Geometry of Quantum States an Introduction to Quantum Entanglement

635 Pages·2017·12.581 MB·English

by  Ingemar Bengtsson, Karol Życzkowski

GEOMETRY OF QUANTUM STATES
AnIntroductiontoQuantumEntanglement
Quantuminformationtheoryisabranchofscienceatthefrontiersofphysics,mathematics
andinformationscience,andoffersavarietyofsolutionsthatareimpossibleusingclassical
theory. This book provides a detailed introduction to the key concepts used in processing
quantum information and reveals that quantum mechanics is a generalisation of classical
probabilitytheory.
The second edition contains new sections and entirely new chapters: the hot topic of
multipartite entanglement; in-depth discussion of the discrete structures in finite dimen-
sionalHilbertspace,includingunitaryoperatorbases,mutuallyunbiasedbases,symmetric
informationallycompletegeneralisedmeasurements,discreteWignerfunctionsandunitary
designs;theGleasonandKochen–Speckertheorems;theproofoftheLiebconjecture;the
measureconcentrationphenomenon;andtheHastings’non-additivitytheorem.
This richly-illustrated book will be useful to a broad audience of graduates and
researchers interested in quantum information theory. Exercises follow each chapter,
withhintsandanswerssupplied.
INGEMAR BENGTSSON is a professor of physics at Stockholm University. After
gaining a Ph.D. in theoretical physics from the University of Göteborg (1984), he held
post-doctoral positions at CERN, Geneva, and Imperial College, London. He returned to
Göteborg in 1988 as a research assistant at Chalmers University of Technology, before
taking up a position as Lecturer in Physics at Stockholm University in 1993. He was
appointedProfessorofPhysicsin2000.ProfessorBengtssonisamemberoftheSwedish
Physical Society andaformerboardmember ofitsDivisionsforParticlePhysics andfor
Gravitation.Hisresearchinterestsarerelatedtogeometry,intheformsofclassicalgeneral
relativityandquantumtheory.
KAROL Z˙YCZKOWSKI is a professor at the Institute of Physics, Jagiellonian Univer-
sity, Kraków, Poland and also at the Center for Theoretical Physics, Polish Academy of
Sciences, Warsaw. He gained his Ph.D. and habilitation in theoretical physics at Jagiel-
lonianUniversity,andhasfollowedthiswithaHumboldtFellowshipinEssen,aFulbright
Fellowship at the University of Maryland, College Park, and a visiting research position
at the Perimeter Institute, Waterloo, Ontario. He has been a docent at the Academy of
Sciences since 1999 and a full professor at Jagiellonian University since 2004. Professor
Z˙yczkowskiisamemberofthePolishPhysicalSocietyandAcademiaEuropaea.Heworks
on quantum information, dynamical systems and chaos, quantum and statistical physics,
appliedmathematicsandthetheoryofvoting.

