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A quantum mechanical bound for CHSH-type Bell inequalities MichaelEpping,HermannKampermannandDagmarBruß 5 1 0 2 n a Abstract Many typical Bell experiments can be described as follows. A source J repeatedly distributes particles among two spacelike separated observers. Each of 7 them makes a measurement, using an observable randomly chosen out of several possibleones,leadingtooneoftwopossibleoutcomes.Aftercollectingasufficient ] h amountofdataonecalculatesthevalueofaso-calledBellexpression.Animportant p question in this context is whether the result is compatible with bounds based on - t theassumptionsoflocality,realismandfreedomofchoice.Hereweareinterested n in bounds on the obtained value derived from quantum theory, so-called Tsirelson a u bounds.WedescribeasimpleTsirelsonbound,whichisbasedonasingularvalue q decomposition. This mathematical result leads to some physical insights. In par- [ ticular the optimal observables can be obtained. Furthermore statements about the 1 dimension of the underlying Hilbert space are possible. Finally, Bell inequalities v can be modified to match rotated measurement settings, e.g. if the two parties do 6 notshareacommonreferenceframe. 0 4 1 0 1 Introduction . 1 0 5 Sincetheadventofquantumtheoryphysicistshavebeenstrugglingforadeeperun- 1 derstandingofitsconceptsandimplications.Oneapproachtothisendistocarveout v: the differences between quantum theory and “classical” theories, i.e. to explicitly i point to the conflicts between quantum theory and popular preconceptions, which X evolved in each individual and the scientific community from decoherent macro- r scopicexperiences.Plainformulationsofsuchdiscrepanciesandconvincingexper- a imental demonstrations are crucial to internalizing quantum theory and replacing existing misconceptions. For this reason the double-slit-experiments (and similar MichaelEpping·HermannKampermann·DagmarBruß Heinrich-Heine-University Du¨sseldorf, Universita¨tsstr. 1, 40225 Du¨sseldorf, Germany, e-mail: [email protected] 1 2 MichaelEpping,HermannKampermannandDagmarBruß Fig.1 Twoparties,Alice(A)andBob(B),performaBellexperiment.Bothofthemreceiveparts ofaquantumsystemfromthesource(S).Theyrandomlychooseameasurementsetting,denoted byx=1,2,...,M andy=1,2,...,M ,andwritedowntheiroutcomesa=−1or1andb=−1or1, 1 2 respectively.Theexperimentisrepeateduntiltheaccumulateddataisanalyzedaccordingtothe text.Anglesof45degreesinthespace-time-diagramcorrespondtothespeedoflight.Thefuture lightconesofA,BandSshow,thatthesettingchoiceandoutcomeofonepartycannotinfluence theotherandthatAandBalsocannotinfluenceanyeventinsidethesource. experimentswithopticalgratings)[1,2,3,4,5],whichexposetheroleofstatesu- perpositionsinquantumtheory,aresoveryfascinatingandfamous.Otherexamples of “eye-openers” are demonstrations of tunneling [6, 7, pp. 33-12], the quantum Zeno effect [8] and variations of the Elitzur-Vaidman-scheme [9, 10, 11], to pick justafew. Bellexperiments[12,13,14,15],whichshowentanglementinaparticularlystrik- ing way, belong to this list. Informally, entanglement is the fact that in quantum theorythestateofacompoundsystem(e.g.twoparticles)isnotonlyacollectionof thestatesofthesubsystems.Thisfactcanleadtostrongcorrelationsbetweenmea- surements on different subsystems. Before going into more detail here, we would liketonotethatthedescribeddifferencesbetweentherelativelynewquantumtheory andouroldpreconceptionsareobviousstartingpointswhentolookforinnovative technologieswhichwereevenunthinkablebefore.Thisisinfactahugemotivation forthefieldofquantuminformation,whereBellexperimentsplayacentralrole. 