Quantum Relative Lorenz Curves

TL;DR

This paper introduces quantum relative Lorenz curves to unify majorization concepts in quantum resource theories, defining new divergences and applying them to quantum thermodynamics transformations.

Contribution

It formulates quantum relative Lorenz curves, unifies majorization and thermomajorization, and introduces Hilbert alpha-divergences for quantum information applications.

Findings

Unified framework for majorization and thermomajorization.

Defined Hilbert alpha-divergences and related them to existing divergences.

Applied framework to quantum state transformation problems in thermodynamics.

Abstract

The theory of majorization and its variants, including thermomajorization, have been found to play a central role in the formulation of many physical resource theories, ranging from entanglement theory to quantum thermodynamics. Here we formulate the framework of quantum relative Lorenz curves, and show how it is able to unify majorization, thermomajorization, and their noncommutative analogues. In doing so, we define the family of Hilbert $\alpha$-divergences and show how it relates with other divergences used in quantum information theory. We then apply these tools to the problem of deciding the existence of a suitable transformation from an initial pair of quantum states to a final one, focusing in particular on applications to the resource theory of athermality, a precursor of quantum thermodynamics.