  This paper considers the decidability of fully quantum nonlocal games with
noisy maximally entangled states. Fully quantum nonlocal games are a
generalization of nonlocal games, where both questions and answers are quantum
and the referee performs a binary POVM measurement to decide whether they win
the game after receiving the quantum answers from the players. The quantum
value of a fully quantum nonlocal game is the supremum of the probability that
they win the game, where the supremum is taken over all the possible entangled
states shared between the players and all the valid quantum operations
performed by the players. The seminal work $\mathrm{MIP}^*=\mathrm{RE}$ implies
that it is undecidable to approximate the quantum value of a fully nonlocal
game. This still holds even if the players are only allowed to share
(arbitrarily many copies of) maximally entangled states. This paper
investigates the case that the shared maximally entangled states are noisy. We
prove that there is a computable upper bound on the copies of noisy maximally
entangled states for the players to win a fully quantum nonlocal game with a
probability arbitrarily close to the quantum value. This implies that it is
decidable to approximate the quantum values of these games. Hence, the hardness
of approximating the quantum value of a fully quantum nonlocal game is not
robust against the noise in the shared states.
  This paper is built on the framework for the decidability of non-interactive
simulations of joint distributions and generalizes the analogous result for
nonlocal games. We extend the theory of Fourier analysis to the space of
super-operators and prove several key results including an invariance principle
and a dimension reduction for super-operators. These results are interesting in
their own right and are believed to have further applications.
