 $$\label{ZEqnNum000000} \begin{split}    \omega \left( {{\pi }_{S}} \right)&=\sum\limits_{i,k}{{{\chi }_{ik}}\left( {{\rho }_{A}},{{\sigma }_{k}} \right)Z_{{{R}_{j}}}^{\left( {{\pi }_{S}} \right)}\left( \left( {{R}_{i}},{{\sigma }_{k}} \right) \right)} \\   & =\left\langle \chi \left( {{\pi }_{S}} \right),{{Z}_{{{R}_{j}}}}\left( {{\pi }_{S}} \right) \right\rangle 