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

The assumption of local realism, in a Bell locality scenario, imposes nontrivial conditions on the Shannon entropies of the associated probability distributions, expressed by linear entropic Bell inequalities. 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 paradigmatic nonlocal Popescu-Rohrlich (PR) box is entropically not different from a classically correlated box. In this paper we show that for the $n$-cycle scenario, entropic inequalities completely characterize the set of local correlations. In particular, every nonsignaling box which violates the Clauser-Horne-Shimony-Holt (CHSH) inequality—including the PR box—can be locally modified so that it also violates the entropic version of CHSH 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 $n$-cycle captures equally well both the notion of local realism introduced by Bell and that of noncontextuality presented by the Kochen-Specker theorem, the results are also valid for noncontextuality scenarios.

Figure 1

Graphical representation of the $n$-cycle. The vertices represent different observables and the edges connect observables that are jointly measurable. (a) CHSH scenario, two parties with two measurement settings each. Labeling $X_1=A_0$, $X_3=A_1$, $X_2=B_0$, $X_4=B_1$, we recover the usual picture where two parties, Alice and Bob, perform spacelike separated measurements. Alice measures one out of two possible measurement settings $A_0$ or $A_1$, and similarly for Bob. (b) KCBS scenario, five observables arranged in a cyclic configuration such that each observable is compatible with its neighbors. (c) Generalization of the CHSH/KCBS scenario, with $n$ observables in a cyclic configuration (the $n$-cycle [22]). For dichotomic observables, the set of local and noncontextual correlations is completely characterized by Eq. (4). For a general number of outcomes, the unique nontrivial entropic inequalities are those given by Eq. (5).