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The n-th relative operator entropies and the n-th operator divergences
Annals of Functional Analysis ( IF 1 ) Pub Date : 2020-01-01 , DOI: 10.1007/s43034-019-00004-5
Hiroshi Isa , Eizaburo Kamei , Hiroaki Tohyama , Masayuki Watanabe

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.

中文翻译:

第 n 个相对算子熵和第 n 个算子散度

设 A 和 B 是希尔伯特空间 $${\mathcal {H}}$$ 和 $$n\in {\mathbb {N}}$$ 上的严格正线性算子。我们定义第 n 个相对算子熵 $$\begin{aligned}&S^{[n]}(A|B) \equiv \frac{1}{n!}A^{\frac{1}{2} } (\log A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^n A^{\frac{1}{2}} = \displaystyle \frac {1}{n!} A(A^{-1}S(A|B))^n \end{aligned}$$ 和第 n 个 Tsallis 相对算子熵 $$T^{[n]}_x (A|B)$$ 归纳如下: $$\begin{aligned}&T^{[1]}_x(A|B) \equiv T_x(A|B) \ \mathrm{and} \\&T^{ [n]}_x(A|B) \equiv \frac{T^{[n-1]}_x(A|B)-S^{[n-1]}(A|B)}{x} \ (x\ne 0)\ \mathrm{for}\n\ge 2. \end{aligned}$$通过引入路径$$A\\natural_x\B$$围绕$$\alpha\的泰勒展开式\在 {\mathbb {R}}$$,我们看到 $$(x-\alpha )^k$$-term 的系数是第 k 个广义相对算子熵,残差项除以 $$(x-\alpha )^n$$ 是 n -th 残差相对算子熵。在本文中,我们展示了这些第 n 个相对算子熵的性质以及它们之间的关系。此外,我们引入了第 n 个算子值的散度作为第 n 个相对算子熵之间的差异,并展示了它们的一些性质。
更新日期:2020-01-01
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