1.1 Bellexperimentsbringthreefundamentalcommonsense assumptionstoatest TheideaofBellwastoshowthatsomecommonsenseassumptionsleadtopredic- tionsofexperimentaldatawhichcontradictthepredictionsofquantumtheory.Inthe followingweemployablackboxapproachtoemphasizethatthisideaiscompletely independentofthephysicalrealizationofanexperiment.Forexamplethemeasure- mentapparatusesgetsomeinput(anintegernumberwhichwillinthefollowingbe called“setting”)andproducesomeoutput(the“measurementoutcomes”).Werefer AquantummechanicalboundforCHSH-typeBellinequalities 3 readerspreferringamoreconcretenotiontoSection1.2,wherephysicalimplemen- tationsandconcretemeasurementsareoutlined. In the present paper we consider the following (typical) Bell experiment, see also Figure 1. There are three experimental sites, two of which we call the parties Al- ice (A) and Bob (B), and the third being a preparation site which we call source (S). Alice and Bob have a spatial separation large enough such that no signal can travelfromonepartytotheotheratthespeedoflightduringtheexecutionofour experiment.ThesourceisseparatedsuchthatnosignalcantravelfromAorBtoit atthespeedoflightbeforeitfinishesthestateproduction.Theimportanceofsuch separationswillbecomeclearlater. Thesourceproducesaquantumsystem,andsendsoneparttoAliceandonetoBob. WewillexemplifythisinSection1.2.AandBareinpossessionofmeasurementap- paratuseswithapredefinedsetofdifferentsettings.Ineachruntheychoosetheset- tingrandomly,e.g.theyturnaknoblocatedattheoutsideoftheapparatus,measure thesystemreceivedfromthesourceandlistthesettingandoutcome.Inthepresent paper the measurements are two-valued and the outcomes are denoted by −1 and +1.LetM andM bethenumberofdifferentmeasurementsettingsatsiteAandB, 1 2 respectively.Welabelthembyx=1,2,3,...,M forAliceandy=1,2,3,...,M for 1 2 Bob.Thispreparationandmeasurementprocedureofaquantumsystemisrepeated untiltheamountofdatasufficestoestimatetheexpectationvalueofthemeasured observables,uptothestatisticalaccuracyoneaimsat.Theexpectationvalueofan observableistheaverageofallpossibleoutcomes,here±1,weightedwiththecor- respondingprobabilitytogetthisoutcome. Letussketchthepreconceptionsthatarejointlyinconflictwiththequantumtheo- reticalpredictionsforBelltests.Thesearemainlythreeconcepts:Locality,realism and freedom of choice. This forces us to question at least one of these ideas, be- cause any interpretation of quantum theory, as well as any “postquantum” theory, cannotobeyallofthem.Weinvitethereadertopickonetoabandonwhilereading thefollowingdescriptions.Donotbeconfusedbyourcomparisonwiththetextbook formalismofquantumtheory:sofaryouarefreetochooseanyofthem. Localityistheassumption,thateffectsonlyhavenearbydirectcauses,ortheother wayaround:anyactioncanonlyaffectdirectlynearbyobjects.Ifsomeactionhere hasanimpactthere,thensomethingtraveledfromheretothere.And,accordingto specialrelativity,thespeedofthissignalisatmostthespeedoflight.Inoursetup, thismeansthatwhateverAlicedoescannothaveanyobservableeffectatBob’ssite. Inparticular,themeasurementoutcomeatonesidecannotdependonthechoiceof measurement setting at the other