testing for entanglement with periodic coarse graining d. s. tasca资料.pdf

该【testing for entanglement with periodic coarse graining d. s. tasca资料 】是由【小舍儿】上传分享，文档一共【10】页，该文档可以免费在线阅读，需要了解更多关于【testing for entanglement with periodic coarse graining d. s. tasca资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。PHYSICALREVIEWA97,042312(2018),1,2,*?ukaszRudnicki,3,,,,ísica,UniversidadeFederalFluminense,Niterói,RJ24210-346,Brazil2InstitutodeFísica,UniversidadeFederaldoRiodeJaneiro,CaixaPostal68528,RiodeJaneiro,RJ21941-972,Brazil3Max-Planck-InstitutfürdiePhysikdesLichts,Staudtstra?e2,91058Erlangen,Germany4CenterforTheoreticalPhysics,PolishAcademyofSciences,AlejaLotników32/46,02-668Warsaw,Poland5SUPA,SchoolofPhysicsandAstronomy,UniversityofGlasgow,Glasgow,G128QQ,UnitedKingdom6DepartamentodeFísica,UniversidadeFederaldeSantaCatarina,Florianópolis,SantaCatarina88040-900,Brazil(Received7February2018;published9April2018)Continuous-variablesystems??nite-dimensionalHilbertspace,-?cientevaluationofentanglementusingspatialmasksactingasmodeanalyzersovertheentiretransverse?～60%:.[31,32].EventhoughtypicalCVentanglementTheef?cientpreparationandmanipulationofhigh-andEPRcriteriahavebeenadaptedforcoarse-grainedmea-dimensionalquantumsystemsallowtheprocessingoflargesurements[32,33],,andmomentumcorrelationstypicallyrequirealargenumbersuchasincreasedtransmissionrates,[1,2],Anumberoftechniqueshavebeenappliedinanefforttononlocality[3],andcontextuality[4]eevidentwithreducethenumberofmeasurementsnecessarytoidentifyen-,Howlandetal.[34]haveshownconstituteaninterestingexperimentalplatformforthestudypressivesensingtechniquesallowstheofhigh-dimensionalquantumsystems[5].Inprinciple,spatialreconstructionofthespontaneousparametricdown-conversiondegreesoffreedom(DOF)aredescribedbyanin?nitesetof(SPDC)jointdetectionprobabilitieswithanef?ciencyim-modes:thetransversepositionormomentummodesprovideprovementoverarasterscanningprocedurewithequivalentthecontinuous-variable(CV)descriptionofthespatialDOF,,nevertheless,theidenti?cationofwhilein?nite-dimensionaldiscretebasiscanalsobeexploredentanglementisstillboundtotheapplicationofthetypicalbyusing,forinstance,theorbitalangularmomentum[6,7]andCVentanglementcriteria[35]based,forexample,ontheradialmodes[8,9].Inthelatterapproach,a?nite-dimensionalevaluationofthemoments[23–25]orentropy[36,37]-dimensionalstates,entangle-Inthepresentcontribution,wedevelopandtestexperi-mentdetection[10–16]isperformedwithtoolsdevelopedformentallyconveniententanglementcriteriabasedonperiodic?nite-dimensionalquantumsystems[17,18].coarse--IntheCVregime,ontheotherhand,theobservationofcertaintyrelation(UR)[19–22],fromwhichcriteriadevotedtoCVsystemsaredevelopedURisexpressedintermsofthecross-correlationusedtotestfornonseparability[23–25],Einstein-Podolsky-functionbetweenaperiodicanalyzerandthedistributionsforRosen(EPR)[26–28]correlations,and“steering”[29].,real-worldexperimentsaresubjectbuildentanglementcriteriadevotedtobipartiteCVquantumtothecoarsegrainingimposedbythedetectorresolution[30]systemsandapplythemtotestforspatialentanglementofaswellasthelimiteddetectorrange[31].Bothoftheseissuesphotonpairsfromspontaneousparametricdown-,thedevelopedcriteriaareexperimentallyaccessibleviathejointtransmissionofthephotonsthroughperiodicaperturesplayingtheroleofspatial-modeanalyz-*dan.******@-9926/2018/97(4)/042312(10)042312-1?,042312(2018)wherex=(x,y)?asthe?elddistributionSpatialMaskGeometryatthebackfocalplaneofthelens(theFourierplane).Thetransformationconnectingthesetwowavefunctionsis1ψ?(x)=d2xψ(x)e?ix·(x/α),(2)Lens2παwherex=(x,y)representsthetransversespatialcoordinateattheFourierplaneandα=fλ/2πisaconstantrelatedtotheopticalsystem:?PeriodicSpatialrecognizethattheFourier-transformed?elddistributionψMaskAnalysermapsthetransversestructureoftheinputphotoninmomentumrepresentation,φ(p)=p|ψ:ψ?(x)∝φ(p).