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A quantum information perspective of fermionic quantum many-body systems [Elektronische Ressource] / Christina V. Kraus

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Technische Universit¨at Mu¨nchenMax-Planck-Institut fu¨r QuantenoptikA Quantum InformationPerspective of Fermionic QuantumMany-Body SystemsChristina V. KrausVollst¨andiger Abdruck der von der Fakult¨at fu¨r Physikder Technischen Universit¨at Mu¨nchenzur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender : Univ.-Prof. Dr. R. GrossPru¨fer der Dissertation : 1. Hon.-Prof. I. Cirac, Ph. D.2. Univ.-Prof. Dr. H. FriedrichDie Dissertation wurde am 14.09.2009 bei derTechnischen Universit¨at Mu¨nchen eingereicht unddurch die Fakult¨at fu¨r Physik am 02.11.2009 angenommen.AbstractIn this Thesis fermionic quantum many-body system are theoretically investigatedfrom a quantum information perspective.Quantumcorrelations infermionic many-bodysystems, thoughcentraltomany ofthemostfascinatingeffectsofcondensedmatterphysics, arepoorlyunderstoodfromatheoretical perspective. Even thenotion of”paired”fermions which is widely usedin the theory of superconductivity and has a clear physical meaning there, is not aconceptofasystematicandmathematicaltheorysofar. Applyingconceptsandtoolsfrom entanglement theory, we close this gap, developing a pairing theory allowingto unambiguously characterize paired states. We develop methods for the detectionand quantification of pairing according to our definition which are applicable tocurrent experimental setups.

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Published 01 January 2009
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TechnischeUniversita¨tMu¨nchen

Max-Planck-Institutfu¨rQuantenoptik

AQuantumInformation
PerspectiveofFermionicQuantum
Many-BodySystems

ChristinaV.Kraus

Vollsta¨ndigerAbdruckdervonderFakulta¨tfu¨rPhysik
derTechnischenUniversita¨tMu¨nchen
zurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
genehmigtenDissertation.

Vorsitzender:Univ.-Prof.Dr.R.Gross

Pru¨ferderDissertation:1.Hon.-Prof.I.Cirac,Ph.D.
2.Univ.-Prof.Dr.H.Friedrich

DieDissertationwurdeam14.09.2009beider
TechnischenUniversita¨tMu¨ncheneingereichtund
durchdieFakulta¨tfu¨rPhysikam02.11.2009angenommen.

Abstract

InthisThesisfermionicquantummany-bodysystemaretheoreticallyinvestigated
fromaquantuminformationperspective.
Quantumcorrelationsinfermionicmany-bodysystems,thoughcentraltomanyof
themostfascinatingeectsofcondensedmatterphysics,arepoorlyunderstoodfrom
atheoreticalperspective.Eventhenotionof”paired”fermionswhichiswidelyused
inthetheoryofsuperconductivityandhasaclearphysicalmeaningthere,isnota
conceptofasystematicandmathematicaltheorysofar.Applyingconceptsandtools
fromentanglementtheory,weclosethisgap,developingapairingtheoryallowing
tounambiguouslycharacterizepairedstates.Wedevelopmethodsforthedetection
andquanticationofpairingaccordingtoourdenitionwhichareapplicableto
currentexperimentalsetups.Pairingisshowntobeaquantumcorrelationdistinct
fromanynotionofentanglementproposedforfermionicsystems,givingfurther
understandingofthestructureofhighlycorrelatedquantumstates.Inaddition,we
showtheresourcecharacterofpairedstatesforprecisionmetrology,provingthat
BCS-statesallowphasemeasurementsattheHeisenberglimit.
Next,thepoweroffermionicsystemsisconsideredinthecontextofquantum
simulations,wherewestudythepossibilitytosimulateHamiltoniantimeevolutions
onacubiclatticeundertheconstraintoftranslationalinvariance.Givenasetof
translationallyinvariantlocalHamiltoniansandshortrangeinteractionswedeter-
minetimeevolutionswhichcanandthosewhichcannotbesimulated.Bosonicand
nite-dimensionalquantumsystems(”’spins”)areincludedinourinvestigations.
Furthermore,wedevelopnewtechniquesfortheclassicalsimulationoffermionic
many-bodysystems.First,weintroduceanewfamilyofstates,thefermionicPro-
jectedEntangledPairStates(fPEPS)onlatticesinarbitraryspatialdimension.
ThesearethenaturalgeneralizationofthePEPSknownforspinsystems,andthey
approximateecientlygroundandthermalstatesofsystemswithshort-rangeinter-
action.WegiveanexplicitmappingbetweenfPEPSandPEPS,allowingtoextend
previoussimulationmethodstofermions.Inaddition,weshowthatfPEPSnat-
urallyariseasexactgroundstatesofcertainfermionicHamiltonians,andgivean
examplethatexhibitscriticalitywhilefulllingthearealaw.
Finally,wederivemethodsforthedeterminationofgroundandthermalstates,
aswellasthetimeevolution,ofinteractingfermionicsystemsusinggeneralized
Hartree-Focktheory(gHFT).Withthecomputationalcomplexityscalingpolyno-
miallywiththenumberofparticles,thismethodcandealwithlargesystems.As
abenchmarkweapplyourmethodstothetranslationallyinvariantHubbardmodel
withattractiveinteractionandndexcellentagreementwithknownresults.

