arXiv:1607.05714v1 [quant-ph] 19 Jul 2016
B. Non-local computation games
VI. SINGLE-SYSTEM BINARY OUTCOME INEQUALITIES WITHOUT QUANTUM VIOLATION
The main result we present in this section is tight non-contextuality inequalities that have no quantum viola-tion and cannot be recovered by the CE principle (see definition1). Formally, it is stated as follows
Theorem. There exist tight nontrivial noncontextuality in-equalities (facets of the non-contextual polytope) with no quantum violation (but violated in general non-disturbing theories) that, moreover, are not in the form of a constraint imposed by the Consistent Exclusivity principle.
We focus on dichotomic measurements, where the output labels are taken as+1 and−1, and contextual-ity scenarios where the compatibilcontextual-ity structure has at most two compatible measurements in each context, and therefore can be represented by a compatibility graph. We will first present two results from which the main theorem follows.
Finding all the facets of the noncontextual polytope for a given scenario can be accomplished by algorithms such as PORTA [42] and cdd [43] for small scenarios, whereas for more complex ones the problem becomes computationally intractable. Instead of applying these algorithms, in this work we obtain noncontextuality in-equalities by exploiting the connection between thecut polytope[21] of a compatibility graph and its noncontex-tual polytope. Such a connection was first established by Aviset al[44] in Bell scenarios, and here we make the ex-plicit statement for general contextuality ones. This will allow us to obtain general statements for contextuality by exploring some established results on cut polytopes.
Given a graphG = (V,E), a cut of Gis defined by a subset S ⊂ V, and consists of those edges joining a vertex of S to a vertex not in S. The edge incidence vector of a cut is a binary vector~xS of size|E|. Its components are: ~xS(j) = 0 if edge j is not present in the cut, and
~xS(j) =1 otherwise. The cut polytope of G is the convex hull of the edge incidence vectors:{~xS|S is a cut}.
We first explore a remark made by Aviset althat one can define the cut polytope in terms ofanti-correlations
xij:= p(a6=b|ij), (57) wherexijis short for~xS({i,j}). That is, the FC-NC (full correlation non-contextual) polytope of G and the cut polytope of G are isomorphic, where the bijection be-tween the two sets is given by the affine transformation hMiMji=1−2xij. (58) Now we will show that a similar statement can be made between the NC-polytope ofGand the cut polytope of thesuspension graph of G. Recall that the suspension graph∇G:= (V′,E′)of a graph G is obtained from G by adding one new vertexOwhich is adjacent to all other vertices ofG.
Proposition 8. Let G(V,E) be a compatibility graph rep-resenting a contextuality scenario. The NC polytope of this scenario is isomorphic to the cut polytope of the suspension graph∇G, via the affine transformations:
hMiMji=1−2xij
hMii=1−2xiO. (59) Proof.We will show that every edge incidence vector of∇Gdefines a deterministic non-contextual behaviour and vice-versa, from which the statement about the identity of their convex hulls follows.
First arrange the components of~xSas follows: the first ncomponents relate to the edges of typeE0{{O,v}|v ∈ V}, i.e. are xO i , and the others to those of the type E={{v,w}|v,w∈V}, i.e. arexij.
On the one hand, to show that the edge incidence vec-tor of a cut S provides a deterministic non contextual model for thexij, we only need to prove that the model is noncontextual, that is, that it factorises properly at the level of correlators via the map (59). What we mean by this is that we should prove thathMiMji = hMiihMji, i.e. 1−2xij = (1−2xO i)(1−2xO j). For this note that there are only 8 ways of separating the verticesi,jand Ointo groups of ‘in S’ and ‘outside S’, and a case by case inspection proves the equality.
