Quantum Communication with Correlated Nonclassical States

S.F.Pereira,*Z.Y.Ou,**andH.J.Kimble

NormanBridgeLaboratoryofPhysics12-33

CaliforniaInstituteofTechnology

Pasadena,CA91125(February1,2008)

arXiv:quant-ph/0003094v1 21 Mar 2000Nonclassicalcorrelationsbetweenthequadrature-phaseamplitudesoftwospatiallyseparatedopticalbeamsareex-ploitedtorealizeatwo-channelquantumcommunicationex-perimentwithahighdegreeofimmunitytointerception.Forthisscheme,eitherchannelalonecanhaveanarbitrarilysmallsignal-to-noiseratio(SNR)fortransmissionofacoher-ent“message”.However,whenthetransmittedbeamsarecombinedproperlyuponauthorizeddetection,theencodedmessagecaninprincipleberecoveredwiththeoriginalSNRofthesource.Anexperimentaldemonstrationhasachieveda3.2dBimprovementinSNRoverthatpossiblewithcorre-latedclassicalsources.Extensionsoftheprotocoltoimproveitssecurityagainsteavesdroppingarediscussed.3.67-a,3.67.Hk,42.50

I.INTRODUCTION

Principalmotivationsfortheinvestigationofman-ifestlyquantumornonclassicalstatesoftheelectro-magneticfieldhavebeentheirpossibleexploitationforopticalcommunication[1–3]andforenhancedmeasure-mentsensitivity.[4]Forexample,relativetoacoher-entstate,thereducedquantumfluctuationsassoci-atedwithsqueezedandnumberstatesofferpotentialforimprovingchannelcapacityinthetransmissionofinformation.[3]Squeezedstatesoflighthavebeenwidelyemployedtoachievemeasurementsensitivitybeyondthestandardquantumlimitsinapplicationssuchaspre-cisioninterferometry,[5]thedetectionofdirectlyen-codedamplitudemodulation,[6]atomicspectroscopy,[7]andquantumnoisereductioninopticalamplification.[8]Likewise,nonclassicalcorrelationsfortheamplitudesofspatiallyseparatedbeamshavebeenexploitedindiversesituations,includingdemonstrationsoftheEPRpara-doxforcontinuousvariables,[9]ofquantumnondemoli-tiondetection(QND),[10,11]andofaquantum-opticaltap.[12]

Withinthebroadersettingofquantuminformationscience(QIS),therehasbeengrowinginterestandim-portantprogressconcerningtheprospectsforquan-tuminformationprocessingwithcontinuousquantumvariables,includinguniversalquantumcomputation,[13]quantumerrorcorrection,[14–16]andentanglementpurification.[17,18]Theoriesforquantumteleportationofcontinuousquantumvariablesinaninfinitedimensional

1

Hilbertspacehavebeendeveloped,[19–22]includingforbroadbandwidthteleportation[23]andforteleportationofatomicwavepackets.[24]Thisformalismhasalsobeenappliedtosuper-densequantumcoding.[25]Onanexper-imentalfront,thesedevelopmentsinQISledtothefirstbonafidedemonstrationofquantumteleportation,whichwascarriedoutbyexploitingnonclassicalstatesoflightinconjunctionwithcontinuousquantumvariables.[26,27]Againstthisbackdropthefocusofattentioninthisar-ticleisopticalcommunicationintwochannelswithquan-tumcorrelatedlightfieldsandtheassociatedquadra-tureamplitudes.[28]Thegoalistoexploretheexten-sionofquantumcryptographyfromtheusualsettingofdiscretevariablesaspioneeredbyC.Bennettandcolleagues[29](e.g.,photonpolarizationasintheexperi-mentsofRefs.[30–33])intotherealmofcontinuousquan-tumvariables(e.g.,thecomplexamplitudeoftheelec-tromagneticfield).Apartfromourwork,severalrelatedschemesforquantumcryptographybaseduponcontin-uousvariableshaverecentlybeenanalyzed,includingasingle-beamschemewithsqueezedlight[34]aswelldual-beamschemeswithsharedentanglement.[35,36]However,westressattheoutsetthatneitherforourschemenorforanyoftheseotherprotocols,cananyclaimaboutab-solutesecuritybemade.Rather,wesuggestthattheseprotocols(andsuitableextensionsthereof)areworthycandidatesformoredetailedanalyses.Suchanundertak-ingwouldinvolvevariousimportantmattersofprincipleaswellaspracticeforcontinuousquantumvariables,andmighthopefullyleadtosecurityproofssuchashavere-centlyemergedinthecaseofdiscretevariables.[37–39]AsillustratedinFigure1,thebasicideainourschemeistoconstructa“transmitter”whichcombinesaco-herentsignalofamplitudeǫ/t(the“message”)withthelargefluctuatingfieldsgeneratedinnondegenerateopti-calparametricamplification(the“noise”).[28]The“mes-sage”andthe“noise”aresuperimposedatmirrorMwithtransmissioncoefficientt<<1.Notethatalthougheachofthetwotransmittedbeamsalongchannels(A,B)haslargephaseinsensitivefluctuationsthatareindivid-uallyindistinguishablefromathermalsource,[40]thequadrature-phaseamplitudesofthetwo-beamscanbequantumcopiesofoneanother,[9,41]andinfactformanentangledEPRstate.[42,43]Hencepropersubtrac-tionofthephotocurrentsatthe“receiver”canresultinthefaithfulreconstructionoftheencoded“message”eventhoughthesignal-to-noiseratiosRj(j=A,B)dur-

