  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.
