Majorization theory for quasiprobabilities

TL;DR

This paper extends majorization theory to continuous quasiprobability distributions over infinite measure spaces, providing new tools for quantum resource theories and analyzing quantum states like the Wigner function.

Contribution

It introduces a generalized majorization framework for quasiprobabilities, proving key equivalences and applying them to quantum resource theories and quantum optics.

Findings

Established equivalence of four definitions of majorization for quasiprobabilities.

Developed new resource monotones for quantum resource theories.

Applied framework to analyze the Wigner function in quantum optics.

Abstract

Majorization theory is a powerful mathematical tool to compare the disorder in distributions, with wide-ranging applications in many fields including mathematics, physics, information theory, and economics. While majorization theory typically focuses on probability distributions, quasiprobability distributions provide a pivotal framework for advancing our understanding of quantum mechanics, quantum information, and signal processing. Here, we introduce a notion of majorization for continuous quasiprobability distributions over infinite measure spaces. Generalizing a seminal theorem by Hardy, Littlewood, and P\'olya, we prove the equivalence of four definitions for both majorization and relative majorization in this setting. We give several applications of our results in the context of quantum resource theories, obtaining new families of resource monotones and no-goes for quantum state conversions. A prominent example we explore is the Wigner function in quantum optics. More generally, our results provide an extensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces.