Quantum Theory from First Principles: An Informational Approach

359 Pages·2017·2.075 MB·English

by  Giacomo Mauro D’Ariano, Giulio Chiribella, Paolo Perinotti

QuantumTheoryfromFirstPrinciples
AnInformationalApproach
Quantumtheoryisthesouloftheoreticalphysics.Itisnotjustatheoryofspecificphysical
systems, but rather a new framework with universal applicability. This book shows how
we can reconstruct the theory from six information-theoretical principles, by rebuilding
the quantum rules from the bottom up. Step by step, the reader will learn how to master
the counterintuitive aspects of the quantum world, and how to efficiently reconstruct
quantuminformationprotocolsfromfirstprinciples.Usingintuitivegraphicalnotationto
represent equations, and with shorter and more efficient derivations, the theory can be
understoodandassimilatedwithexceptionalease.Offeringaradicallynewperspectiveon
thefield,thebookcontainsanefficientcourseofquantumtheoryandquantuminformation
for undergraduates. The book is aimed at researchers, professionals, and students in
physics,computerscience,andphilosophy,aswellasthecuriousoutsiderseekingadeeper
understandingofthetheory.
GiacomoMauroD’Ariano is a Professor at Pavia University, where he teaches Quantum
MechanicsandFoundationsofQuantumTheory,andleadsthegroupQUit.HeisaFellow
of the American Physical Society and of the Optical Society of America, a member
of the Academy Istituto Lombardo of Scienze e Lettere, of the Center for Photonic
Communication and Computing at Northwestern IL, and of the Foundational Questions
Institute(FQXi).
GiulioChiribella is Associate Professor and a CIFAR-Azrieli Global Scholar at the Depart-
ment of Computer Science of The University of Hong Kong. He is a Visiting Fellow of
Perimeter Institute for Theoretical Physics, a member of the Standing Committee of the
International Colloquia on Group Theoretical Methods in Physics, and a member of the
Foundational Questions Institute (FQXi). In 2010, he was awarded the Hermann Weyl
Prizeforapplicationsofgrouptheoryinquantuminformationscience.
PaoloPerinottiisAssistantProfessoratPaviaUniversitywhereheteachesQuantumInfor-
mation Theory. His research activity is focused on foundations of quantum information,
quantum mechanics, and quantum field theory. He is a member of the Foundational
Questions Institute (FQXi), and of the International Quantum Structures Association.
In 2016 he was awarded the Birkhoff–von Neumann prize for research in quantum
foundations.
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Quantum Theory from First
Principles
An Informational Approach
GIACOMO MAURO D’ARIANO
UniversitàdegliStudidiPavia,Italy
GIULIO CHIRIBELLA
TheUniversityofHongKong
PAOLO PERINOTTI
UniversitàdegliStudidiPavia,Italy
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Names:D’Ariano,G.M.(GiacomoM.),author.|Chiribella,Giulio,author.|Perinotti,Paolo,author.
Title:Quantumtheoryfromfirstprinciples:aninformationalapproach/GiacomoMauroD’Ariano
(Universita`degliStudidiPavia,Italy),GiulioChiribella(TheUniversityofHongKong),
PaoloPerinotti(Universita`degliStudidiPavia,Italy).
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,2017.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016040126|ISBN9781107043428(hardback;alk.paper)|
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Toourwivesandchildren:
RosannaandGilda,
AmyandFrancesco,
