Schrodinger Cat States for Quantum Information Processing

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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,andmorethanjustconfirmingthefundamen-discriminatedbyasimplemeasurementschemewithatalsofthetheorytheyarealsobeingrecognizedasthe50-50beamsplitter,anauxiliarycoherentfieldofampli-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-entstateswithamplitudessufficientlydifferentthattheyisteraphoton,thefailureisknownfromtheresultwhen-canberesolvedusinghomodynedetection[29,46].Ineveritoccurs,andPfapproachestozeroexponentiallythischapterwediscusshow,beyondtheirfundamentalasαincreases.Notethatthedetectorsdonothavetobeinterest,thesetypesofstatescanbeusedinquantumhighlyefficientforunambiguousdiscrimination.Alter-informationprocessing.Wethenlookattheproblemofnatively,homodynedetectioncanalsobeveryefficientproducingsuchstateswiththerequiredpropertiesforthequbitreadoutbecausetheoverlapbetweenthecoherentstates|αiand|−αiwouldbeextremelysmallforanappropriatevalueofα.II.QUANTUMINFORMATIONPROCESSINGAlternatively,itispossibletoconstructanexactlyor-WITHSCHRODINGERCATSTATES¨arXiv:quant-ph/0509137v120Sep2005thogonalqubitbasiswiththeequalsuperpositionoftwolinearindependentcoherentstates|αiand|−αi.Con-A.Coherent-statequbitssiderthebasisstatesWenowintroducequbitsystemsusingcoherentstates.|ei=N+(|αi+|−αi)→|0Li,(4)Acoherentstatecanbedefinedas[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)numberofphotonsasveryusefultoolinquantumopticsandalaserfieldis2X∞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+|−2αidefi-nitelycontainsanevennumberofphotons,whiletheodd√√catstate|Di∝|2αi−|−2αidefinitelycontainsanB.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(ϕ)=eiϕaa,Bell-statemeasurementistodiscriminatebetweenthewhereaanda†aretheannihilationandcreationoper-fourBell-catstateswhichcanbedefinedwithcoherentators.Whentheoutcomeis|B3i,thetransformationstatesas[20,21,49,50]shouldbeperformedas|αi→|αiand|−αi→−|−αi.Thistransformationismoredifficultbutcanbeachieved|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,canbeperformedbylinearstatesdefinedinourframeworkareaverygoodapprox-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-flipcorrection,aphase-flipcorrection,CSorbothmaybenecessary.ControlSignGate:Thecontrol-signgate(CS)canbedefinedbyitseffectonthetwoFIG.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-gateisthenperformedonthesecondmodeofthefirst√√√tionalbasis,|αi,|−αiandthephasesuperpositionbasisqubitgivingthestateµα/2(α/2+−α/2)+|αi±exp[iǫ]|−αi.Asdescribedintheprevioussection,−α/√α/√−α/√ν2(2−2).IfaBell-measurementcomputationalbasismeasurementscanbeachievedus-isthencarriedoutbetweenthesecondmodeofthefirstingeitherhomodyneorphotoncountingtechniques.ThequbitandoneofthemodesofthesecondqubitaCSphasesuperpositionbasiscanbemeasuredusingphotongatewillbeachieved.TheamplitudereductioncanbecountinginaDolinarreceivertypearrangement[13,55].correctedasbeforeusingteleportation.DependentonThesimplestcaseisforǫ=0whereweneedtodif-theoutcomeofthevariousBell-measurements,bit-flipferentiateonlybetweenoddorevenphotonnumbersincorrections,phase-flipcorrections,orbothmaybedirectdetection.Wealsoassumewecanmaketwoqubitnecessary.Bell-measurementsand,moregenerally,performtelepor-ResourceState:Theresourcestate