  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.