(3)Equation(3)implicitlyassumes(wesethˉ=1)therelationx=ponentp=(px,py)-?elddistributionofthe?photonatthefront(ψ)andback(ψ)’|M|=M,,suchaperturesaredescribedbyperiodicsquarewaveswithtwoanalyzersactingovertheentiretransverse?eldstructureofindependentparameters:theperiodicityTandanextraspatialthephotonsenablesapproximatelyuniformsingle-,wede?nehighercoincidencedetectionratesthanthetraditionalbinningtheperiodicspatialmaskanalyzeraswithsingleapertures,thusyieldingbettersignal-to-noise1,0x(modT)<s,,theperiodicitiesofthespatialmasksusedM(x;T,s)=(4)0,(Tx)andmomentum(Tp)measurementsworkasfreeparametersthatcanbeindependentlytunedtooptimizeForagivenchoiceofTands,,wetested7344uniquelyspeci?edbyEq.(4),providedthatthemask’soriginbinationsofspatialmaskgeometries,achievingis?,weallowextradisplacementparametersessrateinentanglementdetectionofabout60%.–IV,weItwillbeusefultoconsidertheFourierseriesexpansionofprovidethetheoreticalbackgroundnecessaryforthedevelop-Eq.(4):mentofourentanglementcriteria,.inτxOurexperimentalschemeandmeasurementswiththeperiodicM(x;T,s)=cn(τs)e,(5),andtheanalysisofn∈Zourexperimentaldatawiththederivedentanglementcriteriawhereτ=2π/(κ)=(e?inκ?1)(6)?(x).WethusworkwithasinglepairofconjugatevariablessatisfyingTheparaxialpropagationofamonochromaticsinglepho-mutationrelation[x,?p?]=[x,?x?]/α=,eitherinthefrontalongthepositivezaxisandwedenoteψ(x)=x|ψtheinputorbackfocalplaneofthelens,andweallowdistinctparameterstransverse?(Tx,sx)Wethusconsideraninputpuresinglephotonwhosequantumandmomentum(Tp,sp)=≡stateinpositionrepresentationisarrangementwheneverTx/sxTp/spd,butingeneralanycombinationofperiodicanalyzersisallowed;notethatsince|=|thetransversemomentumvariablepismappedtotheFourierψdxψ(x)x,(1)coordinatexbymeansofthescalingfactorα=fλ/2π,the042312-2OARSE…PHYSICALREVIEWA97,042312(2018)individualchoiceoftheperiodicityandbinwidthisirrelevantRef.[44],itisstraightforwardtoshow(seeAppendix)thatinasymmetricarrangementaslongastheratioTx/sx=Tp/spthecross-correlationfunctions(7)=αTpandsx=αspastheTx/2Tp/2physicalspatialparameters(unitsoflength)oftheanalyzer1212dξx|P(ξx)|+dξp|P?(ξp)|Q,(9)usedintheFourierplaneofthelens,whereasTpandspTx?Tx/2Tp?Tp/2expresstheassociatedquantitiesinmomentumdomain(unitsofinverselength).Fromnowon,unlessspeci?ed,weadoptwith=Cmax+CminG(n2ττ)+,(10)Our?gureofmeritforthedevelopmentoftheURandnnxp∈(x)=|ψ(x)|2andwhereP?(p)=|φ(p)|2,thesetransmissionprobabilitiesaremax/min={||2||2}Cnmax/(τxsx),cn(τpsp),(11)P(ξx)=dxM(x?ξx;Tx,sx)P(x),(7a)andthefunctionG(·)reads[45]R√√√2?1?cos(γ)P?=??Gγ=?.(ξp)dpM(pξp;Tp,sp)P(p),(7b)()22+1(12)R1cos(γ)=?=?Notethat0G(γ),thenotationinEq.(9)whereP(x)RdyP(x)andP(p)RdpyP(p)representthemarginalprobabilitydistributionsalongtherelevantdegreeinvolvingthemodulioftheprobabilitydistributions(real,non-.(7),theparametersξxandξpdescribenegative)(7)canbeunderstoodasthecross-plexformulas(8).Q≡QcorrelationfunctionbetweentheprobabilitydensityoftheTheupperbound(Tx,sx,Tp,sp)-correlationtheperiodicanalyzerparametersusedtoprobethe?’,functionaldependenceoftheFouriercoef?cients(6)isonlythespatialmaskofEq.(4)actsasa?lterthatisusedtoanalyzeontheratioT/s,theboundfunction(10)turnsintoasimplerthespatialstructureofthephoton?eldintheposition[Eq.(7a)]formwheneverworkingwithasymmetricarrangement[seeormomentum[Eq.(7b)].(20)].Tobrie?ysummarize,thecross-correlationbetweensincetheconsideredanalyzerisanamplitudemask,nophase-theperiodicanalyzerandthetransverse?eldstructureofaplementarydomainsissubjectEq.(7).ThisisincontrastwithmodeanalyzersbasedonspiraltotheuncertaintyrelationgiveninEq.(9).Thisuncertainty[38–40]ormultisector[41–43]phasemasksthathavebeenrelationconstitutesthe?-angular--worldmeasurementsofaCVdegreeoffreedomarein-[30].HeretheresolutionlimitationappearsduetoAsitisknown,theFouriertransform(2)impliesthatthethenecessarilydiscretizedtransversedisplacementsintheex-photon’stransverse?elddistributioncannotbearbitrarilywellperimentalcharacterizationofthecross-(7)insteadofits(unfeasible)samplingoveracontinuumofplementaritytobuildanuncertaintyrelationdisplacementsξx(p).Realistically,theexperimentalistholdsbasedonthetransmissionprobabilitiesofthephotonthroughdiscretizeddistributionsPk≡P(kξx)andP?k≡P?(kξp),itisconvenienttousethebyemployinga?nitesetofdisplacementswithscanningreso-posit