=\sum_{0{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle + \sum_{k=0}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle.\end{aligned}$$ 