boxes are wired, $i,j \in\{1,2\}$. All possible wirings for boxes with possible inputs and outputs have been characterized [@PhysRevA.73.012101] and form a convex set whose vertices are described according to five different classes in the Table below.

        Potential couplers classes                                                  $\chi(a,a_1,a_2,x_1,x_2)=1$ ($0$ otherwise)
  --------------------------------------- -------------------------------------------------------------------------------------------------------------------------------
              $\chi_{\mu}^D$                                                                   $x_1=x_2$ and $a=\mu$
          $\chi_{\mu\nu\sigma}^O$                                                   $x_1=x_2=\mu$ and $a=a_{\nu+1}\oplus\sigma$
          $\chi_{\mu\nu\sigma}^X$                                             $x_1=\mu$, $x_2=\nu$, and $a=a_1\oplus a_2\oplus\gamma$
   $\chi_{\mu\nu\sigma\delta\epsilon}^A$                          $x_1=\mu$, $x_2=\nu$, and $a=(a_1\oplus\sigma)(a_2\oplus\delta)\oplus\epsilon$
   $\chi_{\mu\nu\sigma\delta\epsilon}^S$   $x_{\mu+1}=\nu$, $x_{(\mu\oplus 1)+1}=a_{\mu+1}\oplus\sigma$, and $a=a_{(\mu\oplus 1)+1}\oplus\delta a_{\mu+1}\oplus\epsilon$

Before moving on to derive a necessary criterion for nonlocality distillation, let us mention that in the CHSH scenario [@PhysRevLett.23.880], some examples of sets closed under wirings are known. The set of local correlations, quantum correlations, and no-signaling correlations as well as several sets of correlations between local and quantum sets as well as all sets of correlations generated by different levels of the NPA hierarchy [@NPA]. See Ref. [@Lang_2014] for a discussion about sets of correlations closed under wirings. Now since some of these sets are very closely spaced (for example, NPA hierarchies), for points in these regions it is very hard to come up with distillation protocols; the only viable option is to find the wirings which generates a flow gazing on the boundary of the 