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

This paper initiates the study of a class of entangled games, mono-state games, 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 small constant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is a noisy EPR state, which is a two-qubit state with completely mixed states as marginals 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 of the proofs is built the recent framework about the decidability of the non-interactive simulation of joint distributions, which is completely different from all previous optimization-based approaches or "Tsirelson's problem"-based approaches. This paper develops a series of new techniques about the Fourier analysis on matrix spaces and proves a quantum invariance principle and a hypercontractive inequality of random operators. This novel approach provides a new angle to study the decidability of the complexity class MIP$^*$, a longstanding open problem in quantum complexity theory.