Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map

TL;DR

This paper strengthens the monotonicity inequality of quantum relative entropy by establishing a lower bound involving a convex combination of rotated Petz recovery maps, with implications for quantum information theory.

Contribution

It introduces a new lower bound on the difference of quantum relative entropies using pinched Petz recovery maps, enhancing understanding of quantum channel reversibility.

Findings

Reproduces recent bounds on conditional mutual information.

Uses elementary properties of pinching maps and operator logarithm.

Provides a strengthened inequality for quantum relative entropy.

Abstract

The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a "recovery map" that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved in [Fawzi and Renner, Commun. Math. Phys., 2015]. Our proof only relies on elementary properties of pinching maps and the operator logarithm.