The resource theory of quantum reference frames ma

arXiv:0711.0043v2[quant-ph]18Mar2008Theresourcetheoryofquantumreferenceframes:manipulationsandmonotonesGiladGour∗InstituteforQuantumInformationScienceandDepartmentofMathematicsandStatistics,UniversityofCalgary,2500UniversityDriveNW,Calgary,Alberta,CanadaT2N1N4RobertW.Spekkens†DepartmentofAppliedMathematicsandTheoreticalPhysics,UniversityofCambridge,CambridgeCB30WA,UnitedKingdom(Dated:Oct.4,2007)Everyrestrictiononquantumoperationsdefinesaresourcetheory,determininghowquantumstatesthatcannotbepreparedundertherestrictionmaybemanipulatedandusedtocircumventtherestriction.Asuperselectionruleisarestrictionthatarisesthroughthelackofaclassicalreferenceframeandthestatesthatcircumventit(theresource)arequantumreferenceframes.Weconsidertheresourcetheoriesthatarisefromthreetypesofsuperselectionrule,associatedrespectivelywithlacking:(i)aphasereference,(ii)aframeforchirality,and(iii)aframeforspatialorientation.Focussingonpureunipartitequantumstates(andinsomecasesrestrictingourattentionevenfurthertosubsetsofthese),weexploresingle-copyandasymptoticmanipulations.Inparticular,weidentifythenecessaryandsufficientconditionsforadeterministictransformationbetweentworesourcestatestobepossibleand,whentheseconditionsarenotmet,themaximumprobabilitywithwhichthetransformationcanbeachieved.Wealsodeterminewhenaparticulartransformationcanbeachievedreversiblyinthelimitofarbitrarilymanycopiesandfindthemaximumrateofconversion.Acomparisonofthethreeresourcetheoriesdemonstratesthattheextenttowhichresourcescanbeinterconverteddecreasesasthestrengthoftherestrictionincreases.Alongtheway,weintroduceseveralmeasuresofframenessandprovethatthesearemonotonicallynonincreasingundervariousclassesofoperationsthatarepermittedbythesuperselectionrule.ContentsI.Introduction2A.Summaryofmainresults41.Z2-SSR52.U(1)-SSR53.SU(2)-SSR6II.Therestrictionoflackingareferenceframe7A.Preliminaries7B.KrausrepresentationforG-invariantoperations8C.Framenessmonotones9D.Themotivationforrequiringmonotonicity10E.Single-copyframemanipulations11F.Asymptoticframemanipulationsandtheuniqueasymptoticmeasureofframeness11III.ResourcetheoryoftheU(1)-SSR12A.Phasereferences12B.U(1)-invariantoperations131.U(1)-invariantunitaries13C.Deterministicsingle-copytransformations14D.Stochasticsingle-copytransformations151.Necessaryandsufficientconditions152.Maximumprobability16E.StochasticU(1)-framenessmonotones16∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]2-framenessmonotones22E.Stochasticsingle-copytransformations24F.StochasticZ2-framenessmonotones24G.Asymptotictransformations24V.ResourcetheoryoftheSU(2)-SSR26A.Framesfororientation26B.SU(2)-invariantoperations271.SU(2)-invariantunitaries28C.ArestrictedsetofSU(2)framestates281.Proofoftheexistenceofinequivalentclasseswithintherestrictedset28D.Restrictingtoafixedquantizationaxis30E.Deterministicsingle-copytransformations31F.Stochasticsingle-copytransformations31G.StochasticSU(2)-framenessmonotones31H.Asymptotictransformations31VI.Conclusions35References362I.INTRODUCTIONForeveryinterestingrestrictiononoperations,thereisaresultingresourcetheory[1].Forinstance,there-strictionoflocaloperationsandclassicalcommunication(LOCC)leadstothetheoryofentanglement.AgainstthebackdropoftheLOCCrestriction(andanimplicitrestrictionthatthepartiesdonotshareanyentanglementattheoutset),asingleentangledpairorasingleuseofanoiselessquantumchannelarebothconsideredresources,onestatic,theotherdynamic[2].Aresourcetheoryspec-ifiesthemannerinwhichonecaninter-convertbetweenvariousresources,forinstance,whetheroneentangledstatecanbetransformedbyLOCCintoanother.In-deed,muchofquantuminformationtheoryissimplyatheoryoftheinter-conversionbetweenresources[3].Wearehereinterestedintherestrictionofasuperselectionrule(SSR).Specifically,weimagineapartythatisre-strictedtooperationsthatareinvariantundertheactionofagroupGandrefertothisasasuperselectionruleforGorsimplyaG-SSR.AlthoughSSRsareoftencon-sideredtobeaxiomatic,itisbettertoconsiderthemasarisingfrompracticalrestrictions.Indeed,mostSSRscanbeliftedifonehasaccesstoareferenceframeforthegroupinquestion,sotherestrictionisultimatelyoneofaccesstoanappropriatereferenceframe[4,5,6].Inthiscontext,theanalogueofanentangledstate–thatwhichcanbeusedtotemporarilyovercometheLOCCrestric-tion–isastatethatcanbeusedtotemporarilyovercometherestrictionoftheSSR.Suchstateshavebeenreferredtoasbounded-sizereferenceframesorsimplyquantumreferenceframes.Inthisarticle,westudythemannerinwhichonecaninterconvertbetweensuchstatesundertheSSR.Wearethereforeexploringtheresourcetheoryofquantumreferenceframes.Justaswesaythatastateisentangledorhasen-tanglementifitcannotbepreparedbyLOCC,onecansaythatastateisG-asymmetricorhasnonzeroG-framenessifitcannotbepreparedbyG-invariantopera-tions.Oneofthegoalsofthisarticleistoprovideopera-tionalmeasuresofframeness.Theminimalrequirementonsucha