Acknowledgment
Overtheyears,Ihavehadthechancetointeractwithmanycolleaguesonthetopicof nonlocality(thosewhohavealsoprovidedpreciousfeedbackondraftsofthisbookare markedbya∗ ).
Twopeopledeserveaveryspecialmention:AntoineSuarezandNicolasGisin∗ . AntoineintroducedmetothisfieldwhileIwasanundergraduatefascinatedbychaos andfractals(alreadyrandomness,inasense).ForvariousreasonsIembarkedina Ph.D.onNuclearMagneticResonance(NMR)experimentsonmagneticnanostructures:1 Whileencouragingmetocarryittogoodcompletion,Antoineaskedmetojoin himinwritingdownhisideaofbefore-beforeexperiment.Laterheintroducedmeto Nicolas.Afteralltheseyears,Iamstillamazedathisundimmeddrivetowardsunityof knowledge.
Nicolasdaredtohiremeasapostdocinquantuminformationtheory,notwithstanding thatcompletelyunrelatedPh.D.topic.Whenhiringme,heleftitclear:“Aveclanonlocalité onnegagnepassavie”.Indeed,IwouldnotbewhereIamifIhadnotworkedalsoin cryptography.Afewyearslater,wheninaboutofindependenceItoldhimthatIwould resignfromhisgroup,hehadtheclassofstayingpatient,insteadoffiringmeonthespot asIwouldhavedeserved.Thisgavemetimetoreconsiderthatsillydecision,getbackto apositivespirit,andresumework,intimetobepartofthepioneeringworkondeviceindependentcertification.Nicolashasjustofficiallyretired,butIamsurethatneitherhis brainnorhiscomputerwillnoticethedifference.
Theothers ... ifIweretotellallthestories,I’dhavetowriteanotherbook,soI’lljust listthemhereinalphabeticalorder:AntonioAcín∗ ,MafaldaAlmeida,RotemArnonFriedmann∗ ,AlainAspect,Jean-DanielBancal∗ ,JonathanBarrett,CyrilBranciard∗ , GillesBrassard, ˇ CaslavBrukner,NicolasBrunner∗ ,JeffreyBub,HarryBuhrman, FrancescoBuscemi∗ ,AdánCabello∗ ,DanielCavalcanti∗ ,GiulioChiribella,Andrea Coladangelo∗ ,RogerColbeck,DanielCollins,MicheleDall’Arno,MauroD’Ariano, GonzalodelaTorre,ArturEkert∗ ,BergeEnglert,ChrisFuchs,OtfriedGuhne,Michael Hall∗ ,LucienHardy,Paweł Horodecki,JedKaniewski∗ ,DagomirKaszlikowski,Adrian Kent,BarbaraKraus,ChristianKurtsiefer,JoséLatorre,Yeong-CherngLiang∗ ,Noah Linden,AlexanderLing,NorbertL ¨ utkenhaus,LluísMasanes,SergeMassar,Matthew McKague,AndréMéthot,MicheleMosca,MiguelNavascués∗ ,TomaszPaterek,Marcin Pawłowski,PaoloPerinotti,StefanoPironio∗ ,MartinPlesch,SanduPopescu,John Preskill,RenatoRenner∗ ,GrégoireRibordy,AnaBelénSainz∗ ,NicolasSangouard,
1 Enpassant,thankstomyPh.D.supervisorJean-PhilippeAnsermetforbeingalwaysencouragingto someonewhohadobviouslylandedatthewrongplace.
x Acknowledgment
R ¨ udigerSchack,RomanSchmied,YaoyunShi,TonyShort,JamieSikora,Christoph Simon,PaulSkrzypczyk∗ ,RobSpekkens,AndréStefanov,SébastienTanzilli,Nicholas Teh,WolfgangTittel,PhilippTreutlein,AntoniosVarvitsiotis∗ ,UmeshVazirani,Tamás Vértesi,ThomasVidick∗ ,PaoloVilloresi,StephanieWehner,ReinhardWerner,Howard Wiseman,AndreasWinter,MichaelWolf,StefanWolf,ElieWolfe,HenryYuen,Hugo Zbinden,AntonZeilinger,Marek ˙ Zukowski.
SinceIbuiltmyownresearchgroup,Ihaveworkedwithmanyexcellentstudents andresearchers.Iproudlykeepthefulllistinthegroup’swebpage.Thosewhodeserve tobementionedinrelationwithnonlocalityare,againinalphabeticalorder:2 CharlesEdouardBardyn,YuCai,WanCong,BenjaminGoh,KoonTongGoh,YunguangHan, AsaphHo,MelvynHo,YunZhiLaw,ThinhLe,ZhiXianLee,XinhuiLi,CharlesLim, RafaelRabelo,LanaSheridan,ErnestTan,YukunWang,XingyaoWu,TzyhHaurYang, andYuqianZhou.Whenfinishingthisbook,Idecidedtoputittothetestandproposed agraduatemodulebasedonit.Amongthestudentswhotookit,specialthankstoAdrian NugrahaUtama,IgnatiusWilliamPrimaatmaja,AngelineShu,andNoahVanHornefor feedbackanddiscussions.
