Introduction to Quantum-State Estimation

396 Pages·2015·13.578 MB·English

by  Yong Siah Teo

INTRODUCTION TO
Quantum-State
Estimation
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INTRODUCTION TO
Quantum-State
Estimation
Yong Siah Teo
Palacky´ University, Czech Republic
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
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Library of Congress Cataloging-in-Publication Data
Teo, Yong Siah, author.
Introduction to quantum-state estimation / Yong Siah Teo, Palacký University, Czech Republic.
pages cm
Includes bibliographical references and index.
ISBN 978-9814678834 (hardcover : alk. paper) -- ISBN 978-9814678841 (softcover : alk. paper)
1. Quantum theory. 2. Estimation theory. I. Title.
QC174.12.T44 2015
530.12--dc23
2015005913
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2016 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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In-house Editor: Ng Kah Fee
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
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To my parents, teachers, colleagues and friends.
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Preface
Quantum-stateestimationisanimportantsubjectinquantuminformation
theory that focuses on the verification of the quality of a quantum source
through statistical inference from data collected using a quantum mea-
surement scheme. The body of literature in this field is still growing, with
a substantial amount addressing practical issues that still lack complete
solutions.Somenovelideasthathaveemergedincluderecentdevelopments
in scalable partial estimation schemes for many-body quantum states
and quantum channels of large Hilbert-space dimensions, as well as the
assignment of error bars for quantum-state estimators.
In order to understand the subject matter well, it is necessary to
start from the basics. The book begins with a background formalism
for quantum-state estimation to lay the foundation. Next, a substantial
amountoftimewillbespentonexploringsomespecificestimationschemes,
directing the reader’s attention to a few common techniques such as the
maximum-likelihoodandmaximum-entropyestimationschemes,tworather
intuitive techniques suitable for an introductory survey on the subject.
After which, the tomographic performance of state estimation schemes
as well as other key experimental aspects of quantum-state tomography
will be investigated, where the ideas of continuous-variable measurements
are introduced. It is also during this time that preparations are made
for the final discussion on quasi-probability distribution functions, which
are natural representations for measurement data of continuous quantum
degrees of freedom.
This book is intended to serve as an instructive and self-contained
medium for advanced undergraduate and postgraduate students to grasp
the basics of quantum-state estimation, so that future independent studies
can be subsequently carriedoutwith relativeease.Any readerwith a solid
vii
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viii Preface
foundation in quantum mechanics, linear algebra and calculus would be
able to follow the book comfortably. The material presented are partly
a reflection of my perspectives and experience gained from carrying out
research in the field, some of which were presented in 2013 at a summer
schoolhostedinPalacky´UniversityinOlomouc,CzechRepublic.Inwriting
this book, I did not follow any other reference material. However, as
the book covers aspects of contemporary research, I have supplied a
short reading list at the end of every chapter as a source for additional
information.
Hints and sample solutions to all problems are included for reference,
although they by no means represent the only approaches, only exempli-
fying ones, since there is typically more than one method to solve a given
problem.
A goal of this book is to demonstrate some of the mathematical
calculations that are typically performed in the field of quantum-state
estimation. Mastering some elementary aspects of complex analysis and
special mathematical functions would certainly expedite the learning
process for later chapters, although the essential ideas from these topics
are reiterated wherever necessary to make the discussions coherent. It is
my hope that, after going through the details, the reader will find these
mathematical techniques, which are commonly employed in other areas in
physics, more easily accessible.
I wish to express my sincere gratitude to my former Honours-Year
and Ph.D. supervisor Prof. Berthold-Georg Englert at the Centre for
Quantum Technologies, National University of Singapore, for his ever
insightful teachings, patience and friendship. His passion to eloquently
convey physical concepts and mathematical techniques has greatly influ-
enced my understanding in physics. Much more progress ensued through
collaborations with my colleagues and supervisors Prof. Jaroslav Rˇeh´aˇcek
andProf.ZdenˇekHradilattheDepartmentofOpticsinPalacky´University,
with whom I have eventually become good friends, and learnt plenty of
technicalskills thatdeepened my knowledge.They representmany experts
whom I have worked with and gained numerous insights from.
The World Scientific Publishing Company has provided its utmost
professionalassistanceandservicesforthecompletionofthisbook.Iwould
like to especially thank the desk editor Kah Fee Ng for his patience,
understanding and proactive contributions throughout the process of
editing this book, the cover artist Lionel Seow for the captivating cover
design, the copy editor Prathima from the subsidiary editorial services
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Preface ix
company ACES for meticulously transferring the contents of the book to
the master copy, as well as the in-house typesetter Yolande Koh and those
fromStallionPressfortypesettingthecontentsandremovingtypographical
artifacts. They have graciously accommodated my periodically demanding
editorial requests.
Most certainly, I thank all my teachers, colleagues and friends who
helped me in any way during my education, and my family members who
havebeenprovidingencouragementandsupportallthistime.Inparticular,
I would like to acknowledge Jibo Dai, Wei Hou Tan, Yink Loong Len,
Prof. Englert, and Prof. Jan Peˇrina for proofreading the book and giving
valuable comments and suggestions to improve the material.
Olomouc, February 2015 Y. S. Teo