3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

TL;DR

This paper proves that for 3XOR games with perfect commuting operator strategies, the existence of such strategies can be decided efficiently, and these strategies are equivalent to perfect tensor product strategies using a small GHZ state, bounding quantum advantage.

Contribution

It establishes polynomial-time decidability of perfect strategies in 3XOR games and links commuting operator strategies to tensor product strategies with small entanglement.

Findings

Deciding perfect strategies is polynomial-time solvable.

Perfect commuting operator strategies are equivalent to tensor product strategies with a 3-qubit GHZ state.

Quantum advantage in perfect 3XOR games is bounded.

Abstract

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.