Afterspeakingofthefuture,aglancetothepast.Amongmymanyteachers,Iwantto singleoutAlbertoTorresani,whotaughtmephilosophyinLiceoScientificoArgonne, andFran¸coisReuse,whoseopenedmyeyestoaproperunderstandingofquantum theoryduringmyyearsinEcolePolytechniqueFédéraledeLausanne(EPFL).
Myfamilyandclosestfriendswillhardlyunderstandthetitleofthisbook,letalone itscontent.Butwithouttheirsupport,Iwouldnothavebeenabletocompletethetask.
2 CharlesandRafaelhavesincebecomescolleagues,butarementionedinthislistbecausetheystartedin mygroup.ThisisalsothereasonwhyDanielandJean-Danielwerementionedpreviously:Whentheycameto workwithme,theyhadalreadyacquiredagreatcompetenceinnonlocality.
6.1Device-IndependentCertification:AFirstIntroduction
D.1Overview:LookingintoMeasurements
G.1AnalyticalSolutionoftheFacetsoftheCHSHCorrelationPolytope194
G.2Hardy’sTestfromtheSchmidtDecompositionoftheState195
G.3PitfallsinHandlingSignalingBehaviors197
G.4Jordan’sLemma198
G.5ACaseStudyofRandomnesswithCharacterisedDevices199
G.6SimulationsoftheSingletBehavior201
G.7PropertiesoftheVariationalDistance203
G.8InformationCausalityfromDesiderataonInformationEntropies204
FirstEncounterwithBellNonlocality
MisunderstandingsofBell’stheoremhappensofastthattheyviolatelocality.
R.Munroe, XKCD
Thischapterservesbothasanintroductiontothebook,andasaself-containedfirst presentationofBellnonlocality.
1.1ThreeRolesforBellNonlocality
Fewscientificstatementsaremoreradicalthanoneofthecoretenetsofquantumphysics: Thereisindeterminacyinnature.Ithasaccompaniedquantumtheorysinceitsearliest moments:OnlyafewmonthsafterHeisenbergandSchrödingerindependentlydefined thedefinitiveformalism,MaxBornsuggestedthatthelawsofthenewtheoryshouldbe seenasintrinsicallystatistical.Thiswastobecometheorthodoxview.Sensingthedanger, EinsteinquicklywrotetoBornhisconvictionthatatheorywithstatisticallawscouldonly beatemporaryfix,andthatdeterminismshouldultimatelyberecovered.Thedebate continuedfordecadeswithafewflares,notablythecelebratedEPRpaper(Einstein Podolsky,andRosen,1935)andBohr’simmediatereply,butinanatmosphereofoverall indifferenceamongphysicistsatlarge.Inthoseyears,theexcitementaboutquantum theorywasnotfoundindebatingitsmeaning,butinitsalmostboundlesspredictive power.IthasbecomecommonplacetorefertotheattitudeofthoseyearsbyMermin’s dictum“shutupandcalculate.”
Ultimately,thestatisticallanguagebecamethestandardtowhichgenerationsof physicistsconformedoutofinertia.Ifaskedforevidenceofindeterminacy,stilltoday manywouldrefertoHeisenberg’suncertaintyrelations,thathowevercanonlyvoicefor indeterminacyinquantumtheory,notinnature(seeAppendixA.1).Thisissurprising becausedirectevidencehasbeencompellingsince1964,thankstotheworkofJohn Bell(Bell,1964).Heshowedthatthepossibilityofrecoveringadeterministicmodelis amenabletoexperimentalfalsification,throughtheobservationofaphenomenonthat weshallcall Bellnonlocality.Inafirstapproach,Bell’sargumentismathematicallysimple (seesections1.3–1.4);becauseofitsimportance,ithasbeensubmittedtoathorough scrutiny,fromwhichithasemergedunscathedandactuallystrengthenedbymoresolid foundations(seesection1.5andchapter2).
In1964therewasalreadyahugeamountofexperimentalevidencesupportingthe validityofquantumtheory.Nevertheless,noneofthosedatacouldbeusedtocheck Bell’scriterion.Dedicatedexperimentshadtobedesigned.TheworkofAlainAspect andcoworkersiscreditedasthefirstconclusiveevidenceofBellnonlocality(Aspect etal.1982b;Aspect etal.1982a).Theevidencehasbeensteadilygrowingsincethen; eventually,threeindependentexperimentsreportedin2015areconsidereddefinitive (Hensen etal.,2015;Giustina etal.,2015;Shalm etal.,2015).Themaintextofthis bookdoesnotdescribeexperiments;tofacilitatereadingtheexperimentalliterature,a quickguideisprovidedasAppendixB.
Discoveredthankstoquantumtheory,indeterminacyhasbeenvindicatedasa physicalfact,independentofthetheoryitself.Itcanbecircumventedonlyattheprice ofadoptingevenmoreradicalposturesaboutphysicsandnaturethemselves(seesection 1.6). ThisdirectvindicationofindeterminismistheoriginalmotivationofBellnonlocality. Forafewdecades,itwasheldtobeitssoleroletoo.Thosewho,forvariousreasons,were alreadywontotheindeterministiccausehadtakennoteofitandmovedon.Aseriesof worksthatstartedaround2005haveuncoveredasecondrole: Bellnonlocalityprovides themostcompellingcertificationofthecorrectfunctioningofsomequantumdevices,likethose requiredtoperformquantumcryptographyandquantumcomputation.Thefabrication ofthesedevicesandthedevelopmentofcertificationtoolsbasedonnonlocalitystill constitutetechnicalchallenges,butwe’llhavetogetthere.Farfrombeinganexercise inscientificarchaeology,thisbookcontainsmaterialthatfuturequantumengineerswill havetomaster—innuce atleast,thisisatreatiseinappliedphysics.
