Monogamy of Correlations vs. Monogamy of Entanglement

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by  Michael Seevinck

QuantumInformationProcessingmanuscriptNo.
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Monogamy of Correlations vs. Monogamy of Entanglement
M.P.Seevinck
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J Abstract Afruitfulwayofstudyingphysicaltheoriesisvia 1 Introduction

9 the question whether the possible physical states and dif-
1
ferent kinds of correlations in each theory can be shared Itismoreandmorerealisedthatentanglementisaphysical

to differentparties. Over the past few years it has become resource[1].Thishasbeenthedrivingforcebehindtheex-
]
h clearthatbothquantumentanglementandnon-locality(i.e., plodingfieldofquantuminformationtheory,andhasledto
p
correlationsthatviolateBell-typeinequalities)havelimited many operational and information-theoreticinsights. More
-
t shareabilitypropertiesandcansometimesevenbemonoga- recently, and less well-known, it has been noted that non-
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mous.We givea self-containedreviewof these resultsand locality(i.e.,correlationsthatviolateBell-typeinequalities)
a
u presentnewresultsontheshareabilityofdifferentkindsof isalsoaresourceforinformationtheoretictasks[2].Inthis
q correlations,includinglocal,quantumandno-signallingcor- paper one aspect of their usefulness as a resource will be
[
relations. This includes an alternative simpler proof of the considered,namelythatbothentanglementandnon-locality

2 Toner-Verstraete monogamy inequality for quantum corre- havelimitedshareabilityproperties1,and,infact,cansome-
v
lations, as well as a strengthening thereof. Further, the re- times even be monogamous [3,4,5,6]: consider three par-
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lationship between sharing non-localquantum correlations ties a,b,c each holding a qubit, then if a’s and b’s qubits
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8 and sharing mixed entangled states is investigated, and al- aremaximallyentangled,thenc’squbitmustbecompletely
1 readyforthesimplestcaseofbi-partitecorrelationsandqubits unentangledtoeithera’sorb’s.Similarlyifaandbarecor-
.
8 thisisshowntobenon-trivial.Also,arecentlyproposednew related in such a way that they violate the Clauser-Horne-
0 interpretationofBell’stheorembySchumacherintermsof Shimony-Holt(CHSH)inequality[8](thiswillalsobecalled
9 shareabilityofcorrelationsiscriticallyassessed.Finally,the ‘non-locallycorrelated’),thenneitheranorbbecanbecor-
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relevanceofmonogamyofnon-localcorrelationsforsecure relatedinsuchaway(i.e.,non-locally)tocinanyno-signalling
:
v quantum key distribution is pointed out, and in this regard theory.Ithasbeenshownthatsuchcorrelationscanbeused
i
X itisstressedthatnotallnon-localcorrelationsaremonoga- asaresourcetodistributeasecretkeywhichissecureagainst
r mous. eavesdroppers which are only constrained by the fact that
a
anyinformationaccessibletothemmustbecompatiblewith
Keywords Quantum Mechanics Entanglement Non- no-signalling,whichisroughlytheimpossibilityofarbitrar-
locality Monogamy No-signal·ling Shareabili·ty ilyfastsignalling[9].
· · · ·
Cryptography Classically none of this is possible since one does not
have such monogamy trade-offs for states or correlations:
all classical probability distributionscan be shared [6]. In-
PACS 02.50.Cw 03.67.Ud 03.67.Mn 03.65.Ta
· · · deed,ifpartiesa,bandchavebitsinsteadofquantumbits
M.P.Seevinck 1 Here‘shareability’ismeantkinematicalandshouldnotbethought
-InstituteforTheoreticalPhysics, ofinadynamicalsense.Butnotethatthekinematicalshareabilityre-
UtrechtUniversity,P.O.Box80.195,3508TDUtrecht, sultsareofcourserestrictiveforanydynamicsthatonemaywishto
-CentreforTime,PhilosophyDepartment,UniversityofSydney,Main employtoactuallydescribeasharingprocess. Anysuchdynamicsis
QuadA14,NSW2006,Australia(visitingaddress) boundbytheconstraintssetbythekinematics;forthelatterindicate
E-mail:[email protected] whatstatesofthedesiredkindexistatallinthetheory’sstatespace.
2
(qubits) and if a’s bit is perfectly correlated to b’s bit then count.Finally,wewillendwithashortdiscussioninsection
thereisnorestrictiononhowa’sbitiscorrelatedtoc’sbit. 6.
This difference in shareability of states and in shareability
ofcorrelationsisinfactoneofthefundamentaldifferences
2 Shareabilityandmonogamyofstates
betweenclassicalandquantumphysics,althoughithasonly
recently been properly studied. Fortunately,in the last few
Letusfirstconsidertheshareabilityofstates;thatofcorrela-
yearswehavebeenabletowitnessanumberoffundamen-
tionswillbestudiedinthenextsection.Classicalstatescan
talresultsonboththeshareabilityofquantumstatesandof
besharedamongmanypartiesbecauseonecanjustcopythe
correlations[3,4,5,6,10,11,12].
state.Formallywecanrepresentedsuchaclassicalcopying
Thepurposeofthispaperistwo-fold.Firstly,itintends procedureonaphaseorconfigurationspacewhereoneuses
togiveareviewoftherecentresultsinboththesefields(i.e., theCartesianproductstructuretorelatesystemsandsubsys-
shareabilityofquantumstatesandofcorrelations),and,sec- tems.Thatis,itispossibletoextendanybi-partitepurestate
ondly,itaddsnewresultstothelatter,afterwhichbothfields Sab1 =Sa×Sb1 ofthejointpartyab1 toN−1otherparties
arecompared.Thereviewpartofthepaperdrawsheavilyon b2,...bN byconsideringthestateSa×Sb1×Sb2×...×SbN,
workbyToner[6]andMasanes,Acin&Gisin[10],butitis whereSb1=Sbi,∀i.ThisensuresthatthestatesSabi areiden-
intendedtobeself-containedandtriestouseasimplemathe- tical to the originalstate Sab1. The bi-partite state Sab1 can
maticalframeworkintermsoffamiliarmathematicalobjects thusbesharedindefinitely.Allthisremainstrueundercon-
ofjointprobabilitydistributionsforcorrelations.Itusesthe vexdecompositionsofpurestates,andthusalsoforthecase
well-knownCHSH inequalityforexpectationvaluesofthe ofmixedclassicalstates.
productof localoutcomes;it leaves aside the formulations However,inquantummechanicsthingsaredifferent.Ifa
in terms of information theoretic interactive proof systems purequantumstateoftwosystemsisentangled2,thennone
andnon-localgamessetups[6,7]. of the two systems can be entangled with a third system.
