Abstract
Quantum 9, 1689 (2025) 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.