|HRicanbepro-tation,asdescribedintheprevioussection.ducedinthefollowingway.ConsiderthebeamsplitterHadamardGate:TheHadamardgate(H)canbede-interactiongivenbytheunitarytransformationfinedbyitseffectonthecomputationalstates:H|αi=|αi+|−αiandH|−αi=|αi−|−αiwhereforcon-膆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-flipcorrection,aphase-flipcorrection,(13)wherecos2θ(sin2θ)isthereflectivity(transmissivity)oforbothmaybenecessary.22PhaseRotationGate:Thephaserotationgate(R(θ))thebeamsplitter.Supposetwocatstatesarefedintothecanbedefinedbyitseffectonthecomputationalstates: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-flips:Aftereachgatewehavestateendinguponthe“α”mode.Dependentonthenotedthatbitflipand/orphaseflipcorrectionsmaybe 4A1A2itsquantumcoherenceinadissipativeenvironment.Thisprocessiscalleddecoherenceandhasbeenknownasthemainobstacletothephysicalimplementationofquan-t1t2tuminformationprocessing.Quantumerrorcorrectionρab[11,18,45]andentanglementpurification[9,23]haveBS2beenstudiedforquantuminformationprocessingusing2αcatstatestoovercomethisproblem.HerewediscussanP1fabentanglementpurificationtechnique.BS1BOBAnentanglementpurificationforentangledcoherentALICEfabstates(Bell-catstates)havebeenstudiedbyseveralau-thors[9,23].IthasbeenfoundthatcertaintypesofmixedstatesincludingtheWerner-typemixedstatesρcomposedoftheBell-catstatescanbepurifiedbysimpleablinearopticselementsandinefficientdetectors[23].TheFIG.2:Aschematicoftheentanglementpurificationschemeothertypesofmixedstatesneedtobetransformedtotheformixedentangledcoherentstates.P1testsiftheincidentWernertypestatesbylocaloperations.Thisschemeper-′formsamplificationoftheBell-catstatessimultaneouslyfieldsaandawereinthesamestatebysimultaneousclicksatA1andA2.withentanglementpurification.Thisisanimportantob-servationbecauseBell-catstatesoflargeamplitudesarepreferredforquantuminformationprocessingwhiletheirnecessarysinceourgateoperationsarebasedonthetele-generationishard.Asimilartechniqueisemployedtoportationprotocol.Asdiscussedintheprevioussection,generatesingle-modelargecatstates[32].bitflipscanbeeasilyimplementedusingaphaseshifter,P(π),whilephase-flipsaremoreexpensive.Wenowar-Wefirstexplainthepurification-amplificationprotocolguethatinfactonlyactivecorrectionofbit-flipsisneces-forentangledcoherentstatesbyasimpleexampleandsary.Thisisbecausephase-flipscommutewiththephasethenapplyittoarealisticsituation[23].LetussupposerotationgateandthecontrolsigngatebutareconvertedthatAliceandBobwanttodistillentangledcoherentintobitflipsbytheHadamardgate.Thissuggeststhestates|Φ+ifromatypeofensemblefollowingstrategy:Aftereachgateoperationanybit-flipsarecorrectedwhilstphase-flipsarenoted.AfterthenextHadamardgatethephaseflipsareconvertedtobit-flipswhicharethencorrectedandanynewphase-flipsareρab=F|Φ+ihΦ+|+G|Ψ+ihΨ+|,(15)noted.Byfollowingthisstrategyonlybit-flipsneedtobecorrectedactively,with,atworst,somefinalphase-flipsneedingtobecorrectedinthefinalstepofthecircuit.whereF+G≈1for|α|≫1.Weshallassumethiscondition,|α|≫1,forsimplicity.Thepurification-D.EntanglementpurificationforBell-catstatesamplificationprocesscanbesimplyaccomplishedbyper-formingtheprocessshowninFig.2.AliceandBobItisnotpossibletoperfectlyisolateaquantumstatechoosetwopairsfromtheensemblewhicharerepresentedfromitsenvironment.Aquantumstateinevitablylosesbythefollowingdensityoperator2ρabρa′b′=F|Φ+ihΦ+|⊗|Φ+ihΦ+|+F(1−F)|Φ+ihΦ+|⊗|Ψ+ihΨ+|+F(1−F)|ΨihΨ|⊗|ΦihΦ|+(1−F)2|ΨihΨ|⊗|ΨihΨ|.