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.

翻译：主控关系在数学、量子信息与资源理论以及量子热力学等领域有着广泛应用，它描述了两种物理状态之间允许的转换。在许多情况下，当状态向量$x$不主控状态向量$y$时，仍有可能找到催化剂——另一个向量$z$，使得$x \otimes z$主控$y \otimes z$。判断此类催化转换的可行性通常需要检验无穷多个不等式。本文推导了一个有限的不等式充分集，该集合蕴含催化可行性。将此框架扩展至热力学领域，我们还建立了热操作下催化状态转换的有限充分条件集。针对新示例，我们提供了一个实现这些条件的软件工具箱。