is organized as follows. In Section \[sec:tool\] we introduce the basic concepts and tools necessary to state our results. In Section \[sec:NLdist\] we propose a necessary condition for nonlocality distillation, that is then employed in Section \[sec:NLqvoid\] to prove that all $1$D quantum voids and some $2$D and $3$D quantum voids correlations are distillable. In Sections \[sec:CC\] and \[sec:IC\] we look for the consequences of distillation protocols in the non-trivial communication complexity [@van2013implausible] and information causality principles [@pawlowski2009information]. Finally, in Section \[sec:discussion\] we present our conclusions.

Toolbox {#sec:tool}
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We will restrict our attention here to the simplest possible Bell scenario, also known as the Clauser-Horne-Shimony-Holt (CHSH) scenario [@PhysRevLett.23.880]. It refers to two spatially separated parties, Alice and Bob, which upon receiving their shares of a joint physical system randomly and independently choose one of two possible dichotomic measurements to perform. Their measurement choices are labelled by the random variables $X$ and $Y$ and the measurement outcomes as $A$ and $B$, for Alice and Bob respectively. In the CHSH scenario all these variables are binary, that is, $x,y,a,b=0,1$. The results of a Bell experiment are encoded in a probability distribution of these variables $p(a,b,x,y)$ and since typically we do have 