An entropic approach to local realism and noncontextuality

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by  Rafael Chaves

An entropic approach to local realism and noncontextuality
Rafael Chaves1 and Tobias Fritz1
1ICFO–Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
For any Bell locality scenario (or Kochen-Specker noncontextuality scenario), the joint Shannon
entropiesoflocal(ornoncontextual)modelsdefineaconvexconeforwhichthenon-trivialfacetsare
tight entropic Bell (or contextuality) inequalities. In this paper we explore this entropic approach
and derive tight entropic inequalities for various scenarios. One advantage of entropic inequalities
is that they easily adapt to situations like bilocality scenarios, which haveadditional independence
requirements that are non-linear on the level of probabilities, but linear on the level of entropies.
Another advantage is that, despite the nonlinearity, taking detection inefficiencies into account
turnsouttobeverysimple. Whenjoint measurementsareconductedbyasingledetectoronly,the
detectorefficiency for witnessing quantumcontextuality can bearbitrarily low.
PACSnumbers: 03.67.-a,03.67.Mn,42.50.-p
2 I. INTRODUCTION theinformation-theoreticapproachintroducedbyBraun-
1
stein and Caves [4], it was shown that if local realism
0
holds, then the joint Shannon entropies carried by the
2 Quantum mechanics predicts that experiments per-
measurements on two distant systems must satisfy cer-
formed by space-like separated and independent ob-
n tain inequalities, which can be regarded as entropic Bell
a servers may display nonlocal correlations, which cannot
inequalities. One advantage of this entropic approach
J beexplainedsolelybypastinteractions. Theassumption
is that the inequalities do not depend on the number
0 that physicalquantities have wellestablishedvalues pre-
of outcomes of the measured observables, which implies
3 vioustoanymeasurementandthatsignalscannottravel
faster than the speed of light, as stipulated by special that they can readily be applied to quantum systems of
] arbitrarylocaldimensionandto the considerationofde-
h relativity, entails limits on the correlationsthe observers
tection inefficiencies [5].
p may obtain. Such restrictions, usually expressed as Bell
- inequalities [1], may be surpassed within quantum the-
t
n ory when the observers share entangled quantum states,
a and it is in this sense that the quantum correlations are
u In this paper, our aim is to further develop this en-
nonlocal.
q tropic program to Bell inequalities and also introduce
[ Similarly,noncontextualityis a classicallywell-defined entropic inequalities to the study of (quantum and post-
property of mutually compatible observables. Two ob- quantum) contextuality. As we discuss in Sec. II and in
2
servables A and B are mutually compatible if the result
v more detailin [6], the standardinformation-theoreticin-
for the measurement of A, even if not performed, does
0 equalities for joint Shannon entropies (monotonicity and
not depend on the prior or simultaneous measurement
4 submodularity)defineaconvexcone[7]whoseprojection
3 of B and vice versa. The notion of noncontextuality is tothejointentropiesofjointlymeasurableobservablesis
3 precisely captured by the Kochen-Specker (KS) theorem
another convex cone, whose facets correspond to all the
. [2], stating that no noncontextualhidden variable model
1 optimal Shannon-type entropic Bell inequalities.
0 (NCHV) can reproduce the results of quantum mechan-
2 ics. Interestingly, as opposed to Bell’s theorem, the KS
1 theoremisnotstate-dependentanditholdsforanyphys-
: ical system (composite or not) with a state space of di-
v From this geometrical point of view, we investigate
i mension higher than two.
X the family of chained entropic inequalities derived in
Focusing on the Bell scenario, once the particular sce- Ref. [4]—which we have shown to be the tightest en-
r
a nario is defined—the number of spatially separated par- tropic inequalities in the appropriate scenarios [6]—and
ties,the numberofmeasurementssettingsforeachparty lookforviolationsinquantumandpost-quantumproba-
and the number of outcomes for each setting—the as- bilistic theories. In particular, we show how an entropic
sociated local (realistic) models form a convex set with contextuality inequality can be violated—for joint mea-
a finite number of extremal points, an object known as surements of the commuting observables—even for arbi-
the local polytope [3]. The tight Bell inequalities are the trarilylowdetectionefficiencies. Moreover,inSec.IIIwe
non-trivialfacetsboundingthelocalpolytope. Giventhis analyzea possibleconectionbetweenthe entropicCHSH
geometricpicture,anonlocalmodel isapointoutsidethe inequality and nonlocality distillation [8–13]. For a bilo-
local polytope, or, equivalently, a point which violates a cality scenario [14] (a Bell scenario with two indepedent
Bell inequality. sourceswhich allowentanglementswapping[15]), wede-
It is clear that to properly understand nonlocality, it rive in Sec. IV all the relevant tight (Shannon-type) en-
shouldbeconsideredfromasmanyaspectsaspossible. In tropic inequalities.
2
every jointly measurable set of observables. We use no-
tationlikeP(1,0X ,X )fortheprobabilityofobtaining
3 5
|
theoutcomesX =1andX =0inajointmeasurement
3 5
of X and X (assuming that X ,X ).
3 5 3 5
{ }∈M
If S and S′ S, then one can take the marginal
∈M ⊆
ofthedistributionassignedtoSandobtainadistribution
over the outcomes of the observables in S′. Naturally,
thismarginalizeddistributionshouldbetheoneassigned
FIG. 1: (Color online) Some contextuality and nonlocality toS′. Requiringthispropertyleadstothemarginalmod-
scenarios; the number of outcomes of each observable is ar- els of [6], or, equivalently, to the sheaf condition and the
bitrary. a. CHSH scenario, 2 parties with 2 measurements empirical models of[17]. InthecaseofBellscenarios,the
settingseachb. Klyachkoscenario,5observablesarrangedin marginal models are precisely the no-signaling boxes.
acyclicconfigurationsuchthateachobservableiscompatible Noncontextualhidden variables. Wenowdefinewhen
withitsneighborsc. GeneralizationoftheKlyachkoscenario,
a marginal model is contextual. The most intuitive the-
withnobservablesinacyclicconfiguration(then-cycle[16]).
