ( z ) )+~ )~

= I1 =

(9)

( zA),,

where z E C" and X E R. Of course, we can use this last equation to define the expression R ( z ,A), in which case the equality is trivial. The nontrivial content of Taylor's theorem is that R(z,X) is 0 (A3), that is, R(z,X)/X3 is bounded in a neighborhood of X = 0. But z E C" plays the role of a parameter, and so Taylor's theorem gives no a priori information on how this bound on R ( z ,X)/X3 depends on z. Now to pass to (5) I integrate both sides of (9) with respect to p$. Since the first two terms on the right hand side of (9) are integrable (this following from q5 E Bz),we see that R ( z ,A) is integrable if and only if l+(z)I2log 1 + ( ~ ) 1 ~ is integrable. The latter is a rather weak condition; it is implied by $ E L2+' (Cn, p g ) for some E > 0. (See Ref. [6].) Let's note that (9) comes from these elementary calculus results which follow from the definition (4)of the function g,(X) given above: g:(X)

= (2ReW

+ 2Xl4(Z)l2) log I1 + X W I 2 + ( 2 R e W + 2XI+(z)l2)

and

+

so that g,(O) = 0 and gL(0) = 2Re+(z) and g:(O) = 214(2)l2 4(Req5(~))~. However, even knowing that R(z,X)/X2 -+ 0 as X -+ 0 pointwise for each z in Cn is not sufficient to conclude that

as X -+ 0. And the term called 0 ( X 3 ) in (5) is really just J d p ( z ) R ( z , X ) and the only property we subsequently use of this term is that X-' Jdp(z)R(z, A) -+ 0 as X -+ 0 (to go from (7) to (a)), where p means p g . But Taylor's theorem also allows us to write a formula for the remainder term R ( z , A). Will this help us? Well, Taylor's theorem gives us

370

zyxwvu zyxwvu zy zyxw zyxw zyxw zyx

a formula provided that a certain hypothesis is met. And that hypothesis is that gz be of class C3 in some open interval containing 0 and A. (See Ref. [13].) Now it turns out that gt is C1 on all of R, even in the “singular” case when 1 X+(z) = 0 (i.e., when $ ( z ) = -1/X for some real A) since the singularity of the log term in g: is wiped out by its coefficient being zero. Note also that if z is a zero of 4 (i.e., $ ( z ) = 0), then g,(X) = 0 is Coo. However, the singularity of the log term in :g is not controlled by its coefficient, provided that +(z) # 0. And so, gz is not C2 at -1/4(z), provided this expression makes sense and is a real number, namely that 4 ( z ) E R\{O). In the contrary case (namely that 4 ( z ) E C \ R or 4 ( z ) = 0) we have that gz(X) is C” in X E R. The point here is that

+

provided that gz is of class C3 in some neighborhood containing 0 and A. (See Ref. [13].) But if the singularity -1/4(z) falls between 0 and A, then gz is not even C2 between 0 and X and the remainder term R ( z ,A) (though well-defined and even integrable) can not be analyzed using the formula (lo), since that formula need not apply when the hypothesis of Taylor’s theorem does not hold. But (10) is a valid formula when + ( z ) E C \ R or 4 ( z )= 0. So how do we analyze R(z,X) in (9) and (lo)? If we want to use the dominated convergence theorem (this being the most obvious tool to apply) we have to show that the family XT2R(z,Xi) is uniformly bounded (in i) by an integrable function (in z ) for some sequence X i E R \ (0) that satisfies X i --t 0 as i -+ 00. Now the behavior on the measure zero set of z E C” such that 4 ( z ) E R \ (0) is not the only problem. The set of z such that 4 ( z ) is near the real axis is also a problem, as we are going to see now. Substituting the explicit formula (more elementary calculus),

for gL3’(X) into (lo), where z satisfies $ ( z ) $! R, we find the integrand to be a rational function in the variable t of integration, namely, a degree 5 polynomial in t divided by a degree 4 polynomial in t. Such integrals can be explicitly and exactly evaluated (more elementary calculus) - though the details can be hair raising! But the point is that the denominator of that rational function is

zyxw zyxwvu zyxwv z 371

which is the square of a quadratic polynomial in t. The discriminant of that quadratic polynomial is

