Abstract

In this paper, we study a matrix equation involving the matrix geometric mean \(A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},\) \(t\in(1,2)\). We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean \(A\natural_{t}B\) to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.

中文翻译：

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广义矩阵幂均值和新的量子海林格散度

摘要

在本文中，我们研究涉及矩阵几何均值\(A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^ 的矩阵方程{t}A^{1/2},\) \(t\in(1,2)\)。我们研究了该方程的唯一解的几个性质和不等式。我们还使用加权几何平均值\(A\natural_{t}B\)定义新的量子海林格散度，并证明新的量子散度满足数据处理不等式。