ingtransmissionareindividuallymuchlessthanone.In-deed,inalosslesssystemwithlargeparametricgain,thesignal-to-noiseratioofthereconstructedmessageRtcanapproachthesignal-to-noiseratioR0oftheoriginalmes-sage(ǫ2/t2),whichwaswritteninthetransmitterasacoherentstatebeforethemirrorMinFigure1.NotethattheindividualChannels(A,B)haveahighdegreeofim-munitytounauthorizedinterceptionsincethesignal-to-noiseratiosRA,Binthesechannelsareeachverysmall.Furthermore,anyattempttoextractinformationfromthe(A,B)channelswillrevealitselfeitherbyadecreaseinRt(classicalextraction)orbyanincreaseinthefluctu-ationsoftheorthogonalquadratureamplitude(quantumextraction).

InadditiontoachievingafaithfulreconstructionofthemessagetransmittedthroughMtothereceiver,notethattheschemeofFigure1alsopreservestoahighdegreethesignal-to-noiseratiofortheoriginalmessagebeamthatreflectsfromM.Morespecifically,forhighgainandforlossesdominatedbythetransmissioncoefficientofM,thesignal-to-noiseratioRrforthereflectedbeamcanapproachR0fortheoriginalmessage.Inthislimit,wethenhavethattheinformationtransfercoefficientT≡(Rr+Rt)/R0→2,where0≤T≤1forclassicaldevicesand1Ofcoursesimilarschemesfortwo-channelcommunica-tioncanbeimplementedwithcorrelatedclassicalnoisesources(i.e.,thermallight),eachwithlargefluctuationswhich“hide”themessageǫduringtransmission.How-ever,withclassicalsourcesofwhatevertype,onlyexcessfluctuationscanbesubtracted;thequantumfluctuationsatthevacuum-statelevelwillremainunchangedandwillenforceanoisefloorforinformationtransmissionandex-traction.ForthecaseillustratedinFigure1,thisnoisefloorforthemessageatthereceiverisgivenbythesumofindependentvacuumfluctuationsfromfieldsinchannels(A,B)andsetsafundamentalnoiselevelof“2”(with“1”astheindividualvacuum-statelimitsforthetwochannels).Hereweadopttheusualconventionforthedemarcationbetweenclassicalandnonclassicalcorrela-tionsintermsofthebehaviortheGlauber-Sudarshanphase-spacefunction.[41]HenceforthecaseillustratedinFigure1butwithclassicalinputfields,thesignal-to-noiseratioR′

tforthedetectedmessageatthereceiver

isgivenbyR′

t≃|ǫ|2≪Rt.Infact,forclassicalin-puts,wehavethatT′≡R′t+R′

r≤1,andthesystemnolongerfunctionsasaquantumopticaltap.Further-more,theindividualchannels(A,B)arenotprotectedfromunauthorizedeavesdropping,sinceinformationcan

beextractedfromthesechannelswithimpunityforclas-sicalnoisemuchgreaterthanthevacuum-statelimit.Apartfromtheseconsiderationsrelatedtosecurecom-municationandquantumopticaltapping,theconfigura-tionofFigure1canalsobeviewedasameanstorealizesuper-densequantumcoding[45]forcontinuousquantumvariables.[25]Here,themessageǫ/tisagainencodedatthemirrorM,butnowinasinglechannelcorrespond-ingtoonecomponentoftheentangledEPRstate(e.g.,channelA).ThiscombinationofthemessageandthefluctuationsfromonecomponentoftheNOPAaretrans-mittedtothereceivingstationwheretheyarecombinedwiththesecondcomponentoftheentangledoutputoftheNOPAthathasbeenindependentlytransmitted(e.g.,alongchannelB).Thesignalisthendecodedbycombin-ingtheoutputsofthetwochannelsinafashionsimilartothatshowninFigure1asdiscussedinmoredetailinRef.[25].Theprincipaldistinctionsbetweenthisdensecodingschemeandtheaforementioneddualchannelar-rangementare(1)themessageisencodedinasinglecom-ponentoftheentangledEPRbeaminsteadofsymmet-ricallyinbothand(2)thereceivedbeamsfrompaths(A,B)mustbephysicallyrecombined,withthephasesofthelocaloscillators(A,B)atthereceivingstationoff-setbyπ