++++++++(16)Thefieldsofmodesaanda′areinAlice’spossessionoutput(Inthefollowing,onlythecatpartforacompo-whilebandb′inBob’s.InFig.2(a),weshowthatAlice’snentofthemixedstateisshowntodescribetheactionactiontopurifythemixedentangledstate.ThesameisconductedbyBobonhisfieldsofbandb′.Therearefourpossibilitiesforthefieldsofaanda′incidentontothebeamsplitter(BS1),whichgivesthe 5oftheapparatuses)a′wereinthesamestatebutwheneitherA1orA2doesnotresisteraphoton,aanda′werelikelyindifferent√|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|.Notethatinefficiencyof|−αia|−αia′−→|−2αif|0if′.(20)thedetectorsdoesnotdegradethethequalityofthedis-IntheboxedapparatusP1,Alicechecksifmodesaandtilledentangledcoherentstatesbutdecreasesthesuccessa′wereinthesamestatebycountingphotonsatthepho-probability.todetectorsA1andA2.Ifbothmodesaanda′arein|αior|−αi,f′isinthevacuum,inwhichcasetheoutputItcanbesimplyshownthatthesecondandthirdtermsfieldofthebeamsplitterBS2is|α,−αit1,t2.Otherwise,ofEq.(16)arealwaysdiscardedbytheactionofP1andtheoutputfieldiseither|2α,0it1,t2or|0,2αit1,t2.WhenBob’sapparatussameasP1.Forexample,attheoutputboththephotodetectorsA1andA2registeranypho-portsofBS1andBob’sbeamsplittercorrespondingtoton(s),AliceandBobaresurethatthetwomodesaandBS1,|Φ+iab|Ψ+ia′b′becomes√√√√√√√√
2|Φ+iab|Ψ+ia′b′−→N+|2α,0,0,2αi+|0,2α,2α,0i+|0,−2α,−2α,0i+|−2α,0,0,−2αifgf′g′,(21)wheregandg′aretheoutputfieldmodesfromBob’sdecoheresandbecomesamixedstateofitsdensityop-beamsplittercorrespondingtoBS1.Thefieldsofmodeseratorρab(τ),whereτstandsforthedecoherencetime.f′andg′canneverbein|0iatthesametime;atleast,Bysolvingthemasterequation[42]oneofthefourdetectorsofAliceandBobmustnotclick.∂ρThethirdtermofEq.(16)canbeshowntoleadtothe=Jρˆ+Lρˆ;sameresultbythesameanalysis.∂τXγX(24)ForthecasesofthefirstandfourthtermsinEq.(16),Jρˆ=γaρa†,Lρˆ=−(a†aρ+ρa†a)ii2iiiiallfourdetectorsmayregisterphoton(s).AfterthebeamiisplitterBS1,theketof(|Φ−ihΦ−|)ab⊗(|Φ−ihΦ−|)a′b′ofEq.(16)becomeswhereγistheenergydecayrate,themixedstateρab(τ)canbestraightforwardlyobtainedas|Φi|Φi′′−→|Φ′i|0,0i′′−|0,0i|Φ′i′′,n−ab−ab+fgfgfg+fg(22)ρab(τ)=Ne(τ)|tα,tαihtα,tα|+|−tα,−tαih−tα,−tα|√√√√where|Φ′i=N′(|2α,2αi+|−2α,−2αi)witho++thenormalizationfactorN′.Thenormalizationfactor−Γ(|tα,tαih−tα,−tα|+|−tα,−tαihtα,tα|),+intherighthandsideofEq.(22)isomitted.Thefirst(25)′′termisreducedto(|Φ+ihΦ+|)fgafter(|0,0ih0,0|)f′g′ismeasuredoutbyAliceandBob’sP1’s.Similarly,thewhere|±tα,±tαi=|±tαi|±tαi,t=e−γτ/2,Γ=abfourthtermofEq.(16)yields(|Ψ′ihΨ′|),where|Ψ′i22++fg+exp[−4(1−t)|α|],andNe(τ)isthenormalizationfactor.′isdefinedinthesamewayas|Φ+i,after(|0,0ih0,0|)f′g′Thedecoheredstateρab(τ)mayberepresentedbytheismeasured.ThusthedensitymatrixforthefieldofdynamicBell-catstatesdefinedasfollows:modesfandgconditionedonsimultaneousmeasurementofphotonsatallfourphotodetectorsis|Φe±iab=Ne±(|tαia|tαib±|−tαia|−tαib),(26)ρfg=F′|Φ′+ihΦ′+|+(1−F′)|Ψ′+ihΨ′+|,(23)|Ψe±iab=Ne±(|tαia|−tαib±|−tαia|tαib),(27)22−4t|α|−1/2whereF′=F2/{F2+(1−F)2},andF′isalwayslargerwhereNe±={2(1±e)}.ThedecoheredstatethanFforanyF>1/2.Byreiteratingthisprocess,Al-istheniceandBobcandistillsomemaximallyentangledstates(1+Γ)(1−Γ)|Φ+iofalargeamplitudeasymptotically.Ofcourse,aρab(τ)=Ne(τ)Ne2|Φe−ihΦe−|+Ne2|Φe+ihΦe+|sufficientlylargeensembleandinitialfidelityF>1/2−−arerequiredforsuccessfulpurification[3].