Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality

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by  Rafael Chaves

Entropic inequalities as a necessary and sufficient condition to noncontextuality and
locality
Rafael Chaves1
1Institute for Physics, University of Freiburg, Rheinstrasse 10, D-79104 Freiburg, Germany
The assumption of local realism in a Bell locality scenario imposes non-trivial conditions on the
Shannon entropies of the associated probability distributions, expressed by linear entropic Bell in-
equalities. In principle, these entropic inequalities provide necessary but not sufficient criteria for
the existence of a local hidden variable model reproducing the correlations, as, for example, the
paradigmaticnonlocalPR-boxisentropicallynotdifferentfromaclassically correlated box. Inthis
paper we show that for the n-cycle scenario, entropic inequalities completely characterize the set
of local correlations. In particular, every nonsignalling box which violates the CHSH inequality –
includingthePR-box–canbelocallymodifiedsothatitalsoviolatestheentropicversionofCHSH
inequality. As we show, any nonlocal probabilistic model when appropriately mixed with a local
model, violates an entropic inequality, thus evidencing a very peculiar kind of nonlocality. As the
n-cyclecapturesequallywellboththenotionoflocalrealismintroducedbyBellandthatofnoncon-
3 textualitypresentedbytheKochen-Speckertheorem,theresultsarealsovalidfornoncontextuality
1 scenarios.
0
PACSnumbers: 03.65.Ud,03.67.Mn
2

n
a I. INTRODUCTION of CHSH to more outcomes is provided by the Collins-
J
Gisin-Linden-Massar-Popescu (CGLMP) inequality [8],

4 the CHSH and CGLMP inequalities fully describing the
The quantum nonlocal correlations that may arise in
2
set of local correlations up to 3 outcomes [9]. However,
experiments performed by spacelike separated and inde-
foranumberofoutcomeslargerthan3,acompletechar-
] pendent observers,area key conceptin the foundational
h acterization of the inequalities bounding the set of local
aspectsofquantummechanics. Theexpectedclassicalin-
p correlations is still to be found [9, 10], highlighting the
tuitionthatphysicalquantitieshavewell-establishedval-
-
difficulty and limitations of this approach.
t ues previous to any measurement and that signals can-
n
a not propagate instantaneously, do not suffice to repro- In a conceptually different approach introduced by
u duce the quantum mechanical predictions [1], highlight- Braunstein and Caves [11], it was shown that local re-
q ingaverycounterintuitiveaspectofquantumtheorythat alism imposes non-trivialconditions already on the level
[ has received strong experimental corroboration over the oftheShannonentropies. TheShannonentropiescarried

1 years [2]. From an applied point of view, nonlocality is bythemeasurementsontwodistantsystemsmustsatisfy
v now recognized as a novel physical resource, which en- certain inequalities, which can be regarded as entropic
4 ables protocols such as device-independent quantum key Bell inequalities. It was recently pointed out that the
1 distribution [3], random number generation [4] and the characterizationof the local correlations on the entropic
7 reduction of communication complexity in distributed- level,alsodefinesalinearprogrammingproblem[12,13].
5
computing scenarios [5]. One advantage of this entropic approach is that it can
.