FortheoriginalEPRstate,thereexistperfectcorrela-tionsbothinpositionandmomentumfortwomassiveparticles.Intheopticalcase,thequadratureamplitudesoftheelectromagneticfieldplaytherolesofpositionandmomentumwithafinitedegreeofcorrelationforfiniteNOPAgain,ashasbeenexperimentallydemonstrated[9]andexploitedtorealizequantumteleportation.[26]

Acoherent-state“message”oftotalamplitudeǫ/tisencodedinequalmeasureontotheseentangledEPRdensityofthephotocurrentfluctuationsΦ−(Ω)ispro-d

portionaltoV−(Ω).Hence,theSNRRdfordetectionof

d

themessageviai−isgivenbyRd=2ηǫ2/V−(Ω),whereηaccountsforthepropagationanddetectionefficiencyforthemessagefromthemirrorMtothephotocurrentiA,B.Withoutdiscussingthegeneralcase,herewenotesimplythatforefficientpropagationanddetectionwith(1−ξ)<<1andfornearthresholdoperationwith(analysisfrequencyΩ)<<(cavitylinewidthΓ),then

beamsbyorientingitspolarizationat45◦withrespectGdtothesignalandidlerpolarizationsatthemirrorMofq(Ω)→1+

1

Figure1.Toobtainaquantitativestatementoftheper-formanceofthissystem,wemustincludethefinitegainoftheamplifieraswellasvariouspassivelosses,whichtogetherlimitthedegreeofcorrelationthatcanbeex-ploitedforcommunication.FollowingtheanalysisofRef.[9],wefindthattheSNRRj(Ω)fortheindividualsignalandidlerphotocurrentsforpropagationanddetectioninthepresenceofoverallchannelefficiencyξisgivenbyRj(Ω)=ξǫ2/2Gq(Ω),whereGq(Ω)isthedetectedquantum-noisegainoftheamplifierwhichcanbedeter-minedexperimentallyfrommeasurementsofthespectraldensitiesΨA,B(Ω)forthefluctuationsofphotocurrentsforsignalandidlerbeamsaloneateitherdetector.Rela-tivetothefrequencyoftheopticalcarrierdeterminedbythedown-conversionprocessintheNOPA,thefrequencyΩspecifiestheFouriercomponentsofthequadrature-phaseamplitudesofsignalandidlerfieldsaswellasofthecoherentfieldǫ.[41]Notethatξ(0≤ξ≤1)incor-poratesthecavityescapeefficiencyforourNOPA,thepropagationefficiencyfromtheNOPAtothedetectors,andthehomodyneandquantumefficienciesofthebal-anceddetectorsthemselves.[9]

Althoughtheindividualfluctuationsforchannels(A,B)giverisetoalevelGq(Ω)>1,(thatis,greaterthanthevacuum-statelimitofeitherbeamalone),theselargefluctuationsarecorrelatedinanonclassicalman-nerandhencecanbeeliminatedbyproperchoiceofthequadratureamplitudesdetectedat(A,B).AsshowninRef.[9],thereisacontinuoussetofsuchampli-tudeswithminimumvariancefortheirdifferencerequir-ingonlythatthequadrature-phaseangles(θA,θB)satisfyθA+θB=2pπ(p=integer).Denotingonesuchpairby(XA,XB),wehavethat

󰀋(XA(Ω)−XB(Ω))(XA(Ω′)−XB(Ω′))󰀌

=V−(Ω)δ(Ω+Ω′),

(1)

whereV−(Ω)isavariancewhichquantifiesthedegree

ofcorrelationbetween(XA,XB).ExplicitexpressionsforbothV−(Ω)andGq(Ω)aregiveninRef.[9].Forpropagationanddetectioninthepresenceofloss,wein-troducethequantities(V−d

,Gdgainq)whichrefertothevari-anceandquantumnoiseforfictitiousfieldshavingpropagatedwithtotalloss(1−ξ),wherethespectral

