Quantum teleportation - Wikipedia

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. Quantumteleportation FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Physicalphenomenon Quantumteleportationisatechniquefortransferringquantuminformationfromasenderatonelocationtoareceiversomedistanceaway.Whileteleportationiscommonlyportrayedinsciencefictionasameanstotransferphysicalobjectsfromonelocationtothenext,quantumteleportationonlytransfersquantuminformation.Thesenderdoesnothavetoknowtheparticularquantumstatebeingtransferred.Moreover,thelocationoftherecipientcanbeunknown,butclassicalinformationneedstobesentfromsendertoreceivertocompletetheteleportation.Becauseclassicalinformationneedstobesent,teleportationcannotoccurfasterthanthespeedoflight. Oneofthefirstscientificarticlestoinvestigatequantumteleportationis"TeleportinganUnknownQuantumStateviaDualClassicalandEinstein-Podolsky-RosenChannels"[1]publishedbyC.H.Bennett,G.Brassard,C.Crépeau,R.Jozsa,A.Peres,andW.K.Woottersin1993,inwhichtheyuseddualcommunicationmethodstosend/receivequantuminformation.Itwasexperimentallyrealizedin1997bytworesearchgroups,ledbySanduPopescuandAntonZeilinger,respectively.[2][3] Experimentaldeterminations[4][5]ofquantumteleportationhavebeenmadeininformationcontent-includingphotons,atoms,electrons,andsuperconductingcircuits-aswellasdistancewith1,400 km(870 mi)beingthelongestdistanceofsuccessfulteleportationbythegroupofJian-WeiPanusingtheMiciussatelliteforspace-basedquantumteleportation.[6] Contents 1Non-technicalsummary 2Protocol 3Experimentalresultsandrecords 3.1Quantumteleportationover143km 3.2QuantumteleportationacrosstheDanubeRiver 3.3Deterministicquantumteleportationwithatoms 3.4Ground-to-satellitequantumteleportation 4Formalpresentation 5Alternativenotations 6Entanglementswapping 6.1AlgorithmforswappingBellpairs 7Generalizationsoftheteleportationprotocol 7.1d-dimensionalsystems 7.2Multipartiteversions 8Logicgateteleportation 8.1Generaldescription 8.2Furtherdetails 9Localexplanationofthephenomenon 10Recentdevelopments 10.1Higherdimensions 10.2Informationquality 11Seealso 12References 12.1Specific 12.2General 13Externallinks Non-technicalsummary[edit] DiagramoftheBasicComponentsUsedforQuantumTeleportation Inmattersrelatingtoquantuminformationtheory,itisconvenienttoworkwiththesimplestpossibleunitofinformation:thetwo-statesystemofthequbit.Thequbitfunctionsasthequantumanalogoftheclassiccomputationalpart,thebit,asitcanhaveameasurementvalueofbotha0anda1,whereastheclassicalbitcanonlybemeasuredasa0ora1.Thequantumtwo-statesystemseekstotransferquantuminformationfromonelocationtoanotherlocationwithoutlosingtheinformationandpreservingthequalityofthisinformation.Thisprocessinvolvesmovingtheinformationbetweencarriersandnotmovementoftheactualcarriers,similartothetraditionalprocessofcommunications,astwopartiesremainstationarywhiletheinformation(digitalmedia,voice,text,etc.)isbeingtransferred,contrarytotheimplicationsoftheword"teleport." Themaincomponentsneededforteleportationincludeasender,theinformation(aqubit),atraditionalchannel,aquantumchannel,andareceiver.Aninterestingfactisthatthesenderdoesnotneedtoknowtheexactcontentsoftheinformationthatisbeingsent.Themeasurementpostulateofquantummechanics—whenameasurementismadeuponaquantumstate,anysubsequentmeasurementswill"collapse"orthattheobservedstatewillbelost—createsanimpositionwithinteleportation:ifasendermakesameasurementontheirinformation,thestatecouldcollapsewhenthereceiverobtainstheinformationsincethestatehaschangedfromwhenthesendermadetheinitialmeasurement. Foractualteleportation,itisrequiredthatanentangledquantumstateorBellstatebecreatedforthequbittobetransferred.Entanglementimposesstatisticalcorrelationsbetweenotherwisedistinctphysicalsystemsbycreatingorplacingtwoormoreseparateparticlesintoasingle,sharedquantumstate.Thisintermediatestatecontainstwoparticleswhosequantumstatesaredependentoneachotherastheyformaconnection:ifoneparticleismoved,theotherparticlewillmovealongwithit.Anychangesthatoneparticleoftheentanglementundergoes,theotherparticlewillalsoundergothatchange,causingtheentangledparticlestoactasonequantumstate.Thesecorrelationsholdevenwhenmeasurementsarechosenandperformedindependently,outofcausalcontactfromoneanother,asverifiedinBelltestexperiments.Thus,anobservationresultingfromameasurementchoicemadeatonepointinspacetimeseemstoinstantaneouslyaffectoutcomesinanotherregion,eventhoughlighthasn'tyethadtimetotravelthedistance;aconclusionseeminglyatoddswithspecialrelativity.ThisisknownastheEPRparadox.Howeversuchcorrelationscanneverbeusedtotransmitanyinformationfasterthanthespeedoflight,astatementencapsulatedintheno-communicationtheorem.Thus,teleportationasawholecanneverbesuperluminal,asaqubitcannotbereconstructeduntiltheaccompanyingclassicalinformationarrives. Thesenderwillthenpreparetheparticle(orinformation)inthequbitandcombinewithoneoftheentangledparticlesoftheintermediatestate,causingachangeoftheentangledquantumstate.Thechangedstateoftheentangledparticleisthensenttoananalyzerthatwillmeasurethischangeoftheentangledstate.The"change"measurementwillallowthereceivertorecreatetheoriginalinformationthatthesenderhadresultingintheinformationbeingteleportedorcarriedbetweentwopeoplethathavedifferentlocations.Sincetheinitialquantuminformationis"destroyed"asitbecomespartoftheentanglementstate,theno-cloningtheoremismaintainedastheinformationisrecreatedfromtheentangledstateandnotcopiedduringteleportation. Thequantumchannelisthecommunicationmechanismthatisusedforallquantuminformationtransmissionandisthechannelusedforteleportation(relationshipofquantumchanneltotraditionalcommunicationchannelisakintothequbitbeingthequantumanalogoftheclassicalbit).However,inadditiontothequantumchannel,atraditionalchannelmustalsobeusedtoaccompanyaqubitto"preserve"thequantuminformation.Whenthechangemeasurementbetweentheoriginalqubitandtheentangledparticleismade,themeasurementresultmustbecarriedbyatraditionalchannelsothatthequantuminformationcanbereconstructedandthereceivercangettheoriginalinformation.Becauseofthisneedforthetraditionalchannel,thespeedofteleportationcanbenofasterthanthespeedoflight(hencetheno-communicationtheoremisnotviolated).ThemainadvantagewiththisisthatBellstatescanbesharedusingphotonsfromlasersmakingteleportationachievablethroughopenspacehavingnoneedtosendinformationthroughphysicalcablesoropticalfibers. Quantumstatescanbeencodedinvariousdegreesoffreedomofatoms.Forexample,qubitscanbeencodedinthedegreesoffreedomofelectronssurroundingtheatomicnucleusorinthedegreesoffreedomofthenucleusitself.Thus,performingthiskindofteleportationrequiresastockofatomsatthereceivingsite,availableforhavingqubitsimprintedonthem.[7] Asof2015,[update]thequantumstatesofsinglephotons,photonmodes,singleatoms,atomicensembles,defectcentersinsolids,singleelectrons,andsuperconductingcircuitshavebeenemployedasinformationbearers.[8] Understandingquantumteleportationrequiresagoodgroundinginfinite-dimensionallinearalgebra,Hilbertspacesandprojectionmatrixes.Aqubitisdescribedusingatwo-dimensionalcomplexnumber-valuedvectorspace(aHilbertspace),whicharetheprimarybasisfortheformalmanipulationsgivenbelow.Aworkingknowledgeofquantummechanicsisnotabsolutelyrequiredtounderstandthemathematicsofquantumteleportation,althoughwithoutsuchacquaintance,thedeepermeaningoftheequationsmayremainquitemysterious. Protocol[edit] Diagramforquantumteleportationofaphoton Theresourcesrequiredforquantumteleportationareacommunicationchannelcapableoftransmittingtwoclassicalbits,ameansofgeneratinganentangledBellstateofqubitsanddistributingtotwodifferentlocations,performingaBellmeasurementononeoftheBellstatequbits,andmanipulatingthequantumstateoftheotherqubitfromthepair.Ofcourse,theremustalsobesomeinputqubit(inthequantumstate | ϕ ⟩ {\displaystyle|\phi\rangle} )tobeteleported.Theprotocolisthenasfollows: ABellstateisgeneratedwithonequbitsenttolocationAandtheothersenttolocationB. ABellmeasurementoftheBellstatequbitandthequbittobeteleported( | ϕ ⟩ {\displaystyle|\phi\rangle} )isperformedatlocationA.Thisyieldsoneoffourmeasurementoutcomeswhichcanbeencodedintwoclassicalbitsofinformation.BothqubitsatlocationAarethendiscarded. Usingtheclassicalchannel,thetwobitsaresentfromAtoB.(Thisistheonlypotentiallytime-consumingstepafterstep1sinceinformationtransferislimitedbythespeedoflight.) AsaresultofthemeasurementperformedatlocationA,theBellstatequbitatlocationBisinoneoffourpossiblestates.Ofthesefourpossiblestates,oneisidenticaltotheoriginalquantumstate | ϕ ⟩ {\displaystyle|\phi\rangle} ,andtheotherthreearecloselyrelated.TheidentityofthestateactuallyobtainedisencodedintwoclassicalbitsandsenttolocationB.TheBellstatequbitatlocationBisthenmodifiedinoneofthreeways,ornotatall,whichresultsinaqubitidenticalto | ϕ ⟩ {\displaystyle|\phi\rangle} ,thestateofthequbitthatwaschosenforteleportation. Itisworthnoticingthattheaboveprotocolassumesthatthequbitsareindividuallyaddressable,meaningthatthequbitsaredistinguishableandphysicallylabeled.However,therecanbesituationswheretwoidenticalqubitsareindistinguishableduetothespatialoverlapoftheirwavefunctions.Underthiscondition,thequbitscannotbeindividuallycontrolledormeasured.Nevertheless,ateleportationprotocolanalogoustothatdescribedabovecanstillbe(conditionally)implementedbyexploitingtwoindependentlypreparedqubits,withnoneedofaninitialBellstate.Thiscanbemadebyaddressingtheinternaldegreesoffreedomofthequbits(e.g.,spinsorpolarizations)byspatiallylocalizedmeasurementsperformedinseparatedregionsAandBsharedbythewavefunctionsofthetwoindistinguishablequbits.[9] Experimentalresultsandrecords[edit] Workin1998verifiedtheinitialpredictions,[2]andthedistanceofteleportationwasincreasedinAugust2004to600meters,usingopticalfiber.[10]Subsequently,therecorddistanceforquantumteleportationhasbeengraduallyincreasedto16kilometres(9.9 mi),[11]thento97 km(60 mi),[12]andisnow143 km(89 mi),setinopenairexperimentsintheCanaryIslands,donebetweenthetwoastronomicalobservatoriesoftheInstitutodeAstrofísicadeCanarias.[13]Therehasbeenarecentrecordset(asofSeptember 2015[update])usingsuperconductingnanowiredetectorsthatreachedthedistanceof102 km(63 mi)overopticalfiber.[14]Formaterialsystems,therecorddistanceis21metres(69 ft).[15] Avariantofteleportationcalled"open-destination"teleportation,withreceiverslocatedatmultiplelocations,wasdemonstratedin2004usingfive-photonentanglement.[16]Teleportationofacompositestateoftwosinglequbitshasalsobeenrealized.[17]InApril2011,experimentersreportedthattheyhaddemonstratedteleportationofwavepacketsoflightuptoabandwidthof10 MHzwhilepreservingstronglynonclassicalsuperpositionstates.[18][19]InAugust2013,theachievementof"fullydeterministic"quantumteleportation,usingahybridtechnique,wasreported.[20]On29May2014,scientistsannouncedareliablewayoftransferringdatabyquantumteleportation.Quantumteleportationofdatahadbeendonebeforebutwithhighlyunreliablemethods.