Gaussian states in continuous variable quantum inf

arXiv:quant-ph/0503237v131Mar2005PrefaceGaussianstatesincontinuousvariablequantuminformationAlessandroFerraro,StefanoOlivares,MatteoG.A.ParisPublishedasGaussianstatesinquantuminformationISBN88-7088-483-X(Bibliopolis,Napoli,2005)(MGAP)attheUniversityofNapoliandtheUniversityofMilano.Aquitebroadsetofissuesarecovered,rangingfromelementaryconceptstocurrentresearchtopics,andfromfundamentalconceptstoapplications.AspecialemphasishasbeengiventothephasespaceanalysisofquantumdynamicsandtotheroleofGaussianstatesincontinuousvariablequantuminformation.WethankGiuseppeMarmoforhisinvitationtowritetheselecturenotesandforhiskindassistanceinthevariousstagesofthisproject.MGAPwouldliketothankMauroD’Arianofortheexcitingintroductionhegavemetothisfields,andRodolfoBonifacio,whogavemethepossibilityofestablishingaresearchgroupattheUniversityofMilano.MGAPalsothanksMariaBondaniandAlbertoPorzioforthecontinuingdiscussionsonquantumopticsovertheseyears.Manycolleaguescontributedinseveralwaystothematerialsinthisvolume.InparticularwethankAlessioSerafini,NicolaPiovella,MaryCola,AndreaRossi,FabrizioIlluminati,KonradBanaszek,SalvatoreSolimeno,VirginiaD’Auria,SilvioDeSiena,AlessandraAndreoni,AlessiaAllevi,EmilianoPuddu,AntoninoChiummo,PaoloPerinotti,LorenzoMaccone,PaoloLoPresti,MassimilianoSacchi,JardaˇReh´aˇcek,BergeEnglert,PaoloTombesi,DavidVitali,StefanoMancini,GezaGiedke,JaromirFiur´aˇsekandValentinaDeRenzi.AspecialthanktoAlessioSerafiniforhiscarefulreadingandcommentsonvariousportionsofthemanuscript.Oneofus(SO)wouldliketorememberhereafriend,MarioPorta:myworkduringtheseyearsisalsoduetoyourexampleinfrontofthedifficultiesoflife.Milano,December2004AlessandroFerraroStefanoOlivaresMatteoGAParisContentsPrefaceListofsymbolsiiiIntroductionv1Preliminarynotions11.1Systemsmadeofnbosons......................................11.2Matrixnotationsforbipartitesystems................................31.3Symplectictransformations......................................31.4Linearandbilinearinteractionsofmodes..............................41.4.1Displacementoperator....................................51.4.2Two-modemixing......................................81.4.3Single-modesqueezing....................................91.4.4Two-modesqueezing.....................................101.4.5Multimodeinteractions:SU(p,q)Hamiltonians.......................111.5CharacteristicfunctionandWignerfunction.............................121.5.1Traceruleinthephasespace.................................141.5.2Aremarkaboutparametersκ.................................142Gaussianstates172.1Definitionandgeneralproperties...................................172.2Single-modeGaussianstates.....................................192.3Bipartitesystems...........................................192.4Tripartitesystems...........................................233SeparabilityofGaussianstates253.1Bipartitepurestates..........................................253.2Bipartitemixedstates.........................................263.3Tripartitestates............................................304Gaussianstatesinnoisychannels334.1MasterequationandFokker-Planckequation............................334.1.1Single-modeGaussianstatesinnoisychannels.......................334.1.2n-modeGaussianstatesinnoisychannels..........................354.2Gaussiannoise............................................364.3Single-modeGaussianstates.....................................364.3.1Evolutionofpurity......................................374.3.2Evolutionofnonclassicality.................................384.4Two-modeGaussianstates......................................394.4.1Separabilitythresholds....................................394.5Three-modeGaussianstates.....................................405Quantummeasurementsoncontinuousvariablesystems415.1ObservablesandPOVM.......................................415.2Momentgeneratingfunction.....................................425.3Directdetection............................................425.3.1Photocounting........................................43ii5.3.2On/offphotodetectors....................................445.4Application:de-Gaussificationbyvacuumremoval.........................455.4.1De-GaussificationofTWB:theIPSmap...........................465.4.2De-Gaussificationoftripartitestate:theTWBAstate....................485.5Homodynedetection.........................................495.5.1Balancedhomodynedetection................................495.5.2Unbalancedhomodynedetection...............................515.5.3Quantumhomodynetomography..............................525.6Two-modeentangledmeasurements.................................535.6.1Double-homodynedetector..................................535.6.2Heterodynedetector.....................................545.6.3Six-porthomodynedetector.................................565.6.4Outputstatisticsfromatwo-photocurrentdevice......................5