A finite sufficient set of conditions for catalytic majorization

TL;DR

This paper derives a finite set of conditions to determine when catalytic majorization is possible, simplifying the process of checking state transformations in quantum thermodynamics and resource theory.

Contribution

It introduces a finite set of inequalities that guarantee catalytic majorization, extending to thermodynamics, and provides a software toolbox for practical application.

Findings

Finite inequalities imply catalytic majorization.

Conditions extend to thermal operations in thermodynamics.

Software toolbox implemented for checking conditions.

Abstract

The majorization relation has found numerous applications in mathematics, quantum information and resource theory, and quantum thermodynamics, where it describes the allowable transitions between two physical states. In many cases, when state vector $x$ does not majorize state vector $y$, it is nevertheless possible to find a catalyst - another vector $z$ such that $x \otimes z$ majorizes $y \otimes z$. Determining the feasibility of such catalytic transformation typically involves checking an infinite set of inequalities. Here, we derive a finite sufficient set of inequalities that imply catalysis. Extending this framework to thermodynamics, we also establish a finite set of sufficient conditions for catalytic state transformations under thermal operations. For novel examples, we provide a software toolbox implementing these conditions.