978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From
Uncertainty to Entanglement. Teiko Heinosaari and Mário Ziman. Frontmatter.
Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
T H E M AT H E M AT I C A L L A N G UAG E O F QUA N T U M T H E O RY
For almost every student of physics, their first course on quantum theory raises puzzling questions and creates an uncertain picture of the quantum world. This book presents a clear and detailed exposition of the fundamental concepts of quantum theory: states, effects, observables, channels and instruments. It introduces several up-to-date topics, such as state discrimination, quantum tomography, measurement disturbance and entanglement distillation. A separate chapter is devoted to quantum entanglement. The theory is illustrated with numerous examples, reflecting recent developments in the field. The treatment emphasizes quantum information, though its general approach makes it a useful resource for graduate students and researchers in all subfields of quantum theory. Focusing on mathematically precise formulations, the book summarizes the relevant mathematics. T E I K O H E I N O S A A R I is a researcher in the Turku Centre for Quantum Physics of the Department of Physics and Astronomy at the University of Turku, Finland. His research focuses on quantum measurements and quantum information theory. M Á R I O Z I M A N is a researcher in the Research Center for Quantum Information at the Institute of Physics at the Slovak Academy of Sciences, Bratislava, and lectures at the Faculty of Informatics at the Masaryk University in Brno. His research interests include the foundations of quantum physics, quantum estimations and quantum information theory.
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
The present volume is part of an informal series of books, all of which originated as review articles published in Acta Physica Slovaca. The journal can be accessed for free at www.physics.sk/aps. Vladimir Buzek, editor of the journal.
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
THE MATHEMATICAL LANGUAGE O F QUANT U M T HE ORY From Uncertainty to Entanglement TEIKO HEINOSAARI University of Turku, Finland
MÁRIO ZIMAN Slovak Academy of Sciences, Slovakia
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521195836 c T. Heinosaari and M. Ziman 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-19583-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
Contents
Preface Introduction 1 Hilbert space refresher 1.1 Hilbert spaces 1.1.1 Finite- and infinite-dimensional Hilbert spaces 1.1.2 Basis expansion 1.1.3 Example: L 2 () 1.2 Operators on Hilbert spaces 1.2.1 The C ∗ -algebra of bounded operators 1.2.2 Partially ordered vector space of selfadjoint operators 1.2.3 Orthocomplemented lattice of projections 1.2.4 Group of unitary operators 1.2.5 Ideal of trace class operators 1.3 Additional useful mathematics 1.3.1 Weak operator topology 1.3.2 Dirac notation and rank-1 operators 1.3.3 Spectral and singular-value decompositions 1.3.4 Linear functionals and dual spaces 1.3.5 Tensor product 2 States and effects 2.1 Duality of states and effects 2.1.1 Basic statistical framework 2.1.2 State space 2.1.3 State space for a finite-dimensional system 2.1.4 From states to effects 2.1.5 From effects to states 2.1.6 Dispersion-free states and Gleason’s theorem
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2.2 Superposition structure of pure states 2.2.1 Superposition of two pure states 2.2.2 Interference 2.3 Symmetry 2.3.1 Unitary and antiunitary transformations 2.3.2 State automorphisms 2.3.3 Pure state automorphisms and Wigner’s theorem 2.4 Composite systems 2.4.1 System versus subsystems 2.4.2 State purification 3 Observables 3.1 Observables as positive operator-valued measures 3.1.1 Definition and basic properties of observables 3.1.2 Observables and statistical maps 3.1.3 Discrete observables 3.1.4 Real observables 3.1.5 Mixtures of observables 3.1.6 Coexistence of effects 3.2 Sharp observables 3.2.1 Projection-valued measures 3.2.2 Sharp observables and selfadjoint operators 3.2.3 Complementary observables 3.3 Informationally complete observables 3.3.1 Informational completeness 3.3.2 Symmetric informationally complete observables 3.3.3 State estimation 3.4 Testing quantum systems 3.4.1 Complete versus incomplete information 3.4.2 Unambiguous discrimination of states 3.4.3 How distinct are two states? 3.5 Relations between observables 3.5.1 State distinction and state determination 3.5.2 Coarse-graining 3.6 Example: photon-counting observables 3.6.1 Single-mode electromagnetic field 3.6.2 Nonideal photon-counting observables 4 Operations and channels 4.1 Transforming quantum systems 4.1.1 Operations and complete positivity 4.1.2 Schrödinger versus Heisenberg picture
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Contents
