Perfect Quantum Approximate Strategies for Imitation Games

We prove that an imitation game has a perfect quantum approximate strategy if and only if there exists a bi-tracial state on the minimal tensor product of two universal C${}^{\ast }$-algebras, which induces the perfect correlation. Moreover, we are trying to relate imitation games to the minimal tensor product of two universal C${}^{\ast }$-algebras and demonstrate that an imitation game has a perfect quantum approximate strategy if and only if there exist a von Neumann algebra and an amenable tracial state on it, such that the perfect correlation can be induced by the tracial state. However, We encountered some difficulties regarding continuity in the proof process. In section 2 we get some results for special cases, and in section 3 we list our problems.