A volume inequality for quantum Fisher information

arXiv:0706.0791v1[math-ph]6Jun2007AvolumeinequalityforquantumFisherinformationandtheuncertaintyprinciplePaoloGibilisco∗,DanieleImparato†andTommasoIsola‡February1,2008AbstractLetA1,...,ANbecomplexself-adjointmatricesandletρbeadensitymatrix.TheRobertsonuncertaintyprincipledet{Covρ(Ah,Aj)}≥det−i2Tr(ρ[Ah,Aj])ffgivesaboundforthequantumgeneralizedcovarianceintermsofthecommutators[Ah,Aj].Therightsidematrixisantisymmetricandthereforetheboundistrivial(equaltozero)intheoddcaseN=2m+1.Letfbeanarbitrarynormalizedsymmetricoperatormonotonefunctionandleth·,·iρ,fbetheassociatedquantumFisherinformation.Inthispaperweconjecturetheinequalitydet{Covρ(Ah,Aj)}≥detf(0)2hi[ρ,Ah],i[ρ,Aj]iρ,fffthatgivesanon-trivialboundforanyN∈Nusingthecommutatorsi[ρ,Ah].TheinequalityhasbeenprovedinthecasesN=1,2bythejointeffortsofmanyauthors(seetheIntroduction).Inthispaperweprovethe(real)caseN=3.2000MathematicsSubjectClassification.Primary62B10,94A17;Secondary46L30,46L60.Keywordsandphrases.Generalizedvariance,uncertaintyprinciple,operatormonotonefunctions,matrixmeans,quantumFisherinformation.1IntroductionLet(V,g(·,·))bearealinner-productvectorspaceandsupposethatv1,...,vN∈V.TherealN×NmatrixG:={g(vh,vj)}ispositivesemidefiniteandonecandefineVolg(v1,...,vN):=pdet{g(vh,vj)}.Iftheinnerproductdependsonafurtherparameterinsuchawaythatg(·,·)=gρ(·,·),wewriteVolg(v1,...,vN)=Volgρ(v1,...,vN).Asanexample,consideraprobabilityspace(Ω,G,ρ)andletV=L2R(Ω,G,ρ)bethespaceofsquareintegrablerealrandomvariablesendowedwiththescalarproductgivenbythecovarianceCovρ(A,B):=Eρ(AB)−Eρ(A)Eρ(B).ForA1,...,AN∈L2R(Ω,G,ρ),GisthewellknowncovariancematrixandonehasVolCovρ(A1,...,AN)≥0.(1.1)Theexpressiondet{Covρ(Ah,Aj}isknownasthegeneralizedvarianceoftherandomvector(A1,...,AN)and,ingeneral,onecannotexpectastrongerinequality.Forinstance,whenN=1,(1.1)justreducestoVarρ(A)≥0.∗DipartimentoSEFEMEQ,Facolt`adiEconomia,Universit`adiRoma“TorVergata”,ViaColumbia2,00133Rome,Italy.Email:[e-mail]–URL:†DipartimentodiMatematica,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129Turin,Italy.Email:[e-mail]‡DipartimentodiMatematica,Universit`adiRoma“TorVergata”,ViadellaRicercaScientifica,00133Rome,Italy.Email:[e-mail]–URL:∼isola12Innon-commutativeprobabilitythesituationisquitedifferentduetothepossiblenon-trivialityofthecommutators[Ah,Aj].LetMn,sa:=Mn,sa(C)bethespaceofalln×nself-adjointmatricesandletD1nbethesetofstrictlypositivedensitymatrices(faithfulstates).ForA,B∈Mn,saandρ∈D1ndefinethe(symmetrized)covarianceasCovρ(A,B):=1/2[Tr(ρAB)+Tr(ρBA)]−Tr(ρA)·Tr(ρB).IfA1,...,ANareself-adjointmatricesonehasVolCovρ(A1,...,AN)≥(0,N=2m+1,det{−i2Tr(ρ[Ah,Aj])}12,N=2m.(1.2)Letuscall(1.2)the“standard”uncertaintyprincipletodistinguishitfromotherinequalitieslikethe“entropic”uncertaintyprincipleandsimilarinequalities.Inequality(1.2)isduetoHeisenberg,Kennard,RobertsonandSchr¨odingerforN=2(see[14][16][28][30]).ThegeneralcaseisduetoRobertson(see[29]).Examplesofrecentreferenceswhereinequality(1.2)playsarolearegivenby[31][32][33][4][3][15].Supposeoneislookingforageneralinequalityoftype(1.2)givingaboundalsointheoddcaseN=2m+1.IfoneconsidersthecaseN=1,itisnaturaltoseeksuchaninequalityintermsofthecommutators[ρ,Ah].Oneofthepurposesofthepresentpaperistostateaconjectureregardinganinequalitysimilarto(1.2)butnottrivialforanyN∈N.LetFopbethefamilyofsymmetricnormalizedoperatormonotonefunctions.Toeachelementf∈Foponemayassociateaρ-dependingscalarproducth·,·iρ,fontheself-adjoint(traceless)matrices,whichisaquantumversionoftheFisherinformation(see[25]).LetusdenotetheassociatedvolumebyVolfρ.WeconjecturethatforanyN∈N+(thisisoneofthemaindifferencesfrom(1.2))andforarbitraryself-adjointmatricesA1,...,ANonehasVolCovρ(A1,...,AN)≥f(0)2N2Volfρ(i[ρ,A1],...,i[ρ,AN]).(1.3)ThecasesN=1,2ofinequality(1.3)havebeenprovedbythejointeffortsofanumberofauthorsinseveralpapers:S.Luo,Q.Zhang,Z.Zhang([19][20][24][22][23]);H.Kosaki([17]);K.Yanagi,S.Furuichi,K.Kuriyama(see[34]);F.Hansen([13]);P.Gibilisco,D.Imparato,T.Isola([11][6]).Inthispaperwediscusstheinequality(1.3)whenN=3andproveitintherealcaseandforsomeclassesofcomplexself-adjointmatrices(includingPauliandgeneralizedGell-Mannmatrices).ItiswellknownthatstandarduncertaintyprincipleisasimpleconsequenceoftheCauchy-SchwartzinequalityforN=2.Itisworthtonotethatfortheinequality(1.3)thesameroleisplayedbytheKubo-Andoinequality2(A−1+B−1)−1≤mf(A,B)≤12(A+B)sayingthatanyoperatormeanislargerthantheharmonicmeanandsmallerthanthearithmeticmean.Theschemeofthepaperisasfollows.InSection2wedescribethepreliminarynotionsofoperatormonotonefunctions,matrixmeansandquantumFisherinformation.InSection3wediscussacorre-spondencebetweenregularandnon-regularoperatormonotonefunctionsthatisneededinthesequel.InSection4westateourconjecture,namelytheinequality(1.3);wealsostateothertwoconjecturesconcerninghowtherightsidedependsonf∈Fopandtheconditionstohaveequalityin(1.3).InSection5wediscussthecaseN=1of(1.3)presentingthedifferentavailable