1 Inpractice,nonlocalityiswitnessedthroughthe viola- readily be applied to quantum systems of arbitrarylocal
0 tion of a Bell inequality [1]. Given a certain experimen- dimensionandgeneralmeasurementoperators,since the
3
talscenariodefinedbythe numberofspatiallyseparated inequalities do not depend on the number of outcomes
1
parties, the possible different measurement settings for of the measured observables. That is, while the dimen-
:
v each party, and the possible outcomes for each setting, sion of the set of local correlations in terms of probabil-
i
X local-realistic joint probability distributions form a con- ities grows exponentially with the number of outcomes
vex set to whichBell inequalities, a set of linear inequal- for each observable, the entropic set of local correlations
r
a ities of the probabilities, are the non-trivial facets [6]. is independent of that. Another advantage of the en-
Thisgeometricapproachprovidesageneralframeworkin tropic approach is that it easily adapts to situations of
which Bell inequalities can be derived, since the task to additionalindependence requirements,like the bilocality
findthefacetsofaconvexsetisalinearprogramthatcan scenariosintroduced by a entanglementswapping exper-
be solved efficiently. The problem is that, generally, the iment [14, 15] and general correlation and causal model
sizeofthelinearprogramgrowsveryfastasthenonlocal- scenarios[16–18]. Theindependenceconstraintsarenon-
ityscenariobecomes lesssimple,someclassesevenbeing linear on the level of probabilities, defining a nonconvex
knowntobeaNP-completeproblem[6]. Inspiteofthat, set,whileintermsofentropiessuchconstraintsarelinear
someparticularcharacterizationsarewellknown. Thebi- and still define a convex set that can be solved by linear
partite scenario with two dichotomic measurements per programming. In spite of their attractive properties, en-
party is completely characterized by the Clauser-Horne- tropic Bell inequalities are, in principle, sufficient but
Shimony-Holt(CHSH)inequality[7]andageneralization not necessary conditions to witness nonlocality. That is,
2
there are nonlocal distributions violating a Bell inequal-
ity, that, however, do not violate its entropic counter-
part [13]. However, as we show in this paper, the situa-
tion is more involved than initially thought, as entropic
inequalities can, at least in some scenarios, completely
characterize the set of local correlations.
In this paper we show that in the n-cycle scenario,
any nonlocal distribution when augmented with shared
randomness will also violate a entropic Bell inequality. FIG.1. (Coloronline)Graphicalrepresentationofthen-cycle.
The n-cycle can be seen as generalization of the CHSH Theverticesrepresentdifferentobservablesandtheedgescon-
scenario for an arbitrary number of observables for each nect observables that are jointly measurable. a. CHSH sce-
party, with only a subset of pairwise observables being nario, 2 parties with 2 measurement settings each. Labelling
jointlymeasurable(Fig.1). Then-cyclecapturesequally X1 =A0, X3 =A1, X2 =B0, X4 =B1, we recover theusual
picture where two parties, Alice and Bob, perform space-like
wellboththenotionoflocalrealismintroducedbyBell[1]
separated measurements. Alice measures oneout of twopos-
and that of noncontextuality presented by the Kochen-
sible measurement settings A0 or A1, and similarly for Bob.
Specker (KS) theorem [19]. A complete characterization
b. KCBSscenario,5observablesarrangedinacyclicconfigu-
ofthen-cycleintermsofanexponentialnumberoftight,
rationsuchthateachobservableiscompatiblewithitsneigh-
linear inequalities has recently been found [20]. As we bors c. Generalization of the CHSH/KCBS scenario, with n
show here, the n-cycle can be equivalently described by observables in a cyclic configuration (the n-cycle [22]). For
a polynomial number of entropic inequalities and a list dichotomic observables, the set of local/noncontextual cor-
of local/noncontextual points lying in the facets of the relations is completely characterized by (4). For a general
corresponding set of correlations. numberof outcomes, theuniquenontrivial entropic inequali-
ties are those given by(5).
II. THE N-CYCLE SCENARIO
terministic points define a convex set, to which the non-
The n-cycle scenario is defined for any number n ≥ 3 contextuality/Bellinequalitiesarethenon-trivialbound-
of observables X ,...,X , imposing the restriction that aries.
1 n
only Xi and Xi+1 are pairwise jointly measurable for all Similarly, the set of allowed nondisturb-
i = 1,...,n (with Xn+k = Xk). Any two observables ing/nonsignalling distributions also define a convex-set.
