In this paper, we introduce new quantum divergences of the form $\mathrm{\Phi }(A,B)=\mathrm{Tr}(A\sigma B-A\tau B),A,B>0,$where σ and τ are Kubo–Ando operator means such that $\sigma \ge \tau$. More precisely, we show that $\mathrm{\Phi }(A,B)$ is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function $\sqrt{x}$. We also study the least-squares problem, the data processing inequality, and in-betweenness property of the matrix means with respect to the quantum Hellinger-type divergence.

中文翻译：

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关于新的量子分歧

在这篇论文中，我们引入了新的量子发散形式$\mathrm{\Phi }(A,乙)=\mathrm{Tr}(A\sigma 乙-A\tau 乙),A,乙>0,$其中σ和τ是 Kubo–Ando 算子意味着$\sigma \ge \tau$. 更准确地说，我们表明$\mathrm{\Phi }(A,乙)$是当σ是加权 Kubo-Ando 矩阵幂均值且τ是加权几何均值时的量子散度。此外，我们使用函数的线性逼近构造了一个新的量子 Hellinger 型散度$\sqrt{X}$. 我们还研究了最小二乘问题、数据处理不等式以及矩阵均值与量子 Hellinger 型散度相关的介数性质。