2η|t|2

Ur(Ω)+

−

beamsaregeneratedbyTypeIIdown-conversioninasubthresholdopticalparametricoscillatorformedbyafoldedcavitycontainingana-cutcrystalofpotassiumti-tanylphosphate(KTP)thatprovidesnoncriticalphasematchingat1.08µm.Thecrystalis10mmlong,isanti-reflectioncoatedforboth1.08µmand0.54µm,andhasameasuredharmonicconversionefficiencyof6×10−4/W(single-pass)forthisgeometry.Thetotalintracavitypas-sivelossesat1.08µmare0.3%andthetransmissionco-efficientofmirrorM1is3%.Theamplifierispumpedbygreenlightat0.54µmgeneratedbyexternalfre-quencydoublingofafrequency-stabilized,TEM00-modeNd:YAPlaser.[46]Thesubthresholdoscillatoractsasanarrow-bandamplifier(NOPA)whichislockedtotheoriginallaserfrequencywithaweakcounter-propagatingbeam.Simultaneousresonancefortheorthogonallypo-larizedsignalandidlerfieldsisachievedbyadjustingthetemperatureoftheKTPcrystalaround60◦Cwithmil-liKelvinprecision.Thepumpfieldat0.54µmisitselfresonantinaseparateandindependentlylockedbuild-upcavity(enhancement∼5x).

Aswehavedemonstratedinourpreviousexperiments,[9]

theorthogonallypolarizedsignalandidlerfieldsgen-eratedbytheNOPAindividuallyarefieldsofzeromeanvaluesandexhibitlargephaseinsensitivefluctuations.Itisinthemidstofthisnoisethatwenowhidea“message”,withthiscoherentfieldbeingcombinedwiththesignalandidlerfieldsatthehighlyreflectingmirrorMshowninFigure1((1−t2)≃0.99).Thecoherentbeamisin-jectedat45◦withrespecttosignalandidlerpolarizationsandisfrequencyshiftedbyΩ0/2π=1.1MHz(single-sideband)fromtheprimarylaserfrequencywiththehelpofapairofacoustoopticmodulators,whicharegated“on”and“off”toprovideinformationencodedfortransmis-sion.Thenoisybutcorrelatedsignalandidlerbeamstogetherwiththecoherentinformationarethensepa-ratedbyapolarizerP,transmittedindependentlyoverthetwochannels(A,B),andthendirectedtotwosepa-ratebalancedhomodynedetectorsformeasurementsoftheirindividualquadrature-phaseamplitudesandtheirmutualcorrelations.Thelocaloscillatorsforthetwobalancedhomodynedetectorsoriginatefromthelaserat1.08µm;theirphasescanbeindependentlycontrolledbymirrorsmountedonpiezoelectrictransducers.Thespec-traldensitiesofthephotocurrentsforthetwochannels(A=signal,B=idler)aredefinedby

󰀁

ΨA,B(Ω)=eiΩτdτ(3)andarerecordedbyaRFspectralanalyzer,asisthespectraldensity

󰀁

Φ−(Ω)=eiΩτdτ(4)forthedifferencephotocurrenti−≡iA−iB.

4

InFigures2and3wepresentresultsfromaseriesof

measurementsofthesevariousspectraldensities.Firstofall,inFigure2a,traceigivesthespectraldensityΨAforchannelAalonewithaninjected“message”andwiththeamplifierturnedontogeneratelarge(∼7dB)phaseinsensitivenoiseabovethevacuum-statelevelΨ0A(indi-catedbyadashedlineinFigure2)forthesignalbeam.AsimilartraceisobtainedforthespectraldensityΨB.Bycontrast,traceiiinFigure2agivesthespectraldensityΦ−forthedifferencephotocurrenti−,withthephasesofthelocaloscillatorsadjustedforminimumnoiseandmax-imumcoherentsignal.Inthistrace,thecoherentmes-sagethatwascompletelyobscuredintraceiemergeswithhighsignal-to-noiseratio.Notethatintraceiithecorre-latedquantumfluctuationsforsignalandidlerfieldsaresubtractedtoapproximately0.4dBbelowthevacuum-noiselevelΨ0Aofthesignalbeamalone(andlikewisefortheidler),indicatinganimprovementinSNRoveraconventionalsingle-channelcommunicationschemewithaclassicallightsource.

Tocompletethediscussion,wepresentinFigure2bre-sultsobtainedwiththeamplifierturnedoff(thatis,un-correlatedvacuum-stateinputsforsignalandidlerfieldswhicharecombinedwiththecoherent“message”infor-mationatmirrorM).Traceishowstheresultforthesignalbeamalone(ΨA),whereagainthenoisefloorΨ0Aisfromthevacuumfluctuationsofthesignalbeam;asimilartraceisobtainedfortheidlerbeam(ΨB).TraceiigivesthecorrespondingresultforΦ−forthecom-binedsignalandidlerphotocurrentswhentheamplifierisoff.NotethatthistracerepresentsthebestpossibleSNRwithwhichtheencodedinformationcanberecov-eredwhencorrelatedclassicalnoisesourcesareemployedsinceherethe(uncorrelated)vacuumfluctuationsofsig-nalandidlerbeamssetanultimatenoisefloor3dBaboveΨ0A(thatis,Φ0−=2Ψ0A).[47]OncomparingtracesiiinFigures2aand2b,weseethatthecorrelatedquan-tumfluctuationsofsignalandidlerfieldsbroughtaboutbyparametricamplificationresultinanimprovementinSNRof3.2dBrelativetothatpossiblewithclassicalnoisesources.

TheimprovementinSNRwithcorrelatedquantumfieldsoverclassicalfieldsinourtwo-channelcommunica-tionschemecanbeofutilityespeciallywhenthemessageissoweakthattheSNRispoorfortransmissionwithcorrelatedclassicalsources(thatis,forthecasewherevacuumnoisedominatestheencodedmessage).Thissit-uationisillustratedinFigure3,whereweplotΦ−forthetwocaseswithout(tracei)andwith(traceii)corre-latedquantumfields.[47]RelativetoFigure2,herethecoherentbeamhasbeenattenuatedresultinginasmallerSNRforthe“message”.Indeedintracei,thisinfor-mationis“buried”bythevacuumnoiseΦ0−associatedwithindependentvacuumfluctuationsinchannelsAandB;recoveryoftheencodedinformationispoor.Ontheotherhand,asshownintraceii,whencorrelatedquan-

tumfieldsareemployed,thereisareductioninthenoisefloorbymorethan3dBwhichmakespossibleimprovedrecoveryoftheencodedinformation,withtherecoveryherelimitedbylossesinpropagationanddetection.[9]Asfortheactualperformancewithrespecttoopti-caltapping,oursystemfallsfarshortoftheprojectedpossibilitiesdiscussedintheprecedingsectionbecauseofanunfortunatemismatchbetweenthetransmissiv-ity|t|2formirrorMandtheoverallsystemefficiencyξ.Inquantitativeterms,recallthatthetransfercoeffi-cientTforencodinginformationfromtheinputbeamtothereflectedandtransmittedbeamsatMisgivenbyT≡(Rr+Rt)/R0whereasthetransfercoefficientforthedetectedmessagephotocurrentandthereflectedsig-nalfieldisTd≡(Rr+Rd)/R0asgivenexplicitlyinEq.(2).Forthepropagationanddetectionefficienciesinourexperiment(ξ≃0.65andη≃0.75),thesetransfercoef-ficientsareoptimizedformirrortransmission|t|2∼0.5forM.Inourarrangementwehaveinstead|t|2=0.01,withtheinferredresultthatTd≃1.02,whichisonlymarginallyinthequantumdomain.

Intheexperimentdescribedhere,thereceiverusesalocaloscillator(LO)thatoriginatesfromthefundamen-talfrequencyofthesamelaserthatgeneratedthepumpbeamfortheNOPA.ThisLOisnecessaryforproperdetectionofthequadratureamplitudesofthenonclassi-calbeamsandofthemessage,sinceitprovidesaphasereferencethatfollowsphasefluctuationsoftheNOPA’spumpbeam.Inpractice,asthestabilityoftheavailablelasersimprove,oneshouldconsiderschemesforwhichthemeasurementiscarriedoutwithnominallyindepen-dentlasersfortheLOandforthesource.Forexample,onemightemployastabilizedlaserdiodeasareferencetophaselocklasersbothatthesenderandatthere-ceiver,wherethelaserdiodecouldbewidelydistributedthroughopticalfibers.Alternatively,Ralphhasanalyzedaschemeinwhichthelocaloscillatorsaretransmittedandrecoveredaspartoftheoverallprotocol.[35]

IV.COMPARISONWITHOTHERDUALBEAM

SCHEMES

Itisperhapsobviousthatthedegreeofimmunitytointerceptionforatwochannelschemesuchaswehavedis-cussedisrelatedtothedegreeofexcessfluctuationsforeachindividualbeam.ForthedemonstrationinRef.[48],theexcessnoiseusedto“hide”theencodedinformationineachbeamcomesfromsomeartificialunrelatedsource.Unfortunatelysuchuncorrelatedexcessfluctuationsalsoaddnoisetothecoincidencesignalintherecoveryofthe“message,”eventhoughtheaddednoisescalesdifferentlyasafunctionofphotonnumberforsingle-beammea-surements(linearly)andfordualbeammeasurements(quadratically).Hencelargerbackgroundnoisewhichbetter“hides”theencodedinformationalsobringslarger