Finally,asaphenomenonindependentofquantumtheory,Bellnonlocalityisnot merelyaninstrumentforanegativetask(falsifyingdeterminism):Ithasarightto citizenshipinphysics.Asitsthirdrole, Bellnonlocalitycanbeusedasaprincipleconstraining possiblecandidatesforphysicaltheories.Barringafewpioneeringinsights,thisapproach wasalsostartedaftertheyear2000.Ithasalreadycontributedseveralnewideasand notionstothefieldoffoundationsofphysicsbutisstillveryopentofuturedevelopments.
ThesethreerolesofBellnonlocality—evidenceforindeterminism,certificationtool fordevices,andguidelineforfoundations—correspondtothethreepartsintowhichthis bookisdivided,butpervadethewholetext.Withtheminmind,wecanenterthecore ofthesubject.
1.2IntroducingBellNonlocality
1.2.1SettingBelltests:Laboratoriesandgames
TestsofBellnonlocality,or Belltests forshort,arecurrently experimentalsetupsinphysics laboratories.Forsomeyears,theoristshaveratherchosentopresentBelltestsas games thatbearsomeanalogywithTVquizzes,polls,exams,judicialtrials,andotherfamiliar situations.1 Thegamesettingisdefinitelybettertobringuptheessenceofnonlocality,
1 ThereadermaycomebacktothislistafterbecomingfamiliarwithBelltest,tofindanalogiesand differences.Forinstance,inexams,thecontentoftheanswermatters,andtheverifierwillevaluatethe
Figure1.1 SketchofaBelltestfortwoplayers(thegeneralizationtomoreplayersisstraight-forward) inthe laboratorysetting (top)andinthe gamesetting (bottom).Afterhavingagreedonaprocessfor thatround,eachplayerreceivesaninputandhastoprovideanoutput;thedataofseveralroundsare thensortedtoestablishthecorrelationsbetweentheoutputsaandbforanypair(x,y)ofinputs.Inthe laboratorysetting,wegiveingreytheusualrepresentationoftheprocessinquantumtheory:Aquantum state ρ ispreparedandsomemeasurementsarechosen;whichmeasurementisactuallyperformedin eachroundisdeterminedbytheinput.NoneofthisentersthedefinitionofBellnonlocality.Inthegame setting,weintroduceaverifierVthatqueriestheplayersAliceandBobandcollectstheiranswers.
andweshallmostlyfollowitinthisbook.Nonetheless,asweshallalsosee,many importantdiscussionscannotbefullyappreciatedwithoutgoingbacktothelab.The twosettingsaresketchedandcomparedinFigure1.1.
InaBelltestgame,the players,referredtoalphabeticallyasAlice,Bob,Charlie,etc.,are allonthesameteam.Thegameconsistsofmany rounds.Ineachround,theplayerswillbe separated:Eachwillreceiveaquery(input )andwillhavetoprovideananswer(output ). Itisusefultothinkofa verifier distributingtheinputs2 andcollectingtheoutputs.
Therulesofthegameandthelistofpossiblequeriesareknowninadvance.The playersareallowedtoprepareacommon strategy beforethegame,whichconsistsin decidingwhich process theywilluseineachroundofthegame.Weshallalsospeakof the resources thatareusedintheseprocesses.Iftheplayerswereallowedtocommunicate amongeachotherduringthegame,theywouldactuallynotbeseparatedandcouldeasily winanygameofthistype:Themostpowerfulresourcesare signaling ones.Thecase ismoreinterestingwith no-signalingresources.Themostelementaryexampleofanosignalingresourceisalistofpre-determinedoutputs,oneforeachpossibleinput(that is,theprocessconsistsinproducingtheoutputbyreadingthelist).Itisno-signaling performanceofeachplayerwithoutanyconcernforcorrelations.Injudicialtrials,theplayers’goalistoprovidea consistentversionofthestory(whichmaynotbethetruth),buttheydon’tknowinadvancethesetofquestions thattheymaybeasked;etc.
2 Inactualexperiments,theinputsareusuallygeneratedateachplayer’slocationbya“randomnumber generator”:Theimageoftheverifierallowsustopostponethedelicatediscussionaboutrandomnessandits generationwithphysicalmeans(subsection1.5.3).
FirstEncounterwithBellNonlocality
because,ifAlicedoessomethingtoherlist,theotherplayersobviouslywon’tnotice anything.Inotherwords,Alicecan’tsendamessagetoothersbymanipulatingherlist. Forinstance,considerthreegames,eachdefinedbyoneofthefollowingrules:
(i)Theplayersmustproducethesameansweriftheyreceivethesamequery.
(ii)Theplayersmustproducethesameanswerifandonlyiftheyreceivethesame query.
(iii)Theplayersmustproducedifferentanswersifbothreceivequery“1,”thesame answerotherwise.
