Measurement channel
= Measurement channel
A measurement channel writes a <POVM>'s outcome into an orthogonal classical register:
$$
\Phi(X)=\sum_a\operatorname{Tr}(E_aX)|a\rangle\langle a|.
$$
For any input <orthonormal basis> $|j\rangle$, the <Kraus operators> $K_{a,j}=|a\rangle\langle j|\sqrt{E_a}$ realize this map and satisfy $\sum_{a,j}K_{a,j}^\dagger K_{a,j}=I$. It is therefore a trace-preserving <completely positive map>.