line (see Fig. \[2d\_voids\_distillable\]).

![Schematic representation of the $2$D quantum voids that are distillable up to a PR-box. If there exists one edge between two local vertices, then this set is completely distillable and there is at least one single strategy that can distill the entire void. For instance, the set $\{PR,L_1,L_2\}$ is completely distillable, however the set $\{PR,L_1, L_7\}$ is not.[]{data-label="2d_voids_distillable"}](2Ddistillable.pdf)

As a matter of fact, not only all these $2$D quantum voids are fully distillable but can as well be distilled up to a PR-box. In Table \[2dvoid\_strategies\] we show the example of a wiring strategy for each $2$D quantum void that being applied iteratively can bring any nonlocal correlation up to the PR-box.

$\begin{array}{ |c|c|c|c|c|} 
 \hline
  L_iL_{j} & x=0 & x=1 & y=0 & y=1 \\ 
 \hline
L_1L_2 & \chi ^S{}_{0,0,0,0,0} & \chi ^X{}_{1,1,1} & \chi ^S{}_{0,0,1,0,0} & \chi ^X{}_{1,1,0} \\
 L_1L_3 & \chi ^S{}_{0,0,0,0,0} & \chi ^X{}_{1,1,0} & \chi ^S{}_{0,1,1,0,0} & \chi ^S{}_{0,0,1,0,0} \\
L_1L_5 & \chi ^S{}_{0,0,0,0,0} & \chi ^S{}_{0,1,0,0,0} & \chi ^S{}_{0,0,1,0,0} & \chi ^X{}_{1,1,0} \\
L_2L_4& \chi ^S{}_{0,0,1,0,0} & \chi ^X{}_{1,1,0} & \chi ^S{}_{0,0,0,0,0} & \chi ^S{}_{0,1,0,0,0} \\
L_2L_6 & \chi ^S{}_{0,0,1,0,0} & \chi ^S{}_{0,1,1,0,0} & \chi ^S{}_{0,0,0,0,0} & \chi ^X{}_{1,1,1} \\
L_3L_4 