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Physics Letters A

Entanglement playing a dominating role in quantum games

Jiangfeng Du, Xiaodong Xu, Hui Li, Xianyi Zhou, Rongdian Han

Published Oct 1, 2001• 97 citations
Publisher
DOI
10.1016/s0375-9601(01)00575-8

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We investigate the influence of the Unruh effect on three-qubit quantum games. In particular, we interpret the quantum Prisoners' Dilemma, which is a famous, non-zero sum game both for entangled and unentangled initial states and show that the acceleration of non-inertial frames disturbs the symmetry of the game. Using the various strategies, the novel Nash equilibrium is obtained at infinite acceleration ( r = π/4). As a remarkable point, it is shown that in our three-player system, in contrast to the two-player quantum game in non-inertial frames (see Khan et al 2011 J. Phys. A: Math. Theor. 44 355302), there is not a dominant strategy (even classical strategy) in the game and choosing the quantum strategy by each player can be the dominant strategy depending on the kind of strategy chosen by others. Since the entangled states of particles play an important role in the quantum game, finally we argue that the results of the players depend on the degree of entanglement in the initial state of the game.
We present a quantization scheme for a three-player Prisoner's Dilemma game. It is shown that entanglement plays a dominant role in the three-player quantum game. Four different types of payoffs are identified on the basis of different combinations of initial state and measurement basis entanglement parameters. A relation among these different payoffs is also established. We also study the communication aspects of the three-player game. By exploiting different combinations of initial state and measurement basis entanglement parameters, we establish a relationship for the information shared among the parties. It is seen that the strategies of the players act as carriers of information in quantum games.
We apply a Bayesian agent-based framework inspired by QBism to iterations of two quantum games, the CHSH game and the quantum prisoners' dilemma. In each two-player game, players hold beliefs about an amount of shared entanglement and about the actions or beliefs of the other player. Each takes actions which maximize their expected utility and revises their beliefs with the classical Bayes rule between rounds. We simulate iterated play to see if and how players can learn about the presence of shared entanglement and to explore how their performance, their beliefs, and the game's structure interrelate. In the CHSH game, we find that players can learn that entanglement is present and use this to achieve quantum advantage. We find that they can only do so if they also believe the other player will act correctly to exploit the entanglement. In the case of low or zero entanglement in the CHSH game, the players cannot achieve quantum advantage, even in the case where they believe the entanglement is higher than it is. For the prisoners dilemma, we show that assuming 1-fold rational players (rational players who believe the other player is also rational) reduces the quantum extension [Eisert, Wilkens, and Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)] of the prisoners dilemma to a game with only two strategies, one of which (defect) is dominant for low entanglement, and the other (the quantum strategy Q) is dominant for high entanglement. For intermediate entanglement, neither strategy is dominant. We again show that players can learn entanglement in iterated play. We also show that strong belief in entanglement causes optimal play even in the absence of entanglement -- showing that belief in entanglement is acting as a proxy for the players trusting each other. Our work points to possible future applications in resource detection and quantum algorithm design.