A finite sufficient set of conditions for catalytic majorization

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.