On the correspondence between classical and quantu

arXiv:quant-ph/9907057v116Jul1999OnthecorrespondencebetweenclassicalandquantummeasurementsonabosonicfieldG.M.D’ArianoandM.F.SacchiTheoreticalQuantumOpticsGroupINFM—Unit`adiPavia&DipartimentodiFisica“A.Volta”Universit`adegliStudidiPavia,viaA.Bassi6,I-27100Pavia,ITALYH.P.YuenDepartmentofElectricalandComputerEngineering,DepartmentofPhysicsandAstronomy,NorthwesternUniversity2145NorthSheridanRoad,Evanston,IL60208-3118USAAbstractWestudythecorrespondencebetweenclassicalandquantummeasurementsonaharmonicoscillatorthatdescribesaone-modebosonicfieldwithan-nihilationandcreationoperatorsaanda†withcommutation[a,a†]=1.WeconnectthequantummeasurementofanobservableˆO=ˆO(a,a†)ofthefieldwiththepossibilityofamplifyingtheobservableˆOideallythroughaquantumamplifierwhichachievestheHeisenberg-pictureevolutionˆO→gˆO,wheregisthegainoftheamplifier.The“classical”measurementofˆOcorre-spondstothejointmeasurementofthepositionˆq=12(a†+a)andmomentumˆp=i2(a†−a)oftheharmonicoscillator,withfollowingevaluationofafunc-tionf(α,¯α)oftheoutcomeα=q+ip.Fortheelectromagneticfieldthejointmeasurementisachievedbyaheterodynedetector.Thequantummea-surementofˆOisobtainedbypreamplifyingtheheterodynedetectorthroughanidealamplifierofˆO,andrescalingtheoutcomebythegaing.WegiveageneralcriterionwhichstateswhenthispreamplifiedheterodynedetectionschemeapproachestheidealquantummeasurementofˆOinthelimitofinfi-nitegain.Weshowthatthiscriterionissatisfiedandtheidealmeasurementisachievedforthecaseofthephotonnumberoperatora†aandforthequadra-tureˆXφ=(a†eiφ+ae−iφ)/2,whereonemeasuresthefunctionsf(α,¯α)=|α|2andf(α,¯α)=Re(αe−iφ)ofthefield,respectively.Forthephotonnumberoperatora†atheamplificationschemealsoachievesthetransitionfromthecontinuousspectrum|α|2∈Rtothediscreteonen∈Noftheoperatora†a.Moreover,forbothoperatorsa†aandˆXφthemethodisrobusttononunitquantumefficiencyoftheheterodynedetector.Ontheotherhand,weshowthatthepreamplifiedheterodynedetectionschemedoesnotworkforarbi-traryobservableofthefield.Asacounterexample,weprovethatthesimple1quadraticfunctionofthefieldˆK=i(a†2−a2)/2hasnocorrespondingpolyno-mialfunctionf(α,¯α)—includingtheobviouschoicef=Im(α2)—thatallowsthemeasurementofˆKthroughthepreamplifiedheterodynemeasurementscheme.1999PACSnumber(s):03.65.-w,03.65.Bz,42.50.Dv,42.50-p2I.INTRODUCTIONInthestandardformulationofQuantumMechanicsanabstractconceptofphysicalob-servableisformulatedintermsofrealeigenvaluesandsharpprobabilitydistributions,whichleadstothewellknowncorrespondencebetweenobservablesandself-adjointoperatorsontheHilbertspace[1].AnaturalextensionofthisformulationisbasedonthegeneralconceptofPositiveOperator-ValuedMeasure(POVM)[2,3],whichallowsthedescriptionofjointmeasurementsofnon-commutingobservables,withgenerallycomplexeigenvaluesandprob-abilitydistributionsthatarenotsharpforanyquantumstate.Fromanoperationalpointofview,however,wehavenoprescriptiononhowtoachievetheidealquantummeasurement(i.e.withminimumnoise)ofagenericoperator,andtheproblemoffindingauniversaldetectorisstillanopenone.Quantumhomodynetomography—theonlyknownmethodformeasuringthestateitselfofthefield—canalsoberegardedasakindofuniversaldetection[4],howeveritisfarfrombeingideal,duetotheoccurrenceofstatisticalmeasurementerrorsthatareintrinsicofthemethod.Inthispaperwestudythepossibilityofachievingtheidealmeasurementofanobserv-ableˆO=ˆO(a,a†)ofonemodeoftheelectromagneticfieldbymeansofafixeddetectionscheme—theheterodynedetector—afteridealpreamplificationˆO→gˆOoftheobservableˆO,gdenotingtheamplifiergain,seekingaconnectionbetweentheproblemofmeasuringˆOandthatofamplifyingˆOideally.Asheterodynedetectioncorrespondstotheidealjointmeasurementofthecanonicalpairˆq=12(a†+a)andˆp=i2(a†−a)ofaharmonicoscillatorinthephasespace,inthiswaywealsotrytosetalinkbetweenclassicalandquantummeasurements.WewillgiveanecessaryandsufficientconditionthatestablisheswhenthepreamplifiedheterodynedetectionschemeapproachestheidealquantummeasurementofˆOinthelimitofinfinitegain.Weshowthatsuchconditionissatisfiedforthephotonnumberoperatora†a—correspondingtothefunctionf(α,¯α)=|α|2oftheheterodyneout-comeα∈C—andforthequadratureoperatorˆXφ=(a†eiφ+ae−iφ)/2—correspondingtothefunctionf(α,¯α)=Re(αe−iφ).Forthephotonnumberoperatora†atheamplificationschemealsoachievesthetransitionfromthecontinuousspectrum|α|2∈RtothediscretespectrumSa†a≡Nofa†a.Moreover,forbothoperatorsa†aandˆXφthemethodsisalsorobusttononunitquantumefficiencyoftheheterodynedetector.Ontheotherhand,wewillseethatthepreamplifiedheterodyneschemedoesnotworkforarbitraryobservableofthefield.Asacounterexample,weshowthat,unexpectedly,thesimplequadraticfunctionofthefieldˆK=i(a†2−a2)/2hasnocorrespondingpolynomialfunctionf(α,¯α)—includingtheobviouschoicef=Im(α2)—whichallowsthemeasurementofˆKthroughthepreamplifiedheterodynemeasurementscheme.Thepaperisorganizedasfollows.InSectionIIwederivethePOVMoftheheterodynemeasurementofafunctionfofthefield,forgenerallynonunitquantumefficiency.InSectionIIIweanalyzetheidealamplificationofanobservableˆO,andprovethatitcanbealwaysachievedbyaunitarytransformation.InSectionIVwegiveanecessaryandsufficientconditionforth