≡F(τ)|Φe−ihΦe−|+(1−F(τ))|Φe+ihΦe+|(28)Wenowapplyourschemetoarealisticexampleinadissipativeenvironment.Whentheentangledcoher-where,regardlessofthedecaytimeτ,|Φe−iismaximallyentchannel|Φ−iisembeddedinavacuum,thechannelentangledand|Φe−iand|Φe+iareorthogonaltoeach 6other.Thedecoheredstate(28)isnotinthesameformDetectorADetectorBasEq.(15)sothatweneedsomebilateralunitarytrans-(α+β)22|largeCSS>formationsbeforethepurificationschemeisapplied.At1t2HadamardgateHforcoherent-statequbitscanbeusedtotransformthestate(28)intoadistillableformBS2HHρ(τ)H†H†=F(τ)|ΨeihΨe|+(1−F(τ))|ΦeihΦe|,cababba++++fgγwhichisnowinthesameformasEq.(15).Theensembleofstate(28)canbepurifiedsuccessfullyBS1onlywhenF(τ)islargerthan1/2.BecauseF(τ)isob-abtainedasN2(1+Γ)|smallCSS>(α)|smallCSS>(β)+F(τ)=,(29)N+2(1+Γ)−N−2(1−Γ)FIG.3:Aschematicofthenon-deterministiccat-itisfoundthatpurificationissuccessfulwhenthedeco-amplificationprocess.Seetextfordetails.herencetimeγτ0.99)canbeob-ploysanatomofthreerelevantlevelstrappedinanop-taineduptoα=2.5.Anotherinterestingaspectofthisticalcavitywithacoherent-statepulse|αiincidentontoprocessisthatitissomewhatresilienttothephotonpro-thecavity.Oneoftheatomiclevel,|ei,isitsexcitedductioninefficiencybecauseitsfirstiterationpurifiesthestate,andtheothertwolevels,|g0iand|g1i,arelevelsmixedcatstateswhileamplifyingthem.Forexample,inthegroundstatewithdifferenthyperfinespins.Theiftheinefficiencyofthesinglephotonsourceisabouttransitionfrom|g1ito|eiisresonantlycoupledtoacav-40%,thefidelityoftheinitialcat,whichisamixtureitymodewhile|g0iisdecoupledfromthecavitymode.Inwithasqueezedvacuum,isF≈0.60butitwillbecomesuchapreparation,ifthetrappedatomwaspreparedinF≈0.89bythefirstiteration.state|g0i,theinputfieldbecomes|−αiafteraresonantItisalsoimportanttonotethatthereisanalternativereflectionastheinputpulseisresonantwiththebarecav-methodtoobtainasqueezedsinglephotonevenwithoutitymode.Ontheotherhand,iftheatomwaspreparedinasinglephotonsource.Aninterestingobservationisthatstate|g1i,itremains|αiduetoastrongatom-cavitycou-asqueezedsinglephotoncanbeobtainedbysubtractingpling.Therefore,ifthetrappedatomwaspreparedina√aphotonfromasqueezedvacuum.Thiscanbeshownbysuperpositionstatesuchas(|g0i+|g1i)/2,thereflected√applyingtheannihilationoperatortoasqueezedsinglefieldbecomesanentangledstate(|g0i|αi+|g1i|−αi)/2,photonasˆaS(s)|0i=coshsS(s)|1i.Inarecentexper-whichcanbeprojectedtoasinglemodecatstatebyiment[59],thesinglephotonsubtractionwasapproxi-ameasurementonasuperposedbasis|g0i±|g1i.Anmatedbyabeamsplitteroflowreflectivityandasingleadvantageofthisschemeisweakdependenceondipolephotondetector.Suchanexperimentcouldbeimmedi-coupling,butwavefrontdistortionduetodifferencebe-atelylinkedtooursuggestiontoexperimentallygeneratetweenresonantandnon-resonantinteractionscouldbeaalargercatstate.Onecanthengenerateacatstateofprobleminarealexperiment.Thiswork[58]concludesα>2usingourschemewithoutasinglephotonsource.thatacatstatewithaquitelargeamplitude(α≈3.4)Sincethisschemeusesatleasttwobeamsplitterstocouldbegeneratedinthiswaywitha90%fidelityusingmixpropagatingfields,goodmodematchingisrequiredcurrenttechnology. 8(1)PN2inπ/N√C.SchemesusingweaknonlinearitywhereCn,N(αi)=Nψn=1Cn,NhX|−αie/2iwithNψthenormalizationfactorand|Xitheeigen-√Therehasbeenasuggestiontouserelativelyweaknon-stateofXˆ=(a+a†)/2.Afterthehomodynemea-linearitywithbeamsplittingwithavacuumandcon-surement,thestateisselectedwhenthemeasurementditioningbyhomodynedetectiontogeneratecatstatesresultisincertainvalues.Ifcoefficients|C(1)(α)|N/2,Ni[25].Asbeamsplittingwithavacuumandhomodyne(1)measurementcanbehighlyefficientinquantumopticsand|CN,N(αi)|inEq.(33)havethesamenonzerovalue(1)laboratories,thisshowsthatrelativelyweaknonlinearityandalltheother|Cn,N(αi)|’sarezero,thenthestatebe-canstillbeusefultogeneratecatstates.comesadesiredcatstate.SupposeN=4kwherekisTheHamiltonianofasingle-modeKerrnonlinearapositiveintegernumber.IfX=0ismeasuredinthismediumisH=ωa†a+λ(a†a)2,whereaanda†are(1)NLcase,thecoefficients|Cn,N(αi)|’swillbethelargestwhenannihilationandcreationoperators,ωistheenergyleveln=N/4andn=3N/4,andbecomesmallerasnisfarsplittingfortheharmonic-oscillatorpartoftheHamilto-fromthesetwopoints.ThecoefficientscanbeclosetonianandλisthestrengthoftheKerrnonlinearity[60].zeroforalltheothern’sforanappropriatelylargeαisoUndertheinfluenceofthenonlinearinteractiontheini-thattheresultingstatemaybecomeacatstateofhightialcoherentstate|αievolvestothefollowingstateatfidelity.Usingthistechnique,onemayobserveacon-timeτ=π/λN[30]:spicuoussignatureofacatstateevenwitha1/100timesweakernonliearitycomparedwiththecurrentlyrequiredXNlevel[25].Inparticular,thisapproachcanbeusefulto|ψi=C|−αe2inπ/Ni,(30)Nn,Nproduceacatstatewithasignificantlylargeamplituden=1suchasα≥10.Anotherscheme[37]proposedforlinearopticsquan-wheretumcomputation[27]usesweakcross-Kerrnonlinear-††1NX−1iπkityoftheinteractionHamiltonianH=~χa1a1a2a2toC=(−1)kexp[−(2n−k)].(31)generateacatstate.Theinteractionbetweenacoher-n,NNNk=0entstate,|αi√2,andasingle-photonqubit,e.g.,|ψi1=(|0i1+|1i1)/2,isdescribedasThelengthLofthenonlinearcellcorrespondingtoτisL=vπ/2λN,wherevisthevelocityoflight.ForiHKt/~1UK|ψi1|αi2=e√(|0i1+|1i1)|αi2(34)N=2,weobtainadesiredcatstateofrelativephase2ϕ=π/2.Weagainemphasizethenonlinearcoupling1=√(|0i|αi+|1i|αeiθi),(35)λistypicallyverysmallsuchthatN=2cannotbe12122obtainedinalengthlimitwherethedecoherenceeffectcanbeneglected.where|0i(|1i)isthevacuum(single-photon)state,αIfλisnotaslargeasrequiredtogeneratethecatstate,istheamplitudeofthecoherentstate,andθ=χtthestate(30)withN>2maybeobtainedbychoosingwiththeinteractiontimet.Ifθisπandonemea-√anappropriateinteractiontime.Fromthestate(30),itsuresoutmode1onasuperposedbasis(|0i±|1i)/2,isrequiredtoremovealltheothercoherentcomponentamacroscopicsuperpositionstate(so-calledSchr¨odinge√rstatesexcepttwocoherentstatesofaπphasedifference.catstate),(|αi±|−αi)/2.Usingdualraillogicin-First,itisassumedthatthestate(30)isincidentonasteadofthesuperpositionbetweenthesinglephotonand50-50beamsplitterwiththevacuumontotheotherinputthevacuum,themeasurementonthesuperposedbasisofthebeamsplitter.Theinitialcoherentamplitudeαicanbesimplyrealizedwithabeamsplitterandtwopho-istakentoberealforsimplicity.Thestate(30)withini-todetectors[17].Itisagainextremelydifficulttoob-tialamplitudeαiafterpassingthroughthebeamsplittertainθ=πusingcurrentlyavailablenonlinearmedia.becomesHowever,simplybyincreasingtheamplitudeα,onecangainanarbitrarilylargeseparationbetweenαandαeiθXN√√withinanarbitrarilyshortinteractiontime.Itispossi-2inπ/N2inπ/N|ψNi=Cn,N