OCTNNETSContents

i1Introduction1
2PairinginFermionicSystems7
2.1Fermionicstates..............................9
2.1.1Basicnotation...........................9
2.1.2Quantumcorrelationsoffermionicstates............10
2.1.3FermionicGaussianstates....................13
2.1.4Numberconservingfermionicstates...............15
2.2Pairingtheory...............................16
2.2.1Motivationandstatementofthedenition...........16
2.2.2Relationofpairingandentanglement..............18
2.2.3Methodsfordetectingpairing..................20
2.2.4Pairingmeasures.........................23
2.3PairingforGaussianstates........................23
2.3.1PairingwitnessesforGaussianstates..............23
2.3.2Completesolutionofthepairingproblemforfermionic
Gaussianstates..........................24
2.3.3AngularmomentumalgebraforGaussianstates........25
2.3.4ApairingmeasureforGaussianstates.............26
2.4Pairingofnumberconservingstates...................27
2.4.1PairingofallBCSstatesandgeometryofpairedstates....27
2.4.2Eigenvaluesofthetwo-particlereduceddensitymatrix....33
2.4.3Pairingmeasurefornumberconservingstates.........33
2.5Interferometry...............................35
2.5.1Ramseyinterferometrywithfermions..............36
2.5.2Interferometryinvolvingapair-interactionHamiltonian....42
2.6Applicationtoexperimentsandconclusion...............46
3QuantumSimulationsinTranslationallyInvariantSystems49
3.1Statementoftheproblem........................50
3.2QuadraticHamiltonians.........................51
3.3Simulationsinfermionicsystems.....................54
3.3.1Simulationsinone-dimensionalfermionicsystems.......55
3.3.2Simulationsind-dimensionalfermionicsystems........58
3.4Simulationsinbosonicsystems.....................59

iiOCTNNETS3.4.1Simulationsinone-dimensionalbosonicsystems........61
3.4.2Simulationsind-dimensionalbosonicsystems.........61
3.5Simulationsinspinsystems.......................62
3.6Summaryoftheresultsandconclusion.................65
4FermionicProjectedEntangledPairStates(fPEPS)67
4.1MPSandPEPSforspinsystems....................68
4.2ConstructionoffPEPS..........................71
4.3RelationbetweenfPEPSandPEPS...................72
4.4FermionicGaussianstatesandparentHamiltonians..........78
4.5ExampleofacriticalfPEPS.......................81
4.6Conclusion.................................89
5InteractingFermionicSystemsinGeneralizedHartree-FockThe-
19yro5.1ToolboxofgeneralizedHartree-FockTheory..............93
5.2Realtimeevolution............................94
5.3Groundstates...............................95
5.3.1Minimizationoftheenergy....................96
5.3.2Imaginarytimeevolution.....................97
5.4Thermalstates..............................98
5.5Application:The2d-Hubbard-Model..................100
5.6Conclusion.................................103

6ConclusionsandOutlook

AAstandardformforpurefermionicGaussianstates

BProofofThm.2.23

501

901

111

CInterferometrywithpairedstates117
C.1Quasi-bosoniclimit............................121
C.2Interferometryfarfromthebosoniclimit................122
DDerivationoftheevolutionequation127

Chapter1

Introduction

1Quantummechanicalcorrelationsthathavenoclassicalanalogueareoneofthe
mostcompellingphysicaldiscoveriesofthe20thcentury.Theexistenceofsuch
correlations,calledentanglement(”Verschra¨nkung”),wasrevealedbythefamous
Einstein-Podolsky-Rosen(EPR)gedankenexperimentinthe1930s[1].Considered
bysomeasa”spookyactionatadistance”atthattime,thenotionofentanglement
hastransformedintoawell-establishedconcepttoday.Astartingpointforthisde-
velopmentweretheinequalitiesformulatedbyBellin1964[2].ViolationofaBell
inequalityimpliestheexistenceofquantummechanicalcorrelationsthatcannotbe
simulatedbyanyclassicaltheory.Nearlytwentyyearslater,Aspectmanagedto
performtherstconvincingexperimentprovingtheviolationofaBellinequality
[3,4].Aspect’sstrikingexperimentresultedintheadventofquantuminformation
theoryintheearly1990s,wherethequantumcorrelationsplaytheroleofare-
sourcefortechnologicalapplications.Sincethattimequantuminformationscience
hasdevelopedintoavibrantresearcharea,rangingfromfoundationalquestions
oftheinterpretationonquantummechanicstowardsthesearchfortechnological
applicationofentanglement[5].Usingknowledgefromvariouseldsofphysics,
mathematicsandcomputerscience,theunderstandingandthecontrolofquantum
mechanicalsystemsisattheheartofquantuminformationtheory.
Thecoreofquantuminformationscienceistheuseofquantummechanicalpar-
ticlesasthecarrierofinformation.Anyquantummechanicaltwo-levelsystemcan
encodeoneunitbitofquantuminformation,andsuchasystemiscalledqubitin
analogytothebitofclassicalinformationtheory.Theimmensesuccessofquan-
tuminformationtheoryisduetotheinterplayoftheoryandexperimentwith
thosequbits:Protocolsforquantumcryptography[6],quantumdensecoding[7]
orquantumteleportation[8]couldallbedemonstratedinpioneeringexperiments
[9,10,11,12,13,14,15,16].Furthermore,ithasbeenpredictedthatcertain
computationaltasks,suchasfactoringnumbersorsimulatingquantummechanical
systems,canbecarriedoutexponentiallyfasterusingaquantumcomputerbased
onqubitsthanbyanyknownalgorithmrunningonaclassicalcomputer.However,
theexperimentalrealizationofalarge-scalequantumcomputercapableofaccom-
plishingthosetasksis,despitemajorexperimentalprogress,stillanunsolvedtask.
Nevertheless,thecompellingprogressintheeldofquantuminformationsciences

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