5

addednoiseintheextractionofthe“message.”Becauseofthequadraticdependenceonthetotalphotonnumberfortheextranoiseaddedincoincidencedetection,thisschemeisbestsuitedtolowlightleveltransmission,asdemonstratedinthepioneeringexperimentbyHongetal.[48]

Thesituationisquitedifferentforthequadrature-phaseamplitudesofthecorrelatedsignalandidlerfieldsgeneratedbytheNOPA.AstheNOPAispumpedharderandthethresholdforparametricoscillationisap-proached,thegainoftheamplifierincreases,asdotheexcessfluctuationsofthesignalandidlerfields.How-ever,thecorrelationbetweenthefluctuationsofthesig-nalandidlerbeamsalsoimproves,givingrisetoevenbetterSNRfortherecoveredsignal.Thekeypointisthatthelargefluctuationsinthesignalandidlerbeamsneededforimmunitytointerceptionareintrinsicanddonotaddextranoisetotherecoveredsignalbut,onthecontrary,servetoreducethenoiseini−asthegainoftheamplifierincreases.Intheend,theSNRforthere-coveredmessageisarbitratedbytheimperfectcorrela-tionresultingfromfinitegainandfrompassivelossesinpropagationanddetection.Ontheotherhand,thisdependenceprovidesapowerfulmeanstodetecteaves-droppingbecauseunauthorizedextractionofsignaloridlerfieldsfromchannelsAorBresultsinareductionofthedetectedcorrelationandhenceanincreaseinthenoiseflooroftherecoveredmessage.Notethatunautho-rizedextractionofinformationfrombothchannelsbywayofaquantumopticaltap[44]oraquantumnonde-molitionmeasurement[11]canlikewisebedetectedbe-causeoftheunavoidableincreaseoffluctuationsfortheorthogonalquadrature-phaseamplitudes(θA,B+π/2)ofthetwochannels.Furthermore,thesequantumeaves-droppingschemescanbedefeatedinlargemeasurebyrandomswitchingofthephasesofthemessage,signal,andidlerbeamsasdiscussedbelow.

Oursystemalsooffersadvantageswithrespecttothe(classical)digitalVernancipher,whereamessageisde-composedintwocorrelatedrandomsignalsandtrans-mittedovertwoone-waychannels.Althoughthissystemseemstobesimilartooursinthesensethatisalsose-cureprovidedtheeavesdropperhasaccesstoonechan-nelonly,thesituationisdifferentiftheeavesdroppercansplitasmallfractionofbothchannelssinceintheclas-sicalcase,thiscanbedonewithouttheknowledgeofthereceiver.However,inoursystemtheeavesdroppercannotchoosearbitrarilythereflectivityofany“beam-splitter”usedforextractionfromthetwochannelssinceinthequantumcase,thefractionofthebeamsextractedshouldbebigenoughsothatthesignal-to-noiseratiofortheinterceptedmessageisgreaterthanone.Butifthisisthecase,thenunavoidableextra“noise”addedtothetransmittedbeamsbytheopenportofthe“beamsplit-ter”degradesthesignal-to-noiseratioofthemessageatthelegitimatereceiver,thusrevealingtheunauthorized

interventionduringtransmission.

Onemightattempttocircumventthisdifficultybyem-ployingaquantumextractionprocedure,suchasquan-tumnondemolitiondetection[11]ofthequadratuream-plitudesinChannels(A,B).Althoughthesignal-to-noiseratioRdatthereceiverwouldnotinthiscasebedegradedbyanidealeavesdropper,theunauthorizedinterventioncouldnonethelessbediscoveredbecauseoftheinjectionoflargefluctuations(“back-action”noise)inthequadra-tureorthogonaltothatinwhichsignalinformationisstored,aspreviouslynoted.

V.EXTENSIONSVIARANDOMPHASE

SWITCHING

1.Phase0–Setthequadrature-phaseangles(θA,θB)

000to(θA,θB)andthephasesβA,B=βA,Bfortheco-herentmessagebeam|α󰀌=|α|exp[iβ]correspond-ingtotheXquadraturesof(A,B).2.Phasetures.

π

2

correspondingtotheYquadra-

OnewayaneavesdropperEvecouldaccessthesignal

andidlerbeamswithouttheknowledgeofthelegitimatereceiverisifshecaninterceptbothchannelscompletely,detectinthesamemannerasdoesthelegitimatereceiver(i.e.,Eveshouldalsohaveaccesstoalocaloscillatorphasestablewithrespecttothatofsenderandreceiver)andretransmitthebeamsinthesamewayasthelegit-imatesender.Becauseofthispossibility,ourprotocolasdescribediscertainlynotsecure,incontrasttotheprotocolsfordiscretevariables.[37–39]However,wesug-gestthatsimpleextensionsofourprotocolmightleadtosignificantenhancementsinsecurity.