|−αie/2i|−αie/2i,bletotransformthestateoftheformof|αi±|αiθiton=1thesymmetricformof|α′i±|−α′ibythedisplacement(32)operationwhichcanbesimplyperformedusingabeamwhereall|Cn,N|’shavethesamevalue.Therealpartofsplitterwiththetransmissioncoefficientclosetooneandthecoherentamplitudeinthestate(32)isthenmeasuredastrongcoherentstatebeinginjectedintotheotherinputbyhomodynedetectioninordertoproducethecatstateport.Therefore,weakcross-Kerrnonlinearitycanalsobeintheotherpath.Bythemeasurementresult,thestateusefultogenerateacatstatewithasinglephoton,strongisreducedtocoherentstates,beamsplittersandtwophotodetectors.NRemarkably,itcanbeshownthatthisapproachcanalsoX√|ψ(1)i=C(1)(α)|−αe2inπ/N/2i,(33)reducedecoherenceeffectsbyincreasingtheinitialam-Nn,Niiplitudeα,whichisalsotrueforRef.[25].Itshouldalson=1 9benotedthatthedetectorsandthesinglephotonsourceresulttothatofKnillLaflammeandMilburnforsin-inRef.[17]whichcanbedirectlycombinedwithRef.[37]glephotonqubits[27].However,farfeweroperationsdonothavetobeefficientforaconditionalgenerationofpergateareneededinthecoherentstateschemeandacatstatebecausethesefactorsonlydegradethesuccessshortcutsareavailableforcertaintasks.Ontheotherprobabilitytobelessthanunity.handtheseadvantagesarenotusefulunlessagoodwayofproducingcatstatescanbefound.Thuswehavespentsometimediscussingvariousproposals,bothlinearandIV.CONCLUSIONnon-linear,forproducingcatstates.Webelievetheneartermprospectsfordemonstratingsmalltravelling-waveWehavediscussedquantumprocessingtasksusingcatstatesandbasicprocessingtasksbasedonthemarequbitsbasedoncoherentstatesandSchr¨odingercatgood.Whethercoherentstatequbitsorsinglephotonstatesasresources.Wehaveshownthatauniversalsetqubitswillprovebetterforlargerscalequantumopticalofprocessingtaskscanbeachievedusingonlylinearop-processinginthelongrunremainsanopenquestion.tics,feedforwardandphotoncounting.Thisisasimilar[1]A.Aspect,P.GrangierandG.Roger(1981).Experi-oratoryofElectronics,MIT,QuarterlyProgressReportmentaltestsofrealisticlocaltheoriesviaBell’stheorem,No.111,unpublished,pp.115–120.Phys.Rev.Lett.47,pp.460–.[14]J.P.DowlingandG.J.Milburn(2003).Quantumtech-[2]C.H.Bennett,G.Brassard,C.Cr´epeau,R.Jozsa,A.nology:thesecondquantumrevolution,Phil.Trans.Peres,andW.K.Wootters(1993).Teleportinganun-Roy.Soc.Lond.A361,pp.1655–1674.knownquantumstateviadualclassicalandEinstein-[15]A.Einstein,B.PodolskyandN.Rosen(1935)CanPodolsky-Rosenchannels,Phys.Rev.Lett.70,pp.1895–quantum-mechanicaldescriptionofphysicalrealitybe1899(1993).consideredcomplete?,Phys.Rev.47,pp.777–780.[3]C.H.Bennett,D.P.DiVincenzo,J.A.Smolin,andW.K.[16]S.J.vanEnkandO.Hirota(2001).EntangledcoherentWootters(1996).Mixed-stateentanglementandquan-states:Teleportationanddecoherence,Phys.Rev.A64,tumerrorcorrection,Phys.Rev.A54,pp.3824–3851.pp.022313-1–022313-6.[4]C.H.Bennett,H.J.Bernstein,S.Popescu,andB.Schu-[17]C.C.Gerry(1999).Generationofopticalmacroscopicmacher(1996).Concentratingpartialentanglementbyquantumsuperpositionstatesviastatereductionwithalocaloperations,Phys.Rev.A53,pp.2046–2052.Mach-ZehnderinterferometercontainingaKerrmedium[5]D.Bouwmeester,J.W.Pan,K.Mattle,M.Eibl,H.We-Phys.Rev.A59,pp.4095–4098(1999).infurter,andA.Zeilinger(1997).Experimentalquantum[18]S.Glancy,H.M.Vasconcelos,andT.C.Ralph(2004).teleporation,Nature390,pp.575–579.Transmissionofopticalcoherent-statequbits,Phys.Rev.