ories of physics are those where there exists a certain
Theuniquenon-trivialentropicinequalityinallthesecasesis
“hidden” variable λ, distributed according to probabili-
given by(3).
ties ̺(λ) 0 with ̺(λ) = 1, such that λ determines
≥ λ
the complete futurePbehavior of the system. Here, com-
pleteness means that the distribution P(xX ,λ) of any
i
|
II. MARGINAL SCENARIOS AND ENTROPIC observable X , given a certain value of λ, should be in-
i
INEQUALITIES dependent of the outcome distributions of all other ob-
servables. This implies that when X and X are jointly
i j
In the following, we collect the basic definitions and measurable, then their outcome distribution is given by
results concerning entropic inequalities. For a more de-
tailed discussion, we refer to Ref. [6].
Marginal scenarios. The concept of marginal sce- P(xi,xj Xi,Xj)= ̺(λ)P(xi Xi,λ)P(xj Xj,λ), (1)
| | |
nario subsumes both Bell scenarios and contextuality Xλ
scenarios. A marginalscenario is defined by specifying a
and similarly for cases where more than two observables
set of observables X ,...,X for which certain subsets
1 n
are jointly measured. If there exist conditional distri-
areknowntobecompatibleandcanbejointlymeasured.
butions P(x X ,λ) and a hidden variable distribution
If a subset of these observables can be jointly measured, i| i
̺(λ) such that (1) holds for all jointly measurable pairs
then so can any smaller subset; therefore, the collec-
X ,X and more generally for all S , then
tion of subsets of jointly measurable observables should { i j} ∈ M ∈ M
we say that we have found a noncontextual hidden vari-
be closed under taking smaller subsets. These are the
able model, and the given marginal model P is noncon-
marginal scenarios which we have discussed in [6]; see
textual; otherwise P is called contextual. In the case of
thereformorebackgroundandreferencesonthemarginal
Bellscenarios,thenoncontextualhiddenvariablemodels
problem. The measurement covers of [17] capture pre-
are precisely the local hidden variable models, in which
cisely the same idea in a slightly different way.
casewe alsouse the standardterminologyof“local”and
Definition 1. A marginal scenario is a collection “nonlocal”.
M
= S ,...,S of subsets S X ,...,X such Following [16, Thm. 6] or [17, Thm. 8.1], we note that
1 |M| i 1 n
Mthat if{S and S}′ S, then als⊆o S{′ . } the noncontextuality of P is equivalent to the existence
∈M ⊆ ∈M
of a joint distribution
In this sense, every Bell scenario is a marginal sce-
nario [17, Sec. 2.4.1]: the collection of observables P(x ,...,x X ,...,X )=p(x ,...,x ) (2)
1 n 1 n 1 n
X ,...,X should comprise all observables of all par- |
1 n
ties, and a subset of these observables is jointly mea- which marginalizes to the given distributions for all S
surable if it does not contain two different observables . ∈
of the same party. For example in the bipartite case, MThe main question is: how is it possible to decide
with Alice having access to observables A0,...,Am−1 whetheragivenmarginalmodelP inamarginalscenario
and Bob to B0,...,Bm−1; if we write Xi = Ai−1 and is contextual or noncontextual?
Xi+m =Bi−1 fori=1,...,m,thenthe setofallobserv- MEntropic inequalities. Fromajointprobabilitydistri-
ablesis X ,...,X ,andthesubsetsofjointlymeasur-
1 2m bution (2), one can define the associated Shannon en-
{ }
ableobservablesaretheemptysubset,theone-observable
tropy
subsets, and the two-observable subsets X ,X where
i j
{ }
i m and j >m. H(X ...X )
≤ 1 n
Inaphysicalrealizationofamarginalscenario ,one
mmeeaasnusrtehsastoomneeajossinigtnsstaatjiositnictspfroorbaebvielrityySdis∈triMbuM.tioTnhtios =−x1X,...,xnp(x,...,xn)log2p(x1,...,xn)
3
More generally, marginalizing the joint distribution to a unique normal form when expressed in terms of joint
anysubsetS oftheobservablesgivesajointentropy entropies [21, Sec. 13.2].
∈M
H(X ), where we write X for the tuple of observables Besidesthis,therelevanceofourgeneralframework[6]
S S
(X ) . ThisjointentropyH(X )isalsodefinedinany liesinthefactthatthestandardcomputationalgeometry
i i∈S S
marginal model, since the distribution of X is known methods like Fourier-Motzkin elimination, which have
S
for S . been extensively used to characterize tight Bell inequal-
∈M
As has first been noticed in [4] and as we have de- ities in probability space, can also be applied to derive
veloped more formally in a general framework [6], the tight (Shannon-type) entropic contextuality inequalities
noncontextuality of P, i.e. the existence of a joint distri- and tight (Shannon-type) entropic Bell inequalities. Un-
bution (2), implies that the H(X ), for S , satisfy fortunately,thesecomputationsareverydemanding: ap-
S
∈ M
certain inequalities which may be violated in some con- plying the methods described in [6] to the tripartite Bell
textual models. scenariowithtwoobservablesperparty,wehavenotbeen
able to fully characterize the tight (Shannon-type) en-
Definition 2. An entropic contextuality inequality is a
tropic Bell inequalities in this scenario.
linear inequality in the H(X ) for S which is satis-
S Inthefollowing,weanalyzethecasen=4(the CHSH
∈M
fied whenever P is noncontextual. In the special case of
scenario) and the case n = 5 (the Klyachko scenario) in
a Bell scenario, we usetheterm entropicBellinequality.
somemoredetailandanalyzeinparticulartheirviolation
by marginals models arising from quantum theory.
Ifsomemarginalmodelviolatesacertainentropiccon-
textuality inequality, then this inequality has witnessed
the contextuality of the marginal model.
