  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.

Quantum Physics;Computer Science and Game Theory