\right\}$ and $NL(B^{i}_0)=\left\{0,1 \right\}$ as well as $NL(B_{i}^0)=\left\{0,1 \right\}$. The necessary condition above then reduces to $$\label{necessary}
    c_0(NL(B_0^0)-1)+\sum_{i=0}^{2}\tilde{c}_{i}(i-1)>0$$ where $\sum_{i=0}^{2}\tilde{c}_{i}=\sum_{k=1}^{8} c_{k}$ and $\tilde{c}_i$ is the sum of all coefficients $c_{k}$ for which $NL(B^0_{k})+NL(B^{k}_0)=i$. Notice that this necessary condition can be raised to a sufficient one if one can guarantee that every term in the initial distribution is being mapped under the wiring to a distribution of the same form .

Nonlocality distillation in quantum voids {#sec:NLqvoid}
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In what follows we are going to consider whether entire faces of the nonlocal simplex (in particular, quantum voids) are distillable or not. In this case, since the coefficient $c_0$ in can vary as $0 < c_0 < 1$, a necessary condition for distillation is that $NL(B_0 ^0)=1$ (remember that only assume 3 possible values $NL(B_0 ^0)=\left\{0,1/2 ,1 \right\}$). Furthermore, without loss of generality, we can restrict to those wirings such that $B_0 ^0=PR(ab \vert xy)$. Making this restriction, reduces the number of wirings from $82^4$ to $3152$, making a complete analysis of the wirings amenable. Furthermore, in this case, the necessary condition for distillation is then simply given by $$\label{face_condition}
    \tilde{c}_2>\tilde{c}_0.$$

All correlations in a $1$D quantum void can be distilled to 