Relative operator entropies and Tsallis relative operator entropies in JB-algebras

Inequalities for relative operator entropies and operator means

The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by the use of Hermite-Hadamard inequality. Thus, we obtain alternative bounds for the Tsallis relative operator entropy. In the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the generalized relative operator entropy. In addition, we study the properties on monotonicity for the weight of operator means, and for the parameter of relative operator entropies.

Some remarks on Tsallis relative operator entropy

This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\left( A|B \right)=\frac{A{{\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \subset (0,\infty)$. We prove for any $v\in \left[ -1,0 \right)\cup \left( 0,1 \right]$, $$ (\ln_v t)A+\left( A{{\natural}_{v}}B+tA{{\natural}_{v-1}}B \right)\le {{T}_{v}}\left( A|B \right) \le (\ln_v s)A+{{s}^{v-1}}\left( B-sA \right) $$ where $s,t\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Mea…

Convexity of parameter extensions of some relative operator entropies with a perspective approach

In this paper, we introduce two notions of a relative operator $(α, β)$-entropy and a Tsallis relative operator $(α, β)$-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning $α$ and $β$. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.

Tsallis relative operator entropy in mathematical physics

Tsallis relative operator entropy is defined as a parametric extension of the relative operator entropy. Some properties of the Tsallis relative operator entropy are investigated. Also some operator inequalities related to the Tsallis relative operator entropy are shown.

Generalized Shannon inequalities based on Tsallis relative operator entropy

Tsallis relative operator entropy is defined and then its properties are given. Shannon inequality and its reverse one in Hilbert space operators derived by T.Furuta \cite{Fu:par} are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.

$r$-变形的 $\alpha$-$z$-Rényi 相对熵

$r$-deformed $α$-$z$-Rényi relative entropy

本文引入 $r$-对数定义三参数 Rényi 相对熵族，推广了 $\alpha$-$z$-Rényi 相对熵，证明其满足散度公理，给出数据处理不等式成立的参数范围，并证明该熵是 Tsallis 相对熵的上界，且比现有上界更紧。