understanding quantum mechanics from fundamental perspective quantum nonlocality measurement theory decoherence understanding quantum behavior reected in the properties of matter and applications they are the essential resource for quantum technolo ID: 78030
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Quantumcorrelationsfromentanglementtodiscord MicheleAllegraDipartimentodiFisica,UniversitadiTorino&INFN,I-10124,Torino,ItalyInstituteforScienti cInterchangeFoundation(ISI),I-10126,Torino,ItalySeminariodelIIannodidottorato MicheleAllegra Quantumcorrelations Overview Quantumcorrelations:abriefhistory Quantumentanglement&quantumdiscord Quantumdiscord&condensedmatter Quantumdiscord&quantumoptics Next MicheleAllegra Quantumcorrelations Therelevanceofquantumcorrelations Quantumcorrelations:correlationstypicalofquantumsystems,thatcannotbeexplainedbyclassicalphysicsCrucialfortheory... understandingquantummechanicsfromfundamentalperspective:quantumnonlocality,measurementtheory,decoherence understandingquantumbehaviorre\rectedinthepropertiesofmatter...andapplications theyaretheessentialresourceforquantumtechnology:quantumcommunication,quantumcomputation,quantummetrology MicheleAllegra Quantumcorrelations Quantumcorrelations:averybriefhistory 1920es:birthofquantummechanics 1950es:canweexplainquantummechanicsinclassicalterms?Quantumcorrelationsarerecognizedasakeyconcepttoaddressthisquestion 1960es:researchonentangledparticlesreavealsquantumnonlocality:quantumphysicsisfundamentallydi erentfromclassicalphysics 1980es:correlationsbetweenquantumobjectsandtheirenvironmentcanexplaintheemergenceof"classicalreality"fromtheunderlyingquantumworld MicheleAllegra Quantumcorrelations Quantumcorrelations:averybriefhistory 1990es:riseofquantuminformationtheory:realizationthatquantumcorrelationsopenthedoortonovelcommunicationalandcomputationaltasks(quantumcomputation,quantumteleportation,quantumdensecoding...) 2000es:entanglementandcondensedmattertheory:analysisofquantumcorrelationsessentialincharacterizingandsimulatingquantummatter 2010es:entanglementdoesnotaccountforallquantumcorrelations.Newconceptofquantumdiscord.Researchonroleofdiscordinquantumscienceandtechnology MicheleAllegra Quantumcorrelations Quantummechanics:recap quantummecanicsisabouttheprobabilitiesofobtainingmeasurementresults,giventhatthesystemhasbeeninitialized("prepared")insomeway thepossiblepurestatesofaquantumsystemformavectorspace,theHilbertspaceH Apurestatej irepresentsthe nestpossiblepreparationofaquantumsystem(maximalinformationaboutthesystem) Ingeneral,thepreparationisimperfect:the"mixed"stateofthesystemisdescribedbyadensitymatrix2E(H) representsastatisticalmixtureofstates:=Pjpjj jih jj,i.e.,asortofprobabilitydensityovertheHilbertspace MicheleAllegra Quantumcorrelations Quantummechanics:recap ApropertyofasystemisrepresentedasubspaceoftheHilbertspaceSH(setofstatessatisfyingtheproperty) AsubspaceSisspeci edbyaprojectionoperatorSonthatsubspace An"observable"isrepresentedbyanoperatorA=Piaiisuchthati=jiihijareprojectorsontothethesubspacecorrespondingtoai GivenastateandanobservableA: theprobabilityofmeasuringaiisgivenbypi=Tr[%i]thatrepresentstheoverlapbetweenthestate%andthesubsetSicorrespondingtoai thepost-measurementstatecorrespondingtomeasurementresultaiis~%i=1 pi[i%i]thatrepresentstheprojectionof%ontoSi Theaveragepost-measurementstateis~%=Pi[i%i] MicheleAllegra Quantumcorrelations Compoundness:tensorproductstructures tode necorrelations,weneedade nitionofsubsystems giventwosystemswithHilbertspaceHAandHB,thejointsystemhasHilbertspaceHA\nHB Conversely,givenageneralHilbertspaceH=Cn,itcanbe"divided"inseveralways:forinstanceC8=C2\nC2\nC2andC8=C2\nC4: Thejargonisthatthereareseveralpossibletensorproductstructures(TPS)overH:TPSde nequantumsubsystems MicheleAllegra Quantumcorrelations Correlations Givenastate2HA\nHB,the("reduced")statesofthesubsystemsare:A=TrB,B=TrA Thesystemisuncorrelatedwhen=A\nB Thesystemiscorrelatedwhen=A\nB Thismeansthateachsubsystemhasinformationabouttheother