A finite sufficient set of conditions for catalytic majorization

Understanding when one physical state can be transformed into another is a central problem in quantum information science and thermodynamics. Majorization provides a mathematical tool for describing such transformations. Yet many transitions that are forbidden by majorization can become possible in the presence of a catalyst, an auxiliary system that enables the process without being consumed or altered. Determining the feasibility of such catalytic transformation typically involves checking an infinite set of inequalities involving generalized entropic quantities. 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 further examples, we provide a software toolbox implementing these conditions. Our results rely on the connection between a polynomial representation of ℓp norm with Rényi p entropies for any real value of p.