Unified framework for continuity of sandwiched R\'enyi divergences

TL;DR

This paper establishes new uniform continuity bounds for sandwiched R'enyi divergences using three different mathematical approaches, improving existing bounds and extending their applicability in quantum information theory.

Contribution

It introduces three novel methods to derive continuity bounds for sandwiched R'enyi divergences, including the operator space approach and a mixed approach, expanding their use in resource theories.

Findings

Improved continuity bounds over previous results

Extension of bounds to resource theory contexts

First continuity bounds for certain entropic quantities

Abstract

In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched R\'enyi divergences such as the sandwiched R\'enyi conditional entropy. We follow three different approaches: The first one is the "almost additive approach", which exploits the sub-/ superadditivity and joint concavity/ convexity of the exponential of the divergence. In our second approach, termed the "operator space approach", we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched R\'enyi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.