On generalized entropies and information-theoretic Bell inequalities under decoherence

We consider information-theoretic inequalities of the Bell type in the presence of decoherence. It is natural that too strong coupling with the environment can prevent an observation of quantum correlations. In this regard, the use of various entropic functions may give additional capabilities to reveal desired correlations. It was already shown that the Bell and Leggett-Garg inequalities in terms conditional Tsallis entropies are more sensitive in the cases of detection inefficiencies. In this paper, we study capabilities of generalized conditional entropies of the Tsallis type in analyzing the Bell theorem in decoherence scenarios. Two forms of the conditional Tsallis $q$-entropy are known in the literature. We show that each of them can be used for defining a metric in the probability space of interest. Such metrics can be used in realizing the so-called triangle principle. The triangle principle has recently been proposed as a unifying approach to questions of local realism and non-contextuality. Applying the triangle principle leads to the two families of $q$-metric inequalities of the Bell type. Information-theoretic formulations in terms of the $q$-entropic metrics are first discussed for the CHSH scenario in dephasing environment. Then we also revisit $q$-entropic inequalities of the Leggett-Garg type. An environmental influence is modeled by the phase damping channel and by the depolarizing channel.