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Schr¨odingerCatStatesforQuantumInformationProcessingH.JeongandT.C.RalphDepartmentofPhysics,UniversityofQueensland,StLucia,Qld4072,Australia(Dated:August3,2011)WeextensivelydiscusshowSchr¨odingercatstates(superpositionsofwell-separatedcoherentstates)inopticalsystemscanbeusedforquantuminformationprocessing.I.INTRODUCTIONi.e.,|hα|?αi|2≈0.Weidentifythetwocoherentstatesof±αasbasisstatesforalogicalqubitas|αi→|0LiIntheearlydaysofquantummechanicsmanyofitsand|?αi→|1Li,sothataqubitstateisrepresentedbyfoundersbecameveryworriedbysomeoftheparadoxi-calpredictionsthatemergedfromthoughtexperiments|φi=A|0Li+B|1Li=A|αi+B|?αi.(2)basedonthenewtheory.Now,eightyyearson,someoftheseearlythoughtexperimentsarebeingexperimentallyThebasisstates,|αiand|?αi,canbeunambiguouslyrealized,andmorethanjustcon?rmingthefundamen-discriminatedbyasimplemeasurementschemewithatalsofthetheorytheyarealsobeingrecognizedasthe50-50beamsplitter,anauxiliarycoherent?eldofampli-basisof21stcenturytechnologies[14].Anexampleistudeαandtwophotodetectors[22].Atthebeamsplitter,theEPRparadox,proposedbyEinstein,Podolskyandthequbitstate|φi1ismixedwiththeauxiliarystate|αi2Rosenin1935[15],whichdiscussedthestrangeproper-andresultsintheoutputtiesofquantumentanglement.Today,entanglementhas√√beenobservedinoptical[1,28]andion[48]systemsand|φRiab=A|2αia|0ib+B|0ia|?2αib.(3)isrecognizedasaresourceformanyquantuminformationprocessingtasks[36].Thetwophotodetectoraresetformodesaandbre-AboutthesametimeastheEPRdiscussion,spectively.IfdetectorAregistersanyphoton(s)whileSchr¨odingerproposedhisfamouscatparadox[52]thatdetectorBdoesnot,weknowthat|αiwasmeasured.highlightedtheunusualconsequencesofextendingtheOnthecontrary,ifAdoesnotclickwhileBdoes,theconceptofsuperpositiontomacroscopicallydistinguish-measurementoutcomewas|?αi.Eventhoughthereableobjects.Fromaquantumopticsviewpoint,theisnon-zeroprobabilityoffailureP(φ)=|h00|φi|2=fRR2usualparadigmistoconsidersuperpositionsofcoher-|A+B|2e?2αinwhichbothofthedetectorsdonotreg-entstateswithamplitudessu?cientlydi?erentthattheyisteraphoton,thefailureisknownfromtheresultwhen-canberesolvedusinghomodynedetection[29,46].Ineveritoccurs,andPfapproachestozeroexponentiallythischapterwediscusshow,beyondtheirfundamentalasαincreases.Notethatthedetectorsdonothavetobeinterest,thesetypesofstatescanbeusedinquantumhighlye?cientforunambiguousdiscrimination.Alter-informationprocessing.Wethenlookattheproblemofnatively,homodynedetectioncanalsobeverye?cientproducingsuchstateswiththerequiredpropertiesforthequbitreadoutbecausetheoverlapbetweenthecoherentstates|αiand|?αiwouldbeextremelysmallforanappropriatevalueofα.II.QUANTUMINFORMATIONPROCESSINGAlternatively,itispossibletoconstructanexactlyor-WITHSCHRODINGERCATSTATES¨arXiv:quant-ph/0509137v120Sep2005thogonalqubitbasiswiththeequalsuperpositionoftwolinearindependentcoherentstates|αiand|?αi.Con-A.Coherent-statequbitssiderthebasisstatesWenowintroducequbitsystemsusingcoherentstates.|ei=N+(|αi+|?αi)→|0Li,(4)Acoherentstatecanbede?nedas[8,53]|di=N?(|αi?|?αi)→|1Li,(5)X∞np2?|α|2/2αwhereN±=1/2(1±exp[?2|α|]).Itcanbesimply|αi=e√|ni,(1)n!shownthattheyformanorthonormalbasisashe|di=n=0hd|ei=0andhe|ei=hd|di=1.Thebasisstate|ei(|di)where|niisanumberstateandαisthecomplexam-iscalled“evencatstate”(“oddcatstate”)becauseitplitudeofthecoherentstate.Thecoherentstateisacontainsonlyeven(odd)numberofphotonsasveryusefultoolinquantumopticsandalaser?eldis2X∞2nconsideredagoodapproximationofit.Letusconsider?