Approximate recoverability and the quantum data processing inequality

TL;DR

This paper investigates the quantum data processing inequality in the context of approximate recoverability, disproves a key conjecture, and establishes new inequalities and convexity results related to quantum relative entropies.

Contribution

It disproves a longstanding conjecture and introduces new inequalities for approximate recoverability using the Petz recovery map for specific quantum relative entropies.

Findings

Disproved a conjecture by Seshadreesan et al.

Established inequalities for sandwiched quasi and Rénnyi relative entropies at t=2.

Proved convexity theorems for parametrized relative entropy and fidelity.

Abstract

In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et. al. in J. Phys. A: Math. Theor. 48 (2015) is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and R\'enyi relative entropies for the parameter $t=2$. We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.