[21][22]On26February2015,scientistsattheUniversityofScienceandTechnologyofChinainHefei,ledbyChao-yangLuandJian-WeiPancarriedoutthefirstexperimentteleportingmultipledegreesoffreedomofaquantumparticle.Theymanagedtoteleportthequantuminformationfromensembleofrubidiumatomstoanotherensembleofrubidiumatomsoveradistanceof150metres(490 ft)usingentangledphotons.[23][24][25]In2016,researchersdemonstratedquantumteleportationwithtwoindependentsourceswhichareseparatedby6.5 km(4.0 mi)inHefeiopticalfibernetwork.[26]InSeptember2016,researchersattheUniversityofCalgarydemonstratedquantumteleportationovertheCalgarymetropolitanfibernetworkoveradistanceof6.2 km(3.9 mi).[27]InDecember2020,aspartoftheINQNETcollaboration,researchersachievedquantumteleportationoveratotaldistanceof44 km(27.3 mi)withfidelitiesexceeding90%.[28][29] Researchershavealsosuccessfullyusedquantumteleportationtotransmitinformationbetweencloudsofgasatoms,notablebecausethecloudsofgasaremacroscopicatomicensembles.[30][31] Itisalsopossibletoteleportlogicaloperations,seequantumgateteleportation.In2018,physicistsatYaledemonstratedadeterministicteleportedCNOToperationbetweenlogicallyencodedqubits.[32] Firstproposedtheoreticallyin1993,quantumteleportationhassincebeendemonstratedinmanydifferentguises.Ithasbeencarriedoutusingtwo-levelstatesofasinglephoton,asingleatomandatrappedion–amongotherquantumobjects–andalsousingtwophotons.In1997,twogroupsexperimentallyachievedquantumteleportation.Thefirstgroup,ledbySanduPopescu,wasbasedoutItaly.AnexperimentalgroupledbyAntonZeilingerfollowedafewmonthslater. TheresultsobtainedfromexperimentsdonebyPopescu'sgroupconcludedthatclassicalchannelsalonecouldnotreplicatetheteleportationoflinearlypolarizedstateandanellipticallypolarizedstate.TheBellstatemeasurementdistinguishedbetweenthefourBellstates,whichcanallowfora100%successrateofteleportation,inanidealrepresentation.[33] Zeilinger'sgroupproducedapairofentangledphotonsbyimplementingtheprocessofparametricdown-conversion.Inordertoensurethatthetwophotonscannotbedistinguishedbytheirarrivaltimes,thephotonsweregeneratedusingapulsedpumpbeam.Thephotonswerethensentthroughnarrow-bandwidthfilterstoproduceacoherencetimethatismuchlongerthanthelengthofthepumppulse.Theythenusedatwo-photoninterferometryforanalyzingtheentanglementsothatthequantumpropertycouldberecognizedwhenitistransferredfromonephotontotheother.[3] Photon1waspolarizedat45°inthefirstexperimentconductedbyZeilinger'sgroup.Quantumteleportationisverifiedwhenbothphotonsaredetectedinthe | Ψ − ⟩ 12 {\displaystyle|\Psi^{-}\rangle_{12}} state,whichhasaprobabilityof25%.Twodetectors,f1andf2,areplacedbehindthebeamsplitter,andrecordingthecoincidencewillidentifythe | Ψ − ⟩ 12 {\displaystyle|\Psi^{-}\rangle_{12}} state.Ifthereisacoincidencebetweendetectorsf1andf2,thenphoton3ispredictedtobepolarizedata45°angle.Photon3ispassedthroughapolarizingbeamsplitterthatselects+45°and-45°polarization.Ifquantumteleportationhashappened,onlydetectord2,whichisatthe+45°output,willregisteradetection.Detectord1,locatedatthe-45°output,willnotdetectaphoton.Ifthereisacoincidencebetweend2f1f2,withthe45°analysis,andalackofad1f1f2coincidence,with-45°analysis,itisproofthattheinformationfromthepolarizedphoton1hasbeenteleportedtophoton3usingquantumteleportation.[3] Quantumteleportationover143km[edit] Zeilinger'sgroupdevelopedanexperimentusingactivefeed-forwardinrealtimeandtwofree-spaceopticallinks,quantumandclassical,betweentheCanaryIslandsofLaPalmaandTenerife,adistanceofover143kilometers.Inordertoachieveteleportation,afrequency-uncorrelatedpolarization-entangledphotonpairsource,ultra-low-noisesingle-photondetectorsandentanglementassistedclocksynchronizationwereimplemented.Thetwolocationswereentangledtosharetheauxiliarystate:[12] | Ψ − ⟩ 23 = 1 √ 2 ( ( | H ⟩ 2 | V ⟩ 3 ) − ( | V ⟩ 2 | H ⟩ 3 ) ) {\displaystyle|\Psi^{-}\rangle_{23}={\frac{1}{\surd2}}((|H\rangle_{2}|V\rangle_{3})-(|V\rangle_{2}|H\rangle_{3}))} LaPalmaandTenerifecanbecomparedtothequantumcharactersAliceandBob.AliceandBobsharetheentangledstateabove,withphoton2beingwithAliceandphoton3beingwithBob.Athirdparty,Charlie,providesphoton1(theinputphoton)whichwillbeteleportedtoAliceinthegeneralizedpolarizationstate: | ϕ ⟩ 1 = α | H ⟩ 1 + β | V ⟩ 1 {\displaystyle|\phi\rangle_{1}=\alpha|H\rangle_{1}+\beta|V\rangle_{1}} wherethecomplexnumbers α {\displaystyle\alpha} and β {\displaystyle\beta} areunknowntoAliceorBob. AlicewillperformaBell-statemeasurement(BSM)thatrandomlyprojectsthetwophotonsontooneofthefourBellstateswitheachonehavingaprobabilityof25%.Photon3willbeprojectedonto | ϕ ⟩ {\displaystyle|\phi\rangle} ,theinputstate.AlicetransmitstheoutcomeoftheBSMtoBob,viatheclassicalchannel,whereBobisabletoapplythecorrespondingunitaryoperationtoobtainphoton3intheinitialstateofphoton1.Bobwillnothavetodoanythingifhedetectsthe | ψ − ⟩ 12 {\displaystyle|\psi^{-}\rangle_{12}} state.Bobwillneedtoapplya π {\displaystyle\pi} phaseshifttophoton3betweenthehorizontalandverticalcomponentifthe | ψ + ⟩ 12 {\displaystyle|\psi^{+}\rangle_{12}} stateisdetected.[12] TheresultsofZeilinger'sgroupconcludedthattheaveragefidelity(overlapoftheidealteleportedstatewiththemeasureddensitymatrix)was0.863withastandarddeviationof0.038.Thelinkattenuationduringtheirexperimentsvariedbetween28.1 dBand39.0 dB,whichwasaresultofstrongwindsandrapidtemperaturechanges.Despitethehighlossinthequantumfree-spacechannel,theaveragefidelitysurpassedtheclassicallimitof2/3.Therefore,Zeilinger'sgroupsuccessfullydemonstratedquantumteleportationoveradistanceof143 km.