A Collection of Pinsker-type Inequalities for Quantum Divergences

TL;DR

This paper develops Pinsker-type inequalities for various quantum and classical divergences, providing bounds that relate divergence measures to trace distance, and offers methods to adapt these bounds for smoothed divergences.

Contribution

It introduces new Pinsker-type inequalities for multiple divergences, extending classical bounds to quantum settings and smoothing techniques.

Findings

Derived bounds for quantum and classical divergences

Extended inequalities to smoothed divergences

Unified framework for Pinsker-type inequalities

Abstract

Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including $f$-divergences like Hellinger and $\chi^2$-divergences as well as R\'enyi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.