[6]S.L.BraunsteinandA.Mann(1995).MeasurementofA70,pp.022317-1–022317-7.theBelloperatorandquantumteleportation,Phys.Rev.[19]L.V.Hau,S.E.Harris,Z.Dutton,andC.H.BehrooziA51,pp.R1727–R1730.(1999),Lightspeedreductionto17metrespersecondin[7]M.Brune,E.Hagley,J.Dreyer,X.Maˆitre,A.Maali,anultracoldatomicgas,Nature397,pp.594–598.C.Wunderlich,J.M.Raimond,andS.Haroche(1996).[20]O.HirotaandM.Sasaki(2001).EntangledstatebasedObservingtheprogressivedecoherenceofthe“meter”inonnonorthogonalstate,quant-ph/0101018.aquantummeasurement,Phys.Rev.Lett.77,pp.4887–[21]O.Hirota,S.J.vanEnk,K.Nakamura,M.SohmaandK.4890.Kato(2001).EntangledNonorthogonalStatesandTheir[8]K.E.CahillandR.J.Glauber(1969).OrderedExpan-DecoherenceProperties,quant-ph/0101096.sionsinBosonAmplitudeOperators,Phys.Rev.177,[22]H.JeongandM.S.Kim(2002).Efficientquantumpp.1857–1881.computationusingcoherentstates,Phys.Rev.A65,[9]J.Clausen,M.Dakna,L.Kn¨ollandD.-G.Welsch(2000).pp.042305–042310.Measuringquantumstateoverlapsoftravelingoptical[23]H.JeongandM.S.Kim(2002).Purificationofentangledfields,OpticsCommunications179,pp.189–196.coherentstates,QuantumInformationandComputation[10]J.Clausen,L.Kn¨oll,andD.-G.Welsch(2002).Lossy2,pp.208–221.purificationanddetectionofentangledcoherentstates,[24]H.Jeong,M.S.KimandJ.Lee(2001).Quantum-Phys.Rev.A66,pp.062303-1–062303-12.informationprocessingforacoherentsuperpositionstate[11]P.T.Cochrane,G.J.Milburn,andW.J.Munro(1999).viaamixedentangledcoherentchannel,Phys.Rev.AMacroscopicallydistinctquantum-superpositionstatesas64,pp.052308–052314.abosoniccodeforamplitudedamping,Phys.Rev.A59,[25]H.Jeong,M.S.Kim,T.C.Ralph,andB.S.Hampp.2631–2634.(2004).Conditionalproductionofsuperpositionsofco-[12]M.Dakna,T.Opatrn´y,L.Kn¨ollandD.-G.Welsh(1997).herentstateswithinefficientphotondetection,Phys.GeneratingSchr¨odinger-cat-l,ikestatesbymeansofcon-Rev.A70,pp.020101(R)-1–020101(R)-4(2004).ditionalmeasurementsonabeamsplitterPhys.Rev.A[26]A.P.Lund,H.Jeong,T.C.Ralph(2004),tobepublished55,pp.3184–3194.inPhys.Rev.A,quant-ph/0410022.[13]S.J.Dolinar(1973).AnOptimumReceiverfortheBi-[27]E.Knill,R.Laflamme,andG.J.Milburn(2001),Ef-naryCoherentStateQuantumChannel,ResearchLab-ficientlinearopticsquantumcomputation,Nature409, 10pp.46–52(2001).ProceedingsofSPIE4917,pp.1–12.[28]P.G.Kwiat,K.Mattle,H.Weinfurter,A.Zeilinger,[45]T.C.Ralph,A.Gilchrist,G.J.Milburn,W.J.Munro,A.V.Sergienko,Y.Shih(1995).Newhigh-intensitysourceandS.Glancy(2003).Quantumcomputationwithop-ofpolarization-entangledphotonpairs,Phys.Rev.Lett.ticalcoherentstates,Phys.Rev.A68,pp.042319-1–75,pp.4337–4341.042319-11.[29]A.J.LeggettandA.Garg,(1985).Quantummechanics[46]M.D.Reid,(1997).Macroscopicelementsofrealityandversusmacroscopicrealism:Isthefluxtherewhenno-theEinstein-Podolsky-Rosenparadox,QuantumSemi-bodylooks?,Phys.Rev.Lett.54,pp.857–860.class.Opt.9,pp489–499.[30]K.S.Lee,M.S.Kim,S.-D.LeeandV.Buzˇek(1993).[47]H.Ritsch,G.J.MilburnandT.C.Ralph(2004).De-Squeezedpropertiesofmulticomponentsuperpositionterministicgenerationoftailored-optical-coherent-statestateoflight,J.Kor.Phys.Soc.26,pp.197–204.superpositions,Phys.Rev.A70,pp.033804-1–033804-4.[31]M.D.LukinandA.Imamoˇglu(2000).NonlinearOptics[48]C.A.Sackett,D.Kielpinski,B.E.King,C.Langer,V.andQuantumEntanglementofUltraslowSinglePhotons,Meyer,C.J.Myatt,M.Rowe,Q.A.Turchette,W.M.Phys.Rev.Lett.84,pp.1419–1422.Itano,D.J.Wineland,C.Monroe(2000).Experimental[32]A.P.Lund,H.Jeong,T.C.Ralph,andM.S.Kimentanglementoffourparticles,Nature404,pp.256–259.