A. The CHSH scenario
In Ref. [6], we have classified the entropic contextual-
ity inequalities in the n-cycle marginal scenarios. This
family of scenarios is defined by starting with any num- Inthe usualCHSH scenario[18], there aretwodistant
ber n 3 of observables X ,...,X and assuming that parties A and B, each measuring one of two observables
1 n
≥
Xi and Xi+1 are pairwise jointly measurable for all A0,A1 and B0,B1, respectively. Each of these observ-
i = 1,...,n, where we write X = X for ease of no- ables is takento havetwopossible outcomes, sothat the
n+1 1
tation. Nootherpairsofobservablesareassumedjointly set of outcomes can be taken to be 1,+1 . As usual
{− }
measurable, and no triples of observables are assumed for Bellscenarios,we take observablesto be jointly mea-
jointly measurable. For n = 4, Fig. 1a shows that this surable when they belong to different parties. In our
can be identified with the CHSH Bell scenario [18]. For framework, this can be described by the marginal sce-
n=5(see Fig. 1b)it is the marginalscenarioconsidered nario where the collection of jointly measurable sets of
by Klyachko [19, 20], and hence we call it the Klyachko observables is given by
scenario. Forgeneraln,itcanbevisualizedasann-sided
polygon (Fig. 1c). Our result in [6] is that the inequal- , A0 , A1 , B0 , B1 ,
∅ { } { } { } { }
ities derived in [4] are a complete set of tight entropic A ,B , A ,B , A ,B , A ,B .
0 0 0 1 1 0 1 1
{ } { } { } { }
inequalities inthese scenarios. (Anentropicinequality is
tight when no other entropic inequality can be strictly This is illustrated in Fig. 1a. The CHSH inequality [18]
better than this one, so that a complete set of tight en-
tropic inequalities completely characterizes the regionof CHSH
(4)
noncontextualmarginalmodelsinentropyspace.) Stated = A B + A B + A B A B 2
0 0 0 1 1 0 1 1
h i h i h i−h i≤
more formally:
together with its equivalent variants is a necessary and
Theorem 3 ([6]). A marginal model in this scenario is
sufficient condition for noncontextuality (i.e. Bell local-
entropically noncontextual if and only if the entropic in-
ity) in this scenario.
equality
By taking n = 4 and renaming the observables occur-
ing in (3), we obtain the entropic inequality [4]
H(X X ) + H(X ) H(X X ) (3)
i i+1 j j j+1
≤
j6=Xi,i+1 Xj6=i H(A B )+H(A )+H(B )
1 1 0 0
(5)
H(A B )+H(A B )+H(A B ).
holds for all i=1,...,n. 0 0 0 1 1 0
≤
In principle, one may also want to consider inequal- In order to emphasize the similarity with (4), we can
ities containing derived entropic quantities like condi- rewrite this in terms of mutual information as
tional entropies and mutual information. However,since
thesederivedquantitiesarethemselvesnothingbutlinear I(A0 :B0)+I(A0 :B1)+I(A1 :B0) I(A1 :B1)
−
combinationsofjointentropies,everyentropicinequality H(A ) H(B ) 0.
0 0
− − ≤
containing the former can be rewritten in terms of the
latter. In fact, as there are no linear relations between The last two terms in the left-hand side are analogous
jointentropies,everylinearentropicinequalityturnsinto to the classical bound of 2 in (4). We abbreviate the
4
notation of conditional probabilities,
P(a,bx,y)
|
1+( 1)aA 1+( 1)bB (7)
x y
= ψ − − ψ .
(cid:28) (cid:12)(cid:18) 2 ⊗ 2 (cid:19)(cid:12) (cid:29)
(cid:12) (cid:12)
(cid:12) (cid:12)
For ease of l(cid:12)ater notation, we stipulate that th(cid:12)e observ-
ables A and B are 1-valued, but the outcomes a, b
x y
±
are 0,1 -valued.
{ }
As shown in [4], quantum correlations of the form (7)
do indeed lead to violations of (5). These violations wit-
ness the non-existence of local hidden variable models
for (7). Numerical optimization shows that the maximal
violation of (5) for quantum correlations of the above
formisachievedwhenallmeasurementsettingslieinthe
Y-Z plane of the Bloch sphere, that is, for measurement
operatorsA andB oftheformsinθ σ +cosθ σ . (One
x y y z
· ·
couldaswelltakethemalltolieintheX-Zplane;whatis
important is that they lie in the same plane.) The max-
imal violation is obtained for the maximally entangled
state α=π/4, on which one gets CHSH +0.237.
E
≈
For other values of α, the maximal violation of (5),
whenoptimizedoverthemeasurements,followstheexact
same profile as for the standardcorrelatorinequality (4)
as can be seen in Fig. 2a. However, the measurements
FIG. 2: (Color online) a. Violation profile of the entropic that maximize the violation of CHSH are not the ones
CHSH inequality for a pure state of the form cosα|00i + which give the maximal violation of CHSHE. In fact,
sinα|11i. The violation profile essentially coincides with the wewillsee below thatthosechoicesofobservableswhich
oneofthestandardCHSHinequality,althoughthescalesare producethemaximalCHSH valueforacertainαdonot
different. The curve is obtained by optimizing over the ob- violate (5). In general, for the standard CHSH scenario,
servables for each α. b. Maximal violation of the entropic
theviolationofthestandardinequality(4)isanecessary
inequality (3), achieved on a maximally entangled state of
but not sufficient condition for the violation of (5).
two qubits (α=π/4) and k =2,...,10 measurement settings
We have also considered the inequalities (3) for any
for each party.
even n = 2k as entropic Bell inequalities as fol-
lows. When A chooses between 1-valued observables
±
A ,...,A and B among B ,...,B , then (3) becomes
1 k 1 k
applicable upon taking X = B and X = A .
left-hand side by CHSH , so that the entropic CHSH 2i i 2i−1 i
E Fig. 2bshows our numericalresults for the maximalvio-
inequality is
lation of (3) on a two-qubit state (6).
