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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?

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    $\begingroup$ Are 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 $\hat{\rho}$. $\endgroup$ – Rammus Dec 3 '20 at 15:38

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