[12] QuantumteleportationacrosstheDanubeRiver[edit] In2004,aquantumteleportationexperimentwasconductedacrosstheDanubeRiverinVienna,atotalof600meters.An800-meter-longopticalfiberwirewasinstalledinapublicsewersystemunderneaththeDanubeRiver,anditwasexposedtotemperaturechangesandotherenvironmentalinfluences.AlicemustperformajointBellstatemeasurement(BSM)onphotonb,theinputphoton,andphotonc,herpartoftheentangledphotonpair(photonscandd).Photond,Bob'sreceiverphoton,willcontainalloftheinformationontheinputphotonb,exceptforaphaserotationthatdependsonthestatethatAliceobserved.Thisexperimentimplementedanactivefeed-forwardsystemthatsendsAlice'smeasurementresultsviaaclassicalmicrowavechannelwithafastelectro-opticalmodulatorinordertoexactlyreplicateAlice'sinputphoton.Theteleportationfidelityobtainedfromthelinearpolarizationstateat45°variedbetween0.84and0.90,whichiswellabovetheclassicalfidelitylimitof0.66.[10] Deterministicquantumteleportationwithatoms[edit] Threequbitsarerequiredforthisprocess:thesourcequbitfromthesender,theancillaryqubit,andthereceiver'stargetqubit,whichismaximallyentangledwiththeancillaryqubit.Forthisexperiment, Ca + 40 {\displaystyle{\ce{^{40}Ca+}}} ionswereusedasthequbits.Ions2and3arepreparedintheBellstate | ψ + ⟩ 23 = 1 2 ( | 0 ⟩ 2 | 1 ⟩ 3 + | 1 ⟩ 2 | 0 ⟩ 3 ) {\displaystyle|\psi^{+}\rangle_{23}={\frac{1}{\sqrt{2}}}(|0\rangle_{2}|1\rangle_{3}+|1\rangle_{2}|0\rangle_{3})} .Thestateofion1ispreparedarbitrarily.Thequantumstatesofions1and2aremeasuredbyilluminatingthemwithlightataspecificwavelength.Theobtainedfidelitiesforthisexperimentrangedbetween73%and76%.Thisislargerthanthemaximumpossibleaveragefidelityof66.7%thatcanbeobtainedusingcompletelyclassicalresources.[34] Ground-to-satellitequantumteleportation[edit] Thequantumstatebeingteleportedinthisexperimentis | χ ⟩ 1 = α | H ⟩ 1 + β | V ⟩ 1 {\displaystyle|\chi\rangle_{1}=\alpha|H\rangle_{1}+\beta|V\rangle_{1}} ,where α {\displaystyle\alpha} and β {\displaystyle\beta} areunknowncomplexnumbers, | H ⟩ {\displaystyle|H\rangle} representsthehorizontalpolarizationstate,and | V ⟩ {\displaystyle|V\rangle} representstheverticalpolarizationstate.ThequbitpreparedinthisstateisgeneratedinalaboratoryinNgari,Tibet.ThegoalwastoteleportthequantuminformationofthequbittotheMiciussatellitethatwaslaunchedonAugust16,2016,atanaltitudeofaround500 km.WhenaBellstatemeasurementisconductedonphotons1and2andtheresultingstateis | ϕ + ⟩ 12 = 1 2 ( | H ⟩ 1 | H ⟩ 2 + | V ⟩ 1 | V ⟩ 2 ) ) {\displaystyle|\phi^{+}\rangle_{12}={\frac{1}{\sqrt{2}}}(|H\rangle_{1}|H\rangle_{2}+|V\rangle_{1}|V\rangle_{2}))} ,photon3carriesthisdesiredstate.IftheBellstatedetectedis | ϕ − ⟩ 12 = 1 2 ( | H ⟩ 1 | H ⟩ 2 − | V ⟩ 1 | V ⟩ 2 ) {\displaystyle|\phi^{-}\rangle_{12}={\frac{1}{\sqrt{2}}}(|H\rangle_{1}|H\rangle_{2}-|V\rangle_{1}|V\rangle_{2})} ,thenaphaseshiftof π {\displaystyle\pi} isappliedtothestatetogetthedesiredquantumstate.Thedistancebetweenthegroundstationandthesatellitechangesfromaslittleas500 kmtoaslargeas1,400 km.Becauseofthechangingdistance,thechannellossoftheuplinkvariesbetween41 dBand52 dB.Theaveragefidelityobtainedfromthisexperimentwas0.80withastandarddeviationof0.01.Therefore,thisexperimentsuccessfullyestablishedaground-to-satelliteuplinkoveradistanceof500–1,400 kmusingquantumteleportation.Thisisanessentialsteptowardscreatingaglobal-scalequantuminternet.[6] Formalpresentation[edit] Thereareavarietyofwaysinwhichtheteleportationprotocolcanbewrittenmathematically.Someareverycompactbutabstract,andsomeareverbosebutstraightforwardandconcrete.Thepresentationbelowisofthelatterform:verbose,buthasthebenefitofshowingeachquantumstatesimplyanddirectly.Latersectionsreviewmorecompactnotations. Theteleportationprotocolbeginswithaquantumstateorqubit | ψ ⟩ {\displaystyle|\psi\rangle} ,inAlice'spossession,thatshewantstoconveytoBob.Thisqubitcanbewrittengenerally,inbra–ketnotation,as: | ψ ⟩ C = α | 0 ⟩ C + β | 1 ⟩ C . {\displaystyle|\psi\rangle_{C}=\alpha|0\rangle_{C}+\beta|1\rangle_{C}.} ThesubscriptCaboveisusedonlytodistinguishthisstatefromAandB,below. Next,theprotocolrequiresthatAliceandBobshareamaximallyentangledstate.Thisstateisfixedinadvance,bymutualagreementbetweenAliceandBob,andcanbeanyoneofthefourBellstatesshown.Itdoesnotmatterwhichone. | Φ + ⟩ A B = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B + | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle|\Phi^{+}\rangle_{AB}={\frac{1}{\sqrt{2}}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1\rangle_{A}\otimes|1\rangle_{B})} , | Ψ + ⟩ A B = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B + | 1 ⟩ A ⊗ | 0 ⟩ B ) {\displaystyle|\Psi^{+}\rangle_{AB}={\frac{1}{\sqrt{2}}}(|0\rangle_{A}\otimes|1\rangle_{B}+|1\rangle_{A}\otimes|0\rangle_{B})} , | Ψ − ⟩ A B = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B − | 1 ⟩ A ⊗ | 0 ⟩ B ) {\displaystyle|\Psi^{-}\rangle_{AB}={\frac{1}{\sqrt{2}}}(|0\rangle_{A}\otimes|1\rangle_{B}-|1\rangle_{A}\otimes|0\rangle_{B})} . | Φ − ⟩ A B = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B − | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle|\Phi^{-}\rangle_{AB}={\frac{1}{\sqrt{2}}}(|0\rangle_{A}\otimes|0\rangle_{B}-|1\rangle_{A}\otimes|1\rangle_{B})} , Inthefollowing,assumethatAliceandBobsharethestate | Φ + ⟩ A B . {\displaystyle|\Phi^{+}\rangle_{AB}.} Aliceobtainsoneoftheparticlesinthepair,withtheothergoingtoBob.