this paper we aim to provide somewhat more general statements on nonlocality distillation and their implications for information theoretical principles. Because of the difficulty mentioned above, we focus here on the CHSH scenario and distillation protocols involving two copies of the nonlocal correlations. Within this context, we first obtain a general necessary condition for nonlocality distillation with two copies. Then we employ it to analyze faces of the nonlocal simplex of correlations with no quantum realization, the so called quantum voids [@PhysRevA.99.032106] (if the dimension of the face is $k$ then the quantum void is said to be $k$-dimensional. Using our necessary condition we prove that correlations in all $1$D and some of the $2$D quantum voids are distillable to maximal nonlocality. This allows us to generalize previous results in the literature. First, we show that there are whole faces of the non-signalling set violating the principle of non-trivial communication complexity. Finally, we show how a large class of correlations not violating Uffink’s inequality (a necessary condition for a correlation to be compatible with the principle of information causality) can do so after a distillation protocol.

![Illustration of a wiring protocol between two probability distributions (represented as boxes).[]{data-label="wiring"}](wiring_schem.pdf)

The paper 