A := ( 2 X R e 4 ( ~ )-> ~4 . (X21$(z)I2) . 1= - 4X2 ( r m $ ( ~I) 0. )~

Now the explicit formula for the integral in (10) involves dividing terms with factors of + ( z )and +(z)* (and possible log factors) by powers of lAl. (Check a table of integrals, or do it yourself!) So even when we restrict ourselves to $ ( z ) E C \ R, there is no way to control (10) in terms of integral conditions on to get a function integrable in z , since the denominator goes to zero (as I m $ ( z ) + 0) in a way that the numerator can not cancel in general. So we are stuck with a gap; there is no justification here of the step from (3) to (5). Having tried to plug the gap (and thinking it would be trivial) and finding that some very standard tools of analysis are of no avail, I have come to believe it is because Conjecture 1 is true! Of course, this is not at all a proof of Conjecture 1. And what about Conjecture 2? From Ref. [9] we have 1 I IIAIIP+q < 00 for 1 I p < 2, 1 5 q < 2 and p > 1+ q / 2 . If there is just one such pair p , q with IIAIIp+q = 1, then we would have a proof of Conjecture 2. Is this plausible? One bit of bad news here is that there do exist pairs p0,qo satisfying 1 5 po < 2, 1 I qo < 2 and po > 1 q 0 / 2 such that IIAIIPo+qo2 Cn > 1, where C > 1 does not depend on n (though it may depend on p 0 , q o ) . This is shown by studying the action of A on trial functions of the form fa(z) := exp(-az2), for 2 E R”. For example, po = 1.6 and qo = 1.1is one such pair. (See Ref. [9].) But there are other pairs p l , q1 satisfying 1 5 p l < 2, 1 I q1 < 2 and p l > 1 q1/2 such that the “norm” of A when restricted to the functions fa is 1, i.e.,

+

zyxwv zyxw +

+

where a is restricted so that fa is in Lpl(Rn, &). If these functions fa are the “measure” of the norm of the transform A (i.e., Cpl,ql= IIAllpl-+ql), then such pairs p 1 , q l have the property that IIAllpl+ql = 1. While it is not known if the functions fa have this property, this would not be so different from the results of Lieb [14] on integral kernel transforms with a Gaussian kernel function. Even though the division of the region 1 I p < 2, 1 5 q < 2 and p > 1+ q / 2 into the two subregions where Cp,q= 1 and Cp,q> 1, respectively, is explicit, it is algebraically quite complicated and is not very enlightening (at least not to this author as he is writing this). But this could also be the division between the regions where llA\lp+q = 1and ~ ~ A ~ ~>p1,+respectively. q If this is so, then Conjecture 2 is true. Also, as

372

zyxwv zy zyxwv zyx

noted before, Conjecture 2 is true (independent of the nature of IIAllp+q) provided that Conjecture 1 is true. 3. Concluding Remarks

Despite the fact that these two conjectures remain open problems, the above analysis leads me to believe that they are true. What is needed here apparently is a fresh idea or a new tool to resolve the issue.

References

1. V. Bargmann, “On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part I,” Commun. Pure Appl. Math. 14, 187-214 (1961). 2. B.C. Hall, Holomorphic methods in analysis and mathematical physics, In: First Summer School in Analysis and Mathematical Physics, Eds. S . PBrezEsteva and C. Villegas-Blas, Contemp. Math., Vol. 260, pp. 1-59, Am. Math. SOC.,Providence, 2000. 3. I.E. Segal, Mathematical problems of relativistic physics, In: Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. 11, Ed. M. Kac, Lectures in Applied Mathematics, Providence, Am. Math. SOC.,1963. 4. C.E. Shannon, The Mathematical Theory of Communication, University of Illinois Press, Urbana, 1949. 5. S.B. Sontz, Recent results and open problems in Segal-Bargmann analysis, In: Finite and Infinite Dimensional Analysis in Honor of Leonard Gross, Eds. H.-H. Kuo and A.N. Sengupta, Contemp. Math., Vol. 317, pp. 203-213, Am. Math. SOC.,Providence, 2003. 6. S.B. Sontz, “A reverse log-Sobolev inequality in the Segal-Bargmann space,” J. Math. Phys. 40, 1677-1695 (1999). 7. F. Gala-Fontes, L. Gross, S.B. Sontz, “Reverse hypercontractivity over manifolds,” Ark. Mat. 39,283-309 (2001). 8. L. Gross, “Logarithmic Sobolev Inequalities,” A m . J. Math. 97, 1061-1083 (1975). 9. S.B. Sontz, “Entropy and the Segal-Bargmann transform,” J. Math. Phys. 39, 2402-2417 (1998). 10. 1.1. Hirschman, Jr., “A note on entropy,’’ Am. J. Math. 79,152-156 (1957). 11. O.S. Rothaus, “Diffusions on compact manifolds and logarithmic Sobolev inequalities,” J. Fbnct. Anal. 42, 102-109 (1981). 12. B. Simon, “A remark on Nelson’s best hypercontractive estimates,” Proc. A m . Math. SOC.55, 376-378 (1976). 13. Y. Choquet-Bruhat, et al., Analysis, Manifolds and Physics, Rev. Ed., North Holland, Amsterdam, 1982. 14. E.H. Lieb, “Gaussian kernels have only Gaussian maximizers,” Invent. Math. 102, 17s208 (1990).

