Quantum Stochastics and Information: Statistics, Filtering and Control PDF
Preview Quantum Stochastics and Information: Statistics, Filtering and Control
QUANTUM STOCHASTICS AND INFORMATION Statistics, Filtering and Control TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk QUANTUM STOCHASTICS AND INFORMATION Statistics, Filtering and Control University of Nottingham, UK 15 – 22 July 2006 Editors V. P. Belavkin M. Gut¸a˘ University of Nottingham, UK World Scientiic NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI • • • • • • • Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Quantum stochastics and information : statistics, filtering, and control : University of Nottingham, UK, 15–22 July 2006 / edited by V. Belavkin & M. GuÛ|. p. cm. Includes index. ISBN-13: 978-981-283-295-5 (hardcover : alk. paper) ISBN-10: 981-283-295-5 (hardcover : alk. paper) 1. Quantum statistics--Congresses. 2. Control theory--Congresses. 3. Mathematical physics-- Congresses. I. Belavkin, V. P. II. GuÛ|, M. (Madalin), 1973– QC174.4.A1Q36 2008 530.13'3--dc22 2008024091 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2008 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. Printed in Singapore. v PREFACE Quantum Probability was initiated by von Neumann as a new, intrin- sically probabilistic Hilbert space theory of operator observables. Starting with the 80’s of the last century the field was greatly enlarged and devel- oped into a non-commutative analogue of Kolmogorov’s theory of stochas- tic processes. One of its aims is to clarify the probabilistic foundations of quantum theory and its statistical interpretation. The basic problems cen- tre around non-commutative independence (including freeness), quantum dynamic dependence (entanglement) and quantum stochastic dependence (non-commutative Markov processes). Quantum Probability is now a het- erogeneous subject with many connections to other areas of mathematics andphysics.Onthepuremathematicsside,itinteractswithinfinitedimen- sionalfunctionalandnon-commutativeharmonicanalysis,operatoralgebra and operator space theory, non-commutative dynamics, non-commutative K-theory. ThefirstresultsonQuantumInformationandQuantumStatisticsdate back to Stratonovich in the 60’s and Helstrom, Belavkin and Holevo in the 70’s. In the 80’s Belavkin developed the dynamical theory of Quantum Feedback Control based on quantum probabilistic foundations. Currently, these fields enjoy a rapid expansion and have a direct impact on techno- logical advances in Quantum Physics and Engineering. Significant applica- tions to physics include quantum metrology, control of quantum networks, validation of quantum state preparation through statistical inference. The most recent advances along these lines are based on quantum stochastic filteringtheoryasapplicationofquantumstochasticcalculus,alldeveloped in Nottingham and presented in this volume. Quantum filtering describes the interface of classical events with quantum mechanics in the new Even- tum Mechanics, the information dynamics based on constructive quantum stochastic models of the unified quantum-classical mechanics. It gives the dynamical solution of the long standing quantum measurement problem, resolving many famous paradoxes of quantum physics. vi TheQuantum Probability, Information and Control Symposium wasor- ganised during July 15-24, 2006 as the reunion of two closely related con- ferences. Quantum Probability and Applications - QP 27, was the last of a series of network events of the EU Research Training Network ‘Quantum Probability and Applications to Physics, Information and Biology’. The second event was dedicated to Quantum Information, Quantum Statistics, Filtering and Control - QIC and was supported financially by the London Mathematical Society and the Engineering and Physical Sciences Research Council. The 60’s birthday of Professor Belavkin was celebrated in a spe- cially dedicated session. This proceedings volume contains a broad and representative selection of papers on subjects ranging from Quantum Control, Statistics and Prob- ability,toOperatorAlgebrasandNon-commutativeAnalysis.Theunifying paradigm is the further integration between well established classical fields such as Information Theory, Filtering and Control, Statistical Inference, and their quantum counterparts, using the language of Hilbert space op- erators and the probabilistic interpretation of quantum mechanics. This synthesis is catalysed by recent developments and promising applications in Quantum Engineering. WethankJohnGough,MartinLindsayandJoachimZachariasfortheir supportintheorganisationoftheSymposium.Wealsothankthemembers of the Scientific Committee for their help in drawing up the conference program. V.P. Belavkin Nottingham, U.K. M. Gu¸t˘a 1 May 2008 vii ORGANIZING COMMITTEES Scientific Advisory Committee L. Accardi, S. Albeverio, M. D’Ariano, O. Barndorff-Nielsen, M. Boz˙ejko, U. Franz, O. Hirota, R.L. Hudson, M. James, A. Khrennikov, H. Maassen, D. Petz, M. Schu¨rmann, H. Yuen Local organisers V.P. Belavkin, J. Gough, M. Gu¸t˘a, M. Lindsay, J. Zacharias TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk ix CONTENTS Preface v Organizing Committees vii Part A Quantum Probability and Analysis 1 ApproximationviaToyFockSpace—TheVacuum-Adapted Viewpoint 3 A. C. R. Belton Regular Solutions of Quantum Stochastic Differential Equations 23 F. Fagnola From Algebraic to Analytic Double Product Integrals 34 R. Hudson Product Systems; a Survey with Commutants in View 47 M. Skeide Clifford Algebras, Random Graphs, and Quantum Random Variables 87 R. Schott & G. S. Staples The Set of Density Operators Modelled on an Orlicz Space 99 R. F. Streater Quantum Extensions of the Classical Domination Principle 110 V. Umanit`a Analysis in Operator Spaces 121 B. Zegarlin´ski
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