No-signaling violations of CHSH . In the CHSH
E
CHSH 0. scenario, there is a special class of marginal models
E
≤
known as isotropic boxes, which we would now like to
study. Tobegin,thePopescu-Rohrlich box (PRbox)[22]
NotethattheentropicCHSHinequalitydoesnotonlyap-
is defined to be the marginal model
plyto 1-valuedobservablesA andB ;itiscompletely
x y
±
irrelevant how many outcomes each observable has—as PPR(a,bx,y)= 1 1+( 1)a⊕b⊕xy , (8)
longas the Shannonentropiesconverge—andwhichout- | 4h − i
comes these are.
It is the unique marginal model which maximally vio-
Quantum violations of CHSHE. We now assume lates (4). Similarly, the isotropic box with parameter
that the two parties A and B share a pure entangled C [0,1] is defined to be the marginal model
two-qubit state. For any such state, one can choose the ∈
1
local bases such that it has the form Piso(a,bx,y)= 1+C( 1)a⊕b⊕xy . (9)
| 4h − i
ψ =cosα00 +sinα11 , (6) It corresponds to a probabilistic mixture of PPR with
| i | i | i weightC anduniformwhitenoisePw withweight1 C.
−
Equivalently, an isotropic box can be characterizedby
forsomeα (0,π/2). WritingA andB alsoforthe 1-
x y having uniformly random marginals, that is
∈ ±
valued quantum observables measured on ψ , we obtain
| i
the joint distributions, written in terms of the standard A = B =0,
x y
h i h i
5
together with of contextuality: if a probabilistic model violates an en-
tropic inequality, then so does every other probabilistic
A0B0 = A0B1 = A1B0 = A1B1 0. model obtained by permuting the outcome probabilities
h i h i h i −h i≥
ofajointmeasurement,providedthatthepermutedjoint
TheparameterCisdeterminedfromthisbyC = A B .
h 0 0i distribution has the same marginals. For example, this
ItsrelationtotheCHSHvalueofthe boxissimplygiven
leads to the phenomenon observed above that the PR-
by CHSH(Piso)=4C. box PPR is entropically indistinguishable from classical
Any marginal model in the CHSH scenario can be correlationPc.
transformed into an isotropic box through a local depo-
Alongsimilarlines,theonlysymmetryoperationsthat
larization process, keeping the CHSH value (4) invari-
can be applied in order to transforman entropic contex-
ant [23]. Therefore, for many purposes it is enough to
tuality inequality into an equivalent one are permuta-
consider isotropic boxes only.
tions of the observables which map jointly measurable
Interestingly, no isotropic box violates the entropic
sets to jointly measurable sets. In the case of a Bell sce-
CHSH inequality. In particular, this applies to the PR-
nario, these permutations are either permutations of the
box, although it maximally violates the standard CHSH
parties, permutations of the observables of some party,
inequality. One can understand this by noting that en-
or arbitrary combinations thereof. Relabelings of the
tropy only probes the probability values occurring in a
outcomes of an observable do not change the inequality,
distribution, but not which probability values get as-
again due to the fact that entropies are invariant under
signed to which outcomes. This means that the PR-box
outcome permutations.
is,asfarastheentropiesareconcerned,equivalenttothe
marginal model Pc describing classical correlations,
B. The Klyachko scenario
1
Pc(a,bx,y)= 1+( 1)a⊕b . (10)
| 4h − i
A very simple state-dependent proof of the Kochen-
Specker theorem with only five two-outcome observables
Entropic quantities cannot distinguish between the per- was given by Klyachko et al. in [19, 20]. The marginal
fect anti-correlation of A1 and B1 as it appears in PPR, scenariointhiscaseistheonedepictedinfigure1b: there
and the perfect correlation of A1 and B1 as in Pc. All are five 1-valued observables X1,X2,X3,X4,X5 such
joint entropies of the PR-box coincide with those of Pc. that X a±nd X (modulo 5) are compatible.
i i+1
Fortwo-outcomemeasurements,themaximalviolation The so-called Klyachko inequality is a necessary and
of(5)is+1forthefollowingreason: anymarginalmodel sufficient condition for noncontextuality in this sce-
with two-outcome measurements will satisfy H(A0) nario [19, 20]. It is given by
≤
H(A B )andH(B ) H(A B ); similarly,H(A B )
0 0 0 1 0 1 1
1+ H(B ) H(A B≤) +1. Taking these inequalitie≤s 5
1 0 1
tmoogdetehleinr tshheo≤wCsHtShHatscCeHnaSriHo.ET≤hi1sbfoorunadnyonsutchhemvioalragtiinoanl K5 =Xi=1hXiXi+1i≥−3 (11)
can indeed by achieved by the no-signaling box
Due to Theorem 3, the only non-trivial entropic in-
equalityintheKlyachkoscenarioisgivenby,uptocyclic
Pmax = 1PPR+ 1Pc,
2 2 permutations of the observables, the entropic Klyachko
inequality
which is an equal mixture of the PR-box with classical
correlations. Pmaxcanbeunderstoodastheprobabilistic
H(X X )+H(X )+H(X )+H(X )
1 5 2 3 4
modelinwhicheachofthethreepairs(A ,B ),(A ,B )
0 0 0 1 H(X X ) H(X X ) H(X X ) H(X X ) 0
and(A1,B0)displaysperfectcorrelation,whilethefourth − 1 2 − 2 3 − 3 4 − 4 5 ≤
(12)
pair(A ,B )isuncorrelated;seealso[6,Prop.4.3]. Note
1 1
thatPmax achievesavalueof3onCHSH,andtherefore
To investigate the quantum violations of the entropic
does not have a quantum-mechanical realization since it
Klyachkoinequality, we choose two-outcome observables
is beyond Tsirelson’s bound of 2√2. on C3 of the form
This example shows that a convex combinationof two
non-violating marginal models may violate an entropic X =2v v (13)
i i i
contextuality inequality. This highlights the strongly | ih |−1
non-linear character of entropic inequalities (see also with the vectors vi given by
| i
Fig. 4).