4.2 Physical model of quantum channels 4.2.1 Isolated versus open systems 4.2.2 Stinespring’s dilation theorem 4.2.3 Operator-sum form of channels 4.3 Elementary properties of quantum channels 4.3.1 Mixtures of channels 4.3.2 Concatenating channels 4.3.3 Disturbance and noise 4.3.4 Conjugate channels 4.4 Parametrizations of quantum channels 4.4.1 Matrix representation 4.4.2 The χ -matrix representation 4.4.3 Choi–Jamiolkowski isomorphism 4.5 Special classes of channels 4.5.1 Strictly contractive channels 4.5.2 Random unitary channels 4.5.3 Phase-damping channels 4.6 Example: qubit channels 5 Measurement models and instruments 5.1 Three levels of description of measurements 5.1.1 Measurement models 5.1.2 Instruments 5.1.3 Compatibility of the three descriptions 5.2 Disturbance caused by a measurement 5.2.1 Conditional output states 5.2.2 No information without disturbance 5.2.3 Disturbance in a rank-1 measurement 5.2.4 Example: BB84 quantum key distribution 5.3 Lüders instruments 5.3.1 Von Neumann’s measurement model 5.3.2 Lüders instrument for a discrete observable 5.3.3 Lüders’ theorem 5.3.4 Example: mean king’s problem 5.4 Repeatable measurements 5.4.1 Repeatability 5.4.2 Wigner–Araki–Yanase theorem 5.4.3 Approximate repeatability 5.5 Programmable quantum processors 5.5.1 Programming of observables and channels
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5.5.2 Universal processor for channels 5.5.3 Probabilistic programming 6 Entanglement 6.1 Entangled bipartite systems 6.1.1 Entangled vectors 6.1.2 Entangled positive operators 6.1.3 Nonlocal channels 6.2 Entanglement and LOCC 6.2.1 LOCC ordering and separable states 6.2.2 Maximally entangled states 6.2.3 Majorization criterion for LOCC 6.3 Entanglement detection 6.3.1 Entanglement witnesses 6.3.2 Quantum nonlocal realism 6.3.3 Positive but not completely positive maps 6.3.4 Negative partial transpose (NPT) criterion 6.3.5 Range criterion 6.4 Additional topics in entanglement theory 6.4.1 Entanglement teleportation and distillation 6.4.2 Multipartite entanglement 6.4.3 Evolution of quantum entanglement 6.5 Example: Werner states Symbols References Index
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
Preface
Quantum theory is not an easy subject to master. Trained in the everyday world of macroscopic objects like locomotives, elephants and watermelons, we are insensitive to the beauty of the quantum world. Many quantum phenomena are revealed only in carefully planned experiments in a sophisticated laboratory. Some features of quantum theory may seem contradictory and inconceivable in the framework set by our experience. Rescue comes from the language of mathematics. Its mighty power extends the limits of our apprehension and gives us tools to reason systematically even if our practical knowledge fails. Mastering the relevant mathematical language helps us to avoid unnecessary quantum controversies. Quantum theory, as we understand it in this book, is a general framework. It is not so much about what is out there, but, rather, determines constraints on what is possible and what is impossible. This type of constraint is familiar from the theory of relativity and from thermodynamics. We will see that quantum theory is also a framework, and one of great interest, where these kinds of question can be studied. What are the main lessons that quantum theory has taught us? The answer, of course, depends on who you ask. Two general themes in this book reflect our answer: uncertainty and entanglement. Uncertainty. Quantum theory is a statistical theory and there seems to be no way to escape its probabilistic nature. The intrinsic randomness of quantum events is the seed of this uncertainty. There are various different ways in which it is manifested in quantum theory. We will discuss many of these aspects, including the nonunique decomposition of a mixed state into pure states, Gleason’s theorem, the no-cloning theorem, the impossibility of discriminating nonorthogonal states and the unavoidable disturbance caused by the quantum measurement process. Entanglement. The phenomenon of entanglement provides composite quantum systems with a very puzzling and counterintuitive flavour. Many of its consequences dramatically contradict our classical experience. Measurement outcomes observed by different observers can be entangled in a curious way, allowing for the ix
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violation of local realism or the teleportation of quantum states. It is fair to say that entanglement is the key resource for quantum information-processing tasks. It should be noted that both these themes, uncertainty and entanglement, have puzzled quantum theorists since the beginning of the quantum age. Uncertainty can be traced back to Werner Heisenberg, while the word ‘entanglement’ was coined by Erwin Schrödinger. After many decades uncertainty and entanglement are still under active research, probably more so than ever before. The evolution of this book had several stages, quite similar to the life cycle of a frog. Its birth dates back to a series of lectures the authors gave in 2007–8 in the Research Center for Quantum Information, Bratislava. The audience consisted of Ph.D. students working on various subfields of quantum theory. The principal aim of the lectures was to introduce a common language for the core