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I have an evolved quantum composite state $\hat{\rho}^{\otimes{N}}$ that I retrieved from a quantum channel $\mathcal{E}$, Now I do know how to define a POVM for the evolved states $\hat{\rho}$ that the channel outputs one by one. But how can I define POVM for composite quantum state.

Suppose I embed $X = [1,0,1,1]$ classical information into $\rho_i$ as follows:

$\rho_0 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\rightarrow for \; x_0=1, \\ \rho_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \rightarrow for \; x_1=0, \\\rho_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\rightarrow for \; x_2=1, \\\rho_3 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \rightarrow for \; x_3=1,$

Now each of the $\rho_i^N$ evolves when $\mathcal{E}$ is applied and we get respective $\hat{\rho_i}^N$, I tensor the evolved states and retrieve a composite quantum state as $\hat{\rho_i}^{\otimes{N}}$ with $dXd$ square matrix where $d=2^N$. Now, how can I define POVM measurements to get the classical information back?

1Are you wanting to measure all of the individual qubits separately? If so, then you define your POVMs for each system separately (say $M_{i,j}$ is the $i$-th POVM element for the $j$-th system). Then to compute the probability that you get the joint outcome $(a_1, a_2, \dots, a_N)$ you measure the joint POVM element $M_{a_1,1} \otimes M_{a_2,2} \otimes \dots \otimes M_{a_N, N}$ on the joint system $\hat{\rho}$. – Rammus – 2020-12-03T15:38:58.630