Betting on the Outcomes of Measurements A Bayesian

arXiv:quant-ph/0208121v118Aug2002BettingontheOutcomesofMeasurements:ABayesianTheoryofQuantumProbabilityItamarPitowskyDepartmentofPhilosophy,theHebrewUniversity,MountScopus,Jerusalem91905,[e-mail]1,2008AbstractWedevelopasystematicapproachtoquantumprobabilityasatheoryofrationalbettinginquantumgambles.Inthesegamesofchancetheagentisbettinginadvanceontheoutcomesofseveral(finitelymany)incompatiblemeasurements.Oneofthemeasurementsissubsequentlychosenandperformedandthemoneyplacedontheothermeasurementsisreturnedtotheagent.Weshowhowtherulesofrationalbettingimplyalltheinterestingfeaturesofquantumprobability,eveninsuchfinitegambles.TheseincludetheuncertaintyprincipleandtheviolationofBell’sinequalityamongothers.Quantumgamblesarecloselyrelatedtoquantumlogicandprovideanewsemanticstoit.Weconcludewithaphilosophicaldiscussionontheinterpretationofquantummechanics.1QuantumGambles1.1TheGambleTheBayesianapproachtakesprobabilitytobeameasureofignorance,reflect-ingourstateofknowledgeandnotmerelythestateoftheworld.ItfollowsRamsey’scontentionthat“wehavetheauthoritybothofordinarylanguageandofmanygreatthinkersfordiscussingundertheheadingofprobability...thelogicofpartialbelief”(Ramsey1926,p.55).Hereweshallassume,further-more,thatprobabilisticbeliefsareexpressedinrationalbettingbehavior:“Theold-establishedwayofmeasuringaperson’sbelief...byproposingabet,andseewhatarethelowestoddswhichhewillaccept,isfundamentallysound”1.Myaimistoprovideanaccountofthepeculiaritiesofquantumprobabilityinthisframework.Theapproachisintimatelyrelatedtothefoundationalworkon1Ramsey,1926,p.68.Thissimpleschemesuffersfromvariousweaknesses,andbetterwaystoassociateepistemicprobabilitieswithgamblinghavebeendeveloped(deFinetti,1972).AnyoneofdeFinetti’sschemescanserveourpurpose.ForamoresophisticatedwaytoassociateprobabilityandutilityseeSavage(1954)1quantuminformationbyBarnumetal(2000),Fuchs(2001),Schack,BrunandCaves(2001)andCaves,FuchsandSchack(2002)..Forthepurposeofanalyzingquantumprobabilityweshallconsiderquantumgambles.Eachquantumgamblehasfourstages:1.Asinglephysicalsystemispreparedbyamethodknowntoeverybody.2.AfinitesetMofincompatiblemeasurementsisannouncedbythebookie,andtheagentisaskedtoplacebetsonpossibleoutcomesofeachoneofthem.3.OneofthemeasurementsinthesetMischosenbythebookieandthemoneyplacedonallothermeasurementsispromptlyreturnedtotheagent.4.Thechosenmeasurementisperformedandtheagentgainsorloosesinaccordancewithhisbetonthatmeasurement.Wedonotassumethattheagentwhoparticipatesinthegameknowsquan-tumtheory.Wedoassumethatafterthesecondstage,whenthesetofmea-surementsisannounced,theagentisawareofthepossibleoutcomesofeachoneofthemeasurements,andalsooftherelations(ifany)betweentheout-comesofdifferentmeasurementsinthesetM.Letmemaketheseassumptionsprecise.Forthesakeofsimplicityweshallonlyconsidermeasurementswithafinitesetofpossibleoutcomes.LetAbeanobservablewithnpossibledistinctoutcomesa1,a2,...,an.WitheachoutcomecorrespondsaneventEi={A=ai},i=1,2,...,n,andtheseeventsgenerateaBooleanalgebrawhichweshallde-notebyB=hE1,E2,...,Eni.SubsequentlyweshallidentifytheobservableAwiththisBooleanalgebra.Notethatthisisanunusualidentification.ItmeansthatweequatetheobservablesAandf(A),wheneverfisaone-onefunctiondefinedontheeigenvaluesofA.Thisstepisjustifiedsinceweareinterestedinoutcomesandnottheirlabels,hencethescalefreeconceptofobservable.WiththisMisafinitefamilyoffiniteBooleanalgebras.OurfirstassumptionisthattheagentknowsthenumberofpossibledistinctoutcomesofeachmeasurementinthesetM.OurnextassumptionconcernsthecasewheretwomeasurementsinthesetMsharesomepossibleelements.Forexample,letA,B,Cbethreeobservablessuchthat[A,B]=0,[B,C]=0,but[A,C]6=0.Considerthetwoincompatiblemeasurements,thefirstofAandBtogetherandthesecondofBandCtogether.IfB1istheBooleanalgebrageneratedbytheoutcomesofthefirstmeasurementandB2ofthesecond,thenM={B1,B2}andtheevents{B=bi}areelementsofbothalgebras,thatisofB1∩B2.Weassumethattheagentisawareofthesefactswhenheisplacinghisbets.Thesmallestnontrivialcaseofthiskindisdepictedinfigure1.ThegraphrepresentstwoBooleanalgebrasB1=hE1,E2,E3i,B2=hE1,E4,E5icorre-spondingtotheoutcomesoftwoincompatiblemeasurementsandtheyshareacommoneventE1.ThecomplementofE1denotedbyE1isidentifiedasE2∪E3=E4∪E5.Theedgesinthegraphrepresentthepartialorderrelationsineachalgebrafrombottomtotop.Arealizationoftheserelationscanbeob-tainedbythesystemconsideredinKochenandSpecker(1967):LetS2x,S2x‘,S2y,S2y‘,S2zbethesquaredcomponentsofspininthex,x‘,y,y‘,zdirectionsofaspin-1(massive)particle,wherex,y,zandx‘,y‘,zformtwoorthogonaltriplesofdirectionswiththez-directionincommon.TheoperatorsS2x,S2yandS2zall2Figure1:commute,andhaveeigenvalues0,1.Theycanbemeasuredsimultaneusly,andtheysatisfyS2x+S2y+S2z=2I.Similarrelationsholdintheothertriplex‘,y‘,z.Hence,ifwedefineE1={S2z=0},E2={S2x=0},E3={S2y=0},E4={S2x‘=0},E5={S2y‘=0}weobtainthetwoBooleanalgebrasdepictedinfigure1.Tosum,weassumethatwhenthesetofmeasurementsMisannouncedtheagentisfullyawareofthenumberofoutcomesineachmeasurementandoftherelationsbetweentheBooleanalgebrastheyge