We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time.

翻译：我们考虑具有完美交换算子策略的3XOR博弈。对于任意3XOR博弈，我们证明该博弈是否存在完美交换算子策略可在多项式时间内判定。此前该问题的可判定性尚未可知。我们的证明给出了一种构造，表明3XOR博弈存在完美交换算子策略当且仅当其存在使用三量子比特（8维）GHZ态的完美张量积策略。这意味着完美3XOR博弈中量子策略相对于经典策略的优势（由量子-经典偏差比定义）是有界的。这与一般3XOR情形形成对比——后者最优量子策略可能需高维态且量子优势无界。为证明这些结果，我们首先证明判定XOR博弈取值与求解某类直角型Coxeter群上的子群成员问题的实例等价。随后，在占据本文主要篇幅的证明中，我们表明对应于3XOR博弈的该类实例可在多项式时间内求解。