Tight relations and equivalences between smooth relative entropies

TL;DR

This paper establishes a strong equivalence between hypothesis testing relative entropy and a variant of smooth max-relative entropy, leading to tighter bounds and improved divergence inequalities in quantum information theory.

Contribution

It strengthens the connection between key smooth entropic quantities, introduces new proof techniques, and improves bounds and duality relations in one-shot quantum information settings.

Findings

Proves equivalence between hypothesis testing relative entropy and a measured smooth max-relative entropy.

Introduces a modified proof technique using matrix geometric means.

Provides tighter bounds and refined divergence inequalities.

Abstract

The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smooth max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth max-relative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smooth max-relative entropy, introducing a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, establishing provably tight bounds between them. The results then allow us to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with R\'enyi divergences.