zyxwv zyxwvu zyxwvu

zyxw zyxwvu zyxw

NOTE ON QUANTUM MUTUAL TYPE ENTROPIES AND CAPACITY

NOBORU WATANABE Department of Information Sciences, Tokyo University of Science Noda City, Chiba, 278-8510, Japan E-mail: [email protected]. tus. a c j p

The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983. In this paper, we compare with mutual entropy-type measures and show some resuls for quantum capacity.

1. Introduction The development of communication theory is closely connected with study of entropy theory. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers [5, 61. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya [8] invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. Recently Shor [20] and Bennet et a1 [2, 3, 18, 191 took the coherent entropy and defined the mutual type entropy to discuss a sort of coding

zyxwv 373

374

zy zyxwvu zyx z zyxwvutsr zyxwvuts zyx

theorem for communication processes. In this paper, we compare with mutual entropy-type measures and show some resuls for quantum capacity for the attenuation channel. 7-1 2. Quantum Channels

The concept of channel has been carried out an important role in the progress of the quantum communication theory. In particular, an attenuation channel introduced in [8] is one of the most inportaint model for discussing the information transmission in quantum optical communication. Here we review the definition of the quantum channels. Let 7-11, 'Fl2 be the complex separable Hilbert spaces of an input and an output systems, respectively, and let B ( ' H k ) be the set of all bounded linear operators on x k . We denote the set of all density operators on 'Flk (k = 1,2) by

{ p E B ( a k ) ; p 2 0,trp = I}.

6( x k )

(1)

A map A* from the quantum input system to the quantum output system is called a (fully) quantum channel.

(1) A* is called a linear channel if it satisfies the affine property, i.e.,

2. A* : e(7-11) 4 e('7-l~) is called a completely positive (CP) channel if its dual map A satisfies

zyxwvu n

Bj*A(A;&) Bk 2 0

(2)

j,k=l

for any n E N,any Bj E B('FI1) and any Ak E B(7-12),where the dual map A : B (X2) -+ B (7-11) of A* : 6 (7-11) -+ 6 ( 7 - l ~ )satisfies trpA ( A ) = trA* ( p ) A for any p E 6 (7-11) and any A E B (7-12). 2 .I. A t t e n u a t i o n channel

Let us consider the communication processes including noise and loss systems. Let I c 1 , Ic2 be the complex separable Hilbert spaces for the noise

z zyxwvut zyxwv zyxw zyxwv zyxw 375

and the loss systems, respectively. The quantun communication channel

b = 10) (01 and .rrG (.) = VO (.) vO*

AC(p) = trlc2nG ( p @ t o ) ,

(3)

is called the attenuation channel, where 10) (01 is vacuum state in 7-ll and VOis a linear mapping from 7-l1 @ K1 to 7 - l ~@ K2 given by

for any In) in 7-11 and a,P are complex numbers satisfying Ia12t ]PI2 = 1. 2 r] = la1 is the transmission rate of the channel. nt;is called a beam splittings, which means that one beam comes and two beams appear after passing throughrt;. This attenuation channel is generalized by Ohya and Watanabe such as noisy optical channel [14, 151. After that, Accardi and Ohya [l]reformulated it by using liftings, which is the dual map of the transition expectation by mean of Accardi. It contains the concept of beam splittings, which is extended by Fichtner, Freudenberg and Libsher [4] concerning the mappings on generalized Fock spaces. For the attenuation channel A t , one can obtain the following theorem:

zyx

zyxwv

Theorem 2.1. The attenuation channel At; is described by 00

(P) =

C OiVOQpQ*&*Oi*, i=O

(5)

where Q = CEO (IN) €9 lo)) (yil, Oi = C,"=,Izk) ((a1€9 (il), {Ivi)) is a CONS inX1, { I z k ) } is a CONS in 7 - l ~and { li)} is the set of number states in K2.

