The n-th relative operator entropies and the n-th operator divergences

Abstract

Let A and B be strictly positive linear operators on Hilbert space $${\mathcal {H}}$$ and $$n\in {\mathbb {N}}$$. We define the n-th relative operator entropy $$\begin{aligned}&S^{[n]}(A|B) \equiv \frac{1}{n!}A^{\frac{1}{2}} (\log A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^n A^{\frac{1}{2}} = \displaystyle \frac{1}{n!} A(A^{-1}S(A|B))^n \end{aligned}$$and the n-th Tsallis relative operator entropy $$T^{[n]}_x(A|B)$$ inductively as follows: $$\begin{aligned}&T^{[1]}_x(A|B) \equiv T_x(A|B) \ \mathrm{and} \\&T^{[n]}_x(A|B) \equiv \frac{T^{[n-1]}_x(A|B)-S^{[n-1]}(A|B)}{x} \ (x\ne 0)\ \mathrm{for}\ n\ge 2. \end{aligned}$$By introducing the Taylor’s expansion of the path $$A\ \natural _x\ B$$ around $$\alpha \in {\mathbb {R}}$$, we see the coefficient of the $$(x-\alpha )^k$$-term is the k-th generalized relative operator entropy and the residual term divided by $$(x-\alpha )^n$$ is the n-th residual relative operator entropy. In this paper, we show properties of these n-th relative operator entropies and relations among them. In addition, we introduce the n-th operator valued divergences as the differences between the n-th relative operator entropies and show some properties of them.