量子共同利益博弈中的学习与可分离性问题
Learning in Quantum Common-Interest Games and the Separability Problem
AI总结：
本文引入量子共同利益博弈，建立最佳可分离态问题KKT点与纳什均衡的等价性，并提出非对易复制动力学和最优响应/乘法权重更新算法，从博弈学习视角为BSS问题提供去中心化求解方法。
摘要：
博弈中的学习已成为机器学习中一种强大工具，具有众多应用。量子博弈对能够获取量子资源的策略性玩家之间的交互进行建模，近期若干工作研究了量子零和博弈竞争机制下的学习问题。超越这一设定，我们引入量子共同利益博弈（CIGs），其中玩家以密度矩阵作为策略，且其利益完全一致。我们通过建立最佳可分离态（BSS）问题实例的KKT（一阶驻点）点与其对应量子CIG的纳什均衡之间的等价性，弥合了优化与博弈论之间的鸿沟。这使得量子CIG的学习动力学可被视为BSS问题的去中心化算法。从博弈中学习的视角出发，我们进一步引入连续时间复制动力学以及离散时间最优响应动力学/线性乘法权重更新的非对易扩展，用于量子CIG中的学习。我们证明了这些动力学的经典收敛结果的类似物，并探讨了量子设定中出现的差异。最后，我们通过大量实验验证了理论发现。
英文摘要：
Learning in games has emerged as a powerful tool for machine learning with numerous applications. Quantum games model interactions between strategic players who have access to quantum resources, and several recent works have studied {learning in} the competitive regime of quantum zero-sum games. Going beyond this setting, we introduce quantum common-interest games (CIGs) where players have density matrices as strategies and their interests are perfectly aligned. We bridge the gap between optimization and game theory by establishing the equivalence between KKT (first-order stationary) points of an instance of the Best Separable State (BSS) problem and the Nash equilibria of its corresponding quantum CIG. This allows learning dynamics for the quantum CIG to be seen as decentralized algorithms for the BSS problem. Taking the perspective of learning in games, we then introduce non-commutative extensions of the continuous-time replicator dynamics and the discrete-time best response dynamics/linear multiplicative weights update for learning in quantum CIGs. We prove analogues of classical convergence results of the dynamics and explore differences which arise in the quantum setting. Finally, we corroborate our theoretical findings through extensive experiments.
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