(ThisisimplementedbypreparingtheparticlestogetherandshootingthemtoAliceandBobfromacommonsource.)ThesubscriptsAandBintheentangledstaterefertoAlice'sorBob'sparticle. Atthispoint,Alicehastwoparticles(C,theoneshewantstoteleport,andA,oneoftheentangledpair),andBobhasoneparticle,B.Inthetotalsystem,thestateofthesethreeparticlesisgivenby | ψ ⟩ C ⊗ | Φ + ⟩ A B = ( α | 0 ⟩ C + β | 1 ⟩ C ) ⊗ 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B + | 1 ⟩ A ⊗ | 1 ⟩ B ) . {\displaystyle|\psi\rangle_{C}\otimes|\Phi^{+}\rangle_{AB}=(\alpha|0\rangle_{C}+\beta|1\rangle_{C})\otimes{\frac{1}{\sqrt{2}}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1\rangle_{A}\otimes|1\rangle_{B}).} AlicewillthenmakealocalmeasurementintheBellbasis(i.e.thefourBellstates)onthetwoparticlesinherpossession.Tomaketheresultofhermeasurementclear,itisbesttowritethestateofAlice'stwoqubitsassuperpositionsoftheBellbasis.Thisisdonebyusingthefollowinggeneralidentities,whichareeasilyverified: | 0 ⟩ ⊗ | 0 ⟩ = 1 2 ( | Φ + ⟩ + | Φ − ⟩ ) , {\displaystyle|0\rangle\otimes|0\rangle={\frac{1}{\sqrt{2}}}(|\Phi^{+}\rangle+|\Phi^{-}\rangle),} | 0 ⟩ ⊗ | 1 ⟩ = 1 2 ( | Ψ + ⟩ + | Ψ − ⟩ ) , {\displaystyle|0\rangle\otimes|1\rangle={\frac{1}{\sqrt{2}}}(|\Psi^{+}\rangle+|\Psi^{-}\rangle),} | 1 ⟩ ⊗ | 0 ⟩ = 1 2 ( | Ψ + ⟩ − | Ψ − ⟩ ) , {\displaystyle|1\rangle\otimes|0\rangle={\frac{1}{\sqrt{2}}}(|\Psi^{+}\rangle-|\Psi^{-}\rangle),} and | 1 ⟩ ⊗ | 1 ⟩ = 1 2 ( | Φ + ⟩ − | Φ − ⟩ ) . {\displaystyle|1\rangle\otimes|1\rangle={\frac{1}{\sqrt{2}}}(|\Phi^{+}\rangle-|\Phi^{-}\rangle).} Afterexpandingtheexpressionfor | ψ ⟩ C ⊗   | Φ + ⟩ A B {\textstyle{\begin{aligned}|&\psi\rangle_{C}\otimes\|\Phi^{+}\rangle_{AB}\end{aligned}}} ,oneappliestheseidentitiestothequbitswithAandCsubscripts.Inparticular, α 1 2 | 0 ⟩ C ⊗ | 0 ⟩ A ⊗ | 0 ⟩ B = α 1 2 ( | Φ + ⟩ C A + | Φ − ⟩ C A ) ⊗ | 0 ⟩ B , {\displaystyle\alpha{\frac{1}{\sqrt{2}}}|0\rangle_{C}\otimes|0\rangle_{A}\otimes|0\rangle_{B}=\alpha{\frac{1}{2}}(|\Phi^{+}\rangle_{CA}+|\Phi^{-}\rangle_{CA})\otimes|0\rangle_{B},} andtheothertermsfollowsimilarly.Combiningsimilarterms,thetotalthreeparticlestateofA,BandCtogetherbecomesthefollowingfour-termsuperposition: | ψ ⟩ C ⊗   | Φ + ⟩ A B   = 1 2 [   | Φ + ⟩ C A ⊗ ( α | 0 ⟩ B + β | 1 ⟩ B )   +   | Φ − ⟩ C A ⊗ ( α | 0 ⟩ B − β | 1 ⟩ B )   +   | Ψ + ⟩ C A ⊗ ( α | 1 ⟩ B + β | 0 ⟩ B )   +   | Ψ − ⟩ C A ⊗ ( α | 1 ⟩ B − β | 0 ⟩ B ) ] . {\displaystyle{\begin{aligned}|&\psi\rangle_{C}\otimes\|\Phi^{+}\rangle_{AB}\=\\{\frac{1}{2}}{\Big\lbrack}\&|\Phi^{+}\rangle_{CA}\otimes(\alpha|0\rangle_{B}+\beta|1\rangle_{B})\+\|\Phi^{-}\rangle_{CA}\otimes(\alpha|0\rangle_{B}-\beta|1\rangle_{B})\\\+\&|\Psi^{+}\rangle_{CA}\otimes(\alpha|1\rangle_{B}+\beta|0\rangle_{B})\+\|\Psi^{-}\rangle_{CA}\otimes(\alpha|1\rangle_{B}-\beta|0\rangle_{B}){\Big\rbrack}.\\\end{aligned}}} [35] Notethatallthreeparticlesarestillinthesametotalstatesincenooperationshavebeenperformed.Rather,theaboveisjustachangeofbasisonAlice'spartofthesystem.TheactualteleportationoccurswhenAlicemeasureshertwoqubitsA,C,intheBellbasis AsimplequantumcircuitthatmapsoneofthefourBellstates(theEPRpairinthepicture)intooneofthefourtwo-qubitcomputationalbasisstates.ThecircuitconsistsofaCNOTgatefollowedbyaHadamardoperation.Intheoutputs,aandbtakeonvaluesof0or1. | Φ + ⟩ C A , | Φ − ⟩ C A , | Ψ + ⟩ C A , | Ψ − ⟩ C A . {\displaystyle|\Phi^{+}\rangle_{CA},|\Phi^{-}\rangle_{CA},|\Psi^{+}\rangle_{CA},|\Psi^{-}\rangle_{CA}.} Equivalently,themeasurementmaybedoneinthecomputationalbasis, { | 0 ⟩ , | 1 ⟩ } {\displaystyle\{|0\rangle,|1\rangle\}} ,bymappingeachBellstateuniquelytooneof { | 0 ⟩ ⊗ | 0 ⟩ , | 0 ⟩ ⊗ | 1 ⟩ , | 1 ⟩ ⊗ | 0 ⟩ , | 1 ⟩ ⊗ | 1 ⟩ } {\displaystyle\{|0\rangle\otimes|0\rangle,|0\rangle\otimes|1\rangle,|1\rangle\otimes|0\rangle,|1\rangle\otimes|1\rangle\}} withthequantumcircuitinthefiguretotheright. Giventheaboveexpression,evidentlytheresultofAlice's(local)measurementisthatthethree-particlestatewouldcollapsetooneofthefollowingfourstates(withequalprobabilityofobtainingeach): | Φ + ⟩ C A ⊗ ( α | 0 ⟩ B + β | 1 ⟩ B ) {\displaystyle|\Phi^{+}\rangle_{CA}\otimes(\alpha|0\rangle_{B}+\beta|1\rangle_{B})} | Φ − ⟩ C A ⊗ ( α | 0 ⟩ B − β | 1 ⟩ B ) {\displaystyle|\Phi^{-}\rangle_{CA}\otimes(\alpha|0\rangle_{B}-\beta|1\rangle_{B})} | Ψ + ⟩ C A ⊗ ( α | 1 ⟩ B + β | 0 ⟩ B ) {\displaystyle|\Psi^{+}\rangle_{CA}\otimes(\alpha|1\rangle_{B}+\beta|0\rangle_{B})} | Ψ − ⟩ C A ⊗ ( α | 1 ⟩ B − β | 0 ⟩ B ) {\displaystyle|\Psi^{-}\rangle_{CA}\otimes(\alpha|1\rangle_{B}-\beta|0\rangle_{B})} Alice'stwoparticlesarenowentangledtoeachother,inoneofthefourBellstates,andtheentanglementoriginallysharedbetweenAlice'sandBob'sparticlesisnowbroken.Bob'sparticletakesononeofthefoursuperpositionstatesshownabove.NotehowBob'squbitisnowinastatethatresemblesthestatetobeteleported.ThefourpossiblestatesforBob'squbitareunitaryimagesofthestatetobeteleported. TheresultofAlice'sBellmeasurementtellsherwhichoftheabovefourstatesthesystemisin.ShecannowsendherresulttoBobthroughaclassicalchannel.Twoclassicalbitscancommunicatewhichofthefourresultssheobtained. AfterBobreceivesthemessagefromAlice,hewillknowwhichofthefourstateshisparticleisin.Usingthisinformation,heperformsaunitaryoperationonhisparticletotransformittothedesiredstate α | 0 ⟩ B + β | 1 ⟩ B {\displaystyle\alpha|0\rangle_{B}+\beta|1\rangle_{B}} : IfAliceindicatesherresultis | Φ + ⟩ C A {\displaystyle|\Phi^{+}\rangle_{CA}} ,Bobknowshisqubitisalreadyinthedesiredstateanddoesnothing.Thisamountstothetrivialunitaryoperation,theidentityoperator. Ifthemessageindicates | Φ − ⟩ C A {\displaystyle|\Phi^{-}\rangle_{CA}} ,BobwouldsendhisqubitthroughtheunitaryquantumgategivenbythePaulimatrix σ 3 = [ 1 0 0 − 1 ] {\displaystyle\sigma_{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}} torecoverthestate. IfAlice'smessagecorrespondsto | Ψ + ⟩ C A {\displaystyle|\Psi^{+}\rangle_{CA}} ,Bobappliesthegate σ 1 = [ 0 1 1 0 ] {\displaystyle\sigma_{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}} tohisqubit. Finally,fortheremainingcase,theappropriategateisgivenby σ 3 σ 1 = − σ 1 σ 3 = i σ 2 = [ 0 1 − 1 0 ] . {\displaystyle\sigma_{3}\sigma_{1}=-\sigma_{1}\sigma_{3}=i\sigma_{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.