Thiscanbeeasily seen.Supposethatsystems3 aandbare
Morespecifically,section2reviewsthemonogamyand inapureentangledstate.Thenwhenthesystemabiscon-
shareabilityofentanglement,andsection3isdevotedtothe sideredaspartofalargersystem,thereduceddensityoper-
monogamy and shareability of correlations. In subsection atorforabmustbyassumptionbeapurestate.However,for
3.1fivedifferentkindsofcorrelationareintroducedwhose thecompositesystem ab(or foranyofits subsystemsaor
shareabilityandmonogamyaspectsarereviewedinsubsec- b)tobeentangledwithanothersystem,thereduceddensity
tions3.2and3.3.Herewealsoprovethatbothunrestricted operatorof abmustbea mixedstate. Butsince itis byas-
general correlations and so-called partially-local ones can sumptionpure,no entanglementbetween ab andany other
be shared to any number of parties (called ¥ -shareable). system can exist. This feature is referred to as the mono-
Next,subsection3.4givesanalternativesimplerproofofthe gamyofpurestateentanglement4.
Toner-Verstraetemonogamyinequality[5]forquantumcor- Thismonogamycanalsobeunderstoodasaconsequence
relations,andthis inequalityis strengthenedas well. Some of the linearity of quantum mechanics that is also respon-
oftheresultsreviewedinthissectionhaveleadSchumacher sible for the no-cloningtheorem [16,17]. For suppose that
[13] to argue for a new view on Bell’s theorem, namely partyahasaqubitwhichismaximallypurestateentangled
that it is a theorem about the shareability of correlations, tobothaqubitheldbypartybandaqubitheldbypartyc.
andthat,contrarytocommunisopinio,itsphysicalmessage Party a thus has a single qubit coupled to two perfect en-
doesnotat all dealwith issues of locality or local realism. tangled quantum channels, which this party could exploit
This argument will be presented and critically assessed in to teleport two perfect copies of an unknown input state,
subsection3.5. In section 4 we comparethe results of sec-
2 A stater ofabi-partite system abisentangled ifandonly ifit
tions2and3:weinvestigatetherelationshipbetweenshare-
aobfimlitiyxeodfennotann-glolecdalsqtautaens,tuamndcaolrrerealdaytiofonrsthanedsismhpalreesatbcilaistye raca,bjnb,rwneosittphebce(cid:229)tiiwvperiliyt=t.eFn1o,ar0sth≤aecmpoinuv≤lteixp1asrautnimtdeogrfeian,perrroabjdliusscatattitsoetsna,toewfs:htihrceh6=sius(cid:229)bnisoyptsiratetiama⊗lsl
of bi-partite correlations this relationship will be shown to trivial,see[14].
be non-trivial. For example, shareability of non-local cor- 3 Foreaseofnotationwewillusethesamesymbolstorefertopar-
tiesandthesystemstheypossess,e.g.,partyapossessessystema.
relationsimpliesshareabilityof entanglement,butnotvice
4 Thisissometimesconfusinglyreferredtoastheclaimthatinquan-
versa.Section5addressesthepossibilitiesforcryptography
tumtheoryasystemcanbepurestateentangledwithonlyoneother
of extracting a secure secret key from correlationsthat are system[15].ButwhatabouttheGHZstate( 000 + 111 )/√2?All
| i | i
monogamous.Itwillbepointedoutthatsomenon-localcor- threepartiesareentangledtoeachotherinthispurestate,sothisseems
tobeacounterexampletotheclaim.Whatisactuallymeantisthatif
relationsindeedsufficeforthistask,butthatingeneralnot
apurestateoftwosystemsisentangled,thennoneofthetwosystems
all non-local correlations are monogamous(shown in sec-
canbeentangledwithathirdsystem.Thisistheformulationwewill
tion 4) and that this fact should be crucially taken into ac- use.
3
therebyviolating the no-cloningtheorem,and thus the lin- 3 Shareabilityandmonogamyofcorrelations
earityofquantummechanics[18].
If thestate oftwo systemsisnota pureentangledstate 3.1 Kindsofcorrelations
buta mixedentangledstate, thenit is possible thatboth of
We willreviewfivedifferentkindsofcorrelationsthatwill
thetwosystemsareentangledtoathirdsystem.Forexam-
ple,theso-calledW-state y =( 001 + 010 + 100 )/√3 bestudied,aswellasseveralusefulmathematicalcharacter-
| i | i | i | i isticsofthesecorrelations8.
hasbi-partitereducedstatesthatareallidenticalandentan-
gled.Thisfeatureiscalled‘sharingofmixedstateentangle-
ment’, or ‘promiscuity of entanglement’. Entanglement is
General unrestricted correlations. Consider N parties, la-
thusstrictlyspeakingonlymonogamousinthecaseofpure
beledby 1,2,...,N, each holdinga physicalsystem thatis
entangledstates.Inthecaseofmixedentangledstatesitcan
to be measured using a finite set of different observables.
bepromiscuous.Butthispromiscuityisnotunbounded:al-
Denote by A the observable (random variable) that party
j
though some entangled bi-partite states may be shareable
j chooses (also called the setting A ) and by a the corre-
j j
with some finite number of parties, no entangled bi-partite
sponding measurement outcomes. We assume there to be
statecanbesharedwithaninfinitenumberofparties5.Here
only a finite number of discrete outcomes. The outcomes
abi-partitequantumstater issaidtobeN-shareablewhen
ab canbecorrelatedinanarbitraryway.Ageneralwayofde-
it is possible to find a quantum state r such that
ab1b2...bN scribingthissituation,independentoftheunderlyingphys-
r =r =r =...=r , where r is the reduced
ab ab1 ab2 abN abk ical model, is by a set of joint probability distributions for
state for parties a and b . Consider the following theorem
k the outcomes, conditionedon the settings chosen by the N
[19,20]:Abi-partitequantumstateisN-shareableforallN
parties,wherethecorrelationsarecapturedintermsofthese
(alsocalled¥ -shareable[10])iffitisseparable.Thusnobi-
jointprobabilitydistributions.Theyaredenotedby
partiteentangledstate,pureormixed,isN-shareableforall
N.
P(a ,...,a A ,...,A ). (2)
1 N 1 N
Thelimitedshareabilityofentanglementwasfirstquan- |
tifiedbyCoffman,Kundu&Wootters[3].Theygaveatrade-
Theseprobabilitydistributionsareassumedtobepositive
off relation between how entangled a is with b, and how
entangled a is with c in a three-qubit system abc that is
P(a ,...,a A ,...,A ) 0, (3)
1 N 1 N
in a pure state, using the measure of bi-partite entangle- | ≥
ment called the tangle [4]. It states that t (r )+t (r )
t (r )wheret (r )isthetangle6betweenaabandb,aancalo≤- andobeythenormalizationconditions
a(bc) ab
gousfort (r ac)andt (r a(bc))isthebi-partiteentanglement7 (cid:229) P(a ,...,a A ,...,A )=1. (4)
acrossthebipartitiona-bc.Ingeneral,t canvarybetween0 1 N| 1 N
a1,...,aN
and1,butmonogamyconstrainstheentanglement(asquan-
tifiedbyt )thatpartyacanhavewitheachofpartiesband We need not demand that the probabilities should not be
c. The generalisationto, possibly mixed,multi-qubitstates greater than 1 because this follows from them being posi-
hasbeenrecentlyprovenbyOsborne&Verstraete[4]: tiveandfromthenormalizationconditions.Wewillnowput
furtherrestrictionsbesidesnormalizationontheprobability
distributions (2) that are motivated by physical considera-
t (r )+t (r )+...+t (r ) t (r ). (1)
ab1 ab2 abN ≤ a(b1b2...bN) tions.
