Gamification is an emerging trend in the field of machine learning that presents a novel approach to solving optimization problems by transforming them into game-like scenarios. This paradigm shift allows for the development of robust, easily implementable, and parallelizable algorithms for hard optimization problems. In our work, we use gamification to tackle the Best Separable State (BSS) problem, a fundamental problem in quantum information theory that involves linear optimization over the set of separable quantum states. To achieve this we introduce and study quantum analogues of common-interest games (CIGs) and potential games 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 a BSS instance and the Nash equilibria of its corresponding quantum CIG. Taking the perspective of learning in games, we introduce non-commutative extensions of the continuous-time replicator dynamics and the discrete-time Baum-Eagon/linear multiplicative weights update for learning in quantum CIGs, which also serve as decentralized algorithms for the BSS problem. We show that the common utility/objective value of a BSS instance is strictly increasing along trajectories of our algorithms, and finally corroborate our theoretical findings through extensive experiments.

翻译：博弈化是机器学习领域的一个新兴趋势，它通过将优化问题转化为类博弈场景，为解决优化问题提供了一种新颖方法。这种范式转变使得针对复杂优化问题能够开发出鲁棒、易于实现且可并行化的算法。在我们的工作中，我们利用博弈化来处理最佳可分离态（BSS）问题——这是量子信息理论中的一个基本问题，涉及在可分离量子态集合上的线性优化。为此，我们引入并研究了共同利益博弈（CIG）和势博弈的量子模拟，其中玩家以密度矩阵作为策略，且其利益完全一致。通过建立BSS实例的KKT（一阶平稳）点与其对应量子CIG的纳什均衡之间的等价性，我们弥合了优化与博弈论之间的差距。从博弈学习角度出发，我们引入了量子CIG中学习的连续时间复制动力学和离散时间Baum-Eagon/线性乘法权重更新的非交换推广，这些方法同时也作为BSS问题的去中心化算法。我们证明，沿我们的算法轨迹，BSS实例的共同效用/目标值是严格递增的，最后通过大量实验验证了我们的理论发现。