We apply a Bayesian agent-based framework inspired by a Bayesian interpretation of quantum theory (QBism) to iterations of two quantum games, the Clauser-Horne-Shimony-Holt (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 onefold rational players (rational players who believe the other player is also rational) reduces Eisert et al.'s quantum extension of the prisoners' dilemma to a game with only two strategies, one of which (defect) is dominant for low entanglement, and the other (Eisert et al.'s 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 may point to future applications in resource detection and quantum algorithm design.

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• Quantum algorithms & computation
• Quantum correlations in quantum information
• Quantum foundations
• Quantum information processing
• Quantum information theory
• Quantum simulation