Thisisageneralconstraintondistributedentanglementand
No-signallingcorrelations. Ano-signallingcorrelationisa
which quantifies the frustration of entanglement between
correlationP(a ,...,a A ,...,A )suchthatonesubsetof
1 N 1 N
differentparties.Forfurtherinvestigationsofthemonogamy |
parties,sayparties1,2,...,k,cannotsignaltotheotherpar-
ofentanglement,seealso[11,21,22].
tiesk 1,...,N bychangingtheirmeasurementdeviceset-
−
tingsA ,...,A .Mathematicallythisisexpressedasfollows.
1 k
5 Thisisalsoreferredtoas‘monogamyinanasymptoticsense’by Themarginalprobabilitydistributionforeachsubsetofpar-
[18],butwebelievethatthisfeatureisbettercapturedbytheterm‘no
ties only depends on the corresponding observables mea-
unboundedpromiscuity’.
suredbythepartiesinthesubset,e.g.,foralloutcomesa ,
6 The tangle t (r ab) is the square of the concurrence C(r ab):= 1
osmfyatxthh{ee0m,P√aautlrlii1-xs−pri√anbl(ms2ay−t⊗ri√xslfyo)3rr−ta∗hb√e(slyy-4d⊗}i,rsewcyth)ieoirnne.ntohne-dleicarreeastihnegeoigrdeenrv,awluieths ..8.,Tahki:sPis(aa1m,i.n.i.m,aalkr|eAv1ie,w..l.e,aAviNng)=outPth(ea1d,is.c.u.s,saiokn|Ao1f,t.h.e.c,oArrke)la.-
7 Incaseofthreequbitsthetanglet (r a(bc))isequalto4detr a,with tionsintermsofconvexsetsandfacetsofpolytopes.See[12,10],and
r a=Trbc[ y y ]and y thepurethree-qubitstate. chapter2in[23]forsuchamorecomprehensiveoverview.
| ih | | i
4
These conditionscan all be derivedfromthe following subset. The partially-local correlations can then be written
condition[12].Foreachk 1,...,N themarginaldistri- as
∈{ }
butionthatisobtainedwhentracingouta isindependentof
k
P(a ,...,a A ,...,A )=
whatobservable(A orA )ismeasuredbypartyk: 1 N 1 N
k ′k |
dl p(l )P(a ,...,a A ,...,A ,l )
(cid:229) P(a ,...,a ,...,a A ,...,A ,...,A )= ZL 1 k| 1 k ×
1 k N 1 k N
ak | P(aN k,...,aN AN k,...,AN,l ), (8)
(cid:229) − | −
P(a1,...,ak,...,aN|A1,...,A′k,...,AN), (5) The probabilities on the right hand side need not factorise
ak
anyfurther.Incasetheywouldallfullyfactoriseweretrieve
for all outcomes a ,...,a ,a ,...,a and all settings thesetoflocalcorrelationsdescribedabove.
1 k 1 k+1 N
A ,...,A ,A ,...A . Thiss−et ofconditionsensuresthatall Formulas similar to (8) with different partitions of the
1 k ′k N
marginalprobabilitiesareindependentofthesettingscorre- N-partiesinto two subsets, i.e., for differentchoicesof the
spondingto theoutcomesthatarenolongerconsidered.In composingparties and differentvaluesof k, describe other
particular,(5)thedefinesthemarginal possibilitiestogivepartially-localcorrelations.Convexcom-
binationsofthesepossibilitiesarealsoadmissible.Weneed
notconsiderdecompositionintomorethantwosubsystems
P(a ,...,a ,a ,...,a A ,...,A ,A ,...,A ),
1 k 1 k+1 N 1 k 1 k+1 N
− | − since any two subsystems in such a decomposition can be
(6)
considered jointly as parts of one subsystem still uncorre-
latedwithrespecttotheothers[26].
for the N 1 parties not including party k. No-signalling
−
ensuresthatit is notneededto specifywhetherA or A is
k ′k
Quantumcorrelations. Lastly,weconsidertheclassofcor-
beingmeasuredbypartyk.
relationsthatareobtainedbygeneralmeasurementsonquan-
tum states (i.e., those that can be generated if the parties
Localcorrelations. Localcorrelationsarethosethatcanbe sharequantumstates).Thesecanbewrittenas
obtainedifthepartiesarenon-communicatingandshareclas-
sical information, i.e., they only have local operations and P(a1,...,aN|A1,...,AN)=Tr[MaA11⊗···⊗MaANNr ]. (9)
localhiddenvariables(alsocalledsharedrandomness)asa Here r is a quantum state (i.e., a unit trace semi-definite
resource.Wetakethistomeanthatthesecorrelationscanbe positiveoperator)onaHilbertspaceH =H H ,
1 N
writtenas whereH isthequantumstatespaceofthesys⊗tem···h⊗eldby
j
party j.Thesets MA1,...,MAN definewhatiscalledapos-
P(a ,...,a A ,...,A )= { a1 aN}
1 N| 1 N itiveoperatorvaluedmeasure(POVM),i.e.,asetofpositive
ZL dl p(l )P(a1|A1,l )...P(aN|AN,l ), (7) operators{MaAjj}satisfying(cid:229) ajMaAjj =1,∀Aj.Ofcourse,all
operatorsMAj mustcommutefordifferent jinorderforthe
aj
where l L is the value of the shared local hidden vari- joint probability distribution to be well defined, but this is
able, L t∈he space of all hidden variables and p(l ) is the ensured since for different j the operators are defined for
probability that a particular value of l occurs. Note that differentsubsystems(witheachtheirownHilbertspace)and
p(l )isindependentoftheoutcomesajandsettingsAj,i.e., arethereforecommuting.Notethat(9)islinearinbothMaAjj
thesettingsareassumedtobe‘freevariables’[24].Further- andr ,whichisacrucialfeatureofquantummechanics.
more,P(a A ,l ) is the probabilitythatoutcomea is ob- Quantumcorrelationsareno-signallingandthereforethe
1 1 1
|
tainedbyparty1giventhattheobservablemeasuredwasA marginalprobabilitiesderivedfromsuchcorrelationsarede-
1
andthesharedhiddenvariablewasl ,andsimilarlyforthe finedinthesamewayaswasdoneforno-signallingcorrela-
othertermsP(a A ,l ). A correlationthatisnotlocalwill tions(cf.Eq.(6)).Forexample,themarginalprobabilityfor
k k
becallednon-loc|al. party1 isgivenbyP(a A )=Tr[MA1r 1], wherer 1 isthe
1| 1 a1
reducedstateforparty1.
