Nonlocal games with noisy maximally entangled states are decidable

• URL: http://arxiv.org/abs/2108.09140v1
• Date: Fri, 20 Aug 2021 12:25:55 GMT
• Title: Nonlocal games with noisy maximally entangled states are decidable
• Authors: Minglong Qin and Penghui Yao
• Abstract summary: This paper considers a special class of nonlocal games $(G,psi)$, where $G$ is a two-player one-round game. In the game $(G,psi)$, the players are allowed to share arbitrarily many copies of $psi$. It is feasible to approximately compute $omega*(G,psi)$ to an arbitrarily precision.
• Score: 5.076419064097734
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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