v =(0,0,1)
Discussion. It is a basic feature of Shannon entropy | 1i
that the entropy of a probability distribution is invari- v2 =(sinθ,cosθ,0)
| i
ant under permutations of the sample space. From this v = −1(cosθsinφ, sinθsinφ,sinθcosφ)
3
point of view, we find it surprising that the entropic in- | i N −
v =(0,cosφ,sinφ)
equality (5) canbe violatedatall. Violationsofentropic | 4i
v =(1,0,0),
contextuality inequalities witness a very particular kind 5
| i
6
(0,1) (1,0) (0,0) by
X1,X5 4φ2 4φ2 1−8φ2 H(X1)= 4φ2log(φ2)+O(φ2)
XX21,,XX32 1−92φ52φ2 1−4φ52φ2 21φφ22 H(X2)=−−5φ2log(φ2)+O(φ2)
9
H(X )= φ2log(φ2)+O(φ2)
TABLE I: Joint outcome probabilities for the analytic viola- 3 −2
tion. All valuesare up to O(φ3). H(X X )= 5φ2log(φ2)+O(φ2)
1 2
−
H(X X )= 5φ2log(φ2)+O(φ2)
2 3
−
where the normalization factor is = H(X1X5)= 8φ2log(φ2)+O(φ2)
sin2θ+cos2θsin2φ. Up to choice of bNasis and −
pmultiplying the v by irrelevant phases, every config-
i
uration of 5 uni|t viectors v ,..., v R3 with v With this, the left-hand side of (16) is 5φ2log(φ2)+
| 1i | 5i ∈ | ii −2
orthogonalto v is of this form. O(φ2), which is positive for small enough φ.
i+1
| i
Since each v is orthogonal to v , the observable Detection inefficiencies: single-detector model. One
i i+1
X commutes|wiith X , so that t|heseitwo observables can take advantage of the fact that entropic inequalities
i i+1
arecompatibleandwecantalkabouttheirjointmeasure- can handle any finite number of outcomes and use the
ment. Also thanks to orthogonality, their joint outcome sameapproachtoinvestigatethemorerealisticcasewith
(X =1,X =1) never occurs. detectioninefficiencies. Wewillconsidertwopossiblesce-
i i+1
narios: one where compatible observables are measured
We write v v for a unit vector orthogonal to
i i+1
| × i jointly(one detector),andonewhere compatibleobserv-
both v and v .
i i+1
| i | i ables are measured sequentially (two detectors).
Uponmeasuringtheseobservablesonsomeinitialstate
In the single-detector model with detection efficiency
ψ C3, the non-vanishing joint outcome probabilities
| i ∈ η [0,1],thereisanadditionaloutcome( , )whichrep-
are given by ∈ ∅ ∅
resents the no-click event of the detector for each jointly
measurable pair (X ,X ). The new outcome probabil-
i i+1
P(0,1Xi,Xi+1)=P(1Xi+1)= vi+1 ψ 2 ities Pη are given by
| | |h | i|
P(1,0X ,X )=P(1X )= v ψ 2
| i i+1 | i |h i| i| (14) Pη(xi,xi+1 Xi,Xi+1)=ηP(xi,xi+1 Xi,Xi+1)
P(0,0X ,X )=1 P(1X ) P(1X ) | | (17)
| i i+1 − | i − | i+1 Pη( , Xi,Xi+1)=1 η,
= v v ψ 2. ∅ ∅| −
i i+1
|h × | i| where the measurments are now 1,+1, -valued,and
{− ∅}
x ,x 1,+1 are the “proper” outcomes. In this
Numerical results. Numerical calculations show that i i+1 ∈ {− }
model, a no-detection event always occurs for both ob-
the maximal qutrit violationof (12) with observablesX
i servables simultaneously.
occurs on a qutrit state of the form
The joint probabilities (17) marginalize to the single-
observable distributions
1
ψ = (sinα,cosα,sinα) (15)
P(x X )=ηP(x X )
| i 1+sin2α i| i i| i (18)
p P( Xi)=1 η.
∅| −
with α 0.29736 and θ = φ 0.24131, for which the
Proposition 4. With this model of inefficiencies,
≈ ≈
left-hand side of (12) is +0.091.
Analyticalproofofqua≈ntumviolations. Wewouldlike Hη(Xi)=ηH(Xi)+h(η)
to present an analytic proof showing that quantum vio- Hη(X X )=ηH(X X )+h(η).
i i+1 i i+1
lations of (12) occur with the X and ψ of the above
i
form. We set θ = φ and α = 2φ and ex|paind everything where h(η) = ηlogη (1 η)log(1 η) is the binary
− − − −
for φ 1. Then, by symmetry, the joint outcome distri- entropy.
≪
bution of X and X coincides with the one of X and
4 5 2 Proof. This follows from an application of the grouping
X ; likewise, the joint outcome distribution of X and
1 3 rule of Shannon entropy, see e.g. [24, Sec. 2.179].
X coincides with the one of X and X ; the remaining
4 3 2
joint distributions, up to O(φ3), are listed in Table I. Uponpluggingtheseequationsinto(12),onefindsthat
With this symmetry, (12) is equivalent to thecontributionsofh(η)cancel,sothattheleft-handside
simply scales as a linear function of η. Therefore, the
entropic contextuality inequality (12) has violations for
H(X X )+2H(X )+H(X )
1 5 2 3
(16) any η 0. Moreover, the maximal violation with qutrit
2H(X X ) 2H(X X ) 0. ≥
− 1 2 − 2 3 ≤ measurementsoftheform(12),(14)isgivenbythesame
state and vectors v which maximizes the violation for
i
| i
The corresponding relevant entropy values are given η =0 (which are α 0.29736 and θ =φ 0.24131).
