Quantum resource changes and distributions during catalytic transformations

Abstract

One physical system always evolves from a state of low disorder to another of high disorder. This phenomenon can be identified by increased entropy. As we know, quantum resource theories provide a mathematical framework for examining diverse quantum resources in states, with the primary problem being whether quantum states can be transformed into each other by free operations. Unlike natural transformations, free operations in quantum resource theories never increase the resource measure of a primary system. Generally, the standard approach to this central problem is to study approximate transformations and apply catalysts to broaden the set of possible state transformations. By combining approximate transformations and catalytic transformations, we show that the quantum correlation between the primary system and the catalytic system after approximate catalytic transformations determines quantum resource changes in the primary system as well as the distribution of quantum correlations among different primary systems. We first find and establish that the resource measure of the primary system does not decrease under global free operations on the primary system and the corresponding catalytic system using the universal freezing technology and the catalytic transformation methods. A resource measure is monotonic under free operations, additive on tensor products, and continuous. Using these features, we then access relations between variations in the resource measure of the primary system under global free operations and quantum correlations between the primary and catalytic systems. Therefore, quantum correlations exist between the primary and catalytic systems after catalytic transformations, which are essential for nondecreasing resource measures of the primary system. Finally, we establish relationships among different primary systems after repeatedly applying the same catalyst. The correlation is closely associated with the reuse times of the same catalyst.