site. While the formalism of quantum theory has some “nonlocal features”, e.g. a global state, it is strictly local in the above sense, becauseanylocalquantumoperationononesubsystemdoesnotchangeexpectation valuesoflocalobservablesforadifferentsubsystem. Realism is the concept of an objective world that exists independently of subjects (“observers”). A stronger form of realism is the “value-definiteness” assumption meaningthat theproperties ofobjects alwayshave definitevalues,also ifthey are notmeasuredorevenunaccessibleforanyobserver.Itseemstobeagainstcommon sensetoassumethatobjectsceasetohavedefinitepropertiesifwedonotmeasure 4 MichaelEpping,HermannKampermannandDagmarBruß themanylonger.Inparticularthenaturalscienceswerefoundedontheassumption, thatnatureanditspropertiesexistindependentlyofthescientist.Inoursetuprealism implies,thatthemeasurementoutcomesofunperformedmeasurements(inuncho- sensettings)havesomevalue.Wedonotknowthem,butwecansafelyassumethat they exist, give them a name and use them as variables. If possible outcomes are −1and+1,forexample,wemightusethattheoutcomesquaredis1inanyofour calculations. In general the (usual) formalism of quantum theory does not contain definitevaluesformeasurementoutcomesindependentofameasurement. Freedomofchoice,whichisalsosometimescalledthefreewillassumption,means it is possible to freely choose what experiment to perform and how. Because this ideaiselusive,wearecontentwithadecisionthatisstatisticallyindependentofany quantitywhichissubjectofourexperiment.Theideaoffateseemstobetempting tomanypeople.However,droppingfreedomofchoicemakesscienceuseless.Just imagineyou“want”toinvestigatethequestionwhetherabagcontainsblackballs butyourfateistopickonlywhiteballs(andputthembackafterwards),eventhough therearemanyblackballsinside.Inoursetup,freedomofchoiceimplies,thatA’s andB’schoiceofmeasurementsettingdoesnotdependontheother’schoiceorthe outcomes. In quantum theory, there is freedom of choice in the sense that random measurementoutcomesofsomeotherprocesscanbeusedtomakedecisions. Ifyoudecidedthatyoupreferablytakeleaveoflocalityyouareingoodcompany. Many scientists conclude from Bell’s theorem, that the locality assumption is not sustainable. This is particularly interesting when you consider the above compari- son with the standard textbook formalism of quantum theory, which is apparently notrealisticbutlocalinthedescribedsense.Thefactthatinthiscontextmanyscien- tistsspeakabout“quantumnonlocality”thusleadstocontroversy[16].Wetherefore wanttostressagain,thattheexperimentalcontradictiononlytellsusthatatleastone ofalltheassumptionsthatleadtothepredictionsneedstobewrong.Wecannotde- cidewhichassumptioniswrongfromBell’stheoremalone. Wenowfocusonatooltoshowthecontradictioninthedescribedexperimentbe- tween the above assumptions and quantum theory, the so called Bell inequalities. Theseareinequalitiesofmeasurablequantitieswhichare(mainly)derivedfromlo- cality,realismandfreedomofchoiceandthereforeholdforalltheorieswhichobey these principles, whilethey are violated by the predictionsof quantum theory. We consider a special kind of Bell inequalities which are linear combinations of joint expectation values of Alice’s and Bob’s observables. The joint expectation value ofthetwoobservablesofAliceandBobistheexpectationvalueoftheproductof themeasurementoutcomes,whichagaintakesvalues±1.Itdependsonthesetting choicexatAlice’ssiteandyatBob’ssiteandwedenoteitbyE(x,y).Ifwedenote the (real) coefficient in front of the expectation value E(x,y) as g , then we can x,y writesuchBellinequalitiesas M1 M2 ∑ ∑g E(x,y)≤B , (1) x,y g x=1y=1 AquantummechanicalboundforCHSH-typeBellinequalities 5 wheretheboundB dependsonlyonthecoefficientsg .Thesecoefficientsforma g x,y matrixgwhichhasdimensionM ×M .AnyrealmatrixgdefinesaBellinequality 1 2 viaEq.