≈ ≈
7
Detectioninefficiencies: two-detectormodel. Wenow
assumethatthejointmeasurementofX andX isre-
i i+1
alizedbyonedetectormeasuringX andanotherdetector
i
measuring X . Again, we take each detector to have
i+1
an efficiency of η [0,1], for simplicity the same value
∈
forall5detectors,suchthattheno-clickeventofthefirst
detector is independent of the no-click event of the sec-
ond detector. A physical situtation leading to this kind
of model may be a sequential scheme where the system
passesthroughanon-demolitionmeasurementinthefirst
detector before reaching the second detector. FIG.3: (Color online) General nonlocality distillation proto-
Consequently, the jointly measurable pair (X ,X ) colwithtwoboxes. Theinputsforthesecondboxmaylocally
i i+1
has an outcome distribution, with x 1,+1 , dependon theoutput of thefirst one and thefinal outputof
i
∈{− } thecombined boxesis a function of all inputsand outputs.
Pη(x ,x X ,X )=η2P(x ,x X ,X )
i i+1 i i+1 i i+1 i i+1
| |
Pη(x , X ,X )=(1 η)ηP(x X )
i i i+1 i i
∅| − | (19)
Pη( ,x X ,X )=(1 η)ηP(x X ) In this section, we would like to consider the case of
i+1 i i+1 i+1 i+1
∅ | − |
theCHSHscenarioandinvestigateabit,onapurelyphe-
Pη( , X ,X )=(1 η)2
∅ ∅| i i+1 − nomenological level, one particular property which also
hasanasymptoticflavor: thepropertyofnonlocality dis-
which again marginalize to the single-observable distri-
tillation. However, we barely have a precise hypothesis
butions (18).
to offer—let alone a proof—and the similarities we will
Proposition 5. With this model of inefficiencies, describe in the following may easily turn out to be su-
perficial.
Hη(X )=ηH(X )+h(η) (20) Consideringthe CHSHscenario,itwasrecentlyshown
i i
Hη(X X )=η2H(X X ) that nonlocality can be distilled [9–13]: by locally pro-
i i+1 i i+1
cessing several copies of certain bipartite no-signaling
+η(1 η)[H(X )+H(X )]+h(η).
− i i+1 boxes, one can increase the amount of nonlocality ac-
cording to a particular nonlocality measure. In the fol-
where h(η) = ηlogη (1 η)log(1 η) is the binary
− − − − lowing, we consider bipartite no-signaling boxes with bi-
entropy.
nary inputs and outputs. Each party can wire boxes to-
Proof. Again, this follows from an application of the gether using classical circuitry to produce a new binary-
grouping rule of Shannon entropy [24, Sec. 2.179]. input/binary-output box.
The general distillation protocol with two copies of a
Duetotheadditionaltermsin(20),therequireddetec- no-signalingboxisdisplayedinFig. 3. Acertainprotocol
tion efficiency for witnessing quantum violations in the distillsthenonlocalityiftheno-signalingboxobtainedaf-
two-detector model turns out be very high, η 0.995. ter the wiring is more nonlocal—according to a certain
≈
measure—than the original box. As a measure of non-
locality, we will consider the nonlocal part of the EPR2
III. NONLOCALITY DISTILLATION AND THE decomposition [25], defined as follows. Any no-signaling
ENTROPIC CHSH INEQUALITY box P canbe decomposedinto convexcombinationsof a
purely local box PL and another box PNL,
As we have seen in the previous sections, finding vi-
olations of an entropic contextuality inequality, for ex- P =(1 q)PL+qPNL, (21)
−
ample of the entropic CHSH inequality (5), is not easy,
and there are few quantum-mechanical models which do withsomecoefficientq [0,1]. Theminimalcoefficientq,
∈
violatethem. Therefore,weregardtheviolationofanen- whenminimizedoverallsuchdecompositionsisthe non-
tropic contextuality inequality (entropic Bell inequality) localcontent ofP. Bydefinition,computingthenonlocal
asawitnessofaverystrongformofcontextuality(nonlo- contentisalinearprogram. InthecaseoftheCHSHsce-
cality). Sowhatdoestheviolationofanentropiccontex- nario,there is a linear relationshipbetween the nonlocal
tualityinequalitytellusaboutthe violatingmodel? Can contentandtheviolationofthestandardCHSHinequal-
the violation be regardedas a resource for something? ity [13]; therefore, in the present context we may regard
Since Shannon entropy is an asymptotic quantity the CHSH value as our measure of nonlocality.
which measures the effective size of a probability distri- We now focus on two particular wirings. The first one
bution on the level of many copies (see for example the was proposed in [9] and is given by
Asymptotic EquipartitionProperty [21]), we also expect
any answers to these questions to be concerned with the x =x, x =x, a=a a
1 2 1⊕ 2 (22)
limit of many copies. (cid:26) y1 =y, y2 =y, b=b1 b2,
⊕
8
FIG.5: (Coloronline)Distillednonlocality(increaseinnonlo-
calcontent)ofthefamily (27)uponapplication ofthewiring
FIG. 4: (Color online) Slice of theno-signaling polytopecor- (30) for d=2,4 (redfull line), d=3 (bluelarge dashed line)
responding to the family of boxes (25). The quantum region andd=5(blackdashedline)ascomputedbylinearprogram-
andCHSHE violationsareboundedrespectivelybythethin ming. The inset shows the violation of CHSHE, which does
black and thick black curves. The regions of distillability not depend on d.
by the wirings (22) and (23) correspond respectively to the
dashed-dotted red and dashed blue lines. The region of vio-
lationsoftheentropicCHSHinequalityisstrictlylargerthan
For γ = 0, this is a family of boxes with CHSH = 0.
