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There is a mistake, we have $$\text{Tr}(\sigma_{i_1}\sigma_{j_1}\otimes\sigma_{i_2}\sigma_{j_2}\otimes\sigma_{i_3}\sigma_{j_3}) = \delta_{i_1,j_1}\delta_{i_2,j_2}\delta_{i_3,j_3}\text{Tr}(I) = 8\delta_{i_1,j_1}\delta_{i_2,j_2}\delta_{i_3,j_3}.$$ So that $\sum_{i_1,i_2,i_3}|u_{i_1,i_2,i_3}|^2=1$. The same is true for every $n$. The other equalities that we ...
Beware! If $M$ is an operator describing a measurement, it is not that the output after measurement is $M|\psi\rangle$ for initial state $|\psi\rangle$. Instead, let $\{P_i\}$ be projectors onto the different eigenspaces of $M$. The you get the outcome $i$ with probability $p_i=\langle\psi|P_i|\psi\rangle$ and the state after measurement, if you get that ...