Shannon type inequalities of a relative operator entropy including Tsallis and Rényi ones

Abstract

Let $\mathbb{A}=(A_1,\cdots,A_n)$ and $\mathbb{B}=(B_1,\cdots,B_n)$ be operator distributions, that is, $A_i, B_i>0$ $(1\leq i \leq n)$ and $\sum_{i=1}^n A_i =\sum_{i=1}^n B_i =I$. We give a new relative operator entropy of two operator distributions as follows: For $t,s \in \mathbb{R} \setminus \{0\}$, \[ K_{t,s}(\mathbb{A}|\mathbb{B}) \equiv \dfrac{(\sum_{i=1}^n A_i \sharp_{t} B_i)^s -I}{ts}, \] where \( A \sharp_{t} B = A^{\frac12} (A^{\frac{-1}2} B A^{\frac{-1}2})^{t} A^{\frac12}. \) This includes relative operator entropy $S(\mathbb{A}|\mathbb{B})$, Rényi relative operator entropy $I_{t}(\mathbb{A}|\mathbb{B})$ and Tsallis relative operator entropy $T_{t}(\mathbb{A}|\mathbb{B})$. In this paper, firstly, we discuss fundamental properties of $K_{t,s}(\mathbb{A}|\mathbb{B})$. Secondly, we obtain Shannon type operator inequalities by using $K_{t,s}(\mathbb{A}|\mathbb{B})$, which include previous results by Furuta, Yanagi--Kuriyama--Furuichi and ourselves.