.   $\left|B_{R_{i} } \left(\pi _{S} \right)\right|$                                                        Total number of incoming density matrices in $R_{j} $ per a given $\pi _{S} $, as $\left|B_{R_{j} } \left(\pi _{S} \right)\right|=\sum _{i,k}\left|B\left(R_{i} \left(\pi _{S} \right),\sigma _{k} \right)\right| $