Nonlocal games with noisy maximally entangled states are decidable

This paper considers a special class of nonlocal games \((G,\psi)\), where \(G\) is a two-player one-round game, and \(\psi\) is a bipartite state independent of \(G\). In the game \((G,\psi)\), the players are allowed to share arbitrarily many copies of \(\psi\). The value of the game \((G,\psi)\), denoted by \(\omega^*(G,\psi)\), is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states \(\psi\). For a noisy maximally entangled state \(\psi\), a two-player one-round game \(G\) and an arbitrarily small precision \(\epsilon>0\), this paper proves an upper bound on the number of copies of \(\psi\) for the players to win the game with a probability \(\epsilon\) close to \(\omega^*(G,\psi)\). Hence, it is feasible to approximately compute \(\omega^*(G,\psi)\) 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.