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arXiv:0801.4604v1[quant-ph]30Jan2008QuantuminformationwithGaussianstatesXiang-BinWang1DepartmentofPhysics,TsingghuaUniversity,Beijing100084,China;andImaiQuantumComputationandInformationProject,ERATO-SORST,JST,DainiHongoWhiteBldg.201,5-28-3,Hongo,Bunkyo-ku,Tokyo113-0033,JapanTohyaHiroshima2,AkihisaTomita,MasahitoHayashiImaiQuantumComputationandInformationProject,ERATO-SORST,JST,DainiHongoWhiteBldg.201,5-28-3,Hongo,Bunkyo-ku,Tokyo113-0033,JapanAbstractQuantumopticalGaussianstatesareatypeofimportantrobustquantumstateswhicharemanipulatablebytheexistingtechnologies.Sofar,mostoftheimpor-tantquantuminformationexperimentsaredonewithsuchstates,includingbrightGaussianlightandweakGaussianlight.Extendingtheexistingresultsofquantuminformationwithdiscretequantumstatestothecaseofcontinuousvariablequan-tumstatesisaninterestingtheoreticaljob.ThequantumGaussianstatesplayacentralroleinsuchacase.WereviewthepropertiesandapplicationsofGaussianstatesinquantuminformationwithemphasisonthefundamentalconcepts,thecalculationtechniquesandtheeffectsofimperfectionsofthereal-lifeexperimentalsetups.TopicshereincludetheelementarypropertiesofGaussianstatesandrelevantquantuminformationdevice,entanglement-basedquantumtaskssuchasquantumteleportation,quantumcryptographywithweakandstrongGaussianstatesandthequantumchannelcapacity,mathematicaltheoryofquantumentanglementandstateestimationforGaussianstates.Contents1Introduction31.1Elementsofquantuminformationwith2-levelstates61.2PhasespacerepresentationanddefinitionofGaussianstates91.3Coherentstates151Email:xbwang@mail.tsinghua.edu.cn2Email:tohya@qci.jst.go.jpPreprintsubmittedtoElsevierScience3February20081.4Squeezedstates181.5Beam-splitter231.6Beam-splitterasanentangler312Entanglement-basedquantumtasks332.1Teleportationwithtwo-levelstates332.2CVQTwithGaussianstates362.3Experiment402.4Densecodingwithtwo-modesqueezedstates422.5Quantumerrorcorrectioncodes432.6Gaussiancloningtransformation462.7Entanglement-basedquantumtaskswithweakGaussianstates:post-selectionvsnon-post-selection483Quantumcryptographywithweakcoherentlight543.1Introduction543.2QKDinpractice553.3Eavesdroppingandsecurity673.4Ideasofunconditionalsecurityproof714SecurityproofsandprotocolsofQKDwithweakandstrongGaussianstates734.1EntanglementpurificationandsecurityproofofQKD734.2Securekeydistillationwithaknownfractionoftaggedbits814.3Decoy-statemethod864.4SARG04protocol994.5QKDinposition-momentumspace1014.6SecurityproofofQKDwithsqueezedstates1045MathematicaltheoryofquantumentanglementwithGaussianstates10925.1GeneralPropertiesofQuantumEntanglement1105.2EntanglementPropertiesofGaussianStates1125.3Conclusions1276ClassicalcapacitiesofGaussianchannels1286.1ClassicalCapacitiesofQuantumChannels1286.2GaussianChannels1306.3GaussianChannelswithGaussianInputs1366.4DenseCodingwithGaussianEntanglement1406.5Entanglementmeasure1427EstimationtheoryforGaussianstates1427.1Informationquantities1437.2Measurementtheory1457.3Formulationofestimation1497.4Independentandidenticalcondition1507.5Bayesianmethod1517.6Groupcovariantmethod1537.7Unbiasedmethod1547.8Simplehypothesistesting1568Acknowledgement158References1581IntroductionQuantuminformationprocessing(QIP)isasubjectoninformationprocessingwithquantumstates[1].Intherecentyears,thesubjecthasattractedmuchattentionofscientistsfromvariousareas.Ithasbeenfoundthatinsomeimpor-tantcases,quantuminformationprocessingcanhavegreatadvantagetoanyknownmethodinclassicalinformationprocessing.Aquantumcomputercan3factorizealargenumberexponentiallymoreefficientlythantheexistingclas-sicalmethodsdo[2].Thismeans,givenaquantumcomputer,thewidelyusedRSAsysteminclassicalcommunicationisinsecurebecauseonecanfactorizeahugenumberveryeffectivelybyShor’salhorithm.Interestingly,quantumkeydistribution(QKD)canhelptworemotepartiessharearandombinarystringwhichisinprincipleunknowntoanythirdparty[3].PrivatecommunicationbasedonQKDisprovensecureunderwhatevereavesdroppingincludingquan-tumcomputing.Quantumteleportationcantransferunknownquantumstatetoaremotepartywithoutmovingthephysicalsystemitself[4].Inclassicalinformationprocessing,allinformationarecarriedbyclassicalbits,whicharebinarydigitsofeither0or1.Thephysicalcarrierofaclassicalbitcanbeanyphysicalquantitythathastwodifferentvalues,e.g.,theelectri-calpotential(positiveornegativevoltages).Thesearemacroscopicquantitieswhichcanbemanipulatedrobustlybyourexistingtechnology.Inquantuminformationprocessing,weusequantumstatestocarryeitherquantuminfor-mationorclassicalinformation.Also,weusequantumentangledstatesastheresourcetoassisttheeffectiveprocessingofquantuminformation.Inprinci-ple,lotsofdifferentphysicalsystemscanbeusedtogeneratetherequestedquantumstatesandquantumentanglement.Inthosetasksrelatedtocom-munication,lightseemstobethebestcandidateforthephysicalsystemtocarrythequantuminformationand/orthequantumentanglementduetoitsobviousadvantagethatitcanbetransmittedoveralongdistanceefficiently.Thereforeonemaynaturallyconsidertouseasingle-photonstateasaquan-tumbit(qubit)andatwo-photonentangledstateastheentanglementresourceto