Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between $\chi^2$ and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum $f$-divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds.

翻译：本文给出了达到最大$f$-散度的显式经典态，从而为松本定理提供了简洁证明，并将经典$f$-散度间的任意不等式系统性地推广至量子$f$-散度。我们的方法特别简单，无需任何复杂的矩阵分析工具，仅需基础线性代数知识。该方法同样高效，通过两个改进现有界限的示例得以体现：(i) 推导出$\chi^2$散度与迹范数间改进的量子Pinsker不等式，并用于提升退相干理论中的一个界限；(ii) 针对任意量子$f$-散度推导出新的反向量子Pinsker不等式，并与先前（Audenaert-Eisert及Hirche-Tomamichel）界限进行比较。