Agamebasedonrule(i)istriviallywonbytheplayersagreeingonafixedcommon output.Agamebasedonrulenumber(ii)canbesimilarlywonbyagreeingonapredeterminedoutputforeachinput,providedthatthenumberofinputsisnotlargerthan thenumberofoutputs.Ifthereweremoreinputsthanoutputs,thegamecannotbe wonwithalistofpre-determinedanswers.Finally,nostrategybasedonpre-determined answerscanwinagamebasedonrule(iii).
1.2.2ThedefinitionofBellnonlocality
Belllocality meansthat theprocessbywhicheachplayergeneratestheoutputdoesnottake intoaccounttheotherplayer’sinput.Inotherwords,allcorrelationsbetweentheplayers’ outputsisduetothesharedresource,onwhosenaturenoassumptionismade:Itcanbe anything,fromalistofnumbersonapieceofpapertotwojointlyprogrammedquantum computers.WhenBelllocalitydoesnotholdwespeakof Bellnonlocality.
Thisnotionoflocalitycanbeformalizedasfollows.Denoteby λ theprocess.It doesnotneedtobedeterministic,sowecansaythatAlicegenerates a bysampling fromaprobabilitydistribution P λ (a|x).WhatiscrucialisthatthisdoesnottakeBob’s input y intoaccount.Similarly,Bobgenerates b locallybysamplingfromaprobability distribution P λ (b|y).Ifthisisthecase,thestatisticsobservedbytheverifier(whoisnot privyto λ)willbedescribedby
where Q(λ)istheprobabilitydistributionthatdescribesthestrategy,i.e.,howoften aspecificprocess λ isused.Belllocalityisclearlyarestriction:Notallconceivable P (a, b|x, y)canbewritteninthisform.Theextremecounterexampleisastrategythat winsthegameinwhicheachplayerissupposedtooutputtheotherplayer’sinput.Also, anywinningstrategyforthegamebasedonrule(iii)requiresoneoftheplayerstosample fromadistributionthatdependsontheotherplayer’sinput.
Statisticswillbecalled local iftheycanbewrittenintheform(1.1), nonlocal if theycannot.A Belltest isagamewhosewinningstrategyisdescribedbynonlocal statistics.
1.2.3Onresourcesandsemantics
Ifnonlocalstatisticsareobserved,theverifierknowsthattheplayershaveshareda nonlocalresource.Iflocalstatisticsareobserved,wecan’tsaymuchabouttheresource: Theplayersmighthavesharedapotentiallynonlocalonebuthaveuseditpoorly.This soundslikeelementarylogic,butittriggerstwocrucialremarks:
• ThedefinitionofBelllocalitydoesnotrelyonapriorcharacterizationofthe classof“localresources”;evenlessoneneedstoassumethatthere exist innature resourcesthatareintrinsicallylocalinthissense.3 Theoppositeisthecase:Fromthe definitionofBelllocality,thatstandsitsground,onecan define “localresources”as hypotheticalresourcesthatcouldonlyleadtolocalstatistics.Thetraditionalname forsuchlocalresourcesis“localhiddenvariables” (LHVs) orsimply“localvariables” (LVs),theword“hidden”beingarelicofthediscussionsonquantumtheory.
• We’llseeinchapter2thateverylocalstatistic(1.1)canevenberealizedwitha strategybasedonpre-determinedoutputs.Thisresult,knownasFine’stheorem, isthebasisforthemathematicaltoolsofthefield.Butwecannotinferfrom thistheoremthatallobservedlocalbehaviorsareactuallygeneratedwithpredeterminedoutputs,northatthedefinitionofBelllocalityassumesdeterminism.
Now,wereitnotforquantumtheory,thedefinitionofBellnonlocalitywouldsound bothuninterestinganduncontroversial:Nonlocalresourceswouldbecommunication devices.Quantumtheory4 howeverforcesustoenlarge,atleastinprinciple,thelistof possiblenonlocalresources.Letusthenconsiderthattheplayers sharephysicalsystems inastatethatquantumtheorydescribesas ρ AB ,andlet’sassumethattheprocessthat producestheoutputsis performinglocalmeasurements onthisstate.Specifically,upon receivingherinput x,Aliceperformsameasurementonhersystem,withtheoutput a ofthatmeasurementisassociatedtothepositiveoperator x a .Bobactssimilarly.After severalrounds,allplayedwiththisprocess,quantumtheorypredictsthatthestatistics collectedbytheverifieraregivenby
Ingeneral,thesestatisticscannotbecastintheform(1.1):Thisisthecontentof Bell’s theorem (Bell,1964).Explicitexampleswillfillthisbook,butforthetimebeingletus acceptthat somesharedquantumstatesarenonlocalresources.However,itisalsowell
3 Inthesamevein,notwithstandingthefrequentreplacementof“local”with“classical”inthefield’sjargon, thedefinitionofBellnonlocalitydoesnotrelyonadefinitionofclassicality,andevenlessonassumingthe existenceofintrinsicallyclassicalphysicalsystems.Overall,weshallavoidspeakingofclassical/quantumsystems orphenomena.Itiscorrecttospeakofclassicaltheoryandquantumtheory,becausethesearewell-defined(see AppendixC.1.1fortheessentials).Itisalsocustomarytospeakofclassical/quantum information torefertothe resources,insofarasdescribedwithineachtheory,butwewon’tdoit.
