A finite sufficient set of conditions for catalytic majorization
This work provides a practical, finite criterion for catalytic transformations, which is relevant for resource theory and quantum thermodynamics.
The authors derive a finite set of inequalities that guarantee catalytic majorization, replacing the need to check an infinite set. They extend this to thermal operations and provide a software toolbox.
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.