(1).ThemaybemostfamousexampleistheClauser-Horne-Shimony-Holt (CHSH)[17]inequality,whichreads E(1,1)+E(1,2)+E(2,1)−E(2,2)≤2. (2) Herethecorrespondingmatrixgis (cid:18) (cid:19) 1 1 g= . (3) 1−1 Due to its prominence we call the class of Bell inequalities in the form of Eq. (1) CHSH-type Bell inequalities. For completeness we sketch the derivation of B . It g turnsoutthatitsufficestoconsiderdeterministicoutcomesonly,asaprobabilistic theory,wheretheoutcomesfollowsomeprobabilitydistribution,cannotachievea highervalueinEq.(1):itcanbedescribedasamixtureofdeterministictheoriesand thevalueofEq.(1)isthesumofthevaluesforthedeterministictheoriesweighted with the corresponding probability in the mixture. For deterministic theories the expectationvalueismerelytheproductofthetwo(possiblyunmeasured)outcomes aofAliceandbofBob,whichweareallowedtousewhenassumingrealism.Due tolocalityaonlydependsonthesettingxofAlice,whichhasnofurtherdependence duetofreedomofchoice.AnalogouslybdependsonlyonthesettingyofBob,which inturnhasnofurtherdependence.Thustheexpectationvalueis E(x,y)=a(x)b(y). (4) Now we can calculate B by maximizing Eq. (1) over all possible assignments of g −1and+1valuestoa(x)andb(y).InEq.(2)themaximalvalueisB =2,which g is achieved for a(1)=a(2)=b(1)=b(2)=1, for example.Note that the sign of E(2,2)cannotbechangedindependentlyoftheotherthreeterms,becauseE(1,2) andE(2,1)containb(2)anda(2),respectively. Wepointoutthatanyfunctionthatmapstheprobabilitiesofdifferentmeasurement outcomes to a real number may be used to derive Bell inequalities, and different types of Bell inequalities can be found in the literature (e.g. [18]). However, here wefocusonBellinequalitiesoftheformofEq.(1). 1.2 TheCHSHinequalitycanbeviolatedinexperimentswith entangledphotons We recapitulate some basics of quantum (information) theory. Analogously to a classicalbitthequantumbit,orqubit,canbeintwostates0and1,butadditionally ineverypossiblesuperpositionofthem.Mathematicallythisstateisaunitvectorin 6 MichaelEpping,HermannKampermannandDagmarBruß thetwo-dimensionalHilbertspace(avectorspacewithascalarproduct)C2spanned bythebasisvectors (cid:18) (cid:19) (cid:18) (cid:19) 1 0 0:= and 1:= . (5) 0 1 Anexampleofasuperpositionofthesebasisstatesisψ = √1 (0+1).Anyobserv- 2 ableonaqubitwithoutcomes+1and−1canbewrittenas (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 01 0−i 1 0 A=a +a +a , (6) x 10 y i 0 z 0−1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) σx σy σz where the vector a=(a ,a ,a )T (here T denotes transposition) defines the mea- x y z surementdirectionandthematricesσ ,σ andσ arecalledPaulimatrices.Theex- x y z pectationvalueofthisobservablegivenanystateψ canbecalculatedasE=ψ†Aψ (here†denotesthecomplexconjugatedtranspose),whichisbetween−1and+1. Anyquantummechanicalsystemwith(atleast)twodegreesoffreedomcanbeused asaqubit.Inthepresentcontextthespinofaspin-1-particle,twoenergylevelsof 2 anatomandthepolarizationofaphotonareimportantexamplesofqubits.Thespin