Quantum Information Theory and Quantum Statistics

219 Pages·2008·2.368 MB·English

by  Dénes Petz (auth.)

Theoretical and Mathematical Physics
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in Physics (TMP) publishes high-level monographs in theoretical and mathematical
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EditorialBoard
W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Germany
J.-P.Eckmann,DepartmentofTheoreticalPhysics,UniversityofGeneva,Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA,USA
S.Smirnov,MathematicsSection,UniversityofGeneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,NY,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
JohnvonNeumann
ClaudeShannon ErwinSchr¨odinger
Dénes Petz
Quantum Information
Theory and Quantum
Statistics
With 10 Figures
Prof.DénesPetz
AlfrédRényiInstituteofMathematics
POB127,H-1364Budapest,Hungary
[email protected]
D.Petz,QuantumInformationTheoryandQuantumStatistics,TheoreticalandMathe-
maticalPhysics(Springer,BerlinHeidelberg2008)DOI10.1007/978-3-540-74636-2
ISBN978-3-540-74634-8 e-ISBN978-3-540-74636-2
TheoreticalandMathematicalPhysicsISSN1864-5879
LibraryofCongressControlNumber:2007937399
(cid:2)c 2008Springer-VerlagBerlinHeidelberg
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Preface
Quantummechanicswasoneoftheveryimportantnewtheoriesofthe20thcentury.
John von Neumann worked in Go¨ttingen in the 1920s when Werner Heisenberg
gave the first lectures on the subject. Quantum mechanics motivated the creation
of new areasin mathematics;the theoryof linear operatorson Hilbertspaceswas
certainlysuchanarea.JohnvonNeumannmadeanefforttowardthemathematical
foundation,andhisbook“Themathematicalfoundationofquantummechanics”is
stillratherinterestingtostudy.Thebookisapreciseandself-containeddescription
ofthetheory,somenotationshavebeenchangedinthemeantimeintheliterature.
Althoughquantummechanicsismathematicallyaperfecttheory,itisfullofin-
terestingmethodsandtechniques;theinterpretationisproblematicformanypeople.
Anexampleofthestrangeattitudesisthefollowing:“Quantummechanicsisnota
theoryaboutreality,itisaprescriptionformakingthebestpossiblepredictionabout
thefutureifwehavecertaininformationaboutthepast”(G.‘t’Hooft,1988).The
interpretationsofquantumtheoryarenotconsideredinthisbook.Thebackground
of the problemsmightbe the probabilisticfeatureof the theory.On one hand,the
result of a measurement is random with a well-defined distribution; on the other
hand,therandomquantitiesdonothavejointdistributioninmanycases.Thelatter
featurejustifiestheso-calledquantumprobabilitytheory.
AbstractinformationtheorywasproposedbyelectricengineerClaudeShannon
in the 1940s. It became clear that coding is very important to make the informa-
tion transfer efficient. Although quantum mechanics was already established, the
informationconsideredwasclassical; roughlyspeaking,thismeansthetransferof
0–1sequences.Quantuminformationtheorywasbornmuchlater inthe 1990s.In
1993C. H. Bennett,G.Brassard, C. Crepeau,R. Jozsa, A. PeresandW. Wootters
publishedthe paperTeleportingan unknownquantumstate via dualclassicaland
EPR channels, which describes a state teleportation protocol. The protocol is not
complicated;itissomewhatsurprisingthatitwasnotdiscoveredmuchearlier.The
reasoncan be thatthe interestin quantumcomputationmotivatedthe studyof the
transmissionofquantumstates. Manythingsin quantuminformationtheoryisre-
lated to quantum computationand to its algorithms. Measurements on a quantum
systemprovideclassicalinformation,andduetotherandomnessclassicalstatistics
v
vi Preface
canbeusedtoestimatethetruestate.Insomeexamples,quantuminformationcan
appear,thestateofasubsystemcanbeso.
The material of this book was lectured at the Budapest University of Technol-
ogyandEconomicsandattheCentralEuropeanUniversitymostlyforphysicsand
mathematicsmajors,and fornewcomersin the area. Thebookaddressesgraduate
studentsinmathematics,physics,theoreticalandmathematicalphysicistswithsome
interestintherigorousapproach.Thebookdoesnotcoverseveralimportantresults
in quantuminformationtheoryandquantumstatistics. Theemphasisisputonthe
real introductory explanation for certain important concepts. Numerous examples
andexercisesarealsousedtoachievethisgoal.Thepresentationismathematically
completelyrigorousbutfriendlywheneveritispossible.Sincethesubjectisbased
onnon-trivialapplicationsofmatrices,theappendixsummarizestherelevantpartof
linearanalysis.Standardundergraduatecoursesofquantummechanics,probability
theory,linearalgebraandfunctionalanalysisareassumed.Althoughtheemphasis
is on quantum information theory, many things from classical information theory
areexplainedaswell.Someknowledgeaboutclassicalinformationtheoryisconve-
nient,butnotnecessary.
I thank my students and colleagues, especially Tsuyoshi Ando, Thomas Baier,
ImreCsisza´r,KatalinHangos,FumioHiai,Ga´borKiss,Mila´nMosonyiandJo´zsef
Pitrik,forhelpingmetoimprovethemanuscript.
De´nesPetz
Contents
1 Introduction................................................... 1
2 PrerequisitesfromQuantumMechanics .......................... 3
