Properties of probability measures on the set of q

arXiv:math-ph/0607019v23Aug2006Propertiesofprobabilitymeasuresonthesetofquantumstatesandtheirapplications.M.E.Shirokov∗1IntroductionThenotionofensembleasasetofstateswithcorrespondingsetofprobabil-itiesiswidelyusedinquantuminformationtheory.Inparticular,importantcharacteristicssuchastheHolevocapacityofaquantumchannelandtheentanglementofformationofaquantumstatearedefinedbyoptimizationoftheparticularfunctionalsdependingonensembleofstates[10].Anensembleofquantumstatescanbeconsideredasanatomicprobabilitymeasureonthesetofallstates,whichatomscorrespondtothestatesoftheensemble.So,itisnaturaltoconsideranarbitraryBorelprobabilitymeasureonthesetofallstatesasageneralizedensemble.Thispointofviewisespeciallyusefulindealingwithinfinitedimensionalquantumchannelsandsystemssinceinthiscasetherearenoreasonsforexistenceofensemblescalledoptimal,atwhichextremaofseveralimportantfunctionalsareachieved,butundersomeconditionsitispossibletoshowexistenceofoptimalmeasures[7].Moreover,byconsideringprobabilitymeasuresasgeneralizedensemblesitispossibletoproveresults,whichformallyhavenorelationstoprobabilitymeasures[18].Theadvantageofthisapproachisbasedonapplicationofgeneralresultsofthetheoryofprobabilitymeasuresoncompleteseparablemetricspaces[6],[14].InthispapersomeobservationsconcerningtheChoquetordering[15]onthesetofallprobabilitymeasuresonthesetofquantumstatesareconsidered(proposition1,corollary1andlemma3).Theyimply,inparticular,thatarbitrarymeasuressupportedbypurestatescanbeweaklyapproximatedby∗SteklovMathematicalInstitute,119991Moscow,Russia,e-mail:[e-mail]1asequenceofatomicmeasuressupportedbypurestatesandhavingthesamebarycenter(corollary2).Theimportantpropertyoffiniteensemblesofquantumstatesprovedin[17](lemma3)canbeexpressedfigurativelyspeakingasfollows:anarbitrarycontinuousdeformationoftheaveragestateofanyfiniteensemblecanbere-alizedbyappropriatecontinuousdeformationofthestatesoftheensembleandtheirweights.1Thispropertyisthebasicpointoftheproofoflowersemicontinuityoftheχ-functionofanarbitraryquantumchannel(proposi-tion3in[17]).Inthispaperweshowthatitimpliestheopennesspropertiesofthemapping,whichassociateswithaprobabilitymeasurethebarycenterofthismeasure(propositions2and3).Thesepropertiesandthecompactnesscriterionforsubsetsofmeasuresobtainedin[7]resultininterestingobserva-tionsonpropertiesoffunctionsonthesetofquantumstates(theorems1,2,propositions4,5,corollaries8,10).Inparticular,itisshownthateverycontinuousboundedfunctiononthesetofquantumstateshascontinuousboundedconvexclosure(proposition4).Itisalsoshownthateverycontinuousboundedfunctiononthesetofpurequantumstateshasconvex(concave)continuousboundedextensiononthesetofallstateshavingtheparticularminimality(maximality)property(proposition5).Theabovegeneralobservationshaveseveralapplicationstoquantumin-formationtheory(corollaries6,7and9).Inparticular,theyprovideanec-essaryandsufficientconditionofboundednessandcontinuityoftheconvexclosureoftheoutputentropyofaquantumchannel(corollary9).Thesere-sultscanalsobeusedforconstructionofcontinuousboundedcharacteristicsofquantumstatesastheaboveconvex(concave)extensionsofcontinuousboundedfunctionsdefinedonthesetofpurestates.Asanexample,weconsidertheconstructionofquasimeasureofentanglement,whichisacon-tinuousboundedfunctiononthewholestatespaceofainfinitedimensionalbipartitesystemcloselyrelatedtotheentanglementofformation(remark3).In[18]theopenproblemofcoincidingoftwodefinitionsoftheentan-glementofformationofastateininfinitedimensionalbipartitesystemisdiscussed.Inthispaperweconstructanexampleshowingthatthisproblemcannotbesolvedbyusingonlysimpleanalyticalpropertiesofthequantumentropy(remark2andthenotebelow).1EveninR3thereexistconvexsetsforwhichtheanalogueofthisassertionisnotvalid.22PreliminariesLetHbeaseparableHilbertspace,B(H)-thesetofallboundedoperatorsinHwiththeconeB+(H)ofallpositiveoperators,T(H)-theBanachspaceofalltrace-classoperatorswiththetracenormk·k1andS(H)-theclosedconvexsubsetofT(H)consistingofallpositiveoperatorswiththeunittrace-densityoperatorsinH,whichiscompleteseparablemetricspacewiththemetricdefinedbythetracenorm.EachdensityoperatoruniquelydefinesanormalstateonB(H)[5],so,inwhatfollowswewillalsoforbrevityusetheterm”state”.WedenotebycoA(coA)theconvexhull(closure)ofasetAandbycof(cof)theconvexhull(closure)ofafunctionf[13].WedenotebyextAthesetofallextremepointsofaconvexsetA.LetPbethesetofallBorelprobabilitymeasuresonS(H)endowedwiththetopologyofweakconvergence[6],[14].SinceS(H)isacompleteseparablemetricspacePisacompleteseparablemetricspaceaswell[14].LetbPbetheclosedsubsetofPconsistingofallmeasuressupportedbytheclosedsetextS(H)ofallpurestates.ThebarycenterofthemeasureμisthestatedefinedbytheBochnerintegral¯ρ(μ)=ZS(H)σμ(dσ).ForarbitrarysubsetAofS(H)letPA(corresp.bPA)bethesubsetofP(corresp.bP)consistingofmeasureswiththebarycenterinA.ByusingProkhorov’stheorem[16]thefollowingcompactnesscriterionforsubsetsofPisestablishedin[7](proposition2):ThesetPAiscompactifandonlyifthesetAiscompact.Itfollows,inparticular,thatthesubsetP{ρ}ofP,consistingofallmeasureswiththeb