On the other hand, now we need to prove that ev-ery deterministic non contextual models is mapped to an edge incidence vector of a cut S. That is, for each non contextual deterministic strategy we need to find a cut S. One way to do is it the following: Let S be the set of vertexOplus all the vertices that denote measurements that give outcome−1. It is easy to check that the anti-correlationsp(a6=b|ij)coincide with the edge incidence vector of the cut S.
Considerable effort has been devoted to characteris-ing the cut polytopes of complete graphs. Many general inequalities are known [21], and we have the complete list of tight inequalities for complete graph with seven or less measurements [45]. Moreover, it is conjectured that a complete characterisation is known for the case of eight and nine [46] measurements (all these inequalities can be also found in the website [47]).
Now we move on to presenting our examples. Con-sider the case where the compatibility graph isG=Kn, i.e. a complete graph where all thenmeasurements are pairwise compatible, but not necessarily jointly com-patible. When studying quantum correlations, since we are working with projective measurements, they are in-deed jointly measurable, and therefore these are scenar-ios where the quantum set coincides with the noncon-textual polytope. That is, all the facets of the cut poly-topeCUT(∇Kn)are tight noncontextuality inequalities with no quantum advantage.
The question of interest now is whether these tight in-equalities with no quantum violation are CE inequali-ties. In what follows, we will first translate the
ques-14 tion in the hypergraph framework of [7] and then
char-acterise the constraints imposed by CE.
In Kn, the contexts of compatible measurements have two elements, and the quantity of interest is the be-haviour P(ab|ij), where a,b ∈ {−1, 1} and i 6= j ∈ {M1, . . . ,Mn}. These Mi are the measurements in the contextuality scenario, which in this dichotomic case we will also refer to as observables. This allows to interpret the compatibility graph Kn as a hypergraph
Hn as follows. In the language of [7], each context may be considered as a measurement with output set {(−1,−1),(−1, 1),(1,−1),(1, 1)}. Hence, the set of events that characterize the contextuality scenarioHnis V(H) = {(ab|ij)|a,b ∈ {−1, 1},i,j ∈ {M1, . . . ,Mn} ∧ i < j}. The hyperedges of Hn correspond to either normalization conditions or no-disturbance conditions.
Hence,
E={(−1,−1|i,j),(−1, 1|i,j),(1,−1|i,j),(1, 1|i,j)}|i,j∈ {M1, . . . ,Mn} ∧i<j
∪{(−1,−1|i,j),(−1, 1|i,j),(1,−1|i,k),(1, 1|i,k)}|i,j,k∈ {M1, . . . ,Mn} ∧i6=j6=k .
It is easy to see that for every hyperedge e ∈ E,
∑v∈eP(v) = 1, and that the no-disturbance principle is satisfied. Note also that the (family of) hyperedges of the form{(−1,−1|i,j),(−1, 1|i,j),(1,−1|i,k),(1, 1|i,k)} corresponds (are isomorphic) to the following sequen-tial protocol: first measurei, if the outcome is a = −1 measure then j, and otherwise k. The no-disturbance condition assures that these sequential measurements are complete. These type of hyperedges are the so called
“measurement protocols” in appendix D of [7].
As mentioned in sectionII, two events are exclusive if and only if they belong to a same hyperedgee ∈ E.
In what follows, we characterise the CE inequalities that arise in these scenarios.
Proposition 9. All nontrivial CE1inequalities are given by:
−hMiMji − hMjMki − hMkMii ≤1
Proof. The proof consists of two parts. First, we show that the maximum sets of orthogonal events inHnhave size four, and correspond to the trivial constrains given by normalization and no-disturbance conditions. Fi-nally, we show that the non-trivial maximal sets of or-thogonal events inHnhave size 3 and give rise to such CE inequalities.
Let Sbe a set of mutually orthogonal events on Hn. Note that for every two events inS, the corresponding contexts must have at least one observable in common.
Hence, we divide the proof by cases.
First, consider the case where all the events in S re-late to the same context, which we denote byij. Hence, the maximal set is{(ab|ij) : a,b ∈ {−1, 1}}, where all possible values for the outcomes are allowed.