4 Familiaritywithelementaryquantumtheoryisgivenforgrantedinthisbook;moreadvancedtopicsand specificaspectsofquantuminformationtheoryaresummarizedinAppendixC.
known5 thatsharedquantumstates arenotcommunicationchannels:Byactingonlyon hersystem,AlicecannotlearnanythingaboutwhatBobhasdonewithhis—hecould havemeasuredit,keptit,discardedit,andAlicedoesnotseeanychangeinherstatistics. Inthissense,quantumstatesare no-signalingresources justassharedlistsofnumbers. Bell nonlocalityisinterestingandintriguingbecauseitcanbedemonstratedbysharingno-signaling resources
Orcanit?Famously(ornotoriously),quantumtheorydoesnotprovideanyrecipefor thegenerationofeachround’soutput.Woulditbepossiblethatwhatquantumtheory describesasno-signalingresourcesareactuallysignalingones?Einsteindubbedthis possibility“spookyactionatadistance”.Asweshallseeinsection1.6,itisonepossible interpretation.Amongthosewhoopposeit,somethinkthatthewording“nonlocality” evokestoocloselythisunwelcomeinterpretation.Whatwecalled“locality”insubsection 1.2.2,they’drathercall localrealism or localcausality.Theseareelegantexpressionswith philosophicalappeal:Theyremindusthatwearenotmerelydealingwithoperations andobservations,butwithaprejudiceinour Weltanschauung thathasbeenshattered. However,theyarealsonotexemptfromthedangerofbeingover-interpreted.6
Withalltheirpotentiallimitations,thewordings“nonlocality,”“localrealism,”and “local(hidden)variables”havealreadyenjoyedafewdecadesoftraditionandaremost probablyheretostay.IhopeIhavesaidenoughtopreventtheirmisuse,andIshalluse themfreely.Asforinterpretations,weshallreturntotheminsection1.6.
1.3MyFirstBellTest:Clauser-Horne-Shimony-Holt(CHSH)
Toputthesegeneralconsiderationsonconcretegrounds,weproceedtodescribesome specificexamplesofBelltests.Forthisintroductorychapter,Ihavechosentopresent fiveclassicexamples:Oneinthissectionandfourinthenext;severalotherswillbe presentedlaterinthebook.AnelementaryproofthateachisindeedaBelltestisgiven, exploitingFine’stheorem(subsection2.3.3)thatallowsconsideringonlystrategiesbased onpre-establishedanswers.Therelevanceofeachtestforthecertificationofquantum entanglementismerelystated,leavingallthecalculationsforchapters3–5.
5 Evenifthisshouldbeelementaryknowledge,giventhecentralityoftheclaimforthecontentofthisbook, theexplicitproofisgiveninAppendixC.1.3.
6 Ithasbecomecommonplacetosplittheprejudiceof“localrealism/causality”intotwoseparateprejudices, “locality”and“realism”(or“causality”).Tobeatpeacewiththefactofaviolation,itwouldthenbeenough toabandoneither. Abandoninglocality maylegitimatelymeansignaling:In(1.1),onewouldhave P (a|x, y, λ), P (b|x, y, λ),orboth;andthismodificationisindeedsufficienttogenerateBellnonlocality.Butthemeaning of abandoningrealism/causality isbyfarlessclear(Norsen,2007;Gisin,2012).Itcannotmean“abandoning determinism”:Determinismisnotassumedin(1.1),soabandoningitdoesnotgenerateBellnonlocality. Neithershoulditmean“abandoninganyconnectionwithreality,”reducingphysicstounfoundedspeculations: Ifweabandon“localrealism”it’sbecauseweaccepttheverdictofobservation.Probably,“abandoning realism/causality”isawayofsayingthatonlystatisticsarespeakable,seesubsection1.6.3.Butthen,this alternativeisatadifferentlevelthansignaling:Itisnotamechanism,butthestatementthatnomechanism shouldbelookedfor.
MyFirstBellTest:Clauser-Horne-Shimony-Holt(CHSH) 9
ItshouldbeobviousthataBelltestrequires atleasttwoplayers,otherwisethereis nonotionoflocality.Foreachplayer,theremustbe atleasttwopossibleinputs:Ifsome playerscouldonlyreceiveonequery,thoseinputswouldbeknowntotheotherplayers. Finally,foreachinput,theremustbe atleasttwopossiblevaluesfortheoutput.Westart withthissimplestscenario.
TheinputsofAlicearelabeled x ∈{0,1},heroutputs ax ∈{−1, +1} (labelingisof coursearbitrary,thischoiceisconvenientforthecalculationtocome).Theinputsof Bobarelabeled y ∈{0,1},hisoutputs by ∈{−1, +1}
TheruleofthegameprescribesthatAliceandBobshouldaimatgivingthesame answerwhenever(x, y) ∈ (0,0),(0,1),(1,0),butoppositeanswerswhen(x, y) = (1,1). Weconsiderthescore
wheretheaverageistakenoveranarbitrarilylargenumberofrounds.Themaximalscore isobviously S = 4.
