However, for five dimensions and beyond, no asymptotic distillation to PR-box will be possible for all correlation in the quantum void. This follows from the fact that in these cases there is always a gap between the set of quantum correlations and $Q^{1}$.

nonlocality distillation and trivial communication complexity {#sec:CC}
=============================================================

Among the several principles introduced to try to explain why correlations beyond quantum mechanics are unlikely we have the so called non-trivial communication complexity. The basic setup involves two distant parties which locally receive bit strings $\vec{x}$ and $\vec{y}$ respectively and by exchanging a limited amount of information should compute function $f(\vec{x},\vec{y})$ depending on both bit strings. It seems natural that the amount of communication required should increase with the size of the bit strings. Quantum theory is compatible with that but, as shown by Van Dam [@van2013implausible], PR-boxes can make such communication complexity trivial, since their use with a single bit of exchanged information is enough to make such nonlocal computations. Later on, this result has been extended to show that any correlation achieving the value of $\mathrm{CHSH} \geq 4\sqrt{2/3} \approx 3.266$ would also trivialize communication complexity [@PhysRevLett.96.250401].

Afterwards, by considering nonlocality distillation it has been shown that 