Generalmeasureofcorrelations:quantummutualinformationI=S(A)+S(B) S() vonNeumannentropyS()=Trlog:measureshowbroadlyisspreadovertheHilbertspace ForcorrelatedstatesI�0:theglobalstateisknownwithmoreaccuracythanoseofthesingleparts MicheleAllegra Quantumcorrelations Quantumcorrelations:quantumentanglement Quantumsystemscancontain"nonlocal"correlations jointsystemcomposedofsubsystemsAandB supposewepreparestatesstartingfromanuncorrelatedstatethroughasequenceoflocaloperationsOA\nOB,operatingsepartelyoneitersubsystem thenthestatesmustcontainno"nonlocal"correlations suchstatesarecalledseparable wecanprovethatastateisseparableifandonlyifitcanbewrittenasamixtureofproductstates:=XjpjjA\njB (inthespecialcaseofpurestate,astateisseparableifandonlyifitcanbewrittenasaproduct:j i=j Ai\nj Bi) MicheleAllegra Quantumcorrelations Quantumcorrelations:quantumentanglement inthegeneralcase,=XjpjjA\njB Thesestatesmustbepreparedwithglobal(nonlocal)operations.Theyarecalledentangledstates Entangledstatesdisplaystrongcorrelationsthatareimpossibleinclassicalmechanics(andindeedcanviolate,e.g.,Bell'sinequality) MicheleAllegra Quantumcorrelations Therelevanceofentanglement foundationsofquantummechanics:entanglementnecessaryconditionfornonlocality(violationofBell'sinequalities) quantumcomputation:entanglementnecessaryconditionforspeedupinpurestatequantumcomputation. quantumcommunication:entanglementrelevantforprotocolslikequantumteleportation,quantumdensecoding,quantumkeydistribution condensedmatter:studyofentanglementessentialinunderstandinganddesigningnewecientsimulationstrategiesformany-bodysystemsatlowT MicheleAllegra Quantumcorrelations Quantumcorrelations:quantumdiscord Entanglementisnottheonlykindofgenuinely"quantum"correlation Quantumsystemscanbe(globally)disturbedby(local)measurements Ifameasurementonasinglesubsystemalterstheglobalcorrelations,thenthesubsystemsarequantumlycorrelated thisisaquantumfeaturethathasnoclassicalcounterpart(classically,correlationsremainunalteredbymeasurementsonasubsystem) MicheleAllegra Quantumcorrelations Quantumcorrelations:quantumdiscord jointsystemcomposedofsubsystemsAandB performlocalmeasurementBonsubsystemB Theaveragepost-measurementstateis~=XiiBiBwithpi=Tr[iBiB]beingtheprobabilityofoutcomei. Ingeneral,themeasurementoftheBdisruptscorrelationsbetweenAandB:I(~)I() MicheleAllegra Quantumcorrelations Quantumcorrelations:quantumdiscord Quantumdiscordmeasurestheminimalamountofcorrelationsthathavebeenlost:Q=min(I() I(~)) Theoptimizationisoverallpossiblemeasurements Ingeneral,itisverydiculttoidentifytheoptimalmeasurementdisturbsthesystems-andcorrelations-theleast;thismakesanontrivialtechnicalissueincomputingdiscord Forpurestate,entanglementanddiscordcoincide Formixedstates,entanglementanddiscordaredi erent:inparticular,separablestatescanhavenonzerodiscordand,hence,containquantumcorrelationsthatarenotintheformofentanglement MicheleAllegra Quantumcorrelations Therelevanceofquantumdiscord foundationsofquantummechanics:betterunderstandingthequantum-classicaldi erence quantumdiscordmayallowforquantumcomputation,quantumcommunicationandquantummetrologywithmixedstatesthatarenotentangled entanglementisrapidlydestroyedbynoise,discordisoftenmorerobust MicheleAllegra Quantumcorrelations Quantumdiscordincondensedmatter Whytoinvestigatequantumdiscordincondensedmattersystems: Docondensedmattersystemscontainasigni cantamountofquantumcorrelationsintheformofquantumdiscord? Canquantumdiscordberelated/explainsomephysicalpropertiesofthesesystems? Conversely,canthesesystemsilluminategeneralpropertiesofdiscord? MicheleAllegra Quantumcorrelations QuantumdiscordintheexdendedHubbardmodel M.Allegra,P.Giorda,A.Montorsi,Phys.Rev.B84,245133(2011) Quantumdiscordinthe1-DextendedHubbardmodel Simplestmodelofstronglycorrelatedelectrons:H=Hhopping+H lling+Hon siterepulsionHhopping=Phi;ji;[1 x(ni+ni)]cyiciH lling= PiniHon siterepulsion=u=2Pi(ni 1=2)(ni 1=2) MicheleAllegra Quantumcorrelations QuantumdiscordintheexdendedHubbardmodel Exactlysolvablemodel(simplerversionofHubbardmodel) Thegroundstateappearsasfollows:jGSi=(Xk�Kscyk"cy