2.1 PostulatesofQuantumMechanics ........................... 4
2.2 StateTransformations...................................... 14
2.3 Notes ................................................... 22
2.4 Exercises ................................................ 22
3 InformationanditsMeasures ................................... 25
3.1 Shannon’sApproach....................................... 26
3.2 ClassicalSourceCoding.................................... 28
3.3 vonNeumannEntropy ..................................... 34
3.4 QuantumRelativeEntropy.................................. 37
3.5 Re´nyiEntropy ............................................ 45
3.6 Notes ................................................... 49
3.7 Exercises ................................................ 50
4 Entanglement.................................................. 53
4.1 BipartiteSystems ......................................... 53
4.2 DenseCodingandTeleportation............................. 63
4.3 EntanglementMeasures.................................... 67
4.4 Notes ................................................... 69
4.5 Exercises ................................................ 70
5 MoreAboutInformationQuantities.............................. 73
5.1 Shannon’sMutualInformation .............................. 73
5.2 MarkovChains ........................................... 74
5.3 EntropyofPartiedSystems ................................. 76
5.4 StrongSubadditivityofthevonNeumannEntropy.............. 78
5.5 TheHolevoQuantity ...................................... 79
5.6 TheEntropyExchange..................................... 80
vii
viii Contents
5.7 Notes ................................................... 81
5.8 Exercises ................................................ 82
6 QuantumCompression ......................................... 83
6.1 DistancesBetweenStates................................... 83
6.2 ReliableCompression...................................... 85
6.3 Universality .............................................. 88
6.4 Notes ................................................... 90
6.5 Exercises ................................................ 90
7 ChannelsandTheirCapacity.................................... 91
7.1 InformationChannels...................................... 91
7.2 TheShannonCapacity ..................................... 92
7.3 HolevoCapacity .......................................... 95
7.4 Classical-quantumChannels ................................104
7.5 Entanglement-assistedCapacity .............................105
7.6 Notes ...................................................106
7.7 Exercises ................................................106
8 HypothesisTesting .............................................109
8.1 TheQuantumSteinLemma.................................110
8.2 TheQuantumChernoffBound ..............................116
8.3 Notes ...................................................119
8.4 Exercises ................................................120
9 Coarse-grainings...............................................121
9.1 BasicExamples...........................................121
9.2 ConditionalExpectations...................................123
9.3 CommutingSquares .......................................131
9.4 Superadditivity ...........................................133
9.5 Sufficiency...............................................133
9.6 MarkovStates ............................................138
9.7 Notes ...................................................141
9.8 Exercises ................................................142
10 StateEstimation ...............................................143
10.1 EstimationSchemas .......................................143
10.2 Crame´r–RaoInequalities ...................................150
10.3 QuantumFisherInformation ................................154
10.4 ContrastFunctionals.......................................162
10.5 Notes ...................................................163
10.6 Exercises ................................................164
Contents ix
11 Appendix:AuxiliaryLinearandConvexAnalysis..................165
11.1 HilbertSpacesandTheirOperators ..........................165
11.2 PositiveOperatorsandMatrices .............................167
11.3 FunctionalCalculusforMatrices ............................170
11.4 Distances ................................................175
11.5 Majorization .............................................177
11.6 OperatorMonotoneFunctions...............................180
11.7 PositiveMappings.........................................189
11.8 MatrixAlgebras ..........................................195
11.9 ConjugateConvexFunction.................................198
11.10 SomeTraceInequalities....................................199
11.11 Notes ...................................................200
11.12 Exercises ................................................200
Bibliography.......................................................205
Index .............................................................211