A finite sufficient set of conditions for catalytic majorization

2025-02-27

AI Summary

This work addresses the fundamental challenge in catalytic majorization—namely, verifying infinitely many inequalities to determine whether a catalyst exists. We establish the first finite, computationally tractable sufficient condition for deciding whether a vector (x) can be catalytically transformed into (y) (i.e., whether there exists a catalyst (z) such that (x otimes z) majorizes (y otimes z)). Methodologically, we integrate majorization theory, convex analysis, and quantum resource theory; leveraging inequality system reduction and explicit auxiliary state construction, we derive rigorous, efficiently checkable criteria. The framework is further extended to thermodynamic transformations under thermal operations in quantum thermodynamics. Our contributions are threefold: (i) breaking the infinite-verification bottleneck and reducing computational complexity from intractable to polynomial-time; (ii) releasing an open-source software toolbox for designing and experimentally validating catalytic protocols; and (iii) providing a unifying, operational decision framework for catalysis in quantum information theory and nonequilibrium thermodynamics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Causality

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Determining finite conditions for catalytic majorization in state transitions.
• Establishing sufficient inequalities for catalytic transformations in quantum systems.
• Providing a software toolbox for catalytic state transformations in thermodynamics.

Innovation

• Derived finite sufficient set for catalysis
• Extended framework to thermal operations
• Developed software toolbox for implementation