Nonlocal games with noisy maximally entangled states are decidable

Abstract

This paper considers a special class of nonlocal games $(G,ψ)$, where $G$ is a two-player one-round game, and $ψ$ is a bipartite state independent of $G$. In the game $(G,ψ)$, the players are allowed to share arbitrarily many copies of $ψ$. The value of the game $(G,ψ)$, denoted by $ω^*(G,ψ)$, is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states $ψ$. For a noisy maximally entangled state $ψ$, a two-player one-round game $G$ and an arbitrarily small precision $ε>0$, this paper proves an upper bound on the number of copies of $ψ$ for the players to win the game with a probability $ε$ close to $ω^*(G,ψ)$. Hence, it is feasible to approximately compute $ω^*(G,ψ)$ to an arbitrarily precision. Recently, a breakthrough result by Ji, Natarajan, Vidick, Wright and Yuen showed that it is undecidable to approximate the values of nonlocal games to a constant precision when the players preshare arbitrarily many copies of perfect maximally entangled states, which implies that $\mathrm{MIP}^*=\mathrm{RE}$. In contrast, our result implies the hardness of approximating nonlocal games collapses when the preshared maximally entangled states are noisy. The paper develops a theory of Fourier analysis on matrix spaces by extending a number of techniques in Boolean analysis and Hermitian analysis to matrix spaces. We establish a series of new techniques, such as a quantum invariance principle and a hypercontractive inequality for random operators, which we believe have further applications.