ToprovethatthisgameisaBelltest,weneedtofindwhatscorecanbeachieved withlocalresources.InvokingFine’stheorem,itisenoughtoseewhathappenswhen AliceandBobhavesharedapre-determinedquadruple(a0 , a1 ; b0 , b1 )ineachround. Theexistenceofthesefournumbersentailstheexistenceofawell-definedvalueforthe derivedquantity s = a0 b0 + a
1 + a1 b0 a1 b1 .Sincetheaverageofasumisthesum oftheaverages, S = s holds.Now,foreveryquadruple,either s =+2or s =−2.This isreadilyseenbyrewriting s = a0 (b0 + b1 ) + a1 (b0 b1 ):Indeed,if b0 = b1 thesecond termiszero,if b0 =−b1 thefirsttermiszero.InTable1.1welistexplicitlyallsixteen possibilities:Eightofthemgive s =+2andtheothereight s =−2.Noticehowthreeout offourpairsofinputscontributeinthesameway,butthelastpairpullsthesumdown (orupifitwasnegative).Thisobservationwillbecomehandylater.
Ineachround,theverifierseesonlythepair(ax , by )correspondingtotheinputs(x, y) hehassent,sohecannotestimate s.However,byperformingstatisticsconditionalon eachpairofinputs,hecanestimatethefour ax by andobtain S.Now,if S = s and eachinstanceof s canonlytakethevalues ± 2,itfollowsthat
|S | LV ≤ 2.(1.4)
Supposenowthattheverifierfinds S > 2:Hewillhavetoadmitthattheplayerswerenot sharingpre-establishedquadruples,noranyresourcethatcanbesimulatedwiththem— inotherwords,hewillhavetoadmitthat theyshareanonlocalresource. ThisBelltestiscalledCHSHfrom(Clauser,Horne,Shimony,andHolt,1969). Noticehowthemathematicalexpressionofthetestisan inequality,hereEq.(1.4).The playerswillconvincetheverifierthattheyhaveanonlocalresourceiftheymanageto violatetheinequality.JohnBell’soriginalinequality(Bell,1964)isbasicallythesameas(1.4), butderivedundertheassumptionthatoneofthe ax by isexactlyequalto1(Appendix A.4).Sinceperfectcorrelationscanbepredictedbutcannotbeobserved,thatinequality
FirstEncounterwithBellNonlocality
Table1.1 Allsixteenquadruplesofpre-establishedvalues,andderivativequantities, fortheCHSHtest.Inboldface,thetermthatpullsthesumsintheoppositedirection astheotherthree.Noticethatthesecondhalfofthetableisthemirrorimageofthe firsthalf,sinceflippingallthesignsdoesnotchangetheproducts.
issufficienttoprovethatquantumtheorypredictsnonlocalresourcesbutisuntestablein experiments.
TheCHSHtestistheworkhorseofthefield,we’llstudyitingreatdetail.Let’sjust mentionthatthemaximalscore S = 4canbereachedofcoursebysignaling,7 butalso withahypotheticalno-signalingresourcecalleda PR-box (PopescuandRohrlich,1994) thatwe’llencounterinchapter9.However,PR-boxesexistinmathematicsbutdon’tseem toexistinnature:Withquantumentanglement,themaximalscoreis S = 2√2 ≈ 2.8284
7 Thoughratherobvious,hereisonepossiblewayinwhichtheplayerscanscore S = 4withsignaling.Alice andBobagreeonabit a = b.Uponbeingqueried,Aliceoutputs a andsends x toBob;Boboutputs( 1)xy b NoticethatAlice’sanswerispre-determined,butBob’sisnot:Hehastowaitfor x beforeproducingit.So,the conclusionthatbothoutputscouldnothavebeenpre-determinedholds.
Thisvaluecanbeachievedbysuitablemeasurementsonthemaximallyentangledstate oftwoqubits8
andinfact,inasensetobemadepreciseinchapter7, only bythatstateandthose measurements.
1.4FourmoreClassicBellTests
ThissectionintroducesfourotherexamplesofBellteststhatshouldhelpgainingfurther familiaritywiththesenotions.
1.4.1Mermin’soutreachcriterion
InhisefforttoexplainBellnonlocalityinasimpleway,DavidMermin(1981)conceived aBelltestthathasbecomepopular.Therearetwoplayers,eachwiththreeinputs (x, y ∈{1,2,3})andtwooutputs(a, b ∈{+1, 1}).Let’sassumethattheobservation showsthat ai = bi inallcaseswherethesameinputwaschosen.Weareinterestedin O = x,y P (ax = by ) = 3 + x =y P (ax = by ).Iftheoutputsarepre-determined,(a1 , a2 , a3 ) = (b1 , b2 , b3 )cantakeeightvalues,namely(+1, +1, +1),(+1, +1, 1),etc.For (+1, +1, +1)and( 1, 1, 1),onefinds O = 9;theothersixtriplesgive O = 5.Thus, certainly O ≥ 5forlocalresources.Quantumtheorypredictsthatonecangodownto O = 4.5.Inparticular,notevenwithquantumresourcesonecanwinperfectlythegame basedonrule(ii)definedinsubsection1.2.1,becausethatwouldcorrespondto O = 3. Theinequality O ≥ 5definesaBelltestonlyif ai = bi .Ifthisassumptionisremoved, theinequalitymaybeviolatedwithLVs.Forinstance,thechoiceofpre-determined outputs(a1 , a2 , a3 ) = (+1, +1, +1)and(b1 , b2 , b3 ) = ( 1, 1, 1)gives O = 0.Thisis thesameweaknessofBell’soriginalcriterion,whichhadtobetransformedintoCHSH tobecomerobust.RobustBelltestsfortwoplayers,threeinputsandtwooutputswillbe discussedinsection4.2.