Partially-localcorrelations. Partially-localcorrelationsare
thosethatcanbeobtainedfromanN-partitesysteminwhich 3.2 Shareabilityofcorrelations
subsets of the N parties form extended systems, whose in-
ternalstatescanbecorrelatedinanyway(e.g.,signalling), Generalunrestrictedcorrelationsandlocalcorrelationscan
whichhoweverbehavelocalwithrespecttoeachother[25, be shared. The latter fact is proven by Masanes et al. [10]
26].Supposeprovisionallythatparties1,...,k formsucha andthefirstwewillprovehere.However,firstweneedthe
subset and the remaining parties k+1,...,N form another relevant definitions. Shareability of a general unrestricted
5
probabilitydistributionisdefinedasfollows(wherewithout al.[10]provedthat¥ -shareabilityimpliesthattheN-partite
lossofgeneralitywerestrictourselvestoshareabilityofbi- distribution is local (in the sense of Eq. (7)) for all N and
partitedistributions).Abi-partitedistributionP(a,b A,B ,...,Bth)attheconverseholdsaswell. Localcorrelationscanthus
1 1 N
|
isN-shareablewithrespecttothesecondpartyifan(N+1)- besharedindefinitely,andviceversa.
partite distribution P(a,b ,...,b A,B ,...,B ) exists that
1 N 1 N
|
issymmetricwithrespectto(b ,B ),(b ,B ),...,(b ,B )
1 1 2 2 N N
andwith marginalsP(a,bi A,B1,...,BN) equalto the orig- 3.3 Monogamyofcorrelations
|
inal distribution P(a,b A,B ,...,B ), for all i. For nota-
1 1 N
|
tionalclarityweusebiandBi(insteadofaiandAi)todenote Becausegeneralunrestrictedcorrelationsandlocalonescan
outcomesandobservablesrespectivelyforthepartiesother be shared indefinitely they both will not show any mono-
thanthefirstparty.Ifa distributionisshareableforallN it gamy constraints. This implies that partially-local correla-
iscalled¥ -shareable[10]. tions (see Eq. (8)) also do not show any monogamy,since
Shareability of a no-signalling probability distribution thesearecombinationsoflocalandgeneralunrestrictedcor-
isdefinedanalogously:Ano-signallingP(a,b1 A,B1)isN- relations between subsystems of the N-systems. However,
|
shareable with respect to the second party if there exist an andperhapssurprisingly,quantumandno-signallingcorre-
(N+1)-partitedistributionP(a,b1,...,bN A,B1,...,BN)be- lations are not ¥ -shareable and they must therefore show
|
ingsymmetricwithrespectto(b1,B1),(b2,B2),...,(bN,BN) monogamyconstraints, as will now see 9. First, consider a
with marginals P(a,bi A,Bi) equal to the original distribu- verystrongmonogamypropertyforextremalno-signalling
|
tion P(a,b1 A,B1), for all i. The differencebetween share- correlations,mentionedbyBarrettetal. [12]. Supposeone
|
abilityofunrestrictedcorrelationsandofno-signallingcor- has some no-signalling three-party probability distribution
relationsisthatinthefirstcasethemarginalsdependonall P(a,b,cA,B,C)forpartiesa,bandc.Incasethemarginal
|
N+1 settings, whereasin the latter case they only depend distribution
onthetwosettingsAandBi. P(a,bA,B)ofsystemabisanextremalno-signallingcorre-
|
Supposewearegivenageneralunrestrictedcorrelation lation10thenitcannotbecorrelatedtothethirdsystemc:
P(a,b A,B ,...,B ).Wecanthenconstruct
1 1 N
|
P(a,b,cA,B,C)=P(a,bA,B)P(cC), (11)
P(a,b ,...,b A,B ,...,B )= | | |
1 N 1 N
|
P(a,b1|A,B1,...,BN)d b1,b2···d b1,bN, (10) inotherwords,theextremalcorrelationP(a,b|A,B)iscom-
pletely monogamous.Barrett et al. [12] show that this im-
which has the same marginals P(a,b A,B ,...,B ) equal
i| 1 N pliesthatallBell-typeinequalitiesforwhichthemaximalvi-
totheoriginaldistributionP(a,b A,B ,...,B ).Thisholds
for all i, thereby provingthe ¥ -1sh|area1bility. TNhus an unre- olationconsistentwithno-signallingisattainedbyaunique
correlationhavemonogamyconstraints.An exampleis the
strictedcorrelationcanbesharedforallN.Ifwerestrictthe
CHSHinequality,aswillbeshownbelow.
distributionstobeno-signalling,Masanesetal.[10]proved
Extremalno-signallingcorrelationsthusshowmonogamy,
that if the distribution P(a,b A,B ) is N-shareable then it
1| 1 butwhataboutnon-extremalno-signallingcorrelations?Just
satisfies all Bell-type inequalities with N or less different
aswasthecaseforquantumstateswherenon-extremal(mixed
settingsB (thisextendsa similarresultforquantumstates
1 state)entanglementcanbeshared(SeeEq.(1)),non-extremal
byTerhaletal.[27]andWerner[28]).
no-signalling correlations can be shared as well. This can
Thisresultimpliesthatthereexistsalocalmodelofthe
beshownintermsofthewellknownBell-typeexperimen-
form (7) for correlations P(a,bA,B) where the first party
| talsetupwhereeachofthetwopartiesaandbimplements
hasanarbitrarynumberandthesecondpartyhasNpossible
two possible dichotomous observables, A,A and B,B re-
′ ′
measurements if the correlations are N-shareable [10]. In-
spectively. The CHSH inequality B 2 is the only
ab
deed,supposeP(a,bA,B)isshareabletoNparties(labelled |h i| ≤
| non-triviallocalBell-typeinequalityforthissetup[8].Here
B,i=1,...,N). ThecorrelationsbetweenAperformedon
i
party 1 and Bi on party 2 are thus the same as the correla- 9 Intuitivelythiscanbeunderstoodasfollows.Becausetheclassof
tionsbetweenmeasurementsof A onparty1 and B onthe generalunrestrictedcorrelationsisricherthantheclassofno-signalling
i
extra party B. Therefore, the N measurements B ,...,B correlations,andinfactcontainsallpossiblecorrelations,thereisno
i 1 N
possibilityofleavingthisclassofcorrelationswhensharingthem.To
performedbyparty2canbeviewedasonesinglejointmea-
thecontrary,theclassofno-signallingcorrelationsisratherlimited;in-
surementperformedbyN partiesBi (i=1,...,N),anditis deed,itturnsoutthatnotallsuchcorrelationscanbesharedwhilere-
known that there always exists a local model when all but quiringthatonestaysinthisclass.Whentryingtoconstructtheshared
oneofthepartiesperformonlyonemeasurement.Inpartic- no-signallingstatesonewillendupwithresultingstatesthataresig-
nallingratherthanno-signalling.
ular,thisimpliesthattwo-shareablestatescannotviolatethe
10 Ano-signallingcorrelationisextremaliffitisavertexofthebi-
CHSHinequality;seealsosection3.5thatassessesthefoun-
partiteno-signallingpolytopewhichmeansthatanyconvexdecompo-
dational relevance of this result. Furthermore, Masanes et sitionintermsofno-signallingcorrelationsisunique.
