Information theoretical properties of Tsallis entropies

A chain rule and a subadditivity for the entropy of type $\beta$, which is one of the nonadditive entropies, were derived by Z.Dar\'oczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory.