The Standard Model of Quantum Measurement Theory H

arXiv:quant-ph/9603020v114Mar1996Busch&Lahti—TheStandardModelofQuantumMeasurementTheory1FoundationsofPhysics1996TheStandardModelofQuantumMeasurementTheory:HistoryandApplicationsPaulBuschDepartmentofAppliedMathematicsTheUniversityofHullHull,HU67RX,EnglandE-mail:[e-mail]UniversityofTurkuSF-20500Turku50,FinlandE-mail:pekka.lahti@utu.fiThestandardmodelofthequantumtheoryofmeasurementisbasedonaninteractionHamiltonianinwhichtheobservable-to-be-measuredismultipliedwithsomeobservableofaprobesystem.ThissimpleAnsatzhasprovedextremelyfruitfulinthedevelop-mentofthefoundationsofquantummechanics.Whiletheensuingtypeofmodelshasoftenbeenarguedtoberatherartificial,recentadvancesinquantumopticshavedemonstratedtheirprinicpalandpracticalfeasibility.Abriefhistoricalreviewofthestandardmodeltogetherwithanoutlineofitsvirtuesandlimitationsarepresentedasanillustrationofthemutualinspirationthathasalwaystakenplacebetweenfounda-tionalandexperimentalresearchinquantumphysics.1.Introduction.InChapterThreeofhisclassicThePhilosophyofQuantumMechanics(1),MaxJam-merreviewsthedebateabouttheinterpretationoftheindeterminacyrelationforpositionandmomentum.Hepointsout‘thatofalltheinterpretationslisted...thefollowingtwoprovedmostimportantforthedevelopmentofquantummechanics:1.ThenonstatisticalinterpretationI1accordingtowhichitisimpossible,inprinciple,tospecifypreciselythesimultaneousvaluesofcanonicallyconjugatevari-ablesthatdescribethebehaviorofsingle(individual)physicalsystems,2.ThestatisticalinterpretationI2accordingtowhichtheproductofthestan-darddeviationsoftwocanonicallyconjugatevariableshasalowerboundgivenbyh/4π.’1Busch&Lahti—TheStandardModelofQuantumMeasurementTheory2Hethenproceedstoshowthatunderthemeasurement-theoreticassumptionofthemeasurementsinvolvedbeingofthefirstkind,thatis,repeatable,I1isaconsequenceofI2.TheChapterconcludeswiththeobservation‘thatcertainmoderndevelopmentshaveclearedthewayfortheestablishmentoftheoriesonthesimultaneousmeasurementofincompatibleobservables’,butitispointedoutthat‘littleconsensushasbeenreachedontheverybasicissuesofsuchtheories,primarily,itseems,becauseofdivergingdefinitionsof“simultaneousmeasurements.”Itiscertainlytooearlytoformabalancedjudgementonthelegitimacyandprospectsofsuchtheories.’Sincethetimewhentheselineswerewritten(around1974)muchhashappenedintheoreticalaswellasexperimentalrespects.Experimentally,therehavebeenavarietyofillustrationsofthescatterrelation,forexample,bymeansofdiffractionexperimentsinvolvingbeamsofphotons,electrons,orneutrons.Ontheconceptualside,developmentsthathadstartedinthemid-1960shaveledtoawiderconceptofobservableswhichallowstheformulationofjointobservablesforcollectionsofnon-commutingquantities.Quiteindependently,measurement-theoreticalmodelsweredevelopedwhichwereintuitivelyinterpretedassimultaneousposition-momentummeasurements.Foralongtime,boththeconceptualinnovationsandtheconcretemodelsremainedlargelyunnoticed,mainly,becausetheirintimaterelationshipwasnotmadeexplicit.Infactanewandwidermathematicalrepresentationofobserv-ableswouldhavetobejustifiedintermsof–atleastmodel–applications;andthemodelsofjointmeasurementsdidnotallowforanyreasonableexplanationintermsoftheconventionalquantummechanicalformalism.Meanwhilethesituationhaschangedcompletelyinthatthemodelshavebeensatisfactorilylinkedwiththenewobservableconcept,andsearchhasbegunforpossibleexperimentalrealisationsofthecorresponding‘simultaneous’,orjointmeasurements.Itwasacertaintypeofmeasurementmodelthatplayedanessentialroleinthesedevelopments:inthosemodelstheobservable-to-be-measuredenterstheinter-actionHamiltonianasafactor,multipliedbyanobservableofsomeprobesystem.Inviewofitsimportance,weproposetonamethissortofmodelthestandardmodelofquantummeasurementtheory.Itisthepurposeofthiscontributiontosurvey2Busch&Lahti—TheStandardModelofQuantumMeasurementTheory3themainfeaturesofthestandardmodelandtoshowhowitleads,innaturalways,totherepresentationofobservablesaspositiveoperatorvaluedmeasuresaswellastorealisationsofjointmeasurementsofnoncommutingobservables.2.Abriefhistoryofthestandardmodel.Jammer’sobservationofI2implyingI1showsthatatheoryofsimultaneousmeasurementsofcanonicallyconjugate,ormoregeneralnoncommutingpairsofob-servablesmustbebasedonnon-repeatablemeasurements.WeobservethatlogicallythisgivesroomfortheconsiderationofapositivereformulationofthenonstatisticalinterpretationI1into3.TheindividualinterpretationI3accordingtowhichitispossible,inprinciple,tospecifyunsharplythesimultaneousvaluesofcanonicallyconjugatevariablesthatdescribethebehaviorofsingle(individual)physicalsystems.Thetwofoldtaskisthusdefinedto(1)introduceappropriatenotionsofjointobservableandjointmeasurementfornoncommutingquantitiesinaccordancewiththeprobabilisticstructureofquantummechanics,and(2)demonstratetheprincipalrealizabilityofthesenotionsbyformulatingconcretejointmeasurementmodels.Insections3and4weshallemploythestandardmodeltodemonstratehowthesegoalscanbeandindeedhavebeenreached.Beforehandweshallrecallsomeinstancesinthedevelopmentofqu