6
B =AB+AB +AB AB is called the CHSH polyno- Forlocalcorrelationsnosuchtrade-offasinEq.(12)or
ab ′ ′ ′ ′
−
mial (or CHSH operator in the quantum case) for the bi- (13)holds.Indeed,itispossibletohave12both B =
ab local
|h i |
partitesystemab. 2and B =2,seealsoFigure1.Thisreflectsthefact
ac local
|h i |
No-signallingcorrelationsobeythefollowingtighttrade- thatlocalcorrelationsarealwaysshareable.
offrelationintermsoftheCHSHoperatorsB andB for Letusfinallyconsidercorrelationsthatresultfrommak-
ab ac
partyabandacrespectively,asfirstprovenbyToner[6]: ing local measurementson quantum systems. All quantum
correlationsthat violate the CHSH inequality are monoga-
mousasfollowsfromthefollowingtighttrade-offinequality
Bab ns + Bac ns 4. (12) forathree-partitesystemabcprovenbyToner&Verstraete
|h i | |h i |≤
[5]13:
HereB =AC+AC +AC AC istheCHSHpolynomial
forpartiaecsaandc,an′d B′ab−nsi′st′heexpectationvalue11of hBabi2qm+hBaci2qm≤8, (13)
h i
theCHSHoperatorBab forano-signallingcorrelation(5), where B is the CHSH operator for parties a and b, and
ab
and analogousfor Bac ns. Tightness was shown by Toner analogousforB (here,andinthefollowing,thesubscript
h i ac
[6]: for any pair B , B that obeys (12) there is
ab ns ac ns ’qm’ denotes that the expectation value is determined for
h i h i
a no-signalling correlation with these expectation values.
quantum correlations (9)). The correlations admissible by
A particular multipartite generalisation of (12) for a large
thistrade-offrelationlieintheinteriorofthecircleinFigure
class of linear multi-partite Bell-type inequalities has been
1.TonerandVerstraete[5]haveexplicitlyshowntightness.
recentlyachievedbyPawłowski&Brukner[29].
Thustheinequality(13)givesexactlytheallowedvaluesof
This trade-off relation is depicted in the tilted square
of Figure 1. Extremal no-signalling correlations can attain 12 Here Bab localistheexpectationvalueoftheCHSHoperatorfor
h i
localcorrelations(7)betweenpartyaandb.
B =4, butthen necessarily B =0, and vice
|h abins| |h acins| 13 It should be noted that early results in this direction have been
versa(this is monogamyof extremalno-signallingcorrela-
obtainedbyKrenn&Svozil[30](Section5)andbyScarani&Gisin
tions), whereas non-extremal ones are shareable since the [31](Theorem1).
correlationterms B and B can both be non-
ab ns ac ns
|h i | |h i |
zeroatthesametime.Butnotethatincasetheno-signalling
correlationsare non-localthey can not be shared, i.e., it is
B
not possible that B 2 and also B 2 (a h aci
ab ns ac ns
|h i |≥ |h i | ≥
fact already shown in [10]). This shows that if these non- 4
localcorrelationscanbesharedtheymustbesignalling.Al-
ternatively,this can also be phrased as follows. In orderto 2√2
be non-local with both party b and c, and also remain no-
2
signalling, party a is faced with an unsolvable dilemma in
√2
choosing her measurements, which would need to be dif-
ferent in B and B This feature is termed ‘monogamy
ab ac
of non-local correlations’. As a corollary it follows that if B
ab
h i
N+1partiesa,b ,b ,...,b sharesomecorrelations(e.g.,
1 2 N
via a quantum state) and each chooses to measure one of
two observables,then a violatesthe CHSH inequalitywith
atmostonepartyb.
i
We have seen that for general unrestricted correlations
nomonogamyconstraintshold.Theycanthusreachthelargest
squareinFigure1,i.e., B and B arenotmutually
ab ac
|h i| |h i|
constrainedandcaneachobtainavalueof4,soastogivethe
absolutemaximumofthelefthandsideof(12)whichisthe
value8.Themonogamybound(12)thereforegivesawayof Fig.1 Thespace B - B ofallowablevaluesfortheCHSHoper-
ab ac
discriminatingno-signallingfromgeneralcorrelations:if it atorsforsystemsahbandiahc.Geineralunrestrictedcorrelationscanreach
isviolatedthecorrelationsmustbesignalling.Theselieout- the absolute maximum which is the largest square with edge points
( 4, 4). All quantum correlations liewithinthe circle, and all no-
sidethetiltedsquarebutinsidethelargestsquareinFigure ± ±
signallingcorrelationsliewithinthetiltedsquare.Forcomparisonthe
1.
localcorrelationsarealsoshown.Theseliewithinthesquarewithedge
points( 2, 2).Thecorrelationsobtainablebyorthogonal measure-
± ±
11 The subscript ‘ns’ indicatesthattheexpectationvalue isfor no- mentsonseparabletwo-qubitstatesliewithinthesmallestsquarewith
signallingcorrelations(5). edgepoints( √2, √2).Figureadaptedfrom[5].
± ±
7
( B , B ). Note thatas a corollarythe Tsirelson ‘qm’fromtheexpectationvalues.Letusfirstwrite
ab qm ac qm
h i h i
inequality[32]follows: B , B 2√2.
Justaswasthecasef|ohrnaob-isqimg|na|lhlingaccioqrmr|el≤ations,quan- Babcosq +Bacsinq =(A+A′)Bcosq +(A−A′)B′cosq
tum correlationsshow an interesting trade-offrelationship: +(A+A′)Csinq +(A A′)Csinq . (15)
−
Incasethequantumcorrelationsbetweenpartyaandbare
Nextweusethefactthatinthiscontextitissufficienttocon-
non-local(i.e.,when B >2)thecorrelationsbetween
ab qm sider qubits and projective measurements that are real and
|h i |
partiesaandccannotbenon-local(i.e.,necessarily B
ac qm tracelessonly[35,5].LetusthusexpressAandA interms
|h i |≤ ′
2),andviceversa(cf.[31]).Thesenon-localquantumcorre- oforthogonalPauliobservabless ,s formeasurementsin
z x
lationscanthusnotbeshared.Furthermore,incasetheyare thezandxdirectionrespectively:A=cosgs +sings and
x z
maximallynon-local,i.e., Bab qm =2√2theothermust A =cosgs sings . This gives A+A =2cosgs , A
|h i | ′ x z ′ x
beuncorrelated,i.e.,itmustbethat Bac qm =0,andvice A =2sings −.Takingtheexpectationvalueof(15)gives −
|h i | ′ z
versa.
