Some properties and applications of the new quantum $f$-divergences

TL;DR

This paper explores new representations and properties of recently introduced quantum $f$-divergences, demonstrating their applications in quantum information theory, including proofs of bounds and inequalities.

Contribution

It provides alternative expressions for quantum $f$-divergences, proves new properties, and applies these to establish bounds like the quantum Chernoff bound and inequalities between divergences.

Findings

New proof of the quantum Chernoff bound's achievability

Inequalities between known Renyi divergences and the new divergence

Monotonicity and convexity properties of the new $f$-divergences

Abstract

Recently, a new definition for quantum $f$-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum $f$-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum $f$-divergences. We leverage these expressions to prove new properties of these $f$-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new $f$-divergences, and prove inequalities between these divergences for various functions.