zyxwv zyxw

3. Ohya Mutual Entropy and Capacity

The quantum entropy was introduced by von Neumann around 1932 [7], which is defined by

s(p) E -trp

log p

for any density operators p in S (XI) . It denotes the amount of information of the quantum state p. In order to define such a quantum mutual entropy, we need the quantum relative entropy and the joint state, which is called a compound state, describing the correlation between an input state p and the output state A*p through a channel A*. For a state p E 6(7-l1), p = EkAkEk,

(6)

376

zyxw zyxw zyx zyx zyxw

is called a Schatten decomposition [17] of p, where Ek is the one-dimensional projection associated with &. This Schatten decomposition is not unique unless every eigenvalue is non-degenerated. For p E B('H1) and A* : B('H1) + B(IFln), the compound states are define by (TE =

C X,E, n

8 A*E,,

a0 = cp

(7)

~*cp.

The first compound state, which is called a Ohya compund state associating the Schatten decomposition p = &&&, generalizes the joint probability in classical dynamical system and it exhibits the correlation between the initial state p and the final state A*p. Ohya mutual entropy with respect to p and A* is defined by

where S ( ( T E ,(TO) is Umegaki's relative entropy [21]. I ( p ;A*) satisfies the Shannon's type inequality : 0 5 I ( p , A*) 5 min { S ( p ) , S ( A * p ) } . 3.1. Q u a n t u m capacity

The capacity means the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of the fully quantum mutual entropy with respect to a certain subset of the initial state space. The capacity of purely quantum channel was studied in Let S be the set of all input states satisfying some physical conditions. Let us consider the ability of information transmission for the quantum channelA*. The answer of this question is the capacity of quantum channel A* for a certain set S c S('H1) defined by 11112113314.

zyx zyxwv C t (A*) = sup { I ( p ; A*) ;p

E S} .

(9)

zyxwv

When S = S ('HI), the capacity of quantum channel A* is denoted by C, ( A * ) . Then the following theorem for the attenuation channel was proved in [16].

Theorem 3.1. For a subset Sn E { p E S ('HI) ;dims ( p ) = n } , the capacity of the attenuation channel A: satisfies

where s ( p ) is the support projection of p.

zyx zyxwvuts zyxwvu zyxwvut zyxw zyxw 377

When the mean energy of the input state vectors {Id,) can } be taken 2 infinite, i.e., limT+m 1 d k l = 00, the above theorem tells that the quantum capacity for the attenuation channel A: with respect to S, becomes logn. It is a natural result, however it is impossible to take the mean energy of input state vector infinite. 3.2. Semi-classical mutual entropy

When the input system is classical, the state cp is a probability distribution and the Schatten-von Neumann decomposition is unique with delta measures 6, such that cp = C X ,S ,., In this case we need to code the classical state cp by a quantum state $, whose process is a quantum coding described by a channel I?* such that r*d, = $, (quantum state) and$ G r*cp= C , An$,. Then Ohya mutual entropy I (cp; A* o r*)becomes Holevo’s one, that is,

I (cp; A* o r*)= S (A*$) -

C XS,

zyxwv (A*$,)

(10)

n

when

C,, XS,

(A*$,) is finite.

4. Q u a n t u m M u t u a l Type Entropies

Recently Shor [20] and Bennet et a1 [2, 31 took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for quantum communication. In this section, we compare these mutual types entropy. Let us discuss the entropy exchange [18]. For a statep, a channel A* is denoted by using an operator valued measure { A j } such as

A* (.)

3

,

3

A; . Aj ,

which is called a Stinespring-Sudarshan-Kraus form. Then one can define a matrix W = (Wij)i,jwith

by which the entropy exchange is defined by Se(p,A*) = -trW log W.

(13)

By using the entropy exchange, two mutual type entropies are defined as follows:

IC (P; A*)

S (A*p) - S e ( P , A*) 7

(14)

378

zyxwvu zyxwv zyxwv zyxw zyxwvu zyxw It ( p ; A * ) E S ( P )

+

(A*p) - S e ( P , A*) *

(15)

zyxwv zyxwvu zy zyxwv

The first one is called the coherent entropy IC ( p ; A*) [19] and the second one is called the Lindblad entropy 11,(p; A*) [3]. By comparing these mutual entropies for quantum information communication processes, we have the following theorem [16]:

Theorem 4.1. Let { A j } be a projection valued measure with dimAj = 1. For arbitrary state p and the quantum channel A* (.) E C jAj .AT, one has (1) 0 I I (p; A*) 5 min { S ( p ) ,S (A*p)}(Ohya mutual entropy), (2) IC ( p ;A*) = 0 (coherent entropy), (3) I t (p; A*) = S ( p ) (Lindblad entropy). For the attenuation channel AT;, one can obatain the following theorems [16]:

Theorem 4.2. For any state p = C, A, In) (nI and the attenuation chan2 2 nel AT; with la1 = [PI = one has (1) 0 5 I ( p ; A:) 5 min { S (p) ,S (AT;p)} (Ohya mutual entropy), ( 2 ) I c ( p ; AT;) = 0 (coherent entropy), (3) 11,(p; AT;) = S ( p ) (Lindblad entropy).

a,

Theorem 4.3. For the attenuation channel AT; and the input statep = A ( 0 )(01 (1 - A) 10) (01, we have (1) 0 5 I ( p ; AT;) 5 min { S ( p ) , S (AT;p)}(Ohyamutual entropy), (2) -S ( p ) 5 I c ( p ; AT;) 5 S ( p ) (coherent entropy), (3) 0 5 II,(p; AT;) 5 2s ( p ) (Lindblad entropy).

+

Therem 4.3 shows that the coherent entropy IC ( p ; A:) takes a minus value for IaI2 < IPI2 and the Lindblad entropy 11,( p ; AT;) is grater than the von Neumann entropy of the input state p for la12 > \PI2. From these theorems, Ohya mutual entropy I (p; A*) only satisfies the inequality held in classical systems, so that Ohya mutual entropy can be a most suitable candidate as quantum extension of the classical mutual entropy. References 1. Accardi, L., and Ohya, M., Compond channnels, transition expectation and liftings, Appl. Math, Optim., 39, 33-59 (1999). 2. Barnum, H., Nielsen, M.A., and Schumacher, B.W., Information transmission through a noisy quantum channel, Physical Review A, 57, No.6, 4153-4175 (1998).

379

3. Bennett, C.H., Shor, P.W., Smolin, J.A., and Thapliyalz, A.V., Entanglement-Assisted Capacity of a Quantum Channel and the Reverse Shannon Theorem, quant-ph/0106052. 4. Fichtner, K.H., Freudenberg, W., and Liebscher, V., Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematik und Informatik, Math/ Inf/96/ 39, Jena, 105 (1996). 5. Holevo, A.S., Some estimates for the amount of information transmittable by a quantum communication channel (in Russian)}, Problemy Peredachi Informacii, 9, 3-11 (1973). 6. Ingarden, R.S., Kossakowski, A., and Ohya, M., Information Dynamics and Open Systems, Kluwer, 1997. 7. von Neumann, J., Die Mathematischen Grundlagen der Quantenmechanik, Springer-Berlin, 1932. 8. Ohya, M., On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770-774 (1983). 9. Ohya, M., Some aspects of quantum information theory and their applications t o irreversible processes, Rep. Math. Phys., 27, 19-47 (1989). 10. Ohya, M., and Petz, D., Quantum Entropy and its Use, Springer, Berlin, 1993. 11. Ohya, M., Petz, D., and Watanabe, N., On capacity of quantum channels, Probability and Mathematical Statistics, 17, 179-196 (1997). 12. Ohya, M., Petz, D., and Watanabe, N., Numerical computation of quantum capacity, International Journal of Theoretical Physics, 37, No.1, 507-510 (1998). 13. Ohya, M., and Watanabe, N., Quantum capacity of noisy quantum channel, Quantum Communication and Measurement, 3, 213-220 (1997). 14. Ohya, M., and Watanabe, N., Foundatin of Quantum Communication Theory (in Japanese), Makino Pub. Co., 1998. 15. Ohya, M., and Watanabe, N., Construction and analysis of a mathematical model in quantum communication processes, Electronics and Communications in Japan, Part 1, 68, No.2, 29-34 (1985). 16. Ohya, M., and Watanabe, N., Comparison of mutual entropy - type measures, TUS preprint. 17. Schatten, R., Norm Ideals of Completely Continuous Operators, SpringerVerlag, 1970. 18. Schumacher, B.W., Sending entanglement through noisy quantum channels, Physical Review A, 54, 2614 (1996). 19. Schumacher, B.W., and Nielsen, M.A., Quantum data processing and error correction, Physical Review A, 54, 2629 (1996). 20. Shor, P., The quantum channel capacity and coherent information, Lecture Notes, MSRI Workshop on Quantum Computation, 2002. 21. Umegaki, H., Conditional expectations in an operator algebra IV (entropy and information), Kodai Math. Sem. Rep., 14, 59-85 (1962).

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