(2004).Conditionalproductionofsuperpostionsofcoher-[49]B.C.Sanders(1992).Entangledcoherentstates,Phys.entstateswithinefficientphotondetection,Phys.Rev.ARev.A45,pp.6811–6815.70,pp.020101(R)-1–020101(R)-4(2004).[50]B.C.Sanders,K.S.LeeandM.S.Kim(1995).Optical[33]N.L¨utkenhaus,J.Calsamiglia,andK.-A.Suominenhomodynemeasurementsandentangledcoherentstates,(1999).Bellmeasurementsforteleportation,Phys.Rev.Phys.Rev.A52,pp.735–741.A59,pp.3295–3300.[51]H.SchmidtandA.Imamoˇglu(1996).GiantKerrnonlin-[34]K.Mattle,H.Weinfurter,P.G.Kwiat,andA,Zeilingerearitiesusingelectromagneticallyinducedtransparency,(1996).DenseCodinginExperimentalQuantumCom-Opt.Lett.21,pp.1936–1938.munication,Phys.Rev.Lett.76,pp.4656–4659.[52]E.Schr¨odinger(1935).DiegegenwartigeSituationinder[35]C.Monroe,D.M.Meekhof,B.E.King,andD.J.Quantenmechanik,Naturwissenschaften.23,pp.807-812;Wineland(1996).A“Schr¨odingerCat”Superposition823-828;844-849.stateofanatom,Science272,pp.1131–1135.[53]E.Schr¨odinger(1926).Lepassagecontinudelamicro-[36]M.NielsenandI.Chuang(2000).Quantumcomputationmecaniquealamecaniquemacroscopique,Naturwis-andquantuminformation(CambridgeUniversityPress,senschaften14,pp.664-666.Cambridge,UK).[54]S.Song,C.M.CavesandB.Yurke(1990).Generation[37]K.NemotoandW.J.Munro(2004),Phys.Rev.Lett.ofsuperpositionsofclassicallydistinguishablequantum93,NearlyDeterministicLinearOpticalControlled-NOTstatesfromopticalback-actionevasion,Phys.Rev.A41,Gate,pp.250502-1–250502-4.pp.R5261–R5264.[38]NguyenBaAn(2003).Teleportationofcoherent-state[55]M.TakeokaandM.Sasaki(2005).Implementationofpro-superpositionswithinanetwork,Phys.Rev.A68,jectivemeasurementswithlinearopticsandcontinuouspp.022321-1–022321-6.photoncounting,Phys.Rev.A71,p.p.022318.[39]NguyenBaAn(2004).Optimalprocessingofquantum[56]Q.A.Turchette,C.J.Hood,W.Lange,H.Mabuchi,andinformationviaW-typeentangledcoherentstates,Phys.H.J.Kimble(1995).MeasurementofconditionalphaseRev.A69,pp.022315-1–022315-9.shiftsforquantumlogic,Phys.Rev.Lett.75,pp.4710–[40]J.Pan,C.Simon,C.Brukner,andA.Zeilinger(2001).ˇ4713.Entanglementpurificationforquantumcommunication,[57]X.Wang(2001).Quantumteleportationofentangledco-Nature410,pp.1067–1070.herentstates,Phys.Rev.A64,pp.022302–022305.[41]M.Paternostro,M.S.Kim,andG.M.Palma(2003),[58]B.Wang,L.-M.Duan(2004).EngineeringSchr¨odingerGenerationofentangledcoherentstatesviacross-phase-catstatesthroughcavity-assistedinteractionofcoherentmodulationinadoubleelectromagneticallyinducedopticalpulses,quant-ph/0411139.transparencyregime,Phys.Rev.A67,pp.023811-1–[59]J.Wenger,R.Tualle-Brouri,P.Grangier(2004).Non-023811-15.GaussianStatisticsfromIndividualPulsesofSqueezed[42]S.J.D.Phoenix(1990),WavepacketevolutionintheLight,Phys.Rev.Lett.92,pp.153601-1–153601-4.dampedoscillator,Phys.Rev.A.41,pp.5132–5138.[60]B.YurkeandD.Stoler(1986).Generatingquantumme-[43]T.B.PittmanandJ.D.Franson(2003),Phys.Rev.Lett.chanicalsuperpositionsofmacroscopicallydistinguish-90,pp.240401-1–240401.ablestatesviaamplitudedispersion,Phys.Rev.Lett.57,[44]T.C.Ralph,W.J.Munro,andG.J.Milburn(2002).pp.13–16.Quantumcomputationwithcoherentstates,linearin-teractionsandsuperposedresources,quant-ph/0110115