Quantum f-divergences and error correction

TL;DR

This paper explores quantum f-divergences, establishing their monotonicity under certain maps, extending Petz' reversibility theorem, and applying these results to quantum error correction with a focus on operator convex functions.

Contribution

It unifies and extends monotonicity results for quantum f-divergences, and applies these to develop a framework for quantum error correction based on distinguishability measures.

Findings

Quantum f-divergences are monotonic under dual Schwarz maps with operator convex functions.

Extended Petz' reversibility theorem for a broad class of f-divergences.

Provided a canonical construction for reversing stochastic maps in quantum error correction.

Abstract

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.