A generalization of majorization that characterizes Shannon entropy

TL;DR

This paper introduces a new relation on probability distributions that generalizes majorization, providing a Shannon entropy-based characterization relevant to thermodynamics and quantum information.

Contribution

It defines a generalized majorization relation for probability distributions, characterized entirely by Shannon entropy, linking thermodynamics, information theory, and resource transformations.

Findings

The relation is fully characterized by Shannon entropy.

It offers a resource-theoretic interpretation of Shannon entropy.

Provides a simplified proof of Shannon entropy's characterization.

Abstract

We introduce a binary relation on the finite discrete probability distributions which generalizes notions of majorization that have been studied in quantum information theory. Motivated by questions in thermodynamics, our relation describes the transitions induced by bistochastic maps in the presence of additional auxiliary systems which may become correlated in the process. We show that this relation is completely characterized by Shannon entropy H, which yields an interpretation of H in resource-theoretic terms, and admits a particularly simple proof of a known characterization of H in terms of natural information-theoretic properties.