Entanglement and coherence in quantum prisoner's dilemma

Entanglement and coherence are quantum resources widely used in several tasks in quantum information processing. In particular, the emergence of quantum game theory arises the question whether entanglement would be more useful than coherence for quantum players. In this paper, we address such question from a two-person quantum game, namely the quantum version of the prisoner's dilemma. We discuss the players payoffs (i) when an entangled initial game state is provided and (ii) when the system is started in a separable superposition state. As the main result, when an entangled state is provided to players, we find a situation where a non-maximally entangled state is preferable by a quantum player concerning the maximally entangled state. Thus, our first result suggests that we can establish a trade-off between maximum expected payoff and an amount of entanglement required by a quantum player. As a second result, when we provide a non-entangled initial state (but we have coherence), the payoff of a classical player is enhanced concerning the previous case. We discuss how the phase-transition-like behavior emerges from entanglement in the game considered here, so that we could design a game where no change in optimal strategies would be required.