A quantum mechanical bound for CHSH-type Bell inequalities

0.9 MB·English

by  Michael Epping

A quantum mechanical bound for CHSH-type
Bell inequalities
MichaelEpping,HermannKampermannandDagmarBruß
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a Abstract Many typical Bell experiments can be described as follows. A source
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repeatedly distributes particles among two spacelike separated observers. Each of
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them makes a measurement, using an observable randomly chosen out of several

possibleones,leadingtooneoftwopossibleoutcomes.Aftercollectingasufficient
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h amountofdataonecalculatesthevalueofaso-calledBellexpression.Animportant
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question in this context is whether the result is compatible with bounds based on
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t theassumptionsoflocality,realismandfreedomofchoice.Hereweareinterested
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in bounds on the obtained value derived from quantum theory, so-called Tsirelson
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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
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1 Introduction
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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
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evolved in each individual and the scientific community from decoherent macro-
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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]
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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