Optimal Universal Bounds for Quantum Divergences

TL;DR

This paper establishes a universal geometric principle for smoothing classical divergences, leading to optimal bounds for various quantum divergences, including R'enyi and hypothesis testing divergences, improving upon previous bounds.

Contribution

It introduces a divergence-independent structural principle for smoothing, enabling derivation of optimal universal bounds for quantum divergences with state-independent correction terms.

Findings

Derived divergence-independent bounds for smoothed divergences.

Proved the optimality of correction terms in the bounds.

Improved and unified previous bounds for quantum divergences.

Abstract

We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum R'enyi divergences of arbitrary order and the hypothesis testing divergence. Our inequalities relate divergences of different orders through bounds of the form $D_\beta^{\varepsilon} \le D_\alpha + \mathrm{correction}$ and $D_\beta^{\varepsilon} \ge D_\alpha + \mathrm{correction}$, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis testing divergence.