Note that if ab and ac each share a maximally entan- Babcosq ab+ Bacsinq ac = (16)
|h i h i |
gled pair, there are sets of measurements such that either 2 s B cosg cosq + s B sing cosq
x ab z ′ ab
|h i h i
dhBoeasbniqomtcoornhtBradaciicqtm(1i3s)2:√it2is.rBeuqtu,iaresdntohtaetdthbeymToenaesurr[e6m],ethnitss +hs xCiaccosg sinq +hs zC′iacsing sinq |
performedbypartyaarethesameinbothB andinB . The right hand side can be considered to be twice the ab-
ab ac
Thisisanalogoustotherequirementthatwasneededinsec- solute value of the inproduct of the two four-dimensional
tion 2 to show monogamyof entanglement, namely that b vectors aaa=( s xB ab, s zB′ ab, s xC ac, s zC′ ac) and bbb=
andcareentangledwiththesamequbitofa. (cosg cosq ,sihng coisq ,hcosg isinqh,singisinhq ).Iifwenowap-
ply the Cauchy-Schwartzinequality (aaa,bbb) aaa bbb we
Thecorrelationsthatseparablequantumstatesallowfor
areshareable.Indeed,inthe B - B planeofFig- find,forallq : | |≤|| |||| ||
ab qm ac qm
h i h i
ure 1 such correlationscan reachthe fullsquare with edge B cosq + B sinq
ab ab ac ac
length 2. However, when considering qubits and measure- |h i h i |
mentsthatarerestrictedtoorthogonalonesonly(e.g.,Pauli ≤2qhs xBi2ab+hs zB′i2ab+hs xCi2ac+hs zC′i2ac×
spin observables s ,s ,s in the x,y,z-directions) one ob-
x y z
tainstighterbounds;see[33].Insuchacasethepossibleval- cos2g (cos2q +sin2q )+sin2g (cos2q +sin2q )
q
uesarerestrictedtothesmallestsquareofFigure1: B ,
Bac qm √2.Butagainthereisnotrade-offont|hheshabairqem-| ≤2q2(hs xi2a+hs zi2a)
|h i |≤
ability of the correlationsin separable states since this full 2√2 1 s 2 (17)
squarecanbereached. ≤ q −h yia
2√2. (18)
≤
Thisproves(14).Herewehaveusedthat s 2 + s 2 +
h xiqm h yiqm
3.4 Astrongermonogamyrelationfornon-localquantum s 2 1forallsinglequbitquantumstates,andforclarity
h ziqm≤
correlations we have used the subscripts ab, ac and a to indicate with
respecttowhichsubsystemsthequantumexpectationvalues
Wewillnowgiveanalternativesimplerproofoftheinequal- aretaken.Using(17)weobtain
ity (13) thanwas givenbyToner& Verstraete[5], andone
B 2 + B 2 8(1 s 2), (19)
thatalsoallowsustostrengthenthisresultaswell.Theproof h abiqm h aciqm≤ −h yia
usestheideathat(13),whichdescribestheinteriorofacir- whichstrengthenstheoriginalmonogamytrade-offinequal-
cleinthe Bab - Bac plane,isequivalenttotheinteriorof ity(13).Analternative,butsimilarstrengtheningof(13)was
h i h i
thesetoftangentstothiscircle.Itisthusacompactwayof alreadyfoundin[5]: B 2 + B 2 8(1 s s 2 ).
writingthefollowinginfinitesetoflinearequalities So far we have ohnlyabfoiqcmusedhonacsiuqmbs≤ystem−sahbyanydibacc,
andnotonthe subsystembc.One couldthusalso consider
S =max Sq qm 2√2, (14) thequantity Bbc qm.Theabovemethodwouldgivethein-
q h i ≤ h i
tersection of the three cylinders B 2 + B 2 8,
h abiqm h aciqm ≤
B 2 + B 2 8, B 2 + B 2 8.But itis
wherewehaveused x2+y2=maxq (cosq x+sinq y),and h abiqm h bciqm ≤ h aciqm h bciqm ≤
known[5]thatthisboundisnottight.
whereSq =cosq Bpab+sinq Bac. It might be tempting to think that because of these re-
We will now prove this by showing that B cosq +
B sinq 2√2forallq ,usingamethod|hpreasbentedby sultswecouldhavethefollowinginequality,whichiseven
ac iqm|≤ strongerthan(13):
[34] in a different context. In this proof we only consider
quantum correlations, so for brevity we drop the subscript B 2 + B 2 + B 2 8. (20)
h abiqm h aciqm h bciqm≤
8
However, this is not true. For a pure separable state (e.g., physicalmessageofBell’stheoremisthatquantummechan-
000 ) the left hand side has a maximum of 12, which vi- icalcorrelationsareingeneralnot2-shareable;andnotthat
| i
olates (20). But inequality (20) is true for the exceptional quantummechanicsisnon-localinsomewayoranother.
case that we have maximal violation for one pair, say ab, Beforeassessingthisargumentletusseewhy2-shareability
sinceweknowfrom(13)that B and B forthe impliestheCHSHinequality.Considertwoparties,denoted
ac qm bc qm
h i h i
other two pairs must then be zero. We can see the mono- by 1 and 2. Assume that all possible correlations between
gamytrade-offatwork:incaseofmaximalviolationofthe parties1and2are2-shareabletotwootherparties,denoted
CHSH inequality (i.e., for maximal entanglement) the left 1 and2,thatconceivablyexist.Eachpartyhasasinglesys-
′ ′
hand side of (20) has a maximumof 8, whereasin case of tem and subjects it to measurementof a single observable,
no violation of the CHSH inequality it allows for a maxi- which we denote by A,C,B,D respectively.See Figure 2.
′ ′
mumvalueof12,whichcanbeobtainedbypureseparable Then for the four possible outcomes of the measurements
states.Thusweseetheoppositebehaviorfromwhatishap- we get a(c+d )+b(c d )= 2 which implies for the
′ ′ ′
− ±
peningintheordinaryCHSHinequality:fortheexpression expectationvaluesoftheproductofthelocaloutcomes
consideredhere (i.e., the left-hand side of (20)), separabil-
AC + AD + BC BD 2. (22)
itygiveshighervalues,andentanglementnecessarilylower ′ ′ ′ ′
|h i h i h i−h i|≤
values.
Acorrectboundisobtainedfrom(17)andthetwosim-
ilar onesforthe othertwo expressionshBabi2qm+hBbci2qm A C
andhBaci2qm+hBbci2qm.Thisgives: 1 2
B 2 + B 2 + B 2
h abiqm h aciqm h bciqm≤
12 4( s 2+ s 2+ s 2). (21)
− h yia h yib h yic 1′ 2′
B′ D′
However,itisunknownifthisinequalityistight.