measurement can be performed using a Stern-Gerlach-Apparatus [19], the energy levelofanatommaybemeasuredusingresonantlaserlight,orthepolarizationof aphotoncanbemeasuredusingpolarizationfiltersorpolarizingbeamsplitters. The Hilbert space of two qubits is constructed using the tensor product, i.e. C2⊗ C2 = C4. The tensor product of two matrices (of which vectors are a special case) is formed by multiplying each component of the first matrix with the com- plete second matrix, such that a bigger matrix arises. The state of the compos- ite system of two qubits in states φA =(φA,φA)T and φB =(φB,φB)T then reads 1 2 1 2 φAB=φA⊗φB=(φAφB,φAφB,φAφB,φAφB)T.Thestatesofsuchcompositesys- 1 1 1 2 2 1 2 2 temsmightbesuperposed,whichleadstothenotionofentanglement. Out of several physical implementations of the CHSH experiment we sketch the oneswithpolarizationentangledphotons(see[20]).Weidentify0withthehorizon- taland1withtheverticalpolarizationofaphoton.Nonlinearprocessesinspecial opticalelementscanbeusedtocreatetwophotonsinthestate 1 φ = √ (1,0,0,1)T, (7) + 2 i.e.anequalsuperpositionoftwohorizontallypolarizedphotonsandtwovertically polarizedphotons.ThemeasurementsofAliceandBobinsetting1and2are A =cos(2×22.5◦)σ +sin(2×22.5◦)σ , (8) 1 x z A =cos(−2×22.5◦)σ +sin(−2×22.5◦)σ , (9) 2 x z B =cos(2×0◦)σ +sin(2×0◦)σ (10) 1 x z andB =cos(2×45◦)σ +sin(2×45◦)σ , (11) 2 x z AquantummechanicalboundforCHSH-typeBellinequalities 7 respectively.Heretheanglesaretheanglesofthepolarizerandthefactor2isdueto thefactthatincontrasttotheStern-Gerlach-Apparatusarotationofthepolarizerof 180◦ correspondstothesamemeasurementagain.Onecannowcalculatethevalue ofEq.2: E(1,1)+E(1,2)+E(2,1)−E(2,2)= φ†(A ⊗B )φ +φ†(A ⊗B )φ + 1 1 + + 1 2 + +φ†(A ⊗B )φ −φ†(A ⊗B )φ + 2 1 + + 2 2 + (cid:18) (cid:19) 1 1 1 1 = √ +√ +√ − −√ 2 2 2 2 √ =2 2. (12) √ Thevalue2 2≈2.82islargerthan2andthereforetheCHSHinequalityisviolated. Onecanaskwhetheritispossibletoachieveanevenhighervalue,e.g.whenusing higher-dimensionalsystemsthanqubits,becauseatthefirstglanceavalueofupto fourseemstobepossible.Thisquestionisaddressedinthefollowingsections(the answer,whichisnegative,isgiveninSection2.2). 1.3 ThequantumanalogtoclassicalboundsonBellInequalities areTsirelsonBounds Analogously to the “classical” bound one can ask for bounds on the maximal value of a Bell inequality obtainable within quantum theory, so-called Tsirelson bounds[21],andtheobservablesthatshouldbemeasuredtoachievethisvalue.In other words: which observables are best suited to show the contradiction between quantumtheoryandtheconjunctionofthethreediscussedcommonsenseassump- tions.Thisquestion,whichisalsoofsomeimportanceforapplicationsofBellin- equalities,isthemainsubjectofthepresentessay. The scientific literature contains several approaches to derive Tsirelson bounds, some of which we want to mention. The problem of finding the Tsirelson bound ofEq.