} Teleportationisthusachieved.Theabove-mentionedthreegatescorrespondtorotationsofπradians(180°)aboutappropriateaxes(X,YandZ)intheBlochspherepictureofaqubit. Someremarks: Afterthisoperation,Bob'squbitwilltakeonthestate | ψ ⟩ B = α | 0 ⟩ B + β | 1 ⟩ B {\displaystyle|\psi\rangle_{B}=\alpha|0\rangle_{B}+\beta|1\rangle_{B}} ,andAlice'squbitbecomesan(undefined)partofanentangledstate.Teleportationdoesnotresultinthecopyingofqubits,andhenceisconsistentwiththeno-cloningtheorem. Thereisnotransferofmatterorenergyinvolved.Alice'sparticlehasnotbeenphysicallymovedtoBob;onlyitsstatehasbeentransferred.Theterm"teleportation",coinedbyBennett,Brassard,Crépeau,Jozsa,PeresandWootters,reflectstheindistinguishabilityofquantummechanicalparticles. Foreveryqubitteleported,AliceneedstosendBobtwoclassicalbitsofinformation.Thesetwoclassicalbitsdonotcarrycompleteinformationaboutthequbitbeingteleported.Ifaneavesdropperinterceptsthetwobits,shemayknowexactlywhatBobneedstodoinordertorecoverthedesiredstate.However,thisinformationisuselessifshecannotinteractwiththeentangledparticleinBob'spossession. Alternativenotations[edit] Quantumteleportationinitsdiagrammaticform.[36]employingPenrosegraphicalnotation.[37]Formally,suchacomputationtakesplaceinadaggercompactcategory.Thisresultsintheabstractdescriptionofquantumteleportationasemployedincategoricalquantummechanics. Quantumcircuitrepresentationforteleportationofaquantumstate,[38][39]asdescribedabove.Thecircuitconsumesthe | Φ + ⟩ {\displaystyle|\Phi^{+}\rangle} Bellstateandthequbittoteleportasinput,andconsistsofCNOT,Hadamard,twomeasurementsoftwoqubits,andfinally,twogateswithclassiccontrol:aPauliX,andaPauliZ,meaningthatiftheresultfromthemeasurementwas | 1 ⟩ {\displaystyle|1\rangle} ,thentheclassicallycontrolledPauligateisexecuted.Afterthecircuithasruntocompletion,thevalueof | ψ ⟩ C {\displaystyle|\psi\rangle_{C}} willhavemovedto,orteleportedto | ψ ⟩ B {\displaystyle|\psi\rangle_{B}} ,and | ψ ⟩ C {\displaystyle|\psi\rangle_{C}} willhaveitsvaluesettoeither | 0 ⟩ {\displaystyle|0\rangle} or | 1 ⟩ {\displaystyle|1\rangle} ,dependingontheresultfromthemeasurementonthatqubit.Thiscircuitcanalsobeusedforentanglementswapping,if | ψ ⟩ C {\displaystyle|\psi\rangle_{C}} isoneofthequbitsthatmakeupanentangledstate,asdescribedinthetext. Thereareavarietyofdifferentnotationsinusethatdescribetheteleportationprotocol.Onecommononeisbyusingthenotationofquantumgates. Intheabovederivation,theunitarytransformationthatisthechangeofbasis(fromthestandardproductbasisintotheBellbasis)canbewrittenusingquantumgates.Directcalculationshowsthatthisgateisgivenby G = ( H ⊗ I ) CNOT {\displaystyleG=(H\otimesI)\operatorname{CNOT}} whereHistheonequbitWalsh-Hadamardgateand CNOT {\displaystyle\operatorname{CNOT}} istheControlledNOTgate. Entanglementswapping[edit] Teleportationcanbeappliednotjusttopurestates,butalsomixedstates,thatcanberegardedasthestateofasinglesubsystemofanentangledpair.Theso-calledentanglementswappingisasimpleandillustrativeexample. IfAliceandBobshareanentangledpair,andBobteleportshisparticletoCarol,thenAlice'sparticleisnowentangledwithCarol'sparticle.Thissituationcanalsobeviewedsymmetricallyasfollows: AliceandBobshareanentangledpair,andBobandCarolshareadifferententangledpair.NowletBobperformaprojectivemeasurementonhistwoparticlesintheBellbasisandcommunicatetheresulttoCarol.TheseactionsarepreciselytheteleportationprotocoldescribedabovewithBob'sfirstparticle,theoneentangledwithAlice'sparticle,asthestatetobeteleported.WhenCarolfinishestheprotocolshenowhasaparticlewiththeteleportedstate,thatisanentangledstatewithAlice'sparticle.Thus,althoughAliceandCarolneverinteractedwitheachother,theirparticlesarenowentangled. AdetaileddiagrammaticderivationofentanglementswappinghasbeengivenbyBobCoecke,[40]presentedintermsofcategoricalquantummechanics. AlgorithmforswappingBellpairs[edit] AnimportantapplicationofentanglementswappingisdistributingBellstatesforuseinentanglementdistributedquantumnetworks.Atechnicaldescriptionoftheentanglementswappingprotocolisgivenhereforpurebellstates. AliceandBoblocallyprepareknownBellpairsresultingintheinitialstate: | ψ ⟩ i n = | Φ + ⟩ A 1 , A 2 | Φ + ⟩ B 1 , B 2 {\displaystyle|\psi\rangle_{\rm{in}}=|\Phi^{+}\rangle_{A_{1},A_{2}}|\Phi^{+}\rangle_{B_{1},B_{2}}} Alicesendsqubit A 1 {\displaystyleA_{1}} toathirdpartyCarol Bobsendsqubit B 1 {\displaystyleB_{1}} toCarol CarolperformsaBellprojectionbetween A 1 {\displaystyleA_{1}} and B 1 {\displaystyleB_{1}} thatbychanceresultsinthemeasurementoutcome: ⟨ Φ + | A 1 , B 1 | ψ ⟩ i n = | Φ + ⟩ A 2 , B 2 {\displaystyle\langle\Phi^{+}|_{A_{1},B_{1}}|\psi\rangle_{\rm{in}}=|\Phi^{+}\rangle_{A_{2},B_{2}}} InthecaseoftheotherthreeBellprojectionoutcomes,localcorrectionsgivenbyPaulioperatorsaremadebyAliceandorBobafterCarolhascommunicatedtheresultsofthemeasurement. ⟨ Φ − | A 1 , B 1 | ψ ⟩ i n = Z ^ B 2 | Φ + ⟩ A 2 , B 2 {\displaystyle\langle\Phi^{-}|_{A_{1},B_{1}}|\psi\rangle_{\rm{in}}={\hat{Z}}_{B_{2}}|\Phi^{+}\rangle_{A_{2},B_{2}}} ⟨ Ψ + | A 1 , B 1 | ψ ⟩ i n = X ^ B 2 | Φ + ⟩ A 2 , B 2 {\displaystyle\langle\Psi^{+}|_{A_{1},B_{1}}|\psi\rangle_{\rm{in}}={\hat{X}}_{B_{2}}|\Phi^{+}\rangle_{A_{2},B_{2}}} ⟨ Ψ − | A 1 , B 1 | ψ ⟩ i n = X ^ B 2 Z ^ B 2 | Φ + ⟩ A 2 , B 2 {\displaystyle\langle\Psi^{-}|_{A_{1},B_{1}}|\psi\rangle_{\rm{in}}={\hat{X}}_{B_{2}}{\hat{Z}}_{B_{2}}|\Phi^{+}\rangle_{A_{2},B_{2}}} AliceandBobnowhaveaBellpairbetweenqubits A 2 {\displaystyleA_{2}} and B 2 {\displaystyleB_{2}} | ψ ⟩ o u t = | Φ + ⟩ A 2 , B 2 {\displaystyle|\psi\rangle_{\rm{out}}=|\Phi^{+}\rangle_{A_{2},B_{2}}} Generalizationsoftheteleportationprotocol[edit] Thebasicteleportationprotocolforaqubitdescribedabovehasbeengeneralizedinseveraldirections,inparticularregardingthedimensionofthesystemteleportedandthenumberofpartiesinvolved(eitherassender,controller,orreceiver). d-dimensionalsystems[edit] Ageneralizationto d {\displaystyled} -levelsystems(so-calledqudits)isstraightforwardandwasalreadydiscussedintheoriginalpaperbyBennettetal.:[41]themaximallyentangledstateoftwoqubitshastobereplacedbyamaximallyentangledstateoftwoquditsandtheBellmeasurementbyameasurementdefinedbyamaximallyentangledorthonormalbasis.AllpossiblesuchgeneralizationswerediscussedbyWernerin2001.[42] Thegeneralizationtoinfinite-dimensionalso-calledcontinuous-variablesystemswasproposedin[43]andledtothefirstteleportationexperimentthatworkedunconditionally.[44] Multipartiteversions[edit] Theuseofmultipartiteentangledstatesinsteadofabipartitemaximallyentangledstateallowsforseveralnewfeatures:eitherthesendercanteleportinformationtoseveralreceiverseithersendingthesamestatetoallofthem(whichallowstoreducetheamountofentanglementneededfortheprocess)[45]orteleportingmultipartitestates[46]orsendingasinglestateinsuchawaythatthereceivingpartiesneedtocooperatetoextracttheinformation.[47]Adifferentwayofviewingthelattersettingisthatsomeofthepartiescancontrolwhethertheotherscanteleport. Logicgateteleportation[edit] Mainarticle:Quantumgateteleportation Ingeneral,mixedstatesρmaybetransported,andalineartransformationωappliedduringteleportation,thusallowingdataprocessingofquantuminformation.Thisisoneofthefoundationalbuildingblocksofquantuminformationprocessing.Thisisdemonstratedbelow. Generaldescription[edit] Ageneralteleportationschemecanbedescribedasfollows.Threequantumsystemsareinvolved.System1isthe(unknown)stateρtobeteleportedbyAlice.Systems2and3areinamaximallyentangledstateωthataredistributedtoAliceandBob,respectively.Thetotalsystemistheninthestate ρ ⊗ ω . {\displaystyle\rho\otimes\omega.} AsuccessfulteleportationprocessisaLOCCquantumchannelΦthatsatisfies ( Tr 12 ∘ Φ ) ( ρ ⊗ ω ) = ρ , {\displaystyle(\operatorname{Tr}_{12}\circ\Phi)(\rho\otimes\omega)=\rho\,,} whereTr12isthepartialtraceoperationwithrespectsystems1and2,and ∘ {\displaystyle\circ} denotesthecompositionofmaps.ThisdescribesthechannelintheSchrödingerpicture. TakingadjointmapsintheHeisenbergpicture,thesuccessconditionbecomes ⟨ Φ ( ρ ⊗ ω ) | I ⊗ O ⟩ = ⟨ ρ | O ⟩ {\displaystyle\langle\Phi(\rho\otimes\omega)|I\otimesO\rangle=\langle\rho|O\rangle} forallobservableOonBob'ssystem.Thetensorfactorin I ⊗ O {\displaystyleI\otimesO} is 12 ⊗ 3 {\displaystyle12\otimes3} whilethatof ρ ⊗ ω {\displaystyle\rho\otimes\omega} is 1 ⊗ 23 {\displaystyle1\otimes23} . Furtherdetails[edit] TheproposedchannelΦcanbedescribedmoreexplicitly.Tobeginteleportation,Aliceperformsalocalmeasurementonthetwosubsystems(1and2)inherpossession.Assumethelocalmeasurementhaveeffects F i = M i 2 . {\displaystyle{F_{i}}={M_{i}^{2}}.} Ifthemeasurementregistersthei-thoutcome,theoverallstatecollapsesto ( M i ⊗ I ) ( ρ ⊗ ω ) ( M i ⊗ I ) . {\displaystyle(M_{i}\otimesI)(\rho\otimes\omega)(M_{i}\otimesI).} Thetensorfactorin ( M i ⊗ I ) {\displaystyle(M_{i}\otimesI)} is 12 ⊗ 3 {\displaystyle12\otimes3} whilethatof ρ ⊗ ω {\displaystyle\rho\otimes\omega} is 1 ⊗ 23 {\displaystyle1\otimes23} .BobthenappliesacorrespondinglocaloperationΨionsystem3.Onthecombinedsystem,thisisdescribedby ( I d ⊗ Ψ i ) ( M i ⊗ I ) ( ρ ⊗ ω ) ( M i ⊗ I ) . {\displaystyle(Id\otimes\Psi_{i})(M_{i}\otimesI)(\rho\otimes\omega)(M_{i}\otimesI).} whereIdistheidentitymaponthecompositesystem 1 ⊗ 2 {\displaystyle1\otimes2} . Therefore,thechannelΦisdefinedby Φ ( ρ ⊗ ω ) = ∑ i ( I d ⊗ Ψ i ) ( M i ⊗ I ) ( ρ ⊗ ω ) ( M i ⊗ I ) {\displaystyle\Phi(\rho\otimes\omega)=\sum_{i}(Id\otimes\Psi_{i})(M_{i}\otimesI)(\rho\otimes\omega)(M_{i}\otimesI)} NoticeΦsatisfiesthedefinitionofLOCC.Asstatedabove,theteleportationissaidtobesuccessfulif,forallobservableOonBob'ssystem,theequality ⟨ Φ ( ρ ⊗ ω ) , I ⊗ O ⟩ = ⟨ ρ , O ⟩ {\displaystyle\langle\Phi(\rho\otimes\omega),I\otimesO\rangle=\langle\rho,O\rangle} holds.Thelefthandsideoftheequationis: ∑ i ⟨ ( I d ⊗ Ψ i ) ( M i ⊗ I ) ( ρ ⊗ ω ) ( M i ⊗ I ) , I ⊗ O ⟩ {\displaystyle\sum_{i}\langle(Id\otimes\Psi_{i})(M_{i}\otimesI)(\rho\otimes\omega)(M_{i}\otimesI),\;I\otimesO\rangle} = ∑ i ⟨ ( M i ⊗ I ) ( ρ ⊗ ω ) ( M i ⊗ I ) , I ⊗ Ψ i ∗ ( O ) ⟩ {\displaystyle=\sum_{i}\langle(M_{i}\otimesI)(\rho\otimes\omega)(M_{i}\otimesI),\;I\otimes\Psi_{i}^{*}(O)\rangle} whereΨi*istheadjointofΨiintheHeisenbergpicture.Assumingallobjectsarefinitedimensional,thisbecomes ∑ i Tr ( ρ ⊗ ω ) ( F i ⊗ Ψ i ∗ ( O ) ) . {\displaystyle\sum_{i}\operatorname{Tr}\;(\rho\otimes\omega)(F_{i}\otimes\Psi_{i}^{*}(O)).} Thesuccesscriterionforteleportationhastheexpression ∑ i Tr ( ρ ⊗ ω ) ( F i ⊗ Ψ i ∗ ( O ) ) = Tr ρ ⋅ O . {\displaystyle\sum_{i}\operatorname{Tr}\;(\rho\otimes\omega)(F_{i}\otimes\Psi_{i}^{*}(O))=\operatorname{Tr}\;\rho\cdotO.