Xi and Xi+1 are jointly measurable, or compatible, if Nondisturbing/nonsignallingdistributions are defined as
the result for the measurement of Xi, even if not per- the ones for which
formed, does not depend on the prior or simultaneous
measurement of X and vice versa. This is the notion p(x |X ) = p(x x |X X )
i+1 i i i i+1 i i+1
of noncontextuality captured by the KS theorem [19]. xXi+1 (2)
It can readily be turned into the notion of Bell’s local- = p(x x |X X ),
i i−1 i i−1
ity [1], where the compatibility and noncontextuality of xXi−1
the observables is assured by spacelike separation of lo-
calmeasurementsofdifferentparticles. Inparticular,for that is, the outcome of X cannot be affected by which
i
n = 4, the n-cycle corresponds to the CHSH Bell sce- compatible observable (X or X ) it is jointly mea-
i−1 i+1
nario [7] (Fig. 1a), while for n = 5 it is the noncontex- sured with. For dichotomic observables x = 0,1 the
i
tuality scenario considered by Klyachko-Can-Binicioglu- vertices of the nondisturbing/nonsignalling set are given
Shumovsky (KCBS) [21] (Fig. 1b). For general n, it can by [20]
be visualized as an n-sided polygon (Fig. 1c).
Any correlation that can be reproduced by means of pγ (x x |X X )= 1/2 , xi⊕xi+1 =δ−1,γi ,
a noncontextual/localhidden variable model is a convex max i i+1 i i+1 (cid:26)0 , otherwise
sum of the deterministic probability distributions, that (3)
is, where ⊕ stands for addition modulo 2, γ ={γ ,...,γ },
1 n
γ = {−1,1} and the total number of γ = −1 is
i i
p(x x |X X )= ̺(λ)p(x |X ,λ)p(x |X ,λ), odd. Note that for n = 4 and γ = {1,1,−1,1} this
i i+1 i i+1 i i i+1 i+1
Xλ represents the maximally nonlocal distribution allowed
(1) by nonsignalling, the so called PR-box [23]. Given a
where x stands for the outcome x =0,··· ,d−1 of the certainnondisturbing/nonsignallingprobabilitydistribu-
i i
correspondingobservableX . Thesumisperformedover tion,any otherdistributionthancanbe achievedfromit
i
allthe dn deterministic distributionsparameterizedby λ usinglocalreversibletransformationsissaidtobe equiv-
(with a distribution ̺(λ)), that is, all the distributions alent. Local reversible operations consist of operations
that given a certain observable X yield with probabil- oftwo types: relabelling ofthe observablesthat preserve
i
ity 1 a certain outcome x . The noncontextual/local de- themutualcompatibilityofthejointobservablesX X ,
i i i+1
3
e.g, Xi → Xi+2, or relabelling of the outputs (possibly where γi′ =γi for i6=k but γk′ =−γk. Note that pγC′ has
conditioned on the observable), e.g, xi →xi⊕δi,1. aneventotalnumberofγi′ =−1anddoesnotviolateany
Recently, the full characterization of the n-cycle has Cγ inequality. Thus,sincepγ isentropicallyequivalent
n max
been found for dichotomic observables [20]. There are to a noncontextual/localdistribution, it does not violate
2n−1 tight equivalent inequalities given by any entropic inequality. In this sense, an entropic non-
contextual/Bell inequality is a necessary but not suffi-
Cnγ = γihXiXi+1i≤(n−2) (4) cient criterion to probe the noncontextual/local behav-
Xi ior a distribution. However, as we show next, entropic
inequalities can be turned into a necessaryandsufficient
wherehX X i=p(00|X X )+p(11|X X )−p(01|X X )−
i j i j i j i j condition,sinceanycontextual/nonlocaldistributionvio-
p(10|X X ) stands for the expectation value of the ob-
i j latingCγ alsoviolatestheentropicBCinequalitieswhen
servable X X and again the total number of γ = −1 n
i j i properly mixed with a noncontextual/local distribution.