1.4.2Greenberger-Horne-Zeilinger(GHZ)test
TheoriginalnonlocalitytestbyDanielGreenberger,MichaelHorne,andAnton Zeilinger(1989)considersfourplayers,buttheargumentcanbemadeforanynumber
8 Thenotation |0 |0 standfor |0 ⊗ |0 .Hereandintherestofthebook,thetensorproductsymbolis usuallyimplicitwhenwritingquantumstates,whileitisoftenexplicitwhenwritingoperators.Ifindthatthis isthechoicethatfacilitatesthereading.
FirstEncounterwithBellNonlocality
ofplayerslargerthantwo.Nowadays,werefertothethree-playersversionasthe“GHZ test”withoutfurtherqualifiers(Mermin,1990b);thisistheonewepresenthere.As intheCHSHcase,eachplayerhastwoinputsandtwooutputs,thoseofCharliebeing labeled z and c.Assumenowthattheverifierobservesthefollowingperfectcorrelations:
Then,
Indeed, ax by cz =+1meansthat ax by cz =+1ineachsingleround.Iftheoutputsare pre-determined,theymustbesuchthat
1ineachround.By multiplyingthethreeconditionsandnoticingthat
Inquantumtheory,however,onecanhavethecorrelations(1.6)alongside
Thisisobtainedwithsuitablemeasurementsonthestate
and,justasforCHSH,thisquantumrealizationisunique(seechapter7).
Aspresented,theGHZtestreliesontheobservationofperfectcorrelationsoranticorrelations.Tocopewithunavoidableimperfectsituations,itsscorecanbeturnedinto theso-called Mermininequality:
Theproofthatinequality(1.10)holdsisleftasExercise1.1.Thus,contrarytowhat happenswithCHSH,themaximalscore M = 4oftheMermininequalitycaninprinciple beattainedinquantumtheory.Weshallcompletethestudyofthisinequalityandits generalizationsformorepartiesinsection5.2.
1.4.3Hardy’stest
Atthispoint,onemayaskifonecanbuildaBelltestfor two playersonextreme correlationsthatquantumresourcescaninprincipleachieve.Thereareindeedsuch examples.ThefirstonewasfoundbyLucienHardy(1992;1993).Theextreme probabilitiesthatareenforcedare:
P (a0 =−1, b1 =+1) = 0(1.12) P (a1 =+1, b0 =−1) = 0.(1.13)
Thefollowinginferencesarethenobvious:
• From(1.12): b1 =+1implies a0 =+1
• From(1.11): a0 =+1implies b0 =−1
• From(1.13): b0 =−1implies a1 =−1.
Byenchainingthesethreeinferencesweareledtoafourthone,namely“b1 =+1implies a1 =−1,”andthusinparticulartotheprediction
(a1 =+1, b1 =+1) LV = 0.(1.14)
AnotherwayofreachingthesameconclusionissuggestedinExercise1.2.
Insubsection4.4.1,weshallseethatinquantumtheoryonemayhavethethree constraints(1.11)–(1.13)andnonetheless P (a1 =+1, b1 =+1) > 0withamaximal valueofapproximately0.11.Thislooksabsurd,sinceenchainingthebulletpointed threeinferenceslooksinnocuous—butitisnot:JustasinthederivationoftheCHSH inequality,theenchainingassumesthatonecanspeakofboth a0 and a1 ,andofboth b0 and b1 .Rigorously,thefirstinferenceshouldread:If b =+1wasfoundfor y = 1,we knowthat a =+1willbefound iftheinputx = 0 iscalled ;andsimilarlyfortheothers.
1.4.4TheMagicSquare
TherulesoftheMagicSquaretest9 (Cabello,2001b;Cabello,2001a;Aravind,2002) areslightlymorecomplex.Thetesthasthreepossibleinputsperplayer, x, y ∈{1,2,3}; theplayersareaskedtooutputthreebitseach: ax = (
).These outputsshouldideallysatisfythefollowingconditions:
Toseethattheseconditionsareimpossibletofulfilperfectlywithpre-determinedoutputs, letusarrangetheninepre-determinedbitsina3 × 3square.Uponbeingqueried,Alice outputsthe x-thlineofhersquare,andBobthe y-thcolumnofhis.Thethirdcondition saysthatthevalueattheintersectionshouldbethesameforeverycall(x, y)ofthe verifier,whichmeansthatthesquaresmustbeidentical.However,thefirstcondition
9 Thesquaremadeitsfirstappearanceasatestforsingle-player“contextuality”(seeAppendixD.3forthis notion)inworksofN.DavidMerminandAsherPeres.Becauseofthis,itisoftencalledMermin-PeresMagic Square.HereIcitethesubsequentworksthatintroduceditasatwo-playernonlocalitytest.
implies x, j a j x =+1,thesecond y,k bk y =−1.Ifthesearetobeenforced,Alice’sand Bob’s9-bitsquaresshoulddifferinatleastinonebit—andthen,iftheverifiercalls preciselythoseinputs,hewillseethatthethirdconditionfails.
