A study of Wigner functions for discrete-time quan

arXiv:1204.1050v1[quant-ph]4Apr2012AstudyofWignerfunctionsfordiscrete-timequantumwalksM.Hinarejos1,M.C.Bañuls2andA.Pérez1¹DepartamentdeFísicaTeòricaandIFIC,UniversitatdeValència-CSICDr.Moliner50,46100-Burjassot,Spainand2Max-Planck-InstitutfürQuantenoptik,Hans-Kopfermann-Str.1,Garching,D-85748,Germany.AbstractWeperformasystematicstudyofthediscretetimeQuantumWalkononedimensionusingWignerfunctions,whicharegeneralizedtoincludethechirality(orcoin)degreeoffreedom.Inparticular,weanalyzetheevolutionofthenegativevolumeinphasespace,asafunctionoftime,fordifferentinitialstates.Thisnegativitycanbeusedtoquantifythedegreeofdepartureofthesystemfromaclassicalstate.Wealsorelatethisquantitytotheentanglementbetweenthecoinandwalkersubspaces.1I.INTRODUCTIONQuantumWalks(QW)areconsideredaasapieceofpotentialimportanceinthedesignofquantumalgorithms[1–5],asitisthecaseofclassicalrandomwalksintraditionalcomputerscience.Asinthecaseofrandomwalks,QW’scanappearbothunderitsdiscrete-time[6]orcontinuous-time[7]form.Moreover,ithasbeenshownthatanyquantumalgorithmcanberecastundertheformofaQWonacertaingraph:QWscanbeusedforuniversalquantumcomputation,thisbeingprovableforboththecontinuous[8]andthediscreteversion[9].ExperimentshavebeendesignedoralreadyperformedtoimplementtheQW[10–19].Inthispaper,weconcentrateonthediscrete-timeQWonaline.Weperformasys-tematicstudymakinguseofWignerfunctions,whicharedefinedforthisproblem.Wignerfunctions[20,21]wereintroducedasanalternativedescriptionofquantumstates.Theyplayanimportantroleinquantummechanics,havingbeenwidelyusedinquantumopticstovisualizelightstates.Fromtheexperimentalpointofview,theyprovideawayforquantumstatereconstructionviatomographyandinverseRadontransformation[22].Wignerfunctionsarequasi-probabilitydistributionsinphasespace,meaningthattheycannotbeinterpretedasaprobabilitymeasureinmomentumandspaceconfigurations.Thisisanobviousfactforanyquantumdescription,andonlymarginaldistributionscanbeassociatedtoprobabilitiesinpositionormomentum(oranylinearcombination,i.e.anyquadrature).Infact,Wignerfunctionscantakenegativevalues,thusinvalidatingadirectlinktoaprobabilitydistribution.Thiscaveat,however,turnsouttobeapotentialadvantage,foritcanbeusedtoidentify“true”quantumstates.Moreprecisely,thevolumeofthenegativepartoftheWignerfunction,itsnegativity,hasbeensuggestedasafigureofmerittoquantifythedegreeofnon-classicality[23].Thisideahasbeenrecentlyexploited[24]todirectlyestimatingnonclassicalityofastatebymeasuringitsdistancefromtheclosestonewithapositiveWignerfunction.WhendealingwiththediscreteQW,onehastoaccountfortheextradegreeoffreedom(inadditiontothespatialmotion):thecoin.Weconsiderthesimplestcaseofatwolevelcoin.Therefore,theWignerfunctionhastoincorporatethisextraindexand,withtheprescriptionweuse,itturnsintoamatrix.WewillproposearatherstraightforwardextendeddefinitionofnegativityforthisWigner“function”.Thenthequestionarises,whatkindofstatesoftheQWarenonclassical?Doesthisquantumnessincreaseintime,asthe2QWevolvesthroughitsunitaryevolution?WewanttoexplorethesequestionsusingtheWignerfunction.Adifferenttopic,althoughitiscloselyrelatedtothepreviousone,iswhetherthisnonclassicalitycanberelatedtotheentanglementbetweenthewalkerandthecoin,sincethisquantityisalsoevolvingduringtheQWevolution.Thispaperisorganizedasfollows.InSect.IIwereviewthemaindefinitionspertinenttotheQWonaline.Sect.IIIintroducestheWignerfunctionforourproblemandthemainassociatedproperties.WepresentsomeexamplesshowingournumericalresultsfortheWignerfunctionevolutioninSect.IV.InSect.VwedefineanextensionofthenegativitytotheQW,basedontheproposalmadein[23]forascalarfunction.WeendinSect.VIbysummarizingourmainresultsandconclusions.II.DISCRETE-TIMEQWWALKONALATTICEThediscrete-timeQWonalineisdefinedastheevolutionofaone-dimensionalquantumsystemfollowingadirectionwhichdependsonanadditionaldegreeoffreedom,thecoin(orchirality),withtwopossiblestates:“left”|Lior“right”|Ri.ThetotalHilbertspaceofthesystemisthetensorproductHs⊗Hc,whereHsistheHilbertspaceassociatedtothelattice,andHcisthecoinHilbertspace.LetuscallT−(T+)theoperatorsinHsthatmovethewalkeronesitetotheleft(right),and|LihL|,|RihR|thechiralityprojectoroperatorsinHc.TheQWisdefinedbyU(θ)=T−⊗|LihL|C(θ)+T+⊗|RihR|C(θ),(1)whereC(θ)=σzcosθ+σxsinθ,andσz,σxarePaulimatricesactingonHc.Forθ=π/4theoperatorC(θ)becomestheHadamardtransformation.TheunitaryoperatorU(θ)transformsthestateinonetimestepas|ψ(t+1)i=U(θ)|ψ(t)i.(2)Thestateattimetcanbeexpressedasthespinor|ψ(t)i=∞Xn=−∞an(t)bn(t)|ni,(3)wheretheupper(lower)componentisassociatedtotheright(left)chirality,and{|ni/n∈Z}isabasisofpositionstatesonthelattice.AbasisinthewholeHilbertspacecanbeconstructedasthesetofstates{|n,αi=|ni⊗|αi/n∈Z;α=L,R}.3III.WIGNERFUNCTIONSFORTHEQUANTUMWALKAnimportanttoolinsomefieldsrelatedtoquantumphysicsistheuseofquasi-probabilitydistributions.Wignerfunctionsconstitutethemajorexample,althoughotherfunctionsastheGlauber-SudarshanPfunction[25,26]ortheHusimiQfunction[27]arecommonlyusedinquantumoptics.Forthecaseofaone-dimensionalsystemwithcontinuouspositionxandconjugatemomentump,theW