  Repeated quantum game theory addresses long term relations among players who
choose quantum strategies. In the conventional quantum game theory, single
round quantum games or at most finitely repeated games have been widely
studied, however less is known for infinitely repeated quantum games.
Investigating infinitely repeated games is crucial since finitely repeated
games do not much differ from single round games. In this work we establish the
concept of general repeated quantum games and show the Quantum Folk Theorem,
which claims that by iterating a game one can find an equilibrium strategy of
the game and receive reward that is not obtained by a Nash equilibrium of the
corresponding single round quantum game. A significant difference between
repeated quantum prisoner's dilemma and repeated classical prisoner's dilemma
is that the classical Pareto optimal solution is not always an equilibrium of
the repeated quantum game when entanglement is sufficiently strong. When
entanglement is sufficiently strong and reward is small, mutual cooperation
cannot be an equilibrium of the repeated quantum game. In addition we present
several concrete equilibrium strategies of the repeated quantum prisoner's
dilemma.