Ifthegoalweretoachievequantumkeydistribution,oneideaistomakestraightforwardadaptationsoftheprotocolsintroducedbyBennettandcolleaguesforthediscretecase,asinRef.[34–36].Here,weproposethatthesendingstation(Alice)andreceivingstation(Bob)makerandomchoicesforthesetofphasesofthecoherentmessagebeam,aswellasforthesignalandidlerbeams.RecallthatthevarianceV−(Ω)ofEq.1istheminimumpossibleandappliesonlyforthechoiceofquadrature-phaseangles(θA,θB)forthesignalandidlerbeamsthatsatisfyθA+θB=2pπ(p=integer).Fordefiniteness,assumethefollowingtwochoices.

00001.(θA,θB),withθA+θB=0andcorrespondingquadratureamplitudes(XA,XB).0

+π2.(θA=θA2)andcorresponding

quadratureamplitudes(YA,YB).

π/2

Theencodedmessage(whichcouldconsistof|α|=[a0,a1]forabinarytransmission)issenttoBob’sre-ceivingstationpreciselyasinFigure1.Bobmustthenchoosetheappropriatephases(φA,φB)forhislocaloscil-lators(LOA,LOB)todetectquadratureamplitudessuchthatthespectraldensityΦ−(Ω)forthedifferencepho-tocurrenti−≡iA−iBisminimizedandthesignalmaxi-mized.InthecasePhase0,denotethelocaloscillatorset-0

tingsas(φ0A,φB),incorrespondencetothedetectionof(XA,XB)withminimumvarianceV−(Ω).Ontheotherhand,forthecasePhaseπ

3π0

2,φB+

Inthefirstcase,theminimumvarianceV−(Ω)resultsforthecombination(XA−XB),whileinthesecondcase,thecombination(YA+YB)hasminimumvariance.This

π/2

isbecauseYB→−YBisequivalenttotheshiftθB→π/2π/2π/2

θB+π,sothatθA+θB+π=2π.

Withthesedefinitions,Aliceatthesendingstation(randomly)makesoneoftwochoices.

Alicewillhavemadeforanygiventransmission.Hence,hemakesarandomselectionbetweenthealter-π/2π/20

natives(φ0A,φB)and(φA,φB),recoveringthemessageinsomecasesbutnotothers.Afteraseriesoftransmis-sions,AliceandBobcommunicatepubliclyabouttheirchoiceofbases,keepingmeasurementresultsonlywhentheirchoicescoincide.

Now,ifaneavesdropperEveattemptstointervene(ei-therbyastrategyofpartialtappingorbyoneofcom-pleteinterceptionandre-broadcast),shewillnecessarilyincreasethenoiselevelanderrorrateatBob’sreceivingstation.Therandomswitchingofthephases(θA,θB)byAliceforcesEvetomakeaguessastothecorrectquadratures(δA,δB)tobedetected.Havingmadeachoice,informationabouttheorthogonalquadratureislost.Ofcourse,ratherthanhomodynedetection,shecouldchoosetoemployheterodynedetectiontogainin-formationaboutthefullcomplexamplitude.However,relativetohomodynedetection,heterodynedetectionbringsawell-knownpenaltyofa3dBreductioninsignal-to-noiseratio.[49]

WhileitisbeyondthescopeofthecurrentpapertomakeanyclaimsaboutthequantitativelimitstotheinformationthatEvemightaccessorabouttheabso-luteabilityofAliceandBobtodetectherpresence,wedosuggestthatthesewouldbeinterestingquestionstoinvestigate.Therearecertainlyinterventionstrategiesbeyondthosethatwehavementionedthatacunning6

2]

Evewouldwanttoconsider,suchasanadaptivestrat-egyforadjustingthephases(δA,δB)duringthedurationofthetransmissionofanygivenmessage.[50]Likewise,inanyreal-worldsetting,overcomingthedeleteriousef-fectsoflossesinpropagationfromAlicetoBobwillbeaoverridingconsideration.ThequestionofpreservingtheentanglementoftheinitialEPRstateinthefaceofsuchlossesisafascinatingoneforcontinuousquantumvariables.Althoughinitialattemptshavebeenmadetodeveloperrorcorrectingquantumcodesforcontinuousvariables,[14–16]noadequatesolutionseemstoyethavebeenfound.Finally,itwouldbeofinteresttoanalyzethecasewhereonlyoneofthetwocorrelatedbeamsissenttoBob,withthenAliceretainingtheother.