|α|α|ei=2N+e2p|2ni,(6)twocoherentstates|αiand|?αi.Thetwocoherent(2n)!n=0statesarenotorthogonaltoeachotherbuttheiroverlap2|α|2X∞α(2n+1)|hα|?αi|2=e?4|α|decreasesexponentiallywith|α|.For?|di=2N?e2p|2n+1i.(7)example,when|α|isassmallas2,theoverlapis≈10?7,(2n+1)!n=0 2Theevenandoddcatstatescanthusbediscriminatedbysplitter,theBell-catstatesbecomeaphotonparitymeasurementwhichcanberepresentedP∞|Φ+iab?→|Eif|0ig,byOΠ=n=0(|2nih2n|?|2n+1ih2n+1|).Asαgoestozero,theoddcatstate|diapproachesasinglephoton|Φ?iab?→|Dif|0ig,state|1iwhiletheevencatstate|eiapproaches|0i.No|Ψ+iab?→|0if|Eig,matterhowsmallαis,thereisnopossibilitythatno|Ψ?iab?→|0if|Dig,(11)photonwillbedetectedfromthestate|diatanideal√√photodetector.wheretheevencatstate|Ei∝|2αi+|?2αide?-nitelycontainsanevennumberofphotons,whiletheodd√√catstate|Di∝|2αi?|?2αide?nitelycontainsanB.Quantumteleportationoddnumberofphotons.BysettingtwophotodetectorsfortheoutputmodesfandgrespectivelytoperformQuantumteleportationisaninterestingphenomenonnumberparitymeasurement,theBell-catmeasurementfordemonstratingquantumtheoryandausefultoolincanbesimplyachieved.Forexample,ifanoddnum-quantuminformationprocessing[2].Byquantumtele-berofphotonsisdetectedformodef,thestate|Φ?iisportation,anunknownquantumstateisdisentangledinmeasured,andifanoddnumberofphotonsisdetectedasendingplaceanditsperfectreplicaappearsatadis-formodeg,then|Ψ?iismeasured.Eventhoughtheretantplaceviadualquantumandclassicalchannels.Theisnon-zeroprobabilityoffailureinwhichbothofthekeyingredientsofquantumteleportationareanentan-detectorsdonotregisteraphotonduetothenon-zero22overlapof|h0|Ei|2=2e?2|α|/(1+e?4|α|),itissmallforgledchannel,aBell-statemeasurementandappropriateunitarytransformations.Inwhatfollowsweshallexplainanappropriatechoiceofαandthefailureisknownfromhowteleporationcanbeperformedforacoherent-statetheresultwheneveritoccurs.qubit[16,24].Tocompletetheteleportationprocess,BobperformsaLetusassumethatAlicewantstoteleportanunknownunitarytransformationonhispartofthequantumchan-coherent-statequbit|φiaviaapureentangledcoherentnelaccordingtothemeasurementresultsentfromAlicechannelviaaclassicalchannel.Therequiredtransformationsareσxandσzonthecoherent-statequbitbasis,whereσ’s|Ψ?ibc=N?(|αib|?αic?|?αib|αic),(8)arePaulioperators.Whenthemeasurementoutcomeis|B4i,Bobobtainsaperfectreplicaoftheoriginalun-whereN?isthenormalizationfactor.Aftersharingtheknownqubitwithoutanyoperation.Whenthemeasure-quantumchannel|Ψ?i,AliceshouldperformaBell-statementoutcomeis|B2i,Bobshouldperform|αi?|?αimeasurementonherpartofthequantumchannelandtheonhisqubit.Suchaphaseshiftbyπcanbedoneusinga?unknownqubit|φiandsendtheoutcometoBob.ThephaseshifterwhoseactionisdescribedbyP(?)=ei?aa,Bell-statemeasurementistodiscriminatebetweenthewhereaanda?aretheannihilationandcreationoper-fourBell-catstateswhichcanbede?nedwithcoherentators.Whentheoutcomeis|B3i,thetransformationstatesas[20,21,49,50]shouldbeperformedas|αi→|αiand|?αi→?|?αi.Thistransformationismoredi?cultbutcanbeachieved|Φ±i=N±(|αi|αi±|?αi|?αi),(9)moststraightforwardlybysimplyteleportingthestate|Ψ±i=N±(|αi|?αi±|?αi|αi),(10)again(locally)andrepeatinguntiltherequiredphaseshiftisobtained.Therefore,bothoftherequiredunitarywhereN±arenormalizationfactors.ThefourBell-cattransformation,σxandσz,canbeperformedbylinearstatesde?nedinourframeworkareaverygoodapprox-opticselements.Whentheoutcomeis|B1i,σxandσzimationoftheBellbasis.Thesestatesareorthogonalshouldbesuccessivelyapplied.toeachotherexcepthΨ|Φi=1/cosh2|α|2,and|Ψi+++and|Φ+irapidlybecomeorthogonalas|α|grows.ABell-statemeasurement,orsimplyBellmeasure-C.Quantumcomputationment,isveryusefulinquantuminformationprocessing.ItwasshownthatacompleteBell-statemeasurementonWenowdescribehowauniversalsetofquantumgatesaproductHilbertspaceoftwotwo-levelsystemsisnotcanbeimplementedoncoherentstatequbitsusingonlypossibleusinglinearelements[33].ABellmeasurementlinearopticsandphotondetection,providedasupplyschemeusinglinearopticalelements[6]hasbeenusedtoofcatstatesisavailableasaresource.TheideawasdistinguishonlyuptotwooftheBellstatesfortelepor-originallyduetoRalph,MunroandMilburn[44]andtation[5]anddensecoding[34].However,aremarkablewaslaterexpandedonbyRalphetal[45].featureoftheBell-catstatesisthateachoneofthemAuniversalsinglequbitquantumgateelementcancanbeunambiguouslydiscriminatedusingonlyabeambeconstructedfromthefollowingsequenceofgates:splitterandphoton-paritymeasurements[23,24].Sup-Hadamard(H);rotationabouttheZ-axisbyangleθposethatthemodes,aandb,oftheentangledstateare(R(θ));Hadamard(H)and;rotationabouttheZ-axisincidentona50-50beamsplitter.Afterpassingthebeambyangleφ(R(φ)).Ifthetwoqubitgate,controlsign 3outcomeofthephasebasismeasurementandtheBell-HR(θ)HR(φ)measurementabit-?ipcorrection,aphase-?ipcorrection,CSorbothmaybenecessary.ControlSignGate:Thecontrol-signgate(CS)canbede?nedbyitse?ectonthetwoFIG.1:AsetofHadamard(H)gates,rotations(R)aboutqubitcomputationalstates:CS|αi|αi=|αi|αi;theZ-axisandcontrolsign(CS)gatescanprovideuniversalCS|αi|?αi=|αi|?αi;CS|?αi|αi=|?αi|αigateoperations.and;CS|?αi|?αi=?|?αi|?αi.Onewaytoachievethisgateisthefollowing:Thetwoarbitraryqubits,μ|αi+ν|?αiandγ|αi+δ|?αiareboth(CS),isalsoavailablethenuniversalprocessingispos-spliton50:50beamsplittersgivingthetwomode√√√√sible(SeeFig.1).Wenowdescribehowthesegatescanstates:μα/2α/2+ν?α/2?α/2and√√√√beimplemented.Wewillassumethatdeterministicsin-γα/2α/2+δ?α/2?α/2.AHadamardglequbitmeasurementscanbemadeinthecomputa-gateisthenperformedonthesecondmodeofthe?rst√√√tionalbasis,|αi,|?αiandthephasesuperpositionbasisqubitgivingthestateμα/2(α/2+?α/2)+|αi±exp[i?]|?αi.Asdescribedintheprevioussection,?α/√α/√?α/√ν2(2?2).IfaBell-measurementcomputationalbasismeasurementscanbeachievedus-isthencarriedoutbetweenthesecondmodeofthe?rstingeitherhomodyneorphotoncountingtechniques.ThequbitandoneofthemodesofthesecondqubitaCSphasesuperpositionbasiscanbemeasuredusingphotongatewillbeachieved.TheamplitudereductioncanbecountinginaDolinarreceivertypearrangement[13,55].correctedasbeforeusingteleportation.DependentonThesimplestcaseisfor?=0whereweneedtodif-theoutcomeofthevariousBell-measurements,bit-?ipferentiateonlybetweenoddorevenphotonnumbersincorrections,phase-?ipcorrections,orbothmaybedirectdetection.Wealsoassumewecanmaketwoqubitnecessary.Bell-measurementsand,moregenerally,performtelepor-ResourceState:Theresourcestate|HRicanbepro-tation,asdescribedintheprevioussection.ducedinthefollowingway.ConsiderthebeamsplitterHadamardGate:TheHadamardgate(H)canbede-interactiongivenbytheunitarytransformation?nedbyitse?ectonthecomputationalstates:H|αi=|αi+|?