3.5 InterpretingBell’stheorem
Fig.2 Parties 1,2,1′ and 2′ measure observables A,C,B′,D′ respec-
tively.Dottedlinesindicatewhichpartiesarejointlyconsideredinthe
In subsection 3.2 it was pointed out that non-local corre- expression(22).FiguretakenfromSchumacher[13].
lations,eitherquantumorno-signalling,canbecompletely
monogamous,whereas¥ -shareabilityandlocalityofcorre-
Wenowinvoke2-shareabilityofthecorrelationstoper-
lationsareequivalentproperties.Furthermore,itwasshown
formthefollowingcounterfactualreasoning.Byassumption
thatthereexistsalocalmodeloftheform(7)forcorrelations
thecorrelationsbetweenparties1and2arethesameasbe-
P(a,bA,B)whenthefirstpartyhasanarbitrarynumberand
| tweenparties1and2.Therefore,ifparty2wouldhavemea-
thesecondpartyhasNpossiblemeasurementsifandonlyif ′
sured the observable that party 2 measured, the observed
thecorrelationsareN-shareable. ′
correlations between the measurements of this observable
ThishasledSchumacher[13]toargueforanewviewon
by party 2 and the measurement results of party 1 would
Bell’stheorem,which,accordingtocommunisopinio,states
thatquantummechanicsisnon-local14:itisatheoremabout be the same as between party 1 and 2′. We can thus set
AD = AD . Analogously we can set BC = BC and
the shareabilityof correlations,anditsphysicalmessageis h ′i h i h ′ i h i
not at all about issues of locality or local realism15. Schu- B′D′ = BD .Thereforeweobtainfrom(22):
h i h i
macherarguesthat2-shareabilityofcorrelationsissufficient
AC + AD + BC BD 2, (23)
togetaconflictwithquantummechanicsbecauseitimplies |h i h i h i−h i|≤
theCHSHinequalityfromwhichwealreadyknowthatsuch which is the CHSH inequality from which one can prove
a conflict follows. He also stresses that the assumption of Bell’stheorem.Notethatcrucialintheargumentisthatthe
2-shareability of correlations is a weaker assumption than 2-shareabilityjustifiesthecounterfactualreasoning.
the assumption of full-blown local realism, since the latter Althoughtheaboveargumentindeedshowsthat2-shareability
implies¥ -shareability.Fromthisheconcludesthatthereal ofcorrelationsalreadyimpliesaconflictwithquantumme-
chanics,webelieveSchumacher’sdismissalofissuesoflo-
14 Tobemorespecific: Bell’stheorem statesthatquantumcorrela- cality or local realism in interpreting Bell’s theorem to be
tionsexistthatcannotbereproducedintermsoflocalcorrelationsof
ratherartificialandwanting.
theform(7).
15 Foranoutlineofthedoctrineoflocalrealismsee[24]orchapter First of all, there is the elementary logical point that,
2of[23]. given the violation of the Bell inequality, anything which
9
wouldimplythatitshouldhold,isfalse.Thus,despiteSchu- ofparties¥ -shareabilityandlocalityareequivalentproper-
macher’s argument,it is indeed still the case that quantum ties17,cf.section3.2.
mechanicsisnon-localinthesensethatsomequantumcor-
relationscannotbegivenafactorisableformintermsoflo-
calcorrelations,asin(7). 4 Monogamyofnon-localquantumcorrelationsvs.
monogamyofentanglement
Schumacherwashowevertryingtoargueforwhatphys-
ical message one should take home from Bell’s theorem.
Two types of monogamy and shareability have been dis-
Thus although logically Bell’s theorem has to do with is-
cussed: of entanglement and of correlations (in sections 2
suesof locality –aswas justpointedout–,Schumacherbe-
and3respectively).Thesearedifferentinprinciple,although
lievesthephysicalmessageistobesoughtelsewheresince
sometimestheygohandinhand.Monogamy(shareability)
hehasanalternative(alledgedly)weakersetofassumptions
of entanglementis a property of a quantum state, whereas
thanlocalrealismthatimplytheCHSHinequality.Weagree
monogamy(shareability)ofcorrelationsisnotsolelydeter-
thattheweakerthesetofassumptionsthatleadtotheBell-
mined by the state of the system under consideration, but
inequality, the more physically relevant the argument be-
itisalsodependentonthespecificsetupusedto determine
comes16.
thecorrelations.Thatis,inthelatercaseitiscrucialtoalso
But we question –and here is our second point of cri- knowthe numberof observablesper party and the number
tiqueagainstSchumacher’sdismissalofissuesoflocalityin ofoutcomesperobservable.Itisthuspossiblethataquan-
interpretingBell’s theorem–whetherSchumacher’sderiva- tum state can give non-localcorrelations that are monoga-
tionisindeedlogicallyweakerthanstandardderivationsof mouswhen obtainedin one setup, butwhich are shareable
Bell’s theorem.For allthatis neededto getBell’s theorem whenobtainedinanothersetup.Anexampleofthiswillbe
is the CHSH inequality, and in order to get this from the given below. This example also shows that shareability of
requirementsof the doctrineof localrealism we onlyneed non-localquantumcorrelationsandshareabilityofentangle-
to assume that localrealism holdsjust for measurementof mentarerelatedinanon-trivialway.
four differentobservables:two for party 1 (e.g., A,A′) and Masanes et al.[10] already remarked (and as was dis-
twoforparty2 (e.g.,B,B′).Onlywith respectto thesetwo cussed above) that, if we consider an unlimited number of
parties and these four observables we need to assume the parties,localityand¥ -shareabilityofbi-partitecorrelations
correlations to be of the local form (7). It is thus not nec- are identical properties. This is analogous to the fact (also
essary to assume full blown local realism for an unlimited discussed above and also obtained by [10]) that quantum
numberofobservablesandparties. separabilityand¥ -shareabilityofaquantumstateareiden-
ticalinthecaseofanunlimitednumberofparties.Butifwe
In conclusion,for the purposesof obtainingthe CHSH
consider shareability with respect to only one other party
inequalitythe assumptionof 2-shareabilitysuffices, and so
the analogy between locality, separability and shareability
does assuming local realism for measurementof only four
breaksdown.Insteadwewillshowthefollowingresult:Share-
observables(twosetsoftwo).Weseenophysicalreasonto
abilityofnon-localquantumcorrelationsimpliesshareabil-
believethatthefirstassumptionis weakerthanthe second.
ity of entanglement of mixed states (that gives rise to the
Onemightobjecttothisreasoningbystatingthatitisunnat-
non-locality),butnotvice versa.Theproofforthepositive
uraltorequirelocalrealismonlyformeasurementsbetween
implicationrunsasfollows.Becausebyassumptionthecor-
fourdifferentobservables.For,afterall,wecanalwaysthink
relations are shareable they are identical for parties a and
of some extra observables that can be measured over and
bandaandc.Thequantumstatesr andr forthejoint
abovethefouralreadyspecified.Butthisobjectionlosesits ab ac
systemsabandacthataresupposedtogiverisetothesecor-
forceonceonerealisesthatrequiring2-shareabilityinstead
of¥ -shareabilityappearstobejustasunnaturalasrequiring relationsmustthereforealsobeidentical,i.e.,theyarethus
shareable. Furthermore, because the correlations are non-
local realism for only four observables and two parties in-
local,thesequantumstatesmustbeentangled.Theyfurther-
steadofforanunlimitednumberofobservablesandparties.
more must be non-pure,i.e., mixed, because entanglement
For,afterall,wecanalwaysthinkofsomeextrapartyover
of pure states can notbe shared. This concludesthe proof.
andabovethetwoalreadyconsidered.Furthermore,webe-
Below we give an example of this and show that the con-
lieveittobetellingthatinthelimitofanunlimitednumber
verse implication does not hold. In order to do so we will
firstdiscussmethodsthatallowonetorevealtheshareabil-
ityofnon-localcorrelations.
