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In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $omega^*$. We show that the value $omega^*$ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of $omega^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $omega_{qow}$. In particular, we show that $omega_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.
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A framework for discussing relationships between different types of games is proposed. Within the framework, quantum simultaneous games, finite quantum simultaneous games, quantum sequential games, and finite quantum sequential games are defined. In addition, a notion of equivalence between two games is defined. Finally, the following three theorems are shown: (1) For any quantum simultaneous game G, there exists a quantum sequential game equivalent to G. (2) For any finite quantum simultaneous game G, there exists a finite quantum sequential game equivalent to G. (3) For any finite quantum sequential game G, there exists a finite quantum simultaneous game equivalent to G.
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