αiandH|?αi=|αi?|?αiwhereforcon-θ??veniencewehavedroppednormalizationfactors.OneUab=exp[i(ab+ab)](12)2waytoachievethisgateistousetheresourcestate|HRi=|α,αi+|α,?αi+|?α,αi?|?α,?αi.Thisstatewhereaandbaretheannihilationoperatorscorrespond-canbeproducednon-deterministicallyfromcatstatere-ingtotwocoherentstatequbits|γiaand|βib,withγsources,aswillbedescribedshortly.Itisstraightfor-andβtakingvaluesof?αorα.ItiswellknownthatwardtoshowthatifaBell-statemeasurementismadetheoutputstateproducedbysuchaninteractionisbetweenanarbitraryqubitstate|σiandoneofthemodesof|HRithentheremainingmodeisprojectedintotheθθθθstateH|σi,wheredependentontheoutcomeoftheBell-Uab|γia|βib=|cosγ+isinβia|cosβ+isinγib2222measurementabit-?ipcorrection,aphase-?ipcorrection,(13)wherecos2θ(sin2θ)isthere?ectivity(transmissivity)oforbothmaybenecessary.22PhaseRotationGate:Thephaserotationgate(R(θ))thebeamsplitter.Supposetwocatstatesarefedintothecanbede?nedbyitse?ectonthecomputationalstates:beamsplitterandbothoutputbeamsarethenteleported,R(θ)|αi=exp[iθ]|αiandR(θ)|?αi=exp[?iθ]|?αi.theoutputstatewillbe:Onewaytoachievethisgateisthefollowing:Thear-2222bitraryqubit,μ|αi+ν|?αiissplitona50:50beam-e?θα/4(eiθα|?αi|?αi±e?iθα|αi|?αi±√√ababsplittergivingthetwomodestate:μα/2α/2+?iθα2iθα2√√e|?αia|αib+e|αia|αib)(14)ν?α/2?α/2.Oneofthemodesisthen√measuredinthephasesuperpositionbasisα/2±wherethe±signsdependontheoutcomeoftheBell√2exp[?2iθ]?α/2,thusprojectingtheothermodeintomeasurements.Ifwechooseφ=2θα=π/2thenthe√√thestateμexp[iθ]α/2±νexp[?iθ]?α/2.Theresultingstateiseasilyshowntobelocallyequivalentto|HRi(relatedbyphaserotations).Preparationofthisamplitudedecreasecanbecorrectedbyteleportationstateisnon-deterministicbecauseofnon-unitoverlapbe-inthefollowingway[45].TheasymmetricBellstate√√tweenthestateofEq.(13)andtheBellstatesusedintheentanglement,α/2|αi+?α/2|?αiisproducedpEpEteleporter.Asaresulttheteleportercanfailbyrecordingbysplittingthecatstate3/2α+?3/2αonaphotonsatbothoutputsintheBell-measurement.The22probabilityofsuccessise?θα/2.Forα=2thisisabout1/3:2/3beamsplitter.TeleportationisthencarriedoutwiththeBellstatemeasurementbeingperformed√92%probabilityofsuccess.betweenthematching“α/2”modesandtheteleportedCorrectionofPhase-?ips:Aftereachgatewehavestateendinguponthe“α”mode.Dependentonthenotedthatbit?ipand/orphase?ipcorrectionsmaybe 4A1A2itsquantumcoherenceinadissipativeenvironment.Thisprocessiscalleddecoherenceandhasbeenknownasthemainobstacletothephysicalimplementationofquan-t1t2tuminformationprocessing.Quantumerrorcorrectionρa(bǔ)b[11,18,45]andentanglementpuri?cation[9,23]haveBS2beenstudiedforquantuminformationprocessingusing2αcatstatestoovercomethisproblem.HerewediscussanP1fabentanglementpuri?cationtechnique.BS1BOBAnentanglementpuri?cationforentangledcoherentALICEfabstates(Bell-catstates)havebeenstudiedbyseveralau-thors[9,23].IthasbeenfoundthatcertaintypesofmixedstatesincludingtheWerner-typemixedstatesρcomposedoftheBell-catstatescanbepuri?edbysimpleablinearopticselementsandine?cientdetectors[23].TheFIG.2:Aschematicoftheentanglementpuri?cationschemeothertypesofmixedstatesneedtobetransformedtotheformixedentangledcoherentstates.P1testsiftheincidentWernertypestatesbylocaloperations.Thisschemeper-′formsampli?cationoftheBell-catstatessimultaneously?eldsaandawereinthesamestatebysimultaneousclicksatA1andA2.withentanglementpuri?cation.Thisisanimportantob-servationbecauseBell-catstatesoflargeamplitudesarepreferredforquantuminformationprocessingwhiletheirnecessarysinceourgateoperationsarebasedonthetele-generationishard.Asimilartechniqueisemployedtoportationprotocol.Asdiscussedintheprevioussection,generatesingle-modelargecatstates[32].bit?ipscanbeeasilyimplementedusingaphaseshifter,P(π),whilephase-?ipsaremoreexpensive.Wenowar-We?rstexplainthepuri?cation-ampli?cationprotocolguethatinfactonlyactivecorrectionofbit-?ipsisneces-forentangledcoherentstatesbyasimpleexampleandsary.Thisisbecausephase-?ipscommutewiththephasethenapplyittoarealisticsituation[23].LetussupposerotationgateandthecontrolsigngatebutareconvertedthatAliceandBobwanttodistillentangledcoherentintobit?ipsbytheHadamardgate.Thissuggeststhestates|Φ+ifromatypeofensemblefollowingstrategy:Aftereachgateoperationanybit-?ipsarecorrectedwhilstphase-?ipsarenoted.AfterthenextHadamardgatethephase?ipsareconvertedtobit-?ipswhicharethencorrectedandanynewphase-?ipsareρa(bǔ)b=F|Φ+ihΦ+|+G|Ψ+ihΨ+|,(15)noted.Byfollowingthisstrategyonlybit-?ipsneedtobecorrectedactively,with,atworst,some?nalphase-?ipsneedingtobecorrectedinthe?nalstepofthecircuit.whereF+G≈1for|α|?1.Weshallassumethiscondition,|α|?1,forsimplicity.Thepuri?cation-D.Entanglementpuri?cationforBell-catstatesampli?cationprocesscanbesimplyaccomplishedbyper-formingtheprocessshowninFig.2.AliceandBobItisnotpossibletoperfectlyisolateaquantumstatechoosetwopairsfromtheensemblewhicharerepresentedfromitsenvironment.Aquantumstateinevitablylosesbythefollowingdensityoperator2ρa(bǔ)bρa(bǔ)′b′=F|Φ+ihΦ+|?|Φ+ihΦ+|+F(1?F)|Φ+ihΦ+|?|Ψ+ihΨ+|+F(1?F)|ΨihΨ|?|ΦihΦ|+(1?F)2|ΨihΨ|?|ΨihΨ|.++++++++(16)The?eldsofmodesaanda′areinAlice’spossessionoutput(Inthefollowing,onlythecatpartforacompo-whilebandb′inBob’s.InFig.2(a),weshowthatAlice’snentofthemixedstateisshowntodescribetheactionactiontopurifythemixedentangledstate.ThesameisconductedbyBobonhis?eldsofbandb′.Therearefourpossibilitiesforthe?eldsofaanda′incidentontothebeamsplitter(BS1),whichgivesthe 5oftheapparatuses)a′wereinthesamestatebutwheneitherA1orA2doesnotresisteraphoton,aanda′werelikelyindi?erent√|αia|αia′?→|2αif|0if′,(17)states.TheremainingpairisselectedonlywhenAlice√andBob’sallfourdetectorsclicktogether.Ofcourse,|αia|?αia′?→|0if|2αif′,(18)√thereisaprobabilitynottoresisteraphotoneventhough|?αia|αia′?→|0if|?2αif′,(19)thetwomodeswereinthesamestate,whichisduetothe√√2nonzerooverlapof|h0|2αi|.Notethatine?ciencyof|?αia|?αia′?→|?2αif|0if′.(20)thedetectorsdoesnotdegradethethequalityofthedis-IntheboxedapparatusP1,Alicechecksifmodesaandtilledentangledcoherentstatesbutdecreasesthesuccessa′wereinthesamestatebycountingphotonsatthepho-probability.todetectorsA1andA2.Ifbothmodesaanda′arein|αior|?αi,f′isinthevacuum,inwhichcasetheoutputItcanbesimplyshownthatthesecondandthirdterms?eldofthebeamsplitterBS2is|α,?αit1,t2.Otherwise,ofEq.(16)arealwaysdiscardedbytheactionofP1andtheoutput?eldiseither|2α,0it1,t2or|0,2αit1,t2.WhenBob’sapparatussameasP1.Forexample,attheoutputboththephotodetectorsA1andA2registeranypho-portsofBS1andBob’sbeamsplittercorrespondingtoton(s),AliceandBobaresurethatthetwomodesaandBS1,|Φ+iab|Ψ+ia′b′becomes