Second, consider the case where two contexts are en-compassed in the events of S, i.e. these may have
ei-ther the form (·|ij) or (·|ik). Without loss of general-ity, suppose (ab|ij) ∈ S. Then, (ac¯ |ik) ∈ S for any c. Hence, the maximal sets in this case have the form {(ab|ij),(ab¯|ij),(ac¯ |ik),(a¯c¯|ik)} for fixeda,b,c, i.e. cor-respond to measurement protocols over Kn.
Third, consider the case where three contexts are en-compassed in the events ofS. One option is to take the contexts asij,ikandjk. Without loss of generality, sup-pose(ab|ij) ∈ S. Then, for an event with ik to be or-thogonal to (ab|ij), it should have outputs (ac¯ |ik), for any value ofc. Similarly forjk, we have(bd¯ |jk), for any value ofd. However, if we want both events to belong toS, they should further satisfy(ac¯ |ik)⊥(bd¯ |jk), which impliesd =c. Hence, the maximal sets¯ S, characterized by the parametersa,b,c,i,j,k, consist of three elements and read
S={(ab|ij),(ac¯ |ik),(b¯¯c|jk)}.
The other option is to take the contexts asij,ik andil, i.e. they all haveias the common observable. It’s easy to check that, since the observables are binary, there is no possible assignment of outputs for which the events are pairwise orthogonal. Hence, this choice of three con-texts does not give rise to any set of mutually exclusive events.
Finally, we are left with the case where four or more contexts are encompassed in the events of S. In this case, since every pair of contexts should have one ob-servable in common, the only possibility is that all of them share the same observable, i.e. ij1, . . . ,ijk. How-ever, since these are binary observables, it is not possi-ble to assign outputs in a way that assures the pairwise orthogonality of every pair of events with different con-texts. Hence, such a situation may never give rise to a set of mutually exclusive events.
Now we move on to unravel the constraints that CE1 imposes over the space of events in Hn. For the first case, those are∑a,bP(ab|ij) ≤ 1, which is trivially sat-isfied due to normalization of the behaviourP. For the second case, the constraints take the form∑bP(ab|ij) +
15
∑cP(ac¯ |ik) ≤ 1, which is guaranteed by the no-disturbance condition. Indeed, the no-no-disturbance con-dition implies ∑cP(ac¯ |ik) = ∑cP(ac¯ |ij), hence the statement follows from normalization. Finally, in the third case the CE1constraint readsP(ab|ij) +P(ac¯ |ik) + P(¯b¯c|jk)≤1, which is a nontrivial CE inequality.
All the possible choices of the outcomes a,b,c will give rise to all the CE inequalities for each choice of three measurements Mi,Mj,Mk. When translated into the correlatorshMiMjithey take the form of eq. (60).
Note that this lemma is a peculiarity of contextuality scenarios which do not arise from Bell Scenarios. In-deed, due to the local structure of the latter, the orthog-onality between (ab|ij) and (b¯c¯|jk) is not defined for i 6= j 6= k. Hence, there is no contradiction with the fact that LO1≡NS for bipartite Bell Scenarios [12].
We have seen that the inequalities that define the set of behaviors that satisfy CE1 have a very simple form.
We remark that the inequalities presented at eq (60) are also known astriangle inequalitiesin the cut polytope lit-erature (see chapter 27 of [21]) and they were already explored in noncontextuality [48]. Also, they represent all tight NC-inequalities in the the complete compatibil-ity graph scenario with three vertices [17,49].
However, the set of noncontextual behavious in Kn
(which coincides with the quantum set) is defined by the facets of the CUT polytope of∇Kn (see [47] for a list of all known cut polytope inequalities for complete graph scenarios with less than ten measurements ). Therefore, facet defining inequalities forCUT(∇Kn)which are not of triangular form are tight noncontextuality inequali-ties with no quantum violation and not in the form of a Consistency Exclusivity constraint.