k#)Nd| {z }YjkjKscyk| {z }j0i-pairsspinlessfermions -pairsareresponsiblefortheappearanceofo -diagonallong-rangeorder(ODLRO),akindoflong-rangecorrelationthatisatthebasisofsuperconductivity MicheleAllegra Quantumcorrelations QuantumdiscordintheexdendedHubbardmodel Themodelhasanontrivialphasediagramwhenthechemicalpotentialandtherepulsionstrengthuvary Somephases(IIandIII)arecharacterizedbythepresenceof-pairs,henceofODLRO MicheleAllegra Quantumcorrelations QuantumdiscordintheexdendedHubbardmodel weevaluatequantumdiscordinallphasesofthemodel(weusenoveltechniquetoevaluatediscordforqutrits) weevaluatethescalingofdiscordinthevininityofthequantumphasetransitions wecomparethebehaviorofentanglementanddiscordinthesystem MicheleAllegra Quantumcorrelations QuantumdiscordintheexdendedHubbardmodel Somephaseshavediscordinabsenceofentanglement WeareabletorelateODLRO,whichisatthebasisofsuperconductivity,toameasureofquantumcorrelations,quantumdiscord Wehighlightageneralpropertyofdiscord:itviolatesthemonogamyproperty(i.e.,asubsystemcanbecorrelatedwitharbitrarilymanyothersubsystems),themaximalviolationbeinginthepresenceofODLRO Webettercharachterizethephysicsofthesystemclosetothecriticallines,inparticulartheapprearance/disappearanceofODLRO MicheleAllegra Quantumcorrelations Quantumdiscordinquantumoptics Whytoinvestigatequantumdiscordinquantumopticalsystems: Dophotonicstatescontainasigni cantamoutofquantumcorrelationsintheformofquantumdiscord?Whichstateshavemaximalcorrelations? Whatarethemasurementsthatdisturbthesystemminimally?Cantheybecaiiredoverwithrealisticsetups? Canwecreatequantumdiscordwithexperimentallyrealisticresources? MicheleAllegra Quantumcorrelations Non-GaussianquantumdiscordforGaussianstates P.Giorda,M.Allegra,M.G.A.Paris,Phys.Rev.A86,052328(2012) AparadigminquantumopticsistoconsiderGaussianstates Gaussianstatescanbepreparedusingonlylinearopticaldevices(beamsplitters,parametricampli ers,etc.) TheyhaveaGaussianstatistics Allinformationaboutthestateisincorrelationmatrix,whoseentriesarequadraticcorrelationfunctionshaiaji,hayiaji,etc. Goal:evaluatequantumdiscordinGaussianstates Problem:whicharethemeasurementsthatdisturbthesystemtheleast? MicheleAllegra Quantumcorrelations Non-GaussianquantumdiscordforGaussianstates itiseasytocomputethee ectofGaussianmeasurements,i.e.measurementsthatpreserveGaussianity.ThetypicalGaussianmeasurementishomodynedetection optimizingonlyoverGaussianmeasurements,analyticalformulafordiscordwasobtainedDG(A:B)=h(p detB) h(d ) h(d+)+minMh(p detP) Question:aretherenonGaussianmeasurementsthatarebetter(theyarelessdisturbing)thanGaussianones?(ThetypicalnonGaussianmeasurementisphotoncounting) MicheleAllegra Quantumcorrelations Non-GaussianquantumdiscordforGaussianstates WeconsideralargeclassofGaussianstates,mixedthermalstates(MTS)andsqueezedthermalstates(STS)andcomparethee ectofGaussianandnonGaussianmeasurements Forthelargeclassofstatesanalyzed,intherangeofparametersconsidered,theGaussianmeasurementsareoptimal WehavethusstrongevidencethattheGaussianmeasurementsarealwaysoptimalforGaussianstates Thismightbeveri edexperimentallyusingphotoncountersandhomodynedetection MicheleAllegra Quantumcorrelations Next:quantumcorrelationsincomplexquantumfermionicnetworks Networksofquasi-freefermions:H=PNij=1Aijcyicj Aistheadjacencymatrixofageneralgraph(notaregularlattice) Goal:ageneralstudyofquantumcorrelations(entanglementanddiscord)inthesemodels Howdothetopologicalpropertiesofthenetworka ectcorrelations? Whichnodesarethemostcorrelated? Howdoesthenatureofmultipartite/bipartitecorrelationsdependonthedensityoflinksinthenetwork? MicheleAllegra Quantumcorrelations Acknowledgements PaoloGiorda,mysupervisor@ISI Allfellowresearchersandcollaborators Thankyouforyourattention MicheleAllegra Quantumcorrelations Publications2011-2012: M.Allegra,P.Giorda,A.Montorsi,Phys.Rev.B84,245133(2011) M.Allegra,P.Giorda,M.G.A.Paris,Phys.Rev.Lett.107,238902(2011) M.Allegra,P.Giorda,Phys.Rev.E85,051917(2012) P.Giorda,M.Allegra,M.G.A.Paris,Phys.Rev.A86,052328(2012) MicheleAllegra Quantumcorrelations