Measuring the entanglement of bipartite pure states

The problem of the experimental determination of the amount of entanglement of a bipartite pure state is addressed. We show that measuring a single observable does not suffice to determine the entanglement of a given unknown pure state of two particles. Po

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aMeasuringtheentanglementofbipartitepurestatesJ.M.G.Sancho1andS.F.Huelga1,21DepartamentodeF´ sica.UniversidaddeOviedo.CalvoSotelos/n.33007Oviedo.Spain.2OpticsSection,BlackettLaboratory,ImperialCollege,LondonSW72BZ,UnitedKingdom(February1,2008)AbstractTheproblemoftheexperimentaldeterminationoftheamountofentangle-mentofabipartitepurestateisaddressed.Weshowthatmeasuringasingleobservabledoesnotsu cetodeterminetheentanglementofagivenunknownpurestateoftwoparticles.Possibleminimallocalmeasuringstrategiesarediscussedandacomparisonismadeonthebasisoftheirbestachievableprecision.PACS-numbers:03.67.-a,03.65.Bz

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The problem of the experimental determination of the amount of entanglement of a bipartite pure state is addressed. We show that measuring a single observable does not suffice to determine the entanglement of a given unknown pure state of two particles. Po

I.INTRODUCTION

Quantummechanicalstatesofmultiparticlesystemscanbeentangled,afactwellknownsincetheearlydaysoftheformulationofthequantumtheory[1].However,thestatusofthispropertyhaschangedsubstantiallyinrecentyears.Entanglementcanalsobeviewedasaresourceandassuchfeaturesindi erentprocessesofpotentialpracticalimportance,forexamplequantumteleportation[2],quantumcryptography[3]orevenhighprecisionmeasurements[4].Fromatheoreticalpointofview,entanglementofbipartitepurestates,anditspropertiesunderlocalquantumoperations,arereasonablywellunderstood.Thereisauniquemeasureofentanglementforthesesystems,providedbythevonNeumannentropy

[5],andoptimalwaysforentanglementmanipulationareknown[6].However,thereisaremainingpracticalquestionwhichhasnotbeenaddressedsofar:Howcanoneoptimallymeasuretheamountofentanglementofanunknownbipartitepurestate?

Ata rstsight,thisquestionmayseemobvious.Reconstructingthereduceddensityoperatorofanyofthetwosubsystemswilldothejob.However,theessentialpointisthatwerequirethedeterminationtobeoptimalandthereconstructionofthereduceddensitymatrixmayprovideredundantinformation,giventhatweareaskingforjustafeatureofthecompositestate,itsentanglement.Thisisasinglenumberandthe rstquestiontobeanswerediswhetherthereexistsasingleoperatorwhoseexperimentalmeasuremayprovideuswithjusttheamountofentanglementofthestate.Notethatfurtherdetailsofthestateitselfarenotofinterestintheproblemweareposinghere[7].Wewillproveinthefollowingthatsuchanoperatordoesnotexist,aconclusionthatcon rmswhatitcouldinitiallybethoughtasaneducatedguess.Knowingtheimpossibilityofatestusingarepeatedmeasurementsofasingleobservable,wewilldiscusspossiblestrategiesaimedtobeminimal,inthesenseofinvolvingthesmallestnumberofobservables.Inordertoavoidanyambiguitywhencountingthenumberofobservablesinvolvedinagivenmeasurementprotocol,wewillde nesuchnumberasthedi erentnumberofmeterseachobserverhastoreadout.Amongdi erentminimalstrategies,thatis,strategiesinvolvingthesamenumberofmeters,wewillcalloptimaltheoneprovidingthebestaccuracywhensuppliedwiththesameresources.Wewillshowthat,infact,measuringthereduceddensitymatrixturnsouttobeaminimalwaytoproceed.Moreover,theprotocolcanbemadeoptimalwheninvolvingprojectionsalongmutuallyorthogonaldirections.Wehaveorganizedthepaperasfollows.InsectionIIwestateformallytheproblem.SectionIIIshowstheimpossibilityof ndingasingleobservablewhosemeasuremayallowtheexperimentaldeterminationoftheamountofentanglement.MinimallocalstrategiesarediscussedinSectionIV.Whensuppliedwiththesamenumberofidenticallypreparedbipartitepurestates,wediscusstheperformanceoftwoclassesofminimalmeasurementsfromthepointofviewoftheachievableprecisionindeterminingtheamountofentanglement.SectionVisdevotedtoconclusions.

II.ANEXPERIMENTALSCENARIO

Letusimaginethefollowingsituation.Weareprovidedwithastatepreparatorwhichcreatespairsoftwo-levelparticles(qubits)inanunknownentangledstate.Theseentangledpairsaredistributedtotworemotelocationswheretwoobservers,AliceandBob,may

The problem of the experimental determination of the amount of entanglement of a bipartite pure state is addressed. We show that measuring a single observable does not suffice to determine the entanglement of a given unknown pure state of two particles. Po

performlocalmeasurementsaswellasinterchangeclassicalcommunication.Theinternaldynamicsofthedeviceisnotspeci edandtheonlythingAliceandBobknowisthat,withhighaccuracy,thestatetheyshareispure.Therefore,thetwo-qubitstatecanbewrittenas

ρ=|ψAB ψAB|,

where

|ψ AB=a0|00 +a1|01 +a2|10 +a3|11 .(2)(1)

Inthisexpresion(|0 ,|1 )refertotheeigenvectorsoftheoperatorsσz,thecomplexco-e cientsai,(i=0,...,3),beingcompletelyunknown.Inaddition,weassumethatthemachinemaysupplyalargenumberofidenticalpairs.Theaimistousetheresultingpairsforaquantuminformationtaskand,therefore,theonlypropertyweareinterestedinisitsamountofentanglement.Moreover,werequirethemeasurementaimedtodeterminetheamountofentanglementtobeoptimalinthefollowingsense.First,theprotocolshouldinvolvethesmallestpossiblenumberofobservables.Suchtestswillbecalledminimal.Andsecondly,amongminimaltests,wewillde neasoptimaltheclassofprotocolsthatyieldthebestresolutionwhensuppliedwiththesameresources,i.e.,thesamenumberofidenticallypreparedtwo-levelsystems.

Theproblemisstillrathergeneraland,forsimplicity,threefurtherassumptionswillbemade:

1.Theexperimentalsituationissuchthatitonlyallowstoactononepairatatime.Inotherwords,werestrictourselvestoincoherentmeasurements.AliceandBobarenotallowedtostoreagivennumberofparticlesandper …… 此处隐藏：23011字，全部文档内容请下载后查看。喜欢就下载吧 ……