How to specify a general quantum measurement? I know that a general quantum measurement is described by a set of measurement operators $\{M_j\}$ satisfying the completeness relation $\sum_j M_j^\dagger M_j = 1$ and that the outcome j occurs with probability $$p(j)=Tr(M_j^\dagger M_j \rho)$$ Furthermore, the post-measurement state after obtaining this outcome is $$\rho_j = \frac{M_j \rho M_j^\dagger}{p(j)}$$
However, in the POVM formalism we describe the measurment using only the POVM elements $\{A_j\}$ that correspond to the operator products $M_J^\dagger M_j$. Obviously, there are different sets of measurement operators that realize the same POVM.
Here's my question: Given (1) the POVM elements of a quantum measurement and (2) the post-measurement states $\{\sigma_j\}$ for some fixed pre-measurement state $\sigma$, are there measurement operators $\{M_j\}$ such that $$A_j = M_j^\dagger M_j$$ and $$\sigma_j = \frac{M_j \sigma M_j^\dagger}{Tr(A_j \sigma )}$$
Moreover: If there is such a measurement, is it unique?
 A: I constructed a counterexample to the uniqueness property of your first question. Take a 3-state system with
$$\rho = \begin{bmatrix}1/4 & 0 & 0 \\ 0 & 1/4 & 0 \\ 0 & 0 & 1/2\end{bmatrix}$$
and our post-measurement states each with probabilities $1/2$ each.
$$\rho_0 = \begin{bmatrix}1/2 & 0 & 0\\ 0 & 1/2 & 0 \\ 0 & 0 & 0\end{bmatrix},~ \rho_1 = \begin{bmatrix}0 & 0 & 0\\ 0 &0&0 \\ 0&0& 1\end{bmatrix}$$
and corresponding POVMs
$$M_0^\dagger M_0 = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix},~ M_1^\dagger M_1 = \begin{bmatrix}0 & 0 & 0\\ 0 &0&0 \\ 0&0& 1\end{bmatrix}$$
There are at least two possible sets of values for $M_0,M_1$ with the same POVM and corresponding final states, namely:
$$M_0 = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}, M_1 = \begin{bmatrix}0 & 0 & 0 \\ 0& 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
and
$$M_0 = \begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, M_1 = \begin{bmatrix}0 & 0 & 0 \\ 0& 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
