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This paper initiates the study of a class of entangled games, mono-stategames, denoted by $(G,\psi)$, where $G$ is a two-player one-round game and$\psi$ is a bipartite state independent of the game $G$. In the mono-state game$(G,\psi)$, the players are only allowed to share arbitrary copies of $\psi$.This paper provides a doubly exponential upper bound on the copies of $\psi$for the players to approximate the value of the game to an arbitrarily smallconstant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is anoisy EPR state, which is a two-qubit state with completely mixed states asmarginals and maximal correlation less than $1$. In particular, it includes$(1-\epsilon)|\Psi\rangle\langle\Psi|+\epsilon\frac{I_2}{2}\otimes\frac{I_2}{2}$,an EPR state with an arbitrary depolarizing noise $\epsilon>0$.The structure ofthe proofs is built the recent framework about the decidability of thenon-interactive simulation of joint distributions, which is completelydifferent from all previous optimization-based approaches or "Tsirelson'sproblem"-based approaches. This paper develops a series of new techniques aboutthe Fourier analysis on matrix spaces and proves a quantum invariance principleand a hypercontractive inequality of random operators. This novel approachprovides a new angle to study the decidability of the complexity class MIP$^*$,a longstanding open problem in quantum complexity theory.

A doubly exponential upper bound on noisy EPR states for binary games