Allinall,withpre-determinedoutputs,AliceandBobcansatisfythethreeconditions (1.15)foratmost8/9oftheroundsonaverage;butthereexistsaquantumstateand measurementsthatcanfulfilthemperfectly,asweshallshowinsubsection4.4.2.
1.5ACloserScrutiny:AddressingLoopholes
Aswehavejustseen,Belltestscanbedescribedinveryelementaryterms.Butisthisnot tooelementary,especiallygiventhestrongconclusionsthatarereached?Overtheyears, Belltestshavebeensubmittedtotightscrutiny,searchingforflaws,or loopholes,inthe reasoningorintheimplementations.
Thefourpossibleloopholesthathavebeenidentifiedturnouttobeverydifferent fromeachother:Somearemeretechnicalfixes(thathavebeenfixed),othersborderon philosophyandcanbeclosedonlyunderreasonableassumptions(inotherwords,there isapricetopayifonewantstobelievethattheyarestillopen).Wereviewthemherein thisorder;theirworkingwillbeillustratedwiththeCHSHtest.
1.5.1The“memoryloophole,”ordoingproperstatistics
Thememoryloopholeisrelatedtostatistics.Basicstatisticsassumesthatroundsofatest are independentandidenticallydistributed(i.i.d.),butthisi.i.d.assumptionisobviously unwarrantedwhenitcomestosuchfundamentaltests.Woulditbepossibleforthe playerstogiveafalsepositiveinaBelltest,i.e.,violateaBellinequalitywithlocal resources,byadoptinganon-i.i.d.strategy,thatis,bychoosingtheprocesstobeusedin round r basedonallthathashappenedinthepreviousrounds?Theanswerisno.
Toappreciatewhy,consideroneroundoftheCHSHtest.Theplayersmustchoose thepre-establishedquadrupleofvaluestobeusedinthatround.Theycanbasetheir choiceonwhateverpieceofinformationfromthepast:Therewillalwaysbeonepair ofinputswhichpullsthesuminthewrongdirection(seeTable1.1),andtheverifier mighthavepickedpreciselythatpair;so |S |≤ 2stillholds.Thissimpleargumentshows thatthereisonlyonewayfortheplayerstogenerateafalsepositive:Avoidthewrong pairofinputs,eitherbyrefusingtoanswerorbycolludingwiththeverifier.Theseare, respectively,thefair-samplingloopholeandthefree-willloopholetobedescribed.
Thus,evenifweinitiallyderivedourBellinequalitythinkingini.i.d.terms,wehave provedthat thereisnomemoryloophole iftheBelltestisinfinitelylongandtheverifier canextractperfectstatistics.Inarealtest,whenonlyfinitelymanyroundsarepossible, theBelltestmustbephrasedashypothesistesting:Howlikelyistheobservedstringof outputsassumingthattheplayersareusingalocalresource?Forsuchlikelihoodbounds, non-i.i.d.estimatorsmustindeedbeusedinsteadofthefamiliari.i.d.-basedGaussian standarddeviations.We’llgetbacktothispointinsection2.6.
1.5.2The“detectionloophole(s),”orthedangers ofpost-selection
Ifinanexamthestudentswereallowedtodeclinetoanswertilltheyareaskedaquestion oftheirliking,theaveragescorewouldbecertainlyincreased.Thesamehappensfor nonlocality.IntheCHSHtestplayedwithlocalstrategies,wehaveseenthatonlyonepair ofinputs(x, y)pullsthevalueof S inthewrongdirection:Iftheplayerswereallowedto declineansweringwhentheyreceivethatspecificquery,theverifierwouldnevercatch thematfault.Thereisonlyasubtlety:Neitheroftheplayersknowsthepair(x, y),sothe decisiontodeclinemustbemadelocally,basedon x oron y alone.Apossiblestrategy inwhichAliceanswersalwayswhileBobisinchargeofdecliningisgiveninTable1.2. WerefertoExercise1.3forathoroughstudyofthisstrategy,andtoAppendixB.3fora morerigorousquantitativeapproach.
Thefixforthisloopholeisclear: Theverifiermustelicitananswerineveryround.This doesnotsoundtobeabigdeal,butitmaybe.Letuslookatthisloopholefromthe perspectiveofanhonestexperimentalistwhopossessesaverygoodnonlocalresource (say,asourceofphotonsentangledinpolarization)andveryaccuratemeasurement devices,butwhosedetectorshavepoorefficiency.Ifsheisobligedtoproduceoutputsin everyround,inmostoftheroundsshe’llhavetoproduceadummyoutputbecausethe detectorswon’thavefired.Nonlocalityisquicklywasheddown,andlikelytheobserved datawillbecompatiblewithalocalresource.
Thesituationisallthemoreannoyingbecausethisloopholehasaveryconspiratorial character.AsJohnBellhimselfstressedandmanyafterhim,quantumtheoryprovides
Table1.2 AstrategyexploitingthedetectionloopholefortheCHSHtest. Ineachround,AliceandBobchooseoneoftheeightquadruplesof pre-establishedvaluesthatwouldgives =+2(c.f.,Table1.1).Alice answersalways,whileBobdeclinestoanswertotheinputindicatedby brackets,insuchawaythattheproblematicoutput(boldface)isnever produced.
