Quantum Teleportation and Entanglement: A Hybrid Approach to Optical Quantum Information Processing

352 Pages·2011·7.26 MB·English

by  Akira Furusawa

AkiraFurusawaandPetervanLoock
QuantumTeleportationand
Entanglement
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Akira Furusawa and Peter van Loock
Quantum Teleportation and Entanglement
A Hybrid Approach to Optical Quantum Information
Processing
WILEY-VCH Verlag GmbH & Co. KGaA
TheAuthors AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.AkiraFurusawa publisherdonotwarranttheinformation
TheUniversityofTokyo containedinthesebooks,includingthisbook,to
DepartmentofAppliedPhysics befreeoferrors.Readersareadvisedtokeepin
Tokyo,Japan mindthatstatements,data,illustrations,
[email protected] proceduraldetailsorotheritemsmay
inadvertentlybeinaccurate.
Dr.PetervanLoock
PhysikInstitutLSfürOptik LibraryofCongressCardNo.:appliedfor
InstitutTheorieI
UniversitätErlangen-Nürnberg BritishLibraryCataloguing-in-PublicationData:
Max-Planck-Institut(MPL) Acataloguerecordforthisbookisavailable
Erlangen,Germany fromtheBritishLibrary.
[email protected]
Bibliographicinformationpublishedbythe
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publicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableonthe
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©2011WILEY-VCHVerlagGmbH&Co.KGaA,
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ISBN 978-3-527-40930-3
V
Contents
Preface IX
PartOne IntroductionsandBasics 1
1 IntroductiontoQuantumInformationProcessing 3
1.1 WhyQuantumInformation? 5
1.2 StatesandObservables 10
1.2.1 Qubit 12
1.2.2 Qumode 13
1.3 Unitaries 18
1.3.1 Qubit 19
1.3.2 Qumode 20
1.4 Non-unitaries 23
1.4.1 Channels 23
1.4.2 Measurements 27
1.4.2.1 POVM 27
1.4.2.2 NaimarkExtension 29
1.5 Entanglement 31
1.5.1 PureStates 31
1.5.1.1 Qubits 33
1.5.1.2 Qumodes 33
1.5.2 MixedStatesandInseparabilityCriteria 34
1.5.3 EntanglementWitnessesandMeasures 35
1.6 QuantumTeleportation 38
1.6.1 DiscreteVariables 39
1.6.2 ContinuousVariables 40
1.7 QuantumCommunication 41
1.7.1 KeyDistribution 42
1.7.2 RepeatersandRelays 45
1.7.3 ShannonTheory 47
1.8 QuantumComputation 49
1.8.1 Models 52
1.8.2 Universality 54
1.8.2.1 Qubits 54
QuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock
Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40930-3
VI Contents
1.8.2.2 Qumodes 57
1.9 QuantumErrorCorrection 62
1.9.1 Discretization 63
1.9.2 StabilizerCodes 67
1.10 Experiment:Non-opticalImplementations 70
2 IntroductiontoOpticalQuantumInformationProcessing 79
2.1 WhyOptics? 80
2.2 QuantumOpticalStatesandEncodings 82
2.2.1 FieldQuantization 83
2.2.1.1 DiscreteModes 84
2.2.1.2 ContinuousModes 87
2.2.2 Quadratures 88
2.2.3 CoherentStates 89
2.2.4 SqueezedStates 90
2.2.5 Phase-SpaceRepresentations 94
2.2.6 PhotonicQubits 98
2.2.7 Experiment:PolarizationQubits 99
2.2.8 PhotonicQumodes 100
2.2.8.1 StabilizerStates 101
2.2.8.2 GaussianStates 102
2.2.9 Experiment:BroadbandQumodes 107
2.3 QuantumOpticalUnitaries 109
2.4 GaussianUnitaries 113
2.5 QuantumOpticalNon-unitaries 114
2.5.1 Channels 114
2.5.2 Measurements 116
2.6 GaussianNon-unitaries 117
2.7 LinearOptics:PossibilitiesandImpossibilities 120
2.8 OpticalQuantumComputation 121
PartTwo FundamentalResourcesandProtocols 125
3 Entanglement 127
3.1 QubitEntanglement 128
3.1.1 CharacterizationandWitnesses 128
3.1.1.1 TwoParties 128
3.1.1.2 ThreeorMoreParties 129
3.1.2 ClusterandGraphStates 131
3.1.3 Experiment:EntangledPhotonicQubits 132
3.1.3.1 EPR/BellState 132
3.1.3.2 GHZState 135
3.1.3.3 ClusterStates 138
3.2 QumodeEntanglement 142
3.2.1 CharacterizationandWitnesses 142
3.2.1.1 TwoParties 142
Contents VII
3.2.1.2 ThreeorMoreParties 149
3.2.2 ClusterandGraphStates 154
3.2.3 Experiment:EntangledPhotonicQumodes 160
3.2.3.1 Frequency-DomainEPR-TypeState 160
3.2.3.2 Time-DomainEPR-TypeState 161
3.2.3.3 GHZ-TypeState 167
3.2.3.4 Cluster-TypeStates 172
4 QuantumTeleportation 179
4.1 QubitQuantumTeleportation 180
4.1.1 Experiment:QubitQuantumTeleportation 180
4.1.2 Experiment:QubitTelecloning 183
4.1.3 Experiment:QubitEntanglementSwapping 185
4.2 QumodeQuantumTeleportation 188