is odd. A noncontextual/Bell inequality is said to be
equivalenttoanotherone,ifitcanbeobtainedfromitby
localreversibleoperationsand/orpermutationofobserv-
III. ENTROPIC INEQUALITIES COMPLETELY
ables. Note that pγ maximally violates Cγ, achieving
max n CHARACTERIZE THE N-CYCLE SCENARIO
Cnγ(pγmax) = n, while not violating any other equivalent WITH DICHOTOMIC OUTCOMES
inequality. In the following we will generally refer to C
n
andp astheoneswithγ =1fori=1,··· ,n−1but
max i First, note that the maximum violation of the BC in-
γ =−1.
n equality for dichotomic observables is given by BCk =
Similarly,thecompleteentropiccharacterizationofthe n
1, since H(X ) ≤ H(X X ) and H(X X ) ≤
noncontextual/local set of probability distributions has j j j+1 k k+1
H(X ) + H(X ) ≤ H(X X ) + H(X ) ≤
been found for the n-cycle [12, 13] (see also [24] for the k k+1 k+n−1 k k+1
H(X X )+1. The maximal violation of BCk = 1
n = 5 case). A probability distribution in this scenario k+n−1 k n
can be achieved by the probability distribution
is entropically noncontextual if and only if the set of n
equivalent Braunstein-Caves (BC) entropic inequalities pγ = 1(pγ +pγ′), (7)
Emax 2 max C
BCk =H(X X )+ H(X )− H(X X )≤0
n k k+1 j j j+1
for all γ such that γ = −1, since H(X ) =
j6=Xk,k+1 Xj6=k k j
H(X X ) = 1 for all j 6= k and H(X X ) = 2.
(5) j j+1 k k+1
Thatis,aconvexcombinationoftwononviolatingdistri-
hold for all k = 1,...,n where H(X X ) =
i j
butions may violate an entropic inequality, highlighting
−p(x x |X X )log p(x x |X X ) is the Shannon
ePntxrio,xpjy of tihej priobajbility2 disitrijbutiionj associated with the stronglynon-linearcharacterofit. NotethatpγC′ can
the measurements X and X . This set of entropic be achieved with 1 bit of shared randomness, that is,
i j
inequalities is said to be maximal in the sense that xi = xi+1 = 0 or xi = xi+1 = 1 with the same proba-
no other entropic inequality can detect the contextual- bility 1/2 for all i=1,...,n. Similarly the convex com-
ity/nonlocality not detectable by it, so that this set of bination in (7) also requires 1 bit of shared randomness,
tight entropic inequalities completely characterizes the outputtingpγ orpγ′,bothwithprobability1/2. When
max C
regionofnoncontextual/localprobabilistic models in en- augmented with some shared randomness, it is possible
tropy space. to turn an entropically noncontextual/local distribution
In order to transform an entropic inequality into an into a contextual/nonlocal one. In a noncontextual/Bell
equivalent one, the only symmetry operations that can scenario, shared randomness is always an available and
be applied are permutations of the observables. It is a validresource. However,forusualnoncontextual/Bellin-
basic feature of the Shannon entropy its invariance un- equalities, the mixing with a local point cannot improve
der permutations of the sample space. Violations of BC the violation of the inequality due to its linearity.
inequalitieswitnessthenaverypeculiarkindofcontextu- As we show next, the mixing with pγ′ is sufficient
C
ality/nonlocality. If a probabilistic distribution violates to entropically detect any contextual/nonlocal distribu-
the BC inequalities, then so does any other distribution tion. First we note that in the CHSH scenario (n = 4
obtained by the permutation of the outcome probabil- in the n-cycle), it is known that any probability dis-
ities, provided that the permutation leads to the same tribution can be transformed into an isotropic distri-
marginal distributions. This leads to the following phe- bution through a local depolarization process, keeping
nomenon: Fromthepointofviewoftheentropicinequal- the C value invariant [25]. The isotropic distribution
4
ityBCnk,themaximallycontextual/nonlocaldistribution has the property of being invariant under the inter-
pγ isnotdifferentfromaclassicallycorrelatedandnon- change of the inputs or outputs and being locally un-
max
contextual/local distribution, biased, that is, p (x x |X X ) = p (x x |X X )
I i i+1 i i+1 I i+1 i i i+1
and p (x |X ) = 1/2. For the n-cycle, the isotropic dis-
I i i
pγC′(xixi+1|XiXi+1)=(cid:26)10/2 ,, xotih⊕erxwi+is1e=δ−1,γi′ , tribution pI corresponds to a probabilistic mixture,
(6) p =ǫp +(1−ǫ)p , (8)
I max w
4
with p (x x |X X ) = 1/4 being pure white noise. (4), however at the cost of introducing a list of classical
w i i+1 i i+1
ThecorrespondingCnvalueisgivenbyCn(pI)=nǫ,that points pγ′.
