Generalizations of operator Shannon inequality based on Tsallis and Rényi relative entropies

Abstract

Recently, we obtained relations among relative operator entropies of sequences. For example, for relative operator entropy S(A|B), Rényi relative operator entropy It(A|B) and Tsallis relative operator entropy Tt(A|B) of sequences of strictly positive operators A=(A1,…,An) and B=(B1,…,Bn) such that ∑i=1nAi=∑i=1nBi=I,S(A|B)⩽It(A|B)⩽Tt(A|B)⩽0 holds for 0<t<1. This is an extension of operator version of Shannon inequality (briefly, operator Shannon inequality) discussed by Furuta and Yanagi–Kuriyama–Furuichi. In this paper, we shall obtain two generalizations of this inequality by considering generalizations of relative operator entropies of sequences.