4.2.1 Experiment:QumodeQuantumTeleportation 189
4.2.2 Experiment:QumodeTelecloning 196
4.2.3 Experiment:QumodeTeleportationNetwork 203
4.2.4 Experiment:QumodeEntanglementSwapping 210
5 QuantumErrorCorrection 217
5.1 TheNine-QubitCode 218
5.2 TheNine-QumodeCode 219
5.3 Experiment:QuantumErrorCorrection 220
5.3.1 Qubits 220
5.3.2 Qumodes 223
5.4 EntanglementDistillation 231
5.5 Experiment:EntanglementDistillation 233
5.5.1 Qubits 233
5.5.2 Qumodes 236
PartThree Measurement-BasedandHybridApproaches 243
6 QuantumTeleportationofGates 245
6.1 TeleportingQubitGates 246
6.1.1 KLM 246
6.1.2 Experiment:QubitGates 248
6.2 TeleportingQumodeGates 252
6.2.1 Experiment:GaussianQumodeGates 252
6.2.1.1 UniversalSqueezer 252
6.2.1.2 QuantumNon-demolition(QND)SumGate 259
6.2.2 UniversalQumodeGates 266
7 Cluster-BasedQuantumInformationProcessing 271
7.1 Qubits 272
7.1.1 ElementaryQubitTeleportations 272
7.1.2 Experiment:QubitClusterComputation 273
7.2 Qumodes 280
VIII Contents
7.2.1 ElementaryQumodeTeleportations 281
7.2.2 GaussianComputation 282
7.2.3 Experiment:GaussianQumodeClusterComputation 283
7.2.3.1 QuadraticPhaseGate 283
7.2.3.2 FourierandSqueezingGates 291
8 HybridQuantumInformationProcessing 299
8.1 HowtoCreateNon-GaussianStates,CatStates 300
8.2 Experiment:CreationofNon-GaussianStates,CatStates 303
8.3 HybridEntanglement 310
8.4 HybridQuantumTeleportation 312
8.4.1 Experiment:BroadbandQumodeTeleportation
ofaNon-GaussianWavepacket 313
8.4.2 Experiment:BroadbandQumodeTeleportation
ofaPolarizationQubit 314
8.5 HybridQuantumComputing 315
8.5.1 HybridHamiltonians 315
8.5.2 EncodingQubitsintoQumodes 317
8.5.3 GKP 318
References 323
Index 333
IX
Preface
The field of quantum information processing has reached a remarkable maturi-
ty in recent years with regard to experimental demonstrations. In particular, to-
wardsanextensionofopticalcommunicationsfromtheclassicalintothequantum
realm,manyproof-of-principleexperimentswereperformedincludingthegenera-
tionanddistributionofphotonicentangledstatesoverfree-spaceorfiberchannels.
Asanapplication,unconditionallysecurequantumkeydistributionsystemshave
emergedandevendevelopedintoacommerciallyavailabletechnology.
Lightsystems,apartfromtheirobvioususefulnessforcommunication,havenow
as well turned out to be a serious contender for approaches to quantumcompu-
tation. A breakthrough in this context was the theoretical discovery of so-called
measurement-basedmodels:quantumalgorithmsnolongerdependonsequences
ofreversiblequantumgates,eachenactedthroughwellcontrolledinteractionsbe-
tween,forinstance,twoormorequbits;instead,sequencesofmeasurementson
partsofanentangledresourcestatepreparedpriortothecomputationwilldothe
trick.Inotherwords,quantumentanglement,alreadyknowntobeauniversalre-
source for quantum communicationin conjunctionwith quantumteleportation,
representsauniversalresourceforquantumcomputationtoo–andagaintheex-
ploitationoftheentangledresourcereliesuponquantumteleportationwhich,in
itsultimateform,achievesarbitraryquantumstatemanipulations.
The aim of this book is to give a fairly general introduction to two com-
plementary approaches to quantum information processing: those based upon
discrete-variable “qubit” systems and those utilizing quantum oscillator systems
(“qumodes”) most naturally represented by continuous quantum variables such
as amplitude and phase. In quantum optics, the corresponding photonic sys-
temswouldconsistofjustafewphotonsortheywouldcorrespondtofieldswith
extremelyhighmeanphotonnumbers,respectively.Thequbitmaythenberepre-
sentedbythepolarizationofasinglephoton,whileaqumodestateisencodedinto
aninfinite-dimensionalphasespace.Entangledstatescanbedefined,formulated,
andexperimentallyrealizedineitherdimension,includingtheiruseforquantum
teleportation.Sinceeitherapproachencounterssomewhatdifferentcomplications
when it comes to more sophisticated quantum information protocols, a recent
trend in optical quantum information is to combine the two approaches and to
exploitatthesametimediscreteandcontinuousdegreesoffreedominaso-called
QuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock
Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40930-3