C
is, p is contextual/nonlocalfor ǫ>(n−2)/n. From the
I
entropicpointofview p isequivalenttothe distribution
I
ǫpC+(1−ǫ)pw and thus no direct violation of entropic IV. CONCLUSION
BC inequalities is possible.
Thefirststepinourproofistoshowthatanydistribu-
In principle, entropic inequalities only provide a nec-
tioninthen-cyclescenariocanbeturnedintoaisotropic
essary but not sufficient criterion for noncontextuality
one without changing the values of C , that is, a gener-
n and local realism. However, we have shown that for the
alization of the depolarization protocol devised in Ref.
n-cycle with dichotomic outcomes, entropic inequalities
[25] (see also [26]) for the n = 4 case. Then, for our
turn also to be sufficient, since any contextual/nonlocal
purposesandto simplify the presentationit is enoughto
probabilistic model will display entropic violations if
consider isotropic boxes only (however as shown in the
properlymixedwithaclassicalmodel. Itisquitesurpris-
Appendix this isnotstrictly necessary). The depolariza-
ingthatapolynomialnumberofnon-linearandnon-tight
tion procedure is done in two steps:
inequalities may completely characterize the set of non-
i) p(xixj|XiXj) is made locally unbiased by flipping contextual/local correlations, that otherwise would re-
both outputs simultaneously with probability 1/2, that quire an exponential number of linear and tight inequal-
is, xi →xi⊕1 and xj →xj ⊕1 ities to do so.
ii)p(x x |X X ) → p(x¯kx¯k|X X ) where x¯k One obvious question is how this result would extend
i j i j i j i⊕nk j⊕nk i
Xk for more complex scenarios, involving more than two-
means flipping the output if i ∈ {1,...,n−k+1} and outcomes and possibly more parties as in a multipartite
⊕ stands for addition modulo n. After the second step Bell test. Even in the bipartite case, n = 4 for the n-
n
the initial distribution is in the isotropic form, however cycle, the complete characterization of the local correla-
maintaining the value of C unchanged. tionsisnotknownforanumberofoutcomeslargerthan3
n
Computing the BC value for the isotropic distribution [9, 10]. Could it be that the BC entropic inequality aug-
mixed with the classically correlated box, that is, vp + mented with shared randomness fully characterizes the
I
(1−v)p , expanding around v =0 we find that n-cycle for a general number of outputs? Another inter-
C
esting question is to understand the role of entropic in-
v equalities in the bilocality scenario, where independence
BC = [f(n,ǫ)−(2−n(1−ǫ))lnv] (9)
n
ln4 constraintsdefine a nonconvexset, difficult to character-
ize in the probability space [15].
