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

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 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 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 nonsignalling box which violates the 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.