the region of correlations distilable with the wiring (23). On
Thefamily(25)formsatriangleintheno-signalingpoly-
the other hand, in a small part of the quantum region, non-
tope extending from the boundary of the local polytope
locality can be distilled by the protocol (22) while not being
detected by CHSHE. up to the PR-box. As displayed in Fig. 4, the subset of
boxes violating CHSH and the subset of boxes distil-
E
lable via (23) or (22) are very similar. We expect that
by going beyond the two-copy level and considering all
while the second one was proposed in [13] and is given possible wiring protocols, the subset of distillable boxes
by will enlarge. It seems reasonable to ask whether it will
contain all boxes violating CHSH . In general, could
x =x,x =x a 1,a=a a 1 E
(cid:26) 1 y1 =21,y2 =⊕yb11,⊕b=b1⊕b12⊕⊕12.⊕ (23) tthilleavbiioliltayt?ionofCHSHE beasufficientconditionfordis-
Moreover, we have investigated the bipartite scenario
Asshownin[13],thesetwowiringscompletelycharacter-
with two d-outcome observables per party, to which
izethedistilabilityintheCHSHscenarioonthetwo-copy
CHSH still applies. We have considered the family
level. E
of no-signaling boxes
Forexample,forthewiring(22),thewiredno-signaling
box is given by P =ξPPR+(1 ξ)Pc, (27)
ξ d − d
P′(a,bx,y)= P (a1,b1 x,y)P(a2,b2 x,y), (24) where PdPR is the generalized PR-box [26]
| | |
ab11++Xab22==ba PdPR(a,b|x,y)=(cid:26)1/0d iofthae−rwbis≡exy mod d , (28)
Asforanywiringprotocolwithtwocopiesoftheoriginal
box,thenewboxP′isaquadraticfunctionoftheoriginal while Pdc is the classically correlated box
one. So the first (superficial) similarity between wirings 1/d if a=b
and the entropic CHSH inequality is nonlinearity. Pdc(a,b|x,y)=(cid:26) 0 otherwise . (29)
Interestingly, in the case of isotropic boxes—which
we found in IIA not to violate the entropic CHSH It follows from the definition (29) and an application of
inequality—distillation is not possible [8]. More gener- the CGLMP inequality [27] that the nonlocal content of
ally, we have considered the family of boxes Pξ issimplyξ. Ontheotherhand,CHSHE = ξlogξ
− −
(1 ξ)log1 ξ turns out to be the binary entropy.
Pγ,ξ =γPPR+ξPc+(1−γ−ξ)Pf, (25) T−o probe−the nonlocality distillation of Pξ, we have
considered the wiring
with parameters γ,ξ [0,1], PPR and Pc given respec-
tively by (8) and (10)∈, as well as x1 =x, x2 =xa1mod2, a=(a1+a2)modd , (30)
(cid:26) y1 =y, y2 =yb1mod2, b=(b1+b2)modd
1
Pf(a,bx,y)= 2+( 1)a⊕b⊕xy , (26) which can be regarded as a generalization of the wiring
| 8h − i
proposed in Ref. [10]. Fig. 5 shows the increase of
nonlocal content achievable with this protocol for any
which is Pf = 1PPR+ 1Pw, half-way between the PR- d=2,...,5 and the value of CHSH ; we observe qual-
2 2 E
box and white noise. itatively identical behavior.
9
means that a bilocal model is one of the form
P(a,b,cx,y,z)
|
= ̺ (λ )̺ (λ )P(ax,λ )P(by,λ ,λ )P(cz,λ ).
1 1 2 2 1 1 2 2
| | |
FIG. 6: (Color online) Swapping experiment scenario, where λX1,λ2
(33)
two independent entangled states are prepared and sent to
three distant parts. If the central part B measures in a en-
tangled basis, thesubsystemsAandC,that hadnoprevious A direct calculation shows that every such model sat-
contact, become nonlocally correlated. isfies, besides the usual no-signaling equations, also the
condition
P(a,b,cx,y,z)=P(ax)P(cz) a,c,x,y,z (34)
| | | ∀
IV. ENTROPIC BILOCALITY INEQUALITIES Xb
where P(ax) and P(cz) are the marginal behaviors of
Bilocality scenarios. In entanglement swapping [15], A and C, r|espectively.|
there are two sources of entangled quantum states. The However, there are no-signaling boxes P(a,b,cx,y,z)
first source (on the left in Fig. 6) sends half of its entan- which satisfy (34), but nevertheless are not |bilocal,
gledstatetoAandtheotherhalftoB;thesecondsource i.e. cannot be written in the form (33). Many exam-
(ontherightinFig.6)sendshalftoBandhalftoC.IfB ples of this form are even local boxes of the form (32).
appliestherightkindofentangledmeasurementbetween Wewillsoonseemoreconcreteexamplesofthisinwhich
the two quantum systems which he receives, then, con- the “non-bilocality” can be witnessed by an entropic in-
ditioned on an outcome of this measurement, the post- equality.
measurement state between A and C will be entangled. Entropic bilocality inequalities. Duetothe nonlinear-
This idea has been used to obtain strong bounds [14] ity of the bilocality condition ̺(λ ,λ ) = ̺ (λ )̺ (λ ),
1 2 1 1 2 2
ontheexistenceoflocalhiddenvariablemodelsofentan- the set of P(a,b,cx,y,z) of the form (33) is not con-
|
gled quantum states augmented by the assumption that vex [14], and it is difficult to determine whether a given
thehiddenvariablesλ andλ describingthetwosources P lies in this bilocal set or not. This is where entropic
1 2
areprobabilisticallyindependent;thisisthebilocality as- inequalities enter: as we will show in the following, they
sumption. give necessary requirements for P to be bilocal in terms
Undertheassumptionoflocalrealismwithoutthebilo- of inequalities linear in the entropies of P. This lin-
earity is already visible on the level of the two sources,
cality assumption, the conditional probabilities for the
scenario of Fig. (6) would have the form whoseprobabilisticindependenceisequivalenttovanish-
ing mutual information, I(λ :λ )=0, which is a linear
1 2
entropic equation
P (a,b,cx,y,z)
|
= ̺(λ1,λ2)P(ax,λ1)P(by,λ1,λ2)P(cz,λ2) (31) H(λ1λ2)=H(λ1)+H(λ2).
