From the Foundations of Quantum Theory to Quantum

FromtheFoundationsofQuantumTheorytoQuantumTehnology{anIntrodutionGernotAlberNowadays,thenewtehnologialprospetsofproessingquantuminforma-tioninquantumryptography[1℄,quantumomputation[2℄andquantumommuniation[3℄attratnotonlyphysiistsbutalsoresearhersfromothersientiommunities,mainlyomputersientists,disretemathematiiansandeletrialengineers.Currentdevelopmentsdemonstratethatharater-istiquantumphenomenawhihappeartobesurprisingfromthepointofviewoflassialphysismayenableonetoperformtasksofpratialinterestbetterthanbyanyotherknownmethod.Inquantumryptography,theno-loningpropertyofquantumstates[4℄orthephenomenonofentanglement[5℄helpsintheexhangeofseretkeysbetweenvariousparties,thusen-suringtheseurityofone-time-padryptosystems[6℄.Quantumparallelism[7℄,whihreliesonquantuminterfereneandwhihtypiallyalsoinvolvesentanglement[8℄,maybeexploitedforaeleratingomputations.QuantumalgorithmsareevenapableoffatorizingnumbersmoreeÆientlythananyknownlassialmethodis[9℄,thushallengingtheseurityofpubli-keyryp-tosystemssuhastheRSAsystem[6℄.Classialinformationandquantuminformationbasedonentangledquantumsystemsanbeusedforquantumommuniationpurposessuhasteleportingquantumstates[10,11℄.Owingtosigniantexperimentaladvanes,methodsforproessingquan-tuminformationhavedevelopedrapidlyduringthelastfewyears.1Basiquantumommuniationshemeshavebeenrealizedwithphotons[10,11℄,andbasiquantumlogialoperationshavebeendemonstratedwithtrappedions[13,14℄andwithnulearspinsoforganimoleules[15℄.Also,avityquantumeletrodynamialsetups[16℄,atomhips[17℄,ultraoldatomsinoptiallatties[18,19℄,ionsinanarrayofmirotraps[20℄andsolid-statedevies[21{23℄arepromisingphysialsystemsforfuturedevelopmentsinthisresearharea.Allthesetehnologiallyoriented,urrentdevelopmentsrelyonfundamentalquantumphenomena,suhasquantuminterferene,themeasurementproessandentanglement.Thesephenomenaandtheirdistin-tivedierenesfrombasioneptsoflassialphysishavealwaysbeenofentralinterestinresearhonthefoundationsofquantumtheory.However,inemphasizingtheirtehnologialpotential,theadvanesinquantuminfor-1Numerousreentexperimentalandtheoretialahievementsaredisussedin[12℄.2GernotAlbermationproessingreetaprofoundhangeinthegeneralattitudetowardsthesefundamentalphenomena.Thus,afteralmosttwodeadesofimpressivesientiahievements,itistimetoretraesomeofthosesigniantearlyde-velopmentsinquantumphysiswhihareattheheartofquantumtehnologyandwhihhaveshapeditspresent-dayappearane.1EarlyDevelopmentsManyoftheurrentmethodsanddevelopmentsintheproessingofquantuminformationhavegrownoutofalongstruggleofphysiistswiththefoun-dationsofmodernquantumtheory.ThefamousonsiderationsbyEinstein,PodolskyandRosen(EPR)[24℄onreality,loalityandompletenessofphys-ialtheoriesareanearlyexampleinthisrespet.Theritialquestionsraisedbytheseauthorsinspiredmanyresearherstostudyquantitativelytheessen-tialdierenebetweenquantumphysisandthelassialoneptsofrealityandloality.ThebreakthroughwasthedisoverybyJ.S.Bell[25℄thatthestatistialorrelationsofentangledquantumstatesareinompatiblewiththepreditionsofanytheorywhihisbasedontheoneptsofrealityandlo-alityofEPR.Theonstraintsimposedonstatistialorrelationswithintheframeworkofaloal,realistitheory(LRT)areexpressedbyBell’sinequality[25℄.Astheoneptofentanglementanditspeuliarorrelationpropertieshavebeenoffundamentalsignianeforthedevelopmentofquantuminfor-mationproessing,itisworthreallingsomeofitsmostelementaryfeaturesinmoredetail.1.1EntanglementandLoal,RealistiTheoriesInordertolarifytheharateristidierenesbetweenquantummehan-ialorrelationsoriginatingfromentangledstatesandlassialorrelationsoriginatingfromloal,realistitheories,letusonsiderthefollowingbasiexperimentalsetup(Fig.1).Aquantummehanialtwo-partilesystem,suhasaphotonpair,isproduedbyasoures.Polarizationpropertiesofeahβiαi2+1+1ABs-1-1ϕϕϕαβαβ112Fig.1.BasiexperimentalsetupfortestingBell’sinequality;thehoiesofthediretionsofpolarizationontheBlohsphereforoptimalviolationoftheCHSHinequality(3)orrespondto’==4forspin-1=2systemsFromtheFoundationsofQuantumTheorytoQuantumTehnology3ofthesepartilesaremeasuredsubsequentlybytwodistantobserversAandB.ObserversAandBperformpolarizationmeasurementsbyrandomlyse-letingoneofthediretions1or2,and1or2,respetively,ineahexperiment.Furthermore,letusassumethatforeahofthesediretionsonlytwomeasurementresultsarepossible,namely+1or1.Intheaseofpho-tonsthesemeasurementresultswouldorrespondtohorizontalorvertialpolarization.WhataretherestritionsimposedonorrelationsofthemeasurementresultsifthephysialproessanbedesribedbyanunderlyingLRTwithunknown(hidden)parameters?Forthispurpose,letusrstofallsummarizetheminimalsetofonditionsanyLRTshouldfulll.1.Thestateofthetwo-partilesystemisdetermineduniquelybyaparame-ter,whihmaydenoteanarbitrarysetofdisreteorontinuouslabels.ThusthemostgeneralobservableofobserverAorBfortheexperimentalsetupdepitedinFig.1isafuntionofthevariables(i;j;).Iftheatualvalueoftheparameterisunknown(hidden),thestateofthetwo-partilesystemhastobedesribedbyanormalizedprobabilitydis-tributionP(),i.e.RdP()=1,whereharaterizesthesetofallpossiblestates.Thes