Relative submajorization and its use in quantum resource theories

TL;DR

This paper introduces relative submajorization, a generalization of majorization, to analyze quantum resource transformations, providing geometric characterizations and bounds relevant to thermodynamics, entanglement, and coherence.

Contribution

It develops the theory of relative submajorization and applies it to quantum resource theories, offering new geometric and analytical tools for resource transformation analysis.

Findings

Characterizes resource transformation probabilities and errors using Lorenz curves.

Provides bounds on reversibility of quantum resource transformations.

Utilizes linear programming duality as a key technical method.

Abstract

We introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics, bipartite entanglement, and quantum coherence. In particular, we show that relative submajorization characterizes both the probability and approximation error that can be obtained when transforming one resource to another, also when assisted by additional standard resources such as useful work or maximally-entangled states. These characterizations have a geometric formulation as the ratios or differences, respectively, between the Lorenz curves associated with the input and output resources. We also find several interesting bounds on the reversibility of a given transformation in terms of the properties of the forward transformation. The main technical tool used to establish these results is linear programming duality.