  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.