(1)canbeformulatedasasemidefiniteprogram.Semidefiniteprogramming isamethodtoobtaintheglobaloptimumoffunctions,undertherestrictionthatthe variableisapositivesemidefinitematrix(i.e.ithasnonegativeeigenvalues).This implies that well developed (mostly numerical) methods can be applied [22, 23]. TheinterestedreadercanfindaMatlabcodesnippettoplayaroundwithintheap- pendix4.FurthermoretherehasbeensomeefforttoderiveTsirelsonboundsfrom firstprinciples,amongstthemthenon-signallingprinciple[24],informationcausal- ity[25]andtheexclusivityprinciple[26]. Thenon-signallingprincipleissatisfiedbyalltheories,thatdonotallowforfaster- than-light communication. Information causality is a generalization of the non- signalling principle, in which the amount of information one party can gain about data of another is restricted by the amount of (classical) communication between 8 MichaelEpping,HermannKampermannandDagmarBruß them.Theexclusivityprinciplestates,thattheprobabilitytoseeoneeventoutofa setofpairwiseexclusiveeventscannotbelargerthanone. 2 Thesingularvaluebound Herewewilldiscussasimplemathematicalboundforthemaximalquantumvalue of a CHSH-type Bell inequality defined via a matrix g, which we derived in [27]. Whileitisnotaswidelyapplicableasthesemidefiniteprogrammingapproach,itis an analytical expression which is easy to calculate and it already enables valuable insights.For“simple”Bellinequalities,liketheCHSHinequalitygivenabove,itis sufficienttousethemethodofthispaper. Wewillmakeuseofsingularvaluedecompositionsofrealmatrices,astandardtool oflinearalgebra,whichwenowshortlyrecapitulate. 2.1 Anymatrixcanbewritteninasingularvaluedecomposition Asingularvaluedecompositionisverysimilartoaneigenvaluedecomposition,in factthetwoconceptsarestronglyrelated.AnyrealmatrixgofdimensionM ×M 1 2 canbewrittenastheproductofthreematricesV,S,WT,i.e. g=VSWT, (13) wherethesethreematriceshavespecialproperties.ThematrixV isorthogonal,i.e. itscolumns,whicharecalledleftsingularvectors,areorthonormal.Ithasadimen- sionofM ×M .ThematrixS isadiagonalmatrixofdimension M ×M ,which 1 1 1 2 is not necessarily a square matrix. Its diagonal entries are positive and have non- increasingorder(fromupperlefttolowerright).Theyarecalledsingularvaluesof g.ThematrixW isagainorthogonal.IthasdimensionM ×M anditscolumnsare 2 2 calledrightsingularvectors. ThelargestsingularvaluecanappearseveraltimesonthediagonalofS.Wecallthe numberofappearancesthedegeneracyd ofthemaximalsingularvalue.Duetothe orderingofS,thesearethefirstddiagonalelementsofS.Herewenotetheconcept ofatruncatedsingularvaluedecomposition:insteadofusingthefulldecomposition one can approximate g by using only parts of the matrices corresponding to, e.g., thefirstd singularvalues(i.e.onlythemaximalones).Thesearethefirstd leftand right singular vectors, and the first part of S, which is just a d×d identity matrix multipliedbythelargestsingularvalue.Sincethesematricesplayanimportantrole inthefollowinganalysiswewillgivethemspecialnames:V(d),S(d) andW(d).All thesematricesaredepictedinFigure2. The matrix g maps a vector v to a vector gv which, in general, has a differ- ent length than v. Here the length is measured by the (usual) Euclidean norm AquantummechanicalboundforCHSH-typeBellinequalities 9 Fig.2 ThematricesinvolvedinthesingularvaluedecompositionofageneralrealM ×M matrix 1 2 g:V andW areorthogonalmatrices,Sisdiagonal.V andW containtheleftandrightsingular vectors,respectively,ascolumns,andScontainsthesingularvaluesonitsdiagonal.Theshaded partsbelongtoatruncatedsingularvaluedecompositionofg.Wedenotethepartscorresponding tothemaximalsingularvalueasV(d),S(d)andW(d). (cid:113) ||v|| = v2+v2+...