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

  • Hiroshi Isa
  • Eizaburo Kamei
  • Hiroaki TohyamaEmail author
  • Masayuki Watanabe
Original Paper

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.

Keywords

Relative operator entropy Tsallis relative operator entropy Operator valued divergence Residual relative operator entropy 

Mathematics Subject Classification

47A63 47A64 94A17 

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  • Hiroshi Isa
    • 1
  • Eizaburo Kamei
    • 2
  • Hiroaki Tohyama
    • 1
    Email author
  • Masayuki Watanabe
    • 1
  1. 1.Maebashi Institute of TechnologyMaebashiJapan
  2. 2.Kitakaturagi-gunJapan

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