16 ForgiventhefactthattheBellinequalityisviolated,wecanex-
cludemoreandmorepossibledescriptionsofnaturefromusingderiva- 17 Notethatlocalityheremeansthatthecorrelationisoftheform(7),
tionsoftheBellinequalitythatuseweakerandweakerassumptions. withoutfurtherqualificationofthenumberofpartiesorthenumberof
i.e.,moreandmorepossiblecandidate-theoriesarethenrejected. possiblemeasurementsperparty.
10
Ingeneralabi-partitequantumstatecanbeinvestigated inequalityreads:
using differentsetups that each have a differentnumberof
C := AB + AB + A B + AB + AB + AB
observablesperpartyandoutcomesperobservable.Ineach local ′ ′′ ′ ′ ′ ′′
h i h i h i h i h i h i h i
suchasetupthemonogamyandshareabilityofthecorrela- A′′B′ A′B′′ + A + A′ B B′ 4,
−h i−h i h i h i−h i−h i≤
tionsthatareobtainableviameasurementsonthestate can (24)
beinvestigated.ThisisgenerallyperformedviaaBell-type
whereforbrevitythe subscript‘local’is omittedin the ex-
inequality that distinguishes local from non-local correla-
pectation values in the middle term. Any local correlation
tionsforthespecificsetupused.
must obey this inequality. Collins & Gisin [36] show that
Letusfirstassumethecaseoftwopartiesthateachmea- thefullyentangledpurethree-qubitstate
sure two dichotomous observables. For this case the only
relevant local Bell-type inequality is the CHSH inequality f =m 000 + (1 m 2)/2( 110 + 101 ) (25)
| i | i q − | i | i
for which we have seen that the Toner-Verstraete trade-off
gives for some values of m bi-partite correlations between
(13)impliesthatallquantumnon-localcorrelationsmustbe
partyaandbofsubsystemabandbetweenaandcofsub-
monogamous:itisnotpossibletohavecorrelationsbetween
system ac such thatthe inequality(24) is violatedfor both
partyaandbofsubsystemabandbetweenaandcofsub-
thesecorrelations: C = C >4.Thisshowsthat
system ac such that both |hBabiqm| and |hBaciqm| violate someofthenon-lochalacboirqrmelatihonsacbieqtmweenpartyaandbcan
thelocalbound.
thusbesharedwithpartyaandc.
Since f isa pureentangledthree-qubitstate thetwo-
It is tempting to think that those entangled states that
qubit redu|ceid states r and r are mixed. Furthermore,
showmonogamyofnon-localquantumcorrelationswillalso ab ac
sincethestate f issymmetricwithrespecttoqubitband
showmonogamyofentanglement.This,however,isnotthe
| i
c these reducedstates are identical. They must also be en-
case. For example, three-party pure entangled states exist
tangled because they violate the two-party inequality (24).
whosereducedbi-partitestatesareidentical,entangledand
abletoviolatetheCHSHinequality(e.g.,theW-state y = Therefore,thetwo-qubitmixedentangledstater abisshare-
( 001 + 010 + 100 )/√3hassuchreducedbi-parti|tesitates). abletoatleastoneotherqubit.Thisshowsthattheinequal-
| i | i | i ity(24)issuitabletorevealshareabilityofentanglementof
These reduced bi-partite states are mixed and their entan-
mixedstates.
glement is shareable, yet, as shown above, they show mo-
Itwouldbeinterestingtoinvestigatethemulti-partiteex-
nogamyofthenon-localcorrelationsobtainablefromthese
tensionoftheseresults.ApreliminaryinvestigationforN=
states in a setup that has two dichotomousobservablesper
3wasperformedbySeevinck[37].Therethemonogamyof
party. Thus we cannot infer from the monogamy of non-
bi-separable three-partite quantum correlations that violate
local correlations that quantum states responsible for such
a three-qubit Bell-type inequality that has two dichotomic
correlationshavemonogamyofentanglement;someofthem
measurements per party was investigated. For the specific
haveshareablemixedstateentanglement.Consequently,the
Bell-type inequality under study it was found that maxi-
study of the non-localityof correlationsin a setup thathas
mal violation by bi-separable three-partite quantum corre-
twodichotomousobservablesperparty,therebyconsidering
lationsismonogamous.Thisistobeexpectedbecausemax-
theCHSHinequality,doesnotallowonetorevealshareabil-
imalquantumcorrelationsare obtainedfrompurestate en-
ityoftheentanglementofbi-partitemixedstates.
tanglement which is monogamous. However, it was found
Nevertheless, it is possible to revealshareability of en- that the correlations that give non-maximal violations can
tanglement of bi-partite mixed states using a Bell-type in- beshared.Thisindicatesshareabilityofthe non-localityof
equality.But for that it is necessary that the non-localcor- bi-separablethree-partitequantumcorrelations.
relations which are obtained from the state in question are
notmonogamous,i.e.,asetupmustbeusedinwhichsome Monogamyofnon-localcorrelationsisnotuniversal. Both
non-localquantumcorrelationsturnouttobeshareable.We the aboveresultsby Collins & Gisin [36] andby Seevinck
havejustseenthatthecaseoftwodichotomousobservables [37]indicatethatnon-localityofcorrelationscanbeshared.
perparty,andthusthe CHSH inequality,was shownnotto However, as we have stated before, Pawłowski & Brukner
suffice.However,addingoneobservableperpartydoessuf- [29] have provena particular multipartite generalisationof
fice.Considerthesetupwhereeachofthetwopartiesmea- the monogamy constraint (12) for a large class of linear
sures three dichotomicobservables, which will be denoted multi-partiteBell-typeinequalities.Butthemonogamycon-
byA,A,A andB,B,B respectively.Collins&Gisin [36] straintsconsideredbyPawłowski&Bruknercontainasmany
′ ′′ ′ ′′
have shown that for this setup only one relevantnew Bell- Bell-type polynomialsB, C, etc. (or Bell-operatorsin the
typeinequalitybesidestheCHSHinequalitycanbeobtained quantumcase)astherearedifferentsettingsforthedifferent
(modulo permutations of observables and outcomes). This parties.Themonogamyconstraint C + C 4used
ab ac
|h i| |h i|≤