From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence

TL;DR

This paper presents explicit classical distributions that achieve maximal quantum $f$-divergence, simplifies proofs of key inequalities, and improves bounds in quantum information theory using elementary linear algebra.

Contribution

It introduces a straightforward method to extend classical $f$-divergence inequalities to quantum cases, with explicit distributions and improved bounds.

Findings

Derived an improved quantum Pinsker inequality between $\\chi^2$ and trace norm.

Established a new reverse quantum Pinsker inequality for quantum $f$-divergences.

Provided explicit classical states achieving maximal $f$-divergence, simplifying proofs.

Abstract

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.