One example of this is thepentagonal inequality
∑
1≤i<j<n
bibjxij≤0, (61)
with b1 = b2 = b3 = 1,b4 = b5 = −1, which is facet defining for the cut polytope of a complete graph with five vertices. Let us associate vertex 5 with the addi-tional vertexOin∇K4. A direct application of the affine map (59) provides us the tight noncontextuality inequal-ity
∑
1≤i<j<n
−bibjhMiMji ≤2, (62)
withb1 = b2 = b3 = 1,b4 = b5 = −1, whereM5 = 11.
Direct inspection shows that this inequality is not in the form of CE, and therefore provides one explicit example of a tight noncontextuality inequality without quantum violation we are searching for.
In addition to the database presented in [47], chapters 28-30 of [21] present a family of inequalities for the cut polytope of any complete graph. One class is given by
thehypermetric inequaltiesthat are respected by any point in the cut polytope ofKn Triangle inequalities and the pentagonal one are partic-ular examples of hypermetric inequalities. For the hy-permetric inequalities to be tight conditions need to be imposed on the coeffientsbi; some of these are known (see chapter 28 of [21]). For instance, the inequalities are tight wheneverb1 = b2 = . . . = bn−2 = 1, bn−1 = −1 and bn = 4−n (theorem 28.2.4 (iiib) of ref [21]). As a remark, note that in general theories that satisfy the no-disturbance condition one can construct boxes where hMiMji = ±1 whilehMii = 0, such that they achieve the maximum algebraic value of these inequalities.
VII. CONCLUSION
In this paper, we presented progress toward the reso-lution of the question whether there are tight two-party Bell inequalities that are also facets of the quantum set of correlations, focusing on the paradigmatic cases of XOR games and linear games. We formulated a sim-ple sufficient condition to identify and construct games with no quantum advantage within these classes. We then showed that the well-known class of Bell inequal-ities corresponding to non-local computation tasks are not tight in both the binary outcome case and in the gen-eralization to arbitrary prime outputs for a special class of functions, i.e., they do not define facets of the corre-sponding Bell polytopes. We then identified correlation inequalities that define faces of the set of quantum cor-relations in the simplest Bell scenario corresponding to the CHSH polytope.
Then we moved on to general contextuality scen-rios, and studied the question in this single-party sce-nario, namely non-contextuality inequalities with bi-nary outcomes. Here we identified the polytope of non-contextual behaviours with the cut polytope of the suspension graph of the compatibility graph represent-ing the measurements in the experiment. Moreover, we characterised the polytope of behaviours that sat-isfy consistent exclusivity inequalities for compatibil-ity graphs that are complete. We then found that, in this particular case (which is not a Bell scenario) there are tight non-contextuality inequalities with no quan-tum advantage that are not in the Consistent Exclusivity form. An open problem is to find an interpretation of these inequalities as an information theoretic task.
Finally, two main questions are still open: whether the two-party correlation polytope shares any facets with the Tsirelson elliptope and whether the two-party Bell polytope shares any facets with the quantum correlation set.
Acknowledgements. We are grateful to P. Horodecki and M. Horodecki for useful discussions, R.R. thanks
J.-16 D. Bancal and Y.-C. Liang for asking the question
an-swered in Section V. R. R. is supported by the ERC AdG grant QOLAPS. M.T.Q. acknowledges financial support from the Swiss National Science Foundation (grant PP00P2 138917, Starting grant DIAQ). A.B.S.
ac-knowledges the support from the ERC AdG NLST. G.
M. acknowledges support from NWO VIDI, ERC Start-ing Grant and the Brazilian agency Fapemig. R. A. ac-knowledges support from the European Union’s Hori-zon 2020 research and innovation programme under the Marie Skłodowska-Curie N-MuQuas Grant No. 705109.
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