SilviaandMarco.
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Contents
Preface pagexiii
Acknowledgments xv
1 Introduction 1
1.1 TheQuestforPrinciples:vonNeumann 2
1.2 QuantumInformationResurrectstheQuest 3
1.3 QuantumTheoryasanOPT 4
1.4 ThePrinciples 5
PartI TheStatusQuo 9
2 QuantumTheoryfromHilbertSpaces 11
2.1 PrimitiveNotions 12
2.2 Hilbert-spacePostulatesforQuantumTheory 14
2.3 DensityMatricesandPOVMs 16
2.4 Causality,ConvexStructure,Discriminability 20
2.5 QuantumStates 25
2.6 EntangledQuantumStatesandEffects 30
2.7 Compression 33
2.8 QuantumTransformations 35
2.9 ClassicalTheoryasaRestrictionofQuantum 49
2.10 Purification 50
2.11 QuantumNoCloning 59
2.12 ThevonNeumannPostulate:DoWeNeedIt? 61
2.13 QuantumTeleportation 63
2.14 InvertingTransformations 66
2.15 Summary 73
Notes 73
Appendix2.1 PolarDecomposition 78
Appendix2.2 TheGoldenRuleforQuantumExtensions 79
Problems 80
SolutionstoSelectedProblemsandExercises 85
vii
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viii Contents
PartII TheInformationalApproach 109
3 TheFramework 111
3.1 TheOperationalLanguage 111
3.2 OperationalProbabilisticTheory 115
3.3 StatesandEffects 116
3.4 Transformations 118
3.5 Coarse-grainingandRefinement 119
3.6 OperationalDistanceBetweenStates 120
3.7 OperationalDistancesforTransformationsandEffects 122
3.8 Summary 124
Notes 124
Problems 126
SolutionstoSelectedProblemsandExercises 126
4 TheNewPrinciples 130
4.1 AtomicityofComposition 130
4.2 PerfectDiscriminability 132
4.3 IdealCompression 133
4.4 APreviewoftheThreeMainPrinciples 135
4.5 Summary 137
Notes 138
5 CausalTheories 139
5.1 Causality:FromCinderellatoPrinciple 139
5.2 NoSignalingfromtheFuture 141
5.3 Conditioning 142
5.4 AUniqueWastebasket 143
5.5 NoSignalingataDistance 150
5.6 CausalityandSpace-Time 151
5.7 TheorieswithoutCausality 151
5.8 Summary 154
Notes 155
SolutionstoSelectedProblemsandExercises 156
6 TheorieswithLocalDiscriminability 157
6.1 EntanglementandHolism 157
6.2 ThePrinciple 159
6.3 ReconcilingHolismwithReductionism 160
6.4 ConsequencesofLocalDiscriminability 163
6.5 DifferentDegreesofHolism 163
6.6 Summary 165
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Contents ix
Notes 165
Problems 166
SolutionstoSelectedProblemsandExercises 167
7 ThePurificationPrinciple 168
7.1 ADistinctiveandFundamentalTrait 168
7.2 ThePurificationPrinciple 171
7.3 Entanglement 172
7.4 ReversibleTransformationsandTwirling 173
7.5 Steering 175
7.6 ProcessTomography 176
7.7 NoInformationWithoutDisturbance 177
7.8 Teleportation 178
7.9 AReversiblePictureofanIrreversibleWorld 179
7.10 DisplacingtheVonNeumann’sCut 181
7.11 TheState-transformationIsomorphism 182
7.12 EverythingNotForbiddenisAllowed 184
7.13 PurificationinaNutshell 186
7.14 Summary 188
Notes 188
Appendix7.1 Carathe´odory’sTheorem 189
SolutionstoSelectedProblemsandExercises 190
PartIII QuantumInformationWithoutHilbertSpaces 191
8 EncodingInformation 193
8.1 ProcessingData=ProcessingEntanglement 193
8.2 IdealEncodings 195
8.3 IdealCompression 198
8.4 TheMinimalPurification 199
8.5 SendingInformationThroughaNoisyChannel 200
8.6 TheConditionforErrorCorrection 201
8.7 Summary 204
SolutionstoSelectedProblemsandExercises 204
9 ThreeNo-goTheorems 206
9.1 NoCloning 207
9.2 NoProgramming 209
9.3 NoBitCommitment 211
9.4 Summary 215
Notes 215
SolutionstoSelectedProblemsandExercises 217
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