# How to define POVM measurement operators for a composite quantum state

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?

• 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 $\hat{\rho}$. Dec 3 '20 at 15:38