| | |
λX1,λ2
For the sake of concreteness, we consider the specific
scenariowhereAandChavetwoavailablemeasurement
where ̺(λ ,λ ) is the probability distribution over the
1 2 settings to choose from, whereas B always applies the
pairsofhiddenvariables. As usual, x,y,z and a,b,c
{ } { } samefixed measurement,sothat x,z 0,1 andy =0.
describe, respectively,the inputs and outputs ateachlo- ∈{ }
This corresponds to 5 observables A ,A ,B,C ,C . A
calpart. SincethetwosystemsreceivedbyBobarebeing 0 1 0 1
subsetofthese 5observablesis jointly measurablewhen-
jointly measured, they can be treated as a single entity.
everit containsatmostone observableofAandatmost
Since ̺ can be chosen such that λ = λ occurs with
1 2 one of B. Following [6], one can visualize the marginal
probability 1, such a model is equivalent to one of the
scenario as in Fig. 7. In addition, the observables A
0
form
and A are independent of the observables C and C ;
1 0 1
therefore, in order to calculate the entropic inequalities
P (a,b,cx,y,z) for this scenario following the procedure of [6], we need
|
(32) to consider the independence constraint
= ̺(λ)P(ax,λ)P(by,λ)P(cz,λ),
| | |
Xλ
I(A A :C C )=0, (35)
0 1 0 1
which is the standard description of local realism in a
which we write out in terms of joint entropies as
three-party Bell scenario.
The bilocality assumption in addition imposes inde- H(A A C C )=H(A A )+H(C C ). (36)
0 1 0 1 0 1 0 1
pendence of λ and λ , so that the distribution ̺ is
1 2
required to factor as ̺(λ ,λ ) = ̺ (λ )̺ (λ ), which Notethatthedataprocessinginequalityimpliesthatthe
1 2 1 1 2 2
10
FIG.7: Thesimplicial complex ofjointly measurable observ-
ables in the bilocality scenario discussed in the main text.
Two observables are jointly measurable whenever they share
an edge; three observables are jointly measurable whenever
they are thevertices of one of thefour triangles.
FIG. 8: (Color online) Parameter space of the family (38).
The red dot-dashed line delimits the region where (39)
and (40), and therefore (38), are local as witnessed by the
CHSH inequality. The region above the blue dashed line
independence constraints
contains violations of the entropic bilocality inequality #7
from Table II. The black line delimits the region where (39)
I(Ax :C0C1)=0, and(40),andtherefore(38),haveaquantumrealizationwith
a single 3-qubit source. There is a small region (inset) in
I(A A :C )=0, (37)
0 1 z
whichourinequalitydetectsthat(38)isnotbilocal,although
I(A :C )=0,
x z thebox is tripartite local.
follow from (35).
We have used the computational approach of [6], aug-
mented by the independence constraints (36) and (37) rameters ξ,γ [0,1],
∈
written out in terms of joint entropies, in order to
calculate all the (Shannon-type) entropic inequalities PNB(a,b,cx,z) (38)
in this bilocality scenario; although including the con- |
1
straints (37) would not have been strictly necessary, it = 1+ξ( 1)a⊕b⊕c⊕xz+(1 ξ γ)( 1)a⊕b⊕c
8 − − − −
helps in speeding up the computation. This computa- (cid:0) (cid:1)
tionhasresultedin4equationsand52tightinequalities.
This box can be understood as follows. The two out-
The 4 equations are precisely the independence condi-
comesb 0,1 bothoccurwithprobability1/2;ifb=0,
tions I(Ax : Cz) = 0. The 52 inequalities fall into 10 then this∈c{reate}s between A and C the bipartite box
symmetry classes which we have listed in Table II. The
first four inequalities are trivial in the sense that they
will hold for any no-signaling box P(a,b,cx,y,z) satis- ξPPR+γPc+(1 ξ γ)Pw (39)
| − −
fying(34),whiletheothersixinequalitiescanpotentially
be violated by non-bilocal boxes. and if b=1, then the resulting box between A and C is
Looking for quantumviolations. Wenowconsiderthe
quantum case. Instead of sending out independent hid- ξPAPR+γPAc+(1 ξ γ)Pw, (40)
− −
den variables λ and λ , the two sources now emit en-
1 2
tangled quantum states. We take these to be given by where PAPR and PAc stand for an “anti-PR-box”, de-
generic partially entangled two-qubit states
fined as in (9) with C = 1, and classical anti-
−
correlations,respectively.
cosθk 00 +sinθkeiφk 11 Depending on the value of ξ and γ, the box PNB can
| i | i
betripartitelocal,tripartitenonlocalbuttripartitequan-
with k = 1,2 indexing the two sources. Then A and tum, or post-quantumly tripartite nonlocal. PNB satis-
C receive one qubit each, while B receives two. Upon fies (34)since the bipartite marginalbetweenAandC is
choosing the two-qubit measurement of B to be in the pure white noise, PNB(a,cx,z)=Pw(a,cx,z)=1/4.
| |
Bell basis and numerically optimizing over all projective PNB violates some of our entropic bilocality inequali-
measurementsforAandC,wehavenotbeenabletofind ties. We focus on the inequality 7 from Table II. Fig. 8
anyquantumviolationofanyofour52entropicbilocality shows the region of violations as a function of the para-
inequalities. rametersξ andγ. Interestingly,eveninthe regionwhere
On the other hand, it is not difficult to design PNB is local as a tripartite box, the entropic bilocal-
some generalno-signalingboxes P(a,b,cx,z)which sat- ity inequality can be violated. This witnesses that in
isfy (34), but violate some of our entro|pic inequalities. those parameter ranges, PNB cannot be written in the
This applies for example to the family of boxes, for pa- form (33), althoughthe box is local and (34) is satisfied.