Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence

TL;DR

This paper explores quantum zero-sum games, demonstrating that quantum replicator dynamics exhibit conservation laws and Poincare recurrence, extending classical game theory results into the quantum domain.

Contribution

It introduces the analysis of conservation laws and recurrence phenomena in quantum zero-sum games under matrix multiplicative weights updates.

Findings

Quantum replicator dynamics conserve quantum information.

System exhibits Poincare recurrence in quantum zero-sum games.

Generalizes classical game results to quantum settings.

Abstract

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.