withf(n,ǫ)=2−n(1−ǫ)(1+ln2)+(n+ǫ−nǫ)ln(1−ǫ)− Finally,violationsoflinearBellinequalitiescanbeun-
ǫln(1+ǫ)+ln(4/(1−ǫ2)). Forany2−n(1−ǫ)>0tak- derstood as a resource, for instance, allowing for higher
ing a sufficiently small v ensures BC to be positive since probability of success in some information tasks [5]. Is
f(n,ǫ) does not depend on v. That is, for any nonlocal there any operational interpretation for the violation of
isotropic distribution ǫ > (n −2)/n the BC inequality an entropic inequality in terms of a relevant physical
canbe violated. This is the same boundobtainedby the task? If thatturns outto be the case,aninterestingsce-
direct calculationof Cn(pI). Given that any distribution nario would arise, where nonlocal but entropically clas-
canbeturnedintotheisotropicboxwithoutchangingits sical correlations could be turned into a useful resource,
Cn value,thismeansthatany contextual/nonlocal distri- being activated by the use of shared randomness.
bution in the n-cycle scenario also violates the entropic
BC inequality when properly mixed with a classically
n
correlated and noncontextual/local distribution. Appendix: Violation of the entropic inequalities
We note that we have analyzed the specific case of without the depolarization procedure
a distribution violating a specific equivalence of Cγ
n
where all γi = 1 but γn = −1. However, the gen- Weshowherethatgivenageneralnonlocalprobability
eralization for distributions violating other equivalences distribution in the CHSH scenario, the mixing with the
of Cnγ is straightforward. Since there is a one-to-one classical correlation is sufficient to violate the entropic
correspondence between the vertices of the nondisturb- inequality, without the need of the depolarizationproce-
ing/nonsignalling set and each of the facets of the non- dure.
bcoenbterxotuugahlt/ltoocatlhseetis,oitfraopdiicstfroirbmutipoγIn=vioplγmaatxes+Cpnγw, iatgcaainn paTrahmeeltoecraizledseatsconsists of 16 extremal points pαde,βt,γ,δ
without changing the value of Cγ. Mixing pγ with the
n I
corresponding classical correlation pγ′ will lead to viola- 1 , a=αx⊕β,
C
tions of BCnk and therefore to the same conclusions as pdet(ab|xy)= b=γy⊕δ , (A.1)
before. It is important to note that the characterization  0 , otherwise
ofthenoncontextual/localsetisachievedwithnentropic 
inequalities, as opposed to the 2n−1 linear inequalities and all the 8 nonlocal extremal points pα,β,γ of the
PR
5
nonsignalling set can be parameterized as 2̺0,0,0−2̺0,0,1>1. Mixingthep(ab|xy)withtheclassi-
PR PR
calcorrelatedboxasbeforeandexpandingaroundv =0,
one finds that BC = (v/ln4)[g+2lnv(1−C )], that
4 4
1/2 , a⊕b=xy+αx+βy+γ violates the entropic inequality for C > 1 (g is a func-
p (ab|xy)= , 4
PR (cid:26)0 , otherwise tion of p(ab|xy) but independent of v).
(A.2) The same argument can be applied to the general n-
here α, β, γ,δ ∈ {0,1}. To simplify the description we cycle,andwehavetestedupton=7thatthesameresult
have employed the common notation to the CHSH sce- holds. For larger n, the procedure becomes unfeasible
nario, that is, X1 and X3 corresponding to x = 0 and giventhatthenumberofextremalpointsincreasesexpo-
x=1 and X and X correspondingto y =0 andy =1, nentially, but we conjecture that the same result should
2 4
while a and b label the corresponding outcomes. For a hold for any n.
general distribution written as a convex combination of
all extreme points
ACKNOWLEDGMENTS
p(ab|xy)= ̺α,β,γpα,β,γ + ̺α,β,γ,δpα,β,γ,δ, I would like to thank D. Gross for helpful discussions
PR PR det det
αX,β,γ α,Xβ,γ,δ and J. B. Brask, T. Fritz and M. T. Quintino for com-
(A.3) ments on the manuscript. This work was supported by
the condition for the violation of the CHSH inequal- theExcellenceInitiativeoftheGermanFederalandState
ity is that C = (−1)(1−α)(β+δ)+α(β+γ+δ)̺α,β,γ,δ+ Governments (Grant ZUK 43).
4 det
α,Xβ,γ,δ
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