Table Of ContentTHE MATHEMATICAL LANGUAGE
OF QUANTUM THEORY
Foralmosteverystudentofphysics,theirfirstcourseonquantumtheoryraisespuz-
zling 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 quan-
tum 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.
TEIKO HEINOSAARI 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.
MÁRIO ZIMAN is a researcher in the Research Center for Quantum Information
at the Institute of Physics at the Slovak Academy of Sciences, Bratislava, and lec-
tures 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.
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
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.
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
THE MATHEMATICAL LANGUAGE
OF QUANTUM THEORY
From Uncertainty to Entanglement
TEIKO HEINOSAARI
UniversityofTurku,Finland
MÁRIO ZIMAN
SlovakAcademyofSciences,Slovakia
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521195836
c T.HeinosaariandM.Ziman2012
!
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-19583-6Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
Contents
Preface page ix
Introduction xi
1 Hilbert space refresher 1
1.1 Hilbert spaces 1
1.1.1 Finite- and infinite-dimensional Hilbert spaces 1
1.1.2 Basis expansion 7
1.1.3 Example: L2(!) 9
1.2 Operators on Hilbert spaces 11
1.2.1 The C -algebra of bounded operators 11
∗
1.2.2 Partially ordered vector space of selfadjoint operators 17
1.2.3 Orthocomplemented lattice of projections 20
1.2.4 Group of unitary operators 25
1.2.5 Ideal of trace class operators 30
1.3 Additional useful mathematics 34
1.3.1 Weak operator topology 34
1.3.2 Dirac notation and rank-1 operators 36
1.3.3 Spectral and singular-value decompositions 38
1.3.4 Linear functionals and dual spaces 40
1.3.5 Tensor product 42
2 States and effects 45
2.1 Duality of states and effects 45
2.1.1 Basic statistical framework 45
2.1.2 State space 50
2.1.3 State space for a finite-dimensional system 60
2.1.4 From states to effects 68
2.1.5 From effects to states 72
2.1.6 Dispersion-free states and Gleason’s theorem 76
v
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
vi Contents
2.2 Superposition structure of pure states 81
2.2.1 Superposition of two pure states 81
2.2.2 Interference 83
2.3 Symmetry 86
2.3.1 Unitary and antiunitary transformations 87
2.3.2 State automorphisms 92
2.3.3 Pure state automorphisms and Wigner’s theorem 97
2.4 Composite systems 98
2.4.1 System versus subsystems 99
2.4.2 State purification 103
3 Observables 105
3.1 Observables as positive operator-valued measures 105
3.1.1 Definition and basic properties of observables 106
3.1.2 Observables and statistical maps 111
3.1.3 Discrete observables 113
3.1.4 Real observables 115
3.1.5 Mixtures of observables 116
3.1.6 Coexistence of effects 121
3.2 Sharp observables 126
3.2.1 Projection-valued measures 126
3.2.2 Sharp observables and selfadjoint operators 129
3.2.3 Complementary observables 134
3.3 Informationally complete observables 138
3.3.1 Informational completeness 138
3.3.2 Symmetric informationally complete observables 144
3.3.3 State estimation 146
3.4 Testing quantum systems 148
3.4.1 Complete versus incomplete information 149
3.4.2 Unambiguous discrimination of states 150
3.4.3 How distinct are two states? 159
3.5 Relations between observables 162
3.5.1 State distinction and state determination 162
3.5.2 Coarse-graining 164
3.6 Example: photon-counting observables 168
3.6.1 Single-mode electromagnetic field 168
3.6.2 Nonideal photon-counting observables 169
4 Operations and channels 173
4.1 Transforming quantum systems 173
4.1.1 Operations and complete positivity 174
4.1.2 Schrödinger versus Heisenberg picture 179
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
Contents vii
4.2 Physical model of quantum channels 181
4.2.1 Isolated versus open systems 181
4.2.2 Stinespring’s dilation theorem 185
4.2.3 Operator-sum form of channels 188
4.3 Elementary properties of quantum channels 191
4.3.1 Mixtures of channels 191
4.3.2 Concatenating channels 194
4.3.3 Disturbance and noise 198
4.3.4 Conjugate channels 201
4.4 Parametrizations of quantum channels 203
4.4.1 Matrix representation 204
4.4.2 The χ-matrix representation 205
4.4.3 Choi–Jamiolkowski isomorphism 207
4.5 Special classes of channels 210
4.5.1 Strictly contractive channels 210
4.5.2 Random unitary channels 212
4.5.3 Phase-damping channels 214
4.6 Example: qubit channels 216
5 Measurement models and instruments 222
5.1 Three levels of description of measurements 222
5.1.1 Measurement models 223
5.1.2 Instruments 226
5.1.3 Compatibility of the three descriptions 228
5.2 Disturbance caused by a measurement 230
5.2.1 Conditional output states 230
5.2.2 No information without disturbance 232
5.2.3 Disturbance in a rank-1 measurement 235
5.2.4 Example: BB84 quantum key distribution 236
5.3 Lüders instruments 241
5.3.1 Von Neumann’s measurement model 241
5.3.2 Lüders instrument for a discrete observable 243
5.3.3 Lüders’ theorem 244
5.3.4 Example: mean king’s problem 245
5.4 Repeatable measurements 247
5.4.1 Repeatability 248
5.4.2 Wigner–Araki–Yanase theorem 250
5.4.3 Approximate repeatability 252
5.5 Programmable quantum processors 254
5.5.1 Programming of observables and channels 254
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
viii Contents
5.5.2 Universal processor for channels 255
5.5.3 Probabilistic programming 258
6 Entanglement 261
6.1 Entangled bipartite systems 261
6.1.1 Entangled vectors 262
6.1.2 Entangled positive operators 267
6.1.3 Nonlocal channels 270
6.2 Entanglement and LOCC 277
6.2.1 LOCC ordering and separable states 277
6.2.2 Maximally entangled states 279
6.2.3 Majorization criterion for LOCC 285
6.3 Entanglement detection 286
6.3.1 Entanglement witnesses 287
6.3.2 Quantum nonlocal realism 292
6.3.3 Positive but not completely positive maps 295
6.3.4 Negative partial transpose (NPT) criterion 299
6.3.5 Range criterion 300
6.4 Additional topics in entanglement theory 302
6.4.1 Entanglement teleportation and distillation 302
6.4.2 Multipartite entanglement 307
6.4.3 Evolution of quantum entanglement 310
6.5 Example: Werner states 313
Symbols 317
References 318
Index 325
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:00 WET 2013.
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781139031103
Cambridge Books Online © Cambridge University Press, 2013
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 insensi-
tive 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 system-
atically 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 man-
ifested 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 conse-
quences dramatically contradict our classical experience. Measurement outcomes
observed by different observers can be entangled in a curious way, allowing for the
ix
Downloaded from Cambridge Books Online by IP 169.229.32.136 on Sun Mar 10 16:53:36 WET 2013.
http://dx.doi.org/10.1017/CBO9781139031103.001
Cambridge Books Online © Cambridge University Press, 2013
