Proceedings of the Conference Quantum Probability and Infinite Dimensional Analysis : Burg (Spreewald), Germany, 15-20 March, 2001

268 Pages·2003·29.526 MB·English

by  Freudenberg W. (ed.)

Quantum probability
Infinite Dimensional
Analysis
QP-PQ: Quantum Probability and White Noise Analysis
Managing Editor: W. Freudenberg
Advisory Board Members: L. Accardi, T. Hida, R. Hudson and
K. R. Parthasarathy
QP-PQ: Quantum Probability and White Noise Analysis
Vol. 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. 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
Vol. 6: Quantum Probability and Related Topics
ed. L. Accardi
QP-PQ
Quantum Probability and White Noise Analysis
Volume XV
Proceedings of the Conference
Quantum probability
and
Infinite f) imensional
Analysis
Burg (Spreewald), Germany 15-20 March 2001
Edited by
W. Freudenberg
Brandenburg Technical University Cottbus, Germany
Vife World Scientific
wl New Jersey • London • Singapore • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
QUANTUM PROBABILITY AND INFINITE-DIMENSIONAL ANALYSIS
QP-PQ: Quantum Probability and White Noise Analysis — Vol. 15
Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, 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 permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-238-288-7
Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
Preface
This volume constitutes the proceedings of the conference on "Quan-
tum Probability and Infinite Dimensional Analysis" held in Burg/Spreewald
(Germany) from 15 to 20 March 2001. The meeting was organized by Bran-
denburg Technical University Cottbus (Germany) in collaboration with
Ernst Moritz Arndt University Greifswald Germany) and Friedrich Schiller
University Jena (Germany). We very much appreciate the financial sup-
port of the Deutsche Forschungsgemeinschaft, the European Union (INTAS
project 9900545) and the three mentioned universities.
Since more than 20 years there are organized regularly conferences on
the above subject. The present conference is the twenty-second meeting
in this series. It was the intention of the conference to bring together
classical and quantum probabilists. It turns out that both disciplines com-
plement one another, and an alliance between them is of benefit to both
parties. About 60 scientists from 10 countries discussed newest develop-
ments in the fields of quantum probability and infinite dimensional analysis.
The present volume consists of 18 research papers reflecting the impressive
progress made in these areas in the latest years. The volume includes new
results on quantum stochastic integration, the stochastic limit, quantum
teleportation and other areas of interest. In the focus of attention are re-
cent results on quantum Markov semigroups and their relation to classical
Markov processes.
Wolfgang Freudenberg
v
This page intentionally left blank
Contents
Markov Property — Recent Developments on the Quantum
Markov Property
L. Accardi and F. Fidaleo 1
Quantum Boltzmann Statistics in Interacting Systems
L. Accardi and S. Kozyrev 21
Stationary Quantum Stochastic Processes from the
Cohomological Point of View
G. G. Amosov 29
The Stochastic Limit and the Quantum Hall Effect: Electrons
and Quons
F. Bagarello 41
The Feller Property of a Class of Quantum Markov Semigroups II
R. Carbone and F. Fagnola 57
Asymptotic Behaviour of Markov Semigroups on
Noncommutative L1- Spaces
E. Yu. Emel'yanov and M. P. H. Wolff 77
Recognition and Teleportation
K.-H. Fichtner, W. Freudenberg and M. Ohya 85
Prediction Errors and Completely Positive Maps
R. Gohm 107
The Q-Product of Generalised Brownian Motions
M. Guta 121
Multiplicative Properties of Double Stochastic Product Integrals
R. L. Hudson 135
VII
viii
Markovianity of Quantum Random Fields in the B(H) Case
V. Liebscher 151
Isometric Cocycles Related to Beam Splittings
V. Liebscher 161
Multiplicativity via a Hat Trick
J. M. Lindsay and S. J. Wills 181
A Relation Between the Gross Laplacian and Time Changes on
Brownian Motion
N. Privault 195
A Note on Bose 3-Independent Random Variables Fulfilling
Q-Commutation Relations
M. Skeide 205
Dilation Theory and Continuous Tensor Product Systems of
Hilbert Modules
M. Skeide 215
Quasi-Free Fermion Planar Quantum Stochastic Integrals
