We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on $(i)$ the size of the shared quantum system used in the quantum strategy, and $(ii)$ structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.

翻译：我们证明，对于编码约束满足问题（CSP）的双证明者博弈，任何完美量子策略均可通过具有额外经典通信信道的完美经典策略进行模拟，该信道的规模仅取决于：$(i)$ 量子策略中所用共享量子系统的规模，以及$(ii)$ CSP模板的结构参数。该结果通过以下方式获得：首先对具有额外通信信道的完美经典策略进行组合刻画，其次对量子策略中涉及的投影值测量进行几何舍入处理。我们证明的关键中间步骤在于确立：当量子策略涉及有限规模的共享量子信息时，图的经典色数与其量子变体之间的差距是有界的。