Constructing Operator Valued Probability Measures in Phase Space

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1、ConstructingOperatorValuedProbabilityMeasuresinPhaseSpaceDemosthenesEllinas∗DepartmentofSciences,DivisionofMathematics,TechnicalUniversityofCrete,GR73100ChaniaCreteGreeceFebruary1,2008AbstractProbabilitymeasures(quasiprobabilitymass),givenintheformofin-tegralsofWignerfunctionoverareaso

2、ftheunderlyingphasespace,giverisetooperatorvaluedprobabilitymeasures(OVM).GeneralconstructionmethodsofOVMs,areinvestigatedintermsofgeometricpositivetraceincreasingmaps(PTI),forgeneral1Ddomains,aswellas2Dshapese.g.circles,disks.SpectralpropertiesofOVMsandoperationalimplementa-tionsofthe

3、irconstructingPITsarediscussed.Keywords:Quantumprobability,POVM,Wignerfunction,PhasespaceQuantumMechanicsPACS:02.,02.50.-r,03.65.Ta,03.67.-aAnOVMisamapK:F→L(H),fromaσ−algebraFofsubsetsofannonemptysetΩ,totheboundedoperatorsL(H)onaHilbertspaceH,suchthatforΨ∈H,andX∈F,thefunctionµΨ(X)≡hΨ

4、K

5、(X)Ψi,isanormalized(i.e.µΨ(Ω)=1),generalized(i.e.negativevalued)measure(i.e.σ−additivesetfunction).IfF(X)≥0,i.e.µΨ(X)>0,∀X∈FwehaveapositiveOVM,furtherifK(X)†=K(X),andK(X)2=K(X),wehaveaprojectivearXiv:quant-ph/0611278v128Nov2006OVM,namelyanobservable(seee.g.[1]).ForthecaseofWignerfuncti

6、on[2][3]W(α)=hΨ

7、D(α)ΠD(α)†

8、Ψi,wehaveΩ≡C,H≡span(

9、ni,n=0,1,...)the

10、ψ>Fockspace,withNthenumberoperator,Π=eiπNtheparityoperator,and†∗D(α)=eαa−αa=e(iqP−ipQ)thedisplacementoperator.IntegralsofWignerRfunction[4][5]W(α)d2α=Tr(

11、ΨihΨ

12、K(X)),physicallydefinethequasiX

13、ψ>RprobabilitymassoverX,interms

14、ofOVMK(X)≡D(α)ΠD(α)†d2α.XProposition1LetXqaregioninq-axisdeterminedbythecharacteristicfunction(cfun),χ(q),q∈R,withFouriertransform(FT)χe(p),thentheasso-ciatedOVMisKq=χe(P)Π.ThesolutionofeigenproblemKq

15、Ψ±i=λ±

16、Ψ±i,determinestheeigenvaluesλ±(p)=±χe(p),andtheeigenvectors

17、Ψ±i=

18、pi±

19、−∗ellinas

20、@science.tuc.gr1a0P∞mπqpi.Iffurtherχ(q)isL−periodicwithFourierseriesχ(q)=2+m=1amcos(L)+mπqbmsin(L),theOVMbecomes"πaX∞mπmπ0iφmKq=Π

21、p=0ihp=0

22、+πrme

23、p=ihp=

24、2LLm=1i−iφmmπmπ+e

25、p=−ihp=−

26、,(1)LLwhererm≡

27、am+ibm

28、,andφm≡arg(am+ibm).Proof:Theregionoperator(convention:alternativenameforOVMrelate