Tests for quantum contextuality in terms of $q$-entropies

The information-theoretic approach to Bell's theorem is developed with use of the conditional $q$-entropies. The $q$-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional $q$-entropies for all $q\geq1$. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicio\v{g}lu-Shumovsky (KCBS) scenarios. An extension to the case of $n$-cycle scenario is briefly mentioned. The new inequalities with conditional $q$-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The $q$-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the $q$-entropic formulation.