part of quantum theory and to formulate some fundamental theorems in a mathematically precise way. The positive response from the students encouraged us to hatch the egg, and the tadpole stage of this book was an article based on the lecture notes. The article appeared in Acta Physica Slovaca in August 2008. The metamorphosis from a tadpole to an adult frog took more than two years. During that stage we benefited greatly from the comments of our friends and colleagues. We hope that this grown-up frog can serve as a guide for students wishing to get an overview of the mathematical formulation of quantum theory. Acknowledgement. This book would have never come into existence without the guidance, encouragement and help of Vladimír Bužek. Many friends and colleagues read parts of earlier versions of the manuscript and gave us valuable comments and support. Our special thanks are addressed to Paul Busch, Andreas Doering, Viktor Eisler, Sergej Nikolajeviˇc Filippov, Stan Gudder, Jukka Kiukas, Pekka Lahti, Leon Loveridge, Daniel McNulty, Harri Mäkelä, Marco Piani, Daniel Reitzner, Tomáš Rybár, Michal Sedlák, Peter Staˇno and Kari Ylinen. Teiko Heinosaari is grateful to the Alfred Kordelin Foundation for financial support.
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Cambridge University Press 978-0-521-19583-6 - The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement Teiko Heinosaari and Mário Ziman Frontmatter More information
Introduction
What is this book about? This book is an introduction to the basic structure of quantum theory. Our aim is to present the most essential concepts and their properties in a pedagogical and simple way but also to keep a certain level of mathematical precision. On top of that our intention is to illustrate the formalism in examples that are closely related to current research problems. As a result, the book has a quantum information flavor although it is not a quantum information textbook. The ideas related to quantum information are presented as consequences or applications of the basic quantum formalism. Many textbooks on quantum physics concentrate either on finite- or infinitedimensional Hilbert spaces. In this book the idea has been to treat finite- and infinite-dimensional Hilbert-space formalisms on the same footing. To keep the book at a reasonable size and so as not to be drawn into mathematical technicalities, we have sometimes compromised and presented a theorem in a general form
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but have given its proof only in the finite-dimensional case. The last chapter, on the mathematical aspects of the phenomenon of entanglement, is written entirely in a finite-dimensional setting. What this book is not about. We should perhaps warn the reader that this book is not about the different interpretations of quantum theory: we mostly avoid discussions of the philosophical consequences of the theory. These issues are important and interesting, but a proper understanding is easier to achieve if one knows the mathematical structure reasonably well first. Naturally, one cannot grasp the theory without some minimal interpretation. We have tried to keep the discussion away from controversial questions that do not (yet) have commonly agreed frameworks. Actually, even though two quantum theorists might disagree on some deeper foundational issues, they would agree on the basic predictions of quantum theory, such as measurement outcome probabilities. The reader will notice that many topics that could (and perhaps should) be included under the title The Mathematical Language of Quantum Theory are missing. For instance, the book Mathematical Concepts of Quantum Mechanics by Gustafson and Sigal has almost the same title as the present book but an almost disjoint content (actually, it can be recommended as a complement to this book). We believe that our book provides a compact and coherent overview of one branch of the mathematical language of quantum theory. Prerequisites. We assume that the reader has already met quantum theory and has perhaps studied an elementary course on quantum mechanics. We also assume a basic knowledge of real analysis and linear algebra. We imagine that a typical reader is a Master’s or Ph.D. student who wants to broaden his or her knowledge and learn a framework into which quantum concepts fit naturally. Clear definitions of basic concepts and their properties, together with a reasonably comprehensive index, should make the book useful as a reference work also. Structure of the book. The book consists of 28 sections, grouped into six thematic chapters. Each section is divided into several subsections, which typically treat one question or topic. In the first chapter we recall the basics of Hilbert spaces. In each of the following five chapters we concentrate on one or two key concepts of quantum formalism. Within the body of the text there are short exercises where the reader is asked to verify a formula. These exercises are easy and straightforward, and hints are often given.
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