On characterizing quantum correlated equilibria

TL;DR

This paper characterizes quantum correlated equilibria in quantum strategic games, providing necessary and sufficient conditions, explicit strategies for gain, and bounds on quantum advantages over classical strategies.

Contribution

It offers the first complete characterization of quantum correlated equilibria, including conditions, explicit POVMs, and bounds on quantum gains.

Findings

Necessary and sufficient condition for a state to be a QCE

Explicit POVM strategies for players to gain advantage

Upper bounds on quantum strategies' gains, tight in some cases

Abstract

Quantum game theory lays a foundation for understanding the interaction of people using quantum computers with conflicting interests. Recently Zhang proposed a simple yet rich model to study quantum strategic games, and addressed some quantitative questions for general games of growing sizes \cite{Zha10}. However, one fundamental question that the paper did not consider is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state $\rho$ being a QCE. In addition, when the condition fails to hold for some player $i$, we give an explicit POVM for that player to achieve a strictly positive gain. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.