Quantum information processing, operational quantu

arXiv:quant-ph/0304159v124Apr20031Quantuminformationprocessing,operationalquantumlogic,convexity,andthefoundationsofphysicsLosAlamosTechnicalReportLA-UR03-1199HowardBarnum∗aaCCS-3:Modelling,Algorithms,andInformatics.MailStopB256LosAlamosNationalLaboratory,LosAlamos,[e-mail]enceisasourceoftask-relatedaxiomswhoseconsequencescanbeexploredingeneralsettingsencompassingquantummechanics,classicaltheory,andmore.Quantumstatesarecompendiaofprobabilitiesfortheoutcomesofpossibleoper-ationswemayperformasystem:“operationalstates.”Idiscussgeneralframeworksfor“operationaltheories”(setsofpossibleoperationalstatesofasystem),inwhichconvexityplayskeyrole.Themaintechnicalcontentofthepaperisinatheoremthatanysuchtheorynaturallygivesrisetoa“weakeffectalgebra”whenoutcomeshavingthesameprobabilityinallstatesareidentified,andintheintroductionofanotionof“operationalgebra”thatalsotakesaccountofsequentialandconditionaloperations.Suchframe-worksareappropriateforinvestigatingwhatthingslooklikefroman“insideview,”i.e.fordescribingperspectivalinformationthatonesubsystemoftheworldcanhaveaboutanother.Understandinghowsuchviewscancombine,andwhetheranoverall“geomet-ric”picture(“outsideview”)coordinatingthemallcanbehad,evenifthispictureisverydifferentinstructurefromtheperspectiveswithinit,isthekeytowhetherwemaybeabletoachieveaunified,“objective”physicalviewinwhichquantummechanicsistheappropriatedescriptionforcertainperspectives,orwhetherquantummechanicsistrulytellinguswemustgobeyondthis“geometric”conceptionofphysics.Keywords:quantuminformation;foundationsofquantummechanics;quantumcom-putation;quantumlogic;convexity;operationaltheoriesPACScodes:1.IntroductionThecentralquestionquantummechanicsraisesforthefoundationsofphysicsiswhethertheattempttogetaphysicalpicture,from“outside”theobserver,oftheobserver’s∗ThankstotheUSDOEforfinancialsupport.2interactionwiththeworld,apicturewhichviewstheobserveraspartofarealitywhichisatleastroughlydescribedbysomemathematicalstructure,whichisinterpretedbypointingoutwhereinthisstructurewe,theobserversandexperimenters,showup,andwhythingsenduplookingastheydotoobserversinourposition,isdoomed.The“relativestate”picturethatariseswhenonetriestodescribethewholeshebangbyanobjectivelyexistingquantumstateisunattractive,andmanyseektointerpretquantumstatesinsteadassubjective,“information”abouthowourmanipulationsoftheworldcouldturnout.Whateverelsetheymaybethequantumstatesofsystemsclearlyarecompendiaofprobabilitiesfortheoutcomesofpossibleoperationswemayperformonthesystems:“operationalstates.”Anoperationaltheoryisaspecificationofthesetofpossibleoperationsonasystemandasetofadmissibleoperationalstates.This“operational”pointofviewcanbeusefulwhetheronewantstoconsidertheoperationaltheoryasforsomereasonallwecanhopefor,orasadescriptionofhowperspectiveslookwithinanoverarchingtheorysuchastherelativestateinterpretation(RSI).Whileithasnotyetmadeadecisivecontributiontowardresolvingthistension,byfo-cussingontheroleofinformationheld(throughentanglementorcorrelation)orobtained(bymeasurement)byonesystemaboutanotherQIPconcentratesone’sattentiononthepracticalimportanceofsuchmeasurements,anddevelopsflexibilityinmovingbetweentheinsideandoutsideviewsofsuchinformation-gatheringprocesses.Itthusprovidestoolsandconcepts,aswellastheever-presentawareness,likelytobeusefulinresolvingthistension,ifthatispossible.Thispaperisdedicatedtothememoriesoftworesearchersinquantumfoundations,whoIknewonlythroughtheircollaboratorsandtheirwork:RobCliftonandGottfriedT.(“Freddy”)R¨uttiman.Theywillcontinuetoinfluenceandinspireforthedurationoftheintellectualadventureofunderstanding,atthedeepestlevel,ourtheoriesoftheworld.Theirworkisparticularlyrelevanttothethemesofthispaper.Algebraicquantumfieldtheoryisanexampleofintegratinglocalperspectives(local∗-algebrasofobservables)intoacoherentoverallstructure;CliftonmadedeepinvestigationsintofoundationalissuesinAQFT—forexample,CliftonandHalvorson(2001)considersentanglementinthissetting.Hewasalsoinvolvedinoneofthemostspectacularsuccessestodateoftheprojectofapplyingquantuminformation-theoreticaxiomstoquantumfoundations(Cliftonetal.,2002).R¨uttiman’sworkinvolved,forexample,linearizationtheoremsforlattice-basedquantumlogics(R¨uttiman,1993)whichparallelandprefiguretheonesdiscussedhereinforconvexeffect-algebras,andinvestigationoftherelationbetweenthepropertylatticeandfacelatticeofastatespace(R¨uttiman,1981).Thepaperisorganizedasfollows.Section2considerssomesalientgeneralimplicationsofQISforfoundationalquestions(irrespectiveofitscontributionstothisproject).Section3discussestherelativestateand“subjective”viewsonthefoundationsofquantumme-chanics.Section4discusseswhetherandhowtheperspectivesofdifferentobserverscanbecombined,viatensorproductsandotherconstructions.Section5constructs“weakeffectalgebras”fromprobabilitycompendiaviaidentificationofprobabilisticallyequiva-lentoutcomes,reviewsoperationalquantumlogic,especiallyconvexeffectalgebras,and3introducesthenotionofoperationalgebrawhichformalizesthenotionofdoingoperationsinsequence