+v2 . The largest possible stretching factor for all vectors 2 1 2 M2 visapropertyofthematrix:itsmatrixnorminducedbytheEuclideannorm.The value of this matrix norm coincides with the maximal singular value S . We can 11 thereforeexpressthemaximalsingularvalueusing ||gv|| S = max 2 =:||g|| . (14) 11 2 v∈RM2 ||v||2 Thenotation||g|| forthemaximalsingularvalueofgismoreconvenientthanS , 2 11 asitcontainsthematrixasanargument. 2.2 ThesingularvalueboundisasimpleTsirelsonbound Itturnsoutthatthematrixnormofg,i.e.itsmaximalsingularvalue,leadstoanup- perboundonthequantumvalueforaBellinequality,definedbygviaEq.(1).Thisis thecentralinsightofthisessay.Itisremarkablethatamathematicalproperty,solely due to the rules of linear algebra, leads to a bound for a physical theory, here the theoryofquantummechanics.Withthedefinitionofthematrixnormgivenabove, wecannowwritethissingularvalueboundofg,asimpleTsirelsonbound[27].It reads M1 M2 √ ∑ ∑g E(x ,x )≤ M M ||g|| , (15) x1,x2 1 2 1 2 2 x=1y=1 where E now denotes the expectation value of a quantum measurement in setting x andx .Eq.(15)isthecentralformulaofthisessay.Notethatthisboundisnot 1 2 alwaystight,i.e.thereexistexampleswheretherighthandsidecannotbereached 10 MichaelEpping,HermannKampermannandDagmarBruß within quantum mechanics. However for many examples it is tight. The proof of thisboundissketchedinAppendix3. WenowcalculatethisboundfortheCHSHinequalitygiveninEq.(2).Wesee,that herethematrixofcoefficientsis (cid:18)1 1 (cid:19) (cid:32)√1 √1 (cid:33)(cid:18)√2 0 (cid:19)(cid:18)10(cid:19) g= = 2 2 √ . (16) 1−1 √1 −√1 0 2 01 2 2 (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) V S WT Itiseasytocheckthatthegivendecompositionofgisasingularvaluedecomposi- tion,i.e.V,SandW havethepropertiesdescribedabove.Fromthisweread,thatthe √ maximalsingularvalueofgis||g|| = 2.ThenEq.(15)tellsus,thatthemaximal 2 value of the CHSH inequality (Eq. (2)) within quantum theory is not larger than √ 2 2,avaluewhichcanalsobeachievedwhenusingappropriatemeasurementsand states(seeSection1.2) 2.3 Tightnessoftheboundcanbecheckedefficiently Wealreadymentionedthattheinequality(15)isnotalwaystight,i.e.sometimesitis notpossibletofindobservablesandaquantumstatesuchthatthereisequality.From thederivationofEq.(15)sketchedinAppendix3oneunderstands,whythisisthe √ case.Thevalue M M ||g|| isachievedifandonlyifthereexistsarightsingular 1 2 2 vectorvtothemaximalsingularvalueandandacorrespondingleftsingularvector wwhichfulfillfurthernormalizationconstraints. It is common to denote the element in the i-th row and j-th column of a matrix A as A . We will extend this notation to denote the whole i-th row by A and the ij i∗ whole j-thcolumnbyA ,i.e.the∗standsfor“all”.Forexample,thel-thM +M ∗j 1 2 dimensionalcanonicalbasisvector,withaoneatpositionl and0everywhereelse, canthenbewrittenas1(M1+M2). ∗,l Withthisnotationathandwewritedownthenormalizationconstraintfromabove asthesystemofequations (cid:13) (cid:13)2 (cid:13)αTV(d)(cid:13) = 1forx=1,2,...,M (17) (cid:13) x∗ (cid:13) 1 (cid:13)(cid:114) (cid:13)2 and (cid:13)(cid:13) M2αTWy(∗d)(cid:13)(cid:13) =1fory=1,2,...,M2, (18) (cid:13) M (cid:13) 1 where the d×d(cid:48) matrix α is the unknown. The bound in Eq. (15) is tight if and onlyifsuchmatrixα solvingthissystemofequationscanbefound.Hered isthe degeneracyofthemaximalsingularvalueofgandd(cid:48),thedimensionofthevectors v =αTV(d) andw =αTV(d),isanaturalnumber.ThestepsleadingtoEqs.(17) x x∗ y y∗ and(18)canbefoundinthesupplementalmaterialof[27].BecauseEqs.(17)and

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