W. J. Spring and I. F. Wilde 243
Antilinearity in Bipartite Quantum Systems and Imperfect
Quantum Teleportation
A. Uhlmann 255
MARKOV PROPERTY - RECENT DEVELOPMENTS ON
THE QUANTUM MARKOV PROPERTY *
LUIGI ACCARDI
Luigi Accardi
Centra Interdisciplinare Vito Volterra
II Universita di Roma "Tor Vergata"
Via di Tor Vergata, 00133 Roma, Italy
email: accardi@@volterra.uniroma2. it
FRANCESCO FIDALEO
Francesco Fidaleo
Dipartimento di Matematica and Centra Interdisciplinare Vito Volterra
II Universita di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy
email: fidaleo<[email protected]
We review recent developments in the theory of quantum Markov states on the
standard Zd-spin lattice. A Dobrushin theory for quantum Markov fields is pro-
posed. In the one—dimensional case where the order plays a crucial role, the struc-
ture arising from a quantum Markov state is fully understood. In this situation
we obtain a splitting of a Markov state into a classical part, and a purely quan-
tum part. This result allows us to provide a reconstruction theorem for quantum
Markov states on chains.
Mathematics Subject Classification: 46L53, 46L60, 60J99, 82B10.
Key words: Non commutative measure, integration and probability; Quantum
Markov processes; Mathematical quantum statistical mechanics.
1. Introduction
The problem of introducing a notion of quantum Markov field, explicit
enough to allow a quantum generalization of Dobrushin's theory, has been
open for several years. Recent advances in the structure theory of Markov
states on chains (3'6) have suggested a natural multi-dimensional general-
ization of the notion of Markov state, see 4. Such a notion has the advantage
*WORK PARTIALLY SUPPORTED BY INTAS N. 991/545
1
2
of being entirely expressible in terms of Umegaki conditional expectations
with additional localization properties. This allows to formulate a quan-
tum Dobrushin theory for Markov fields which exactly parallels the classical
theory, at the basis of equilibrium statistical mechanics.
In the present paper we review recent developments in the theory of
quantum Markov states on the standard Zd-spin lattice.
In the one-dimensional case where the order plays a crucial role, the
structure arising from a quantum Markov state is fully understood. Fol-
lowing previous results of 5'6, a splitting of a Markov state into a classical
part, and a purely quantum part was obtained in 3. This result allowed us
to provide a reconstruction theorem for quantum Markov states on chains.
Further, it emerged that the Markov property for a locally faithful state tf>
on the spin algebra 21 on the chain, can be equivalently established through
properties of generalized conditional expectations defined in 2, which are
canonical objects intrinsically associated to the local structure of the quasi-
local algebra 21, and the state ip under consideration. This was done by
discovering the existence of a very explicit nearest neighbour Hamiltonian
canonically associated to the Markov state <p, which generates on the quasi-
local algebra 21, a one-parameter group of automorphisms admitting (p as
a KMS-state.
Taking into account the suggestion emerging from one-dimensional
models, the intrinsic definition of the Markov property in terms of prop-
erties of generalized conditional expectations, was the starting-point in 4,
in order to investigate the general multi-dimensional case. For these quan-
tum Markov fields (i.e. quantum Markov processes with multi-dimensional
indices), deep connections with the KMS boundary condition, as well as
phenomena of phase transitions and symmetry breaking, naturally emerge,
generalizing the classical situation, see 10'n>12.
Every quantum Markov field is canonically associated to a (non-
commutative) potential. The problem to give a full reconstruction theorem
for these potentials remains still open. However, the conditions on the po-
tential associated to a Markov state, could be explicit enough to allow the
construction of a multiplicity of non trivial examples.
We conclude the introduction by recalling some standard definitions
used in the sequel.
We consider quasi-local algebras obtained in the following way. For
each j in an index set /, a finite-dimensional C*-algebra AP is assigned