Monotonicity of quantum relative entropy and recoverability

TL;DR

This paper refines the understanding of quantum relative entropy's monotonicity by establishing a recovery bound using a rotated Petz map, advancing towards a conjecture with broad implications in quantum information theory.

Contribution

It introduces a remainder term for quantum relative entropy monotonicity using a rotated Petz recovery map, linking multiple entropy inequalities.

Findings

Established a recovery bound for quantum relative entropy

Connected refinements of entropy inequalities through Petz maps

Progressed towards a conjecture involving the Petz recovery map

Abstract

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.