Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements

300 Pages·2022·9.603 MB·English

by  Konstantin A. Makarov, Eduard Tsekanovskii

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Published by
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
THE MATHEMATICS OF OPEN QUANTUM SYSTEMS
Dissipative and Non-Unitary Representations and Quantum Measurements
Copyright © 2022 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,
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is not required from the publisher.
ISBN 978-981-124-122-2 (hardcover)
ISBN 978-981-124-123-9 (ebook for institutions)
ISBN 978-981-124-124-6 (ebook for individuals)
For any available supplementary material, please visit
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In respectful memory of our beloved teachers M. Livˇsic and B. Pavlov.
This book is dedicated to the memory of the remarkable Human
Beings and Mathematicians Michail Samoilovich Livˇsic (M.S.) and Boris
Sergeevich Pavlov (B.S.). Their pioneering research in the theory of non-
self-adjoint operators and applications to scattering problems and system
theoryhaveattractedmanyresearchersandmadethepresentbookpossible.
TheApostolicservicetostudents,colleaguesandthemathematicalcom-
munity providedby M.S. andB.S. was enormousandtheir gooddeeds will
never be forgotten. The light of scientific accomplishments of M.S. and
B.S.shinebrightlyandissuccinctlydescribedbythepoeticwordofGalina
Volchek:
Uhod(cid:2) ostav(cid:3)te Svet! (cid:4)to bol(cid:3)xe, qem ostat(cid:3)s(cid:2)...
(cid:4)to luqxe, qem prowat(cid:3)s(cid:2) i va(cid:5)nei˘, qem dat(cid:3) sovet...
Uhod(cid:2) ostav(cid:3)te Svet - pered nim otstupit holod!
Svet soboi˘ zapolnit gorod... Da(cid:5)e esli vas tam net...
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ACKNOWLEDGMENTS
WeareverygratefultoM.Ashbaugh,W.BanksandS.Belyifortheinvalu-
able help in the preparation of this monograph for publication. K.A.M.
is indebted to A. B. Plachenov for stimulating discussions. The authors
are also grateful to the three referees for their valuable comments and
suggestions.
K.A.M. is grateful to his wife Marina and son Konstantin for their
patience and continuous support.
E.T. is very thankful to his son Vladislav, daughter-in-law Elena,
and granddaughters H and R for overwhelming continuous help, support
and care.
ResearchofK.A.M.waspartiallysupportedbytheSimonscollaboration
grant 00061759.
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PREFACE
The main goal of this monograph is to develop a mathematical framework
that accommodates an adequate description of the results of continuous
observationofquantumsystems. Itiswellknownthataquantumobserva-
tion/measurementalwaysaffectsthesystemsubjecttoit,andthereforethe
system can no longer be considered completely isolated. Instead, it should
be treated as part of a more general system in which the presence of an
observer/measuring device is taken into account. As such, the initial sys-
tem should be regarded as an open quantum system that interacts with a
part of the larger system.
In the abstract setting, dealing with open systems assumes the pres-
ence of communication channels through which the interaction is carried
out, both between parts of the system in question and with the outside
world. A special case of open systems is the class of dissipative systems,
the dynamics of which are governedby a strongly continuous semigroupof
contractions. By applyingthe canonicaldilationprocedure, suchopensys-
temscanalwaysbeviewedasapartofalargerclosedsystem: theresulting
Hamiltonianofthedilatedsystemischosentobetheself-adjointdilatation
of the generator of the semigroup, while the space of (pure) states of the
larger system can be identified with the extended Hilbert space where the
dilated operator has been realized as a self-adjoint operator. Despite its
mathematical attractiveness,this dilationmethodis incompatible withthe
physical requirement that the total energy of the obtained large isolated
system must be bounded from below.
The transition in the opposite direction is usually associated with the
reduction of the unitary evolution onto a subspace, in most cases lacks
the semigroup property and needs a special consideration. The situation
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