} Localexplanationofthephenomenon[edit] AlocalexplanationofquantumteleportationisputforwardbyDavidDeutschandPatrickHayden,withrespecttothemany-worldsinterpretationofquantummechanics.TheirpaperassertsthatthetwobitsthatAlicesendsBobcontain"locallyinaccessibleinformation"resultingintheteleportationofthequantumstate."Theabilityofquantuminformationtoflowthroughaclassicalchannel[…],survivingdecoherence,is[…]thebasisofquantumteleportation."[48] Recentdevelopments[edit] Whilequantumteleportationisinaninfancystage,therearemanyaspectspertainingtoteleportationthatscientistsareworkingtobetterunderstandorimprovetheprocessthatinclude: Higherdimensions[edit] Quantumteleportationcanimprovetheerrorsassociatedwithfaulttolerantquantumcomputationviaanarrangementoflogicgates.ExperimentsbyD.GottesmanandI.L.Chuanghavedeterminedthata"Cliffordhierarchy"[49]gatearrangementwhichactstoenhanceprotectionagainstenvironmentalerrors.Overall,ahigherthresholdoferrorisallowedwiththeCliffordhierarchyasthesequenceofgatesrequireslessresourcesthatareneededforcomputation.Whilethemoregatesthatareusedinaquantumcomputercreatemorenoise,thegatesarrangementanduseofteleportationinlogictransfercanreducethisnoiseasitcallsforless"traffic"thatiscompiledinthesequantumnetworks.[50]Themorequbitsusedforaquantumcomputer,themorelevelsareaddedtoagatearrangement,withthediagonalizationofgatearrangementvaryingindegree.HigherdimensionanalysisinvolvesthehigherlevelgatearrangementoftheCliffordhierarchy.[51] Informationquality[edit] Consideringthepreviouslymentionedrequirementofanintermediateentangledstateforquantumteleportation,thereneedstobeconsiderationplacedontothepurityofthisstateforinformationquality.Aprotectionthathasbeendevelopedinvolvestheuseofcontinuousvariableinformation(ratherthanatypicaldiscretevariable)creatingasuperimposedcoherentintermediatestate.Thisinvolvesmakingaphaseshiftinthereceivedinformationandthenaddingamixingstepuponreceptionusingapreferredstate,whichcouldbeanoddorevencoherentstate,thatwillbe"conditionedtotheclassicalinformationofthesender"creatingatwomodestatethatcontainstheoriginallysentinformation.[52] Therehavealsobeendevelopmentswithteleportinginformationbetweensystemsthatalreadyhavequantuminformationinthem.ExperimentsdonebyFeng,Xu,Zhouetal.havedemonstratedthatteleportationofaqubittoaphotonthatalreadyhasaqubitworthofinformationispossibleduetousinganopticalqubit-ququartentanglinggate.[4]Thisqualitycanincreasecomputationpossibilitiesascalculationscanbedonebasedonpreviouslystoredinformationallowingforimprovementsonpastcalculations. 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Externallinks[edit] Loophole-freeBelltest-KavliInstituteofNanoscience "Spookyactionandbeyond"–InterviewwithProf.Dr.AntonZeilingeraboutquantumteleportation.Date:2006-02-16 QuantumTeleportationatIBM PhysicistsSucceedInTransferringInformationBetweenMatterAndLight Quantumtelecloning:CaptainKirk'scloneandtheeavesdropper Teleportation-basedapproachestouniversalquantumcomputation Teleportationasaquantumcomputation Quantumteleportationwithatoms:quantumprocesstomography EntangledStateTeleportation Fidelityofquantumteleportationthroughnoisychannelsby TelePOVM—Ageneralizedquantumteleportationscheme EntanglementTeleportationviaWernerStates QuantumTeleportationofaPolarizationState TheTimeTravelHandbook:AManualofPracticalTeleportation&TimeTravel letterstonature:Deterministicquantumteleportationwithatoms QuantumteleportationwithacompleteBellstatemeasurement "WelcometothequantumInternet".ScienceNews.16August2008. Quantumexperiments–interactive. A(mostlyserious)introductiontoquantumteleportationfornon-physicists vteQuantuminformationscienceGeneral DiVincenzo'scriteria NISQera Quantumcomputing Timeline Cloud-based Quantuminformation Quantumprogramming Qubit physicalvs.logical Quantumprocessors Theorems Bell's Gleason's Gottesman–Knill Holevo's Margolus–Levitin No-broadcast No-cloning No-communication No-deleting No-hiding No-teleportation PBR Quantumthreshold Solovay–Kitaev Quantumcommunication Classicalcapacity entanglement-assisted Quantumcapacity Entanglementdistillation Monogamyofentanglement LOCC Quantumchannel Quantumnetwork Quantumcryptography Quantumkeydistribution BB84 SARG04 Three-stagequantumcryptographyprotocol QuantumSecretSharing Quantumteleportation Superdensecoding Quantumalgorithms Bernstein–Vazirani Deutsch–Jozsa Grover's Quantumcounting Quantumphaseestimation Shor's Simon's Amplitudeamplification Linearsystemsofequations Quantumannealing QuantumFouriertransform Quantumneuralnetwork Universalquantumsimulator Quantumcomplexitytheory BQP EQP QIP QMA PostBQP Quantumcomputingmodels Adiabaticquantumcomputation Differentiablequantumcomputing One-wayquantumcomputer clusterstate Quantumcircuit Quantumlogicgate QuantumTuringmachine Topologicalquantumcomputer Quantumerrorcorrection Codes CSS Quantumconvolutional stabilizer Shor Steane Toric gnu Entanglement-assistedquantumerrorcorrection PhysicalimplementationsQuantumoptics Bosonsampling CavityQED CircuitQED Linearopticalquantumcomputing KLMprotocol Ultracoldatoms Opticallattice Trappedionquantumcomputer Spin-based KaneQC SpinqubitQC Nitrogen-vacancycenter NuclearmagneticresonanceQC Superconductingquantumcomputing Chargequbit Fluxqubit Phasequbit Transmon Quantumprogramming OpenQASM-Qiskit-IBMQX Quil-Forest/RigettiQCS Cirq Q# libquantum manyothers... 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