  Recent advances in quantum computing and in particular, the introduction of
quantum GANs, have led to increased interest in quantum zero-sum game theory,
extending the scope of learning algorithms for classical games into the quantum
realm. In this paper, we focus on learning in quantum zero-sum games under
Matrix Multiplicative Weights Update (a generalization of the multiplicative
weights update method) and its continuous analogue, Quantum Replicator
Dynamics. When each player selects their state according to quantum replicator
dynamics, we show that the system exhibits conservation laws in a
quantum-information theoretic sense. Moreover, we show that the system exhibits
Poincare recurrence, meaning that almost all orbits return arbitrarily close to
their initial conditions infinitely often. Our analysis generalizes previous
results in the case of classical games.