GEOMETRY OF QUANTUM STATES
An Introduction to Quantum Entanglement
SECOND EDITION
INGEMAR BENGTSSON
StockholmUniversity,Sweden
KAROL Z˙YCZKOWSKI
JagiellonianUniversity,Poland
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107026254
DOI:10.1017/9781139207010
©IngemarBengtssonandKarolZ˙yczkowski2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2006
Secondedition2017
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Contents
Preface pagexi
Acknowledgements xv
1 Convexity,coloursandstatistics 1
1.1 Convexsets 1
1.2 Highdimensionalgeometry 9
1.3 Colourtheory 15
1.4 Whatis‘distance’? 18
1.5 Probabilityandstatistics 25
Problems 28
2 Geometryofprobabilitydistributions 29
2.1 Majorisationandpartialorder 29
2.2 Shannonentropy 35
2.3 Relativeentropy 41
2.4 Continuousdistributionsandmeasures 46
2.5 StatisticalgeometryandtheFisher–Raometric 48
2.6 Classicalensembles 55
2.7 Generalisedentropies 56
Problems 62
3 Muchadoaboutspheres 63
3.1 Spheres 63
3.2 Paralleltransportandstatisticalgeometry 68
3.3 Complex,Hermitian,andKählermanifolds 74
3.4 Symplecticmanifolds 80
3.5 TheHopffibrationofthe3-sphere 82
3.6 Fibrebundlesandtheirconnections 88
3.7 The3-sphereasagroup 95
v
vi Contents
3.8 Cosetsandallthat 99
Problems 101
4 Complexprojectivespaces 103
4.1 Fromarttomathematics 103
4.2 Complexprojectivegeometry 108
4.3 Complexcurves,quadricsandtheSegreembedding 110
4.4 Stars,spinorsandcomplexcurves 114
4.5 TheFubini–Studymetric 117
4.6 CPn illustrated 122
4.7 SymplecticgeometryandtheFubini–Studymeasure 129
4.8 Fibrebundleaspects 131
4.9 Grassmanniansandflagmanifolds 133
Problems 137
5 Outlineofquantummechanics 138
5.1 Quantummechanics 138
5.2 QubitsandBlochspheres 140
5.3 ThestatisticalandtheFubini–Studydistances 143
5.4 Areallookatquantumdynamics 146
5.5 Timereversals 150
5.6 Classicalandquantumstates:aunifiedapproach 154
5.7 GleasonandKochen–Specker 157
Problems 164
6 Coherentstatesandgroupactions 165
6.1 Canonicalcoherentstates 165
6.2 Quasi-probabilitydistributionsontheplane 170
6.3 Blochcoherentstates 178
6.4 FromcomplexcurvestoSU(K)coherentstates 184
6.5 SU(3)coherentstates 186
Problems 190
7 Thestellarrepresentation 191
7.1 Thestellarrepresentationinquantummechanics 191
7.2 Orbitsandcoherentstates 194
7.3 TheHusimifunction 197
7.4 WehrlentropyandtheLiebconjecture 202
7.5 GeneralisedWehrlentropies 205
7.6 Randompurestates 207
7.7 FromthetransportproblemtotheMongedistance 214
Problems 217
Contents vii
8 Thespaceofdensitymatrices 219
8.1 Hilbert–Schmidtspaceandpositiveoperators 220
8.2 Thesetofmixedstates 223
8.3 Unitarytransformations 226
8.4 Thespaceofdensitymatricesasaconvexset 229
8.5 Stratification 234
8.6 Projectionsandcross-sections 239
8.7 Analgebraicafterthought 244
8.8 Summary 247
Problems 248
9 Purificationofmixedquantumstates 249
9.1 Tensorproductsandstatereduction 250
9.2 TheSchmidtdecomposition 252
9.3 StatepurificationandtheHilbert–Schmidtbundle 255
9.4 AfirstlookattheBuresmetric 258
9.5 BuresgeometryforN = 2 261
9.6 FurtherpropertiesoftheBuresmetric 263
Problems 266
10 Quantumoperations 267
10.1 MeasurementsandPOVMs 267
10.2 Algebraicdetour:matrixreshapingandreshuffling 274
10.3 Positiveandcompletelypositivemaps 278
10.4 Environmentalrepresentations 283
10.5 Somespectralproperties 286
10.6 Unitalandbistochasticmaps 288
10.7 Onequbitmaps 291
Problems 295
11 Duality:mapsversusstates 296
11.1 Positiveanddecomposablemaps 296
11.2 Dualconesandsuper-positivemaps 304
11.3 Jamiołkowskiisomorphism 305
11.4 Quantummapsandquantumstates 309
Problems 311
12 DiscretestructuresinHilbertspace 313
12.1 UnitaryoperatorbasesandtheHeisenberggroups 313
12.2 Prime,compositeandprimepowerdimensions 317
12.3 Moreunitaryoperatorbases 322
viii Contents
12.4 Mutuallyunbiasedbases 326
12.5 FinitegeometriesanddiscreteWignerfunctions 333
12.6 Cliffordgroupsandstabilizerstates 338
12.7 Somedesigns 343
12.8 SICs 347
Problems 353
13 Densitymatricesandentropies 355
13.1 Orderingoperators 356
13.2 VonNeumannentropy 359
13.3 Quantumrelativeentropy 365
13.4 Otherentropies 369
13.5 Majorisationofdensitymatrices 372
13.6 ProofoftheLiebconjecture 376
13.7 Entropydynamics 380
Problems 384
14 Distinguishabilitymeasures 386
14.1 Classicaldistinguishabilitymeasures 386
14.2 Quantumdistinguishabilitymeasures 391
14.3 Fidelityandstatisticaldistance 396
Problems 401
15 Monotonemetricsandmeasures 402
15.1 Monotonemetrics 402
15.2 Productmeasuresandflagmanifolds 408
15.3 Hilbert–Schmidtmeasure 410
15.4 Buresmeasure 414
15.5 Inducedmeasures 416
15.6 Randomdensitymatrices 418
15.7 Randomoperations 422
15.8 Concentrationofmeasure 426
Problems 431
16 Quantumentanglement 433
16.1 Introducingentanglement 433
16.2 Two-qubitpurestates:entanglementillustrated 436
16.3 Maximallyentangledstates 441
16.4 Purestatesofabipartitesystem 444
16.5 Afirstlookatentangledmixedstates 454
16.6 Separabilitycriteria 460
Contents ix
16.7 Geometryofthesetofseparablestates 466
16.8 Entanglementmeasures 471
16.9 Two-qubitmixedstates 481
Problems 491
17 Multipartiteentanglement 493
17.1 Howmuchisthreelargerthantwo? 493
17.2 Botanyofstates 494
17.3 Permutationsymmetricstates 498
17.4 Invarianttheoryandquantumstates 503
17.5 Monogamyrelationsandglobalmultipartiteentanglement 515
17.6 Localspectraandthemomentummap 518
17.7 AMEstatesanderror-correctingcodes 525
17.8 Entanglementinquantumspinsystems 532
Problems 542
Epilogue 544
AppendixA Basicnotionsofdifferentialgeometry 545
A.1 Differentialforms 545
A.2 Riemanniancurvature 546
A.3 Akeyfactaboutmappings 547
AppendixB Basicnotionsofgrouptheory 549
B.1 LiegroupsandLiealgebras 549
B.2 SU(2) 550
B.3 SU(N) 551
B.4 Homomorphismsbetweenlow-dimensionalgroups 552
AppendixC Geometry:doityourself 553
AppendixD Hintsandanswerstotheexercises 557
References 572
Index 614