ACKNOWLEDGMENTS

WegratefullyacknowledgethecommentsofJ.H.Shapirowhopointedouttheconnectionofourexperi-menttoRef.[44],ofS.L.BraunsteinandH.Mabuchiforcriticaldiscussions,andofoneoftherefereeswhobroughttoourattentiontheVernoncipher.ThisworkwassupportedbytheOfficeofNavalResearch,bytheNationalScienceFoundation,andbyDARPAviatheQUICadministeredbytheArmyResearchOffice.

[42][43][44][45][46][47]

[48][49][50]

tics,ed.byJ.Dalibard,J.M.Raimond,andJ.ZinnJustin(Elsevier,Amsterdam,1992).545ff.

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M.D.ReidandP.D.Drummond,Phys.Rev.Lett.60,2731(1988);M.D.Reid,Phys.Rev.A40,913(1989).J.H.Shapiro,Opt.Lett.5,351(1980).

C.H.BennettandS.J.Wiesner,Phys.Rev.Lett.69,2881(1992).

Z.Y.Ou,S.F.Pereira,E.S.Polzik,andH.J.Kimble,Opt.Lett.17,640(1992).

InFigs.(2,3),Φ−=1.9Ψ0,A(2.8dBabove)duetoaslightimbalance(0.1dB)betweenthe(A,B)detectorsandtoasmallcontribution(0.1dB)fromdetectorthermalnoise.L.Mandel,J.Opt.Soc.Am.B1,108(1984);C.K.Hong,S.R.Friberg,andL.Mandel,Appl.Opt.24,3877(1985).E.ArthursandJ.L.KellyJr.,Bell.Syst.Tech.J.April,725(1965).

H.M.Wiseman,Phys.Rev.Lett.75,4587(1995);H.M.WisemanandR.B.Killip,Phys.Rev.A57,2169(1998).

FIG.1.Principalcomponentsoftheexperimentshow-ingthe“transmitter”,whereamessageǫiscombinedwithnoisefieldsfromanondegenerateopticalparametricamplifier(NOPA)atmirrorM.TheorthogonallypolarizedsignalandidlerbeamsareseparatedbypolarizerP,andthenpropagatealongindependentchannels(A,B)totwoseparatebalancedhomodynedetectorsthatformthe“receiver”.Inthefigure,arrowsrepresentcoherentamplitudesofvariousfields,whiletheshadedcirclesaremeanttoindicatetheirfluctuations.

FIG.2.(a)Signalrecoverywithcorrelatedquantumstatesinchannels(A,B).TraceigivesthespectraldensityΨA(Ω)ofphotocurrentfluctuationsforchannelAaloneasafunctionoftime.TraceiiisthespectraldensityΦ−(Ω)forthecom-binedphotocurrenti−=iA−iBfrombothchannels;herethe“message”(coherentbeamchoppedonandoff)clearlyemerges.(b)Signalrecoverywithuncorrelatedvacuumfluc-tuations.Again,traceigivesΨA(Ω)(channelAonly)whiletraceiigivesΦ−(Ω)(combinedphotocurrenti−).In(a)and(b)thevacuum-statelevelΨ0AforchannelA(signalbeamonly)andΦ0−forthecombinedphotocurrenti−(dualbeam)areshownasdashedlines.NotethatΨ0Alies15dBabovetheelectronicnoisefloor.Spectrumanalyzeracquisitionparam-etersareasfollows:resolutionbandwidth=100kHz,videobandwidth=100Hz,analysisfrequencyΩ0/2π=1.1MHz,andsweeptime=300ms.

FIG.3.SpectraldensityofthephotocurrentfluctuationsΦ−(Ω)fori−=iA−iBforthecasewhentheon-offmod-ulationofthemessageisencodedwithsmallSNR.Tracei−Uncorrelatedvacuumfluctuations.Traceii−Correlatedquantumfluctuationsinchannels(A,B).Thevacuum-statelimitsΨ0AandΦ0−areindicated.AcquisitionparametersareasinFig.2.

8

\"Message\"\"Transmitter\"M\"Noise\"SignalIdlerPM1KTPPump @ 2ω

NOPAChannel AChannel BLO A\"Receiver\"ABLO BHomodyneDetectors\"Message\"Recoveredi-(a)10

Φ- , ΨA (dB)86420-20

100

200

(i)

(ii)

Φ0-Ψ0A

300

Time (ms)

A (b)8

64(ii)

2Φ0-(i)

0Ψ0A

-20

100

200

300

Time (ms)

Φ-,Ψ(dB)6Φ- (dB)42(i)Φ0-(ii)0-20100Ψ0A

200300Time (ms)