= Solution
Choose an <orthonormal basis> $\{|j\rangle\}_{j=1}^d$ of the input space and define operators from the input to the $k$-dimensional outcome register by
$$
K_{a,j}=|a\rangle\langle j|\sqrt{E_a}.
$$
These give a <Kraus representation>:
$$
\sum_{a,j}K_{a,j}XK_{a,j}^\dagger
=\sum_a\operatorname{Tr}(E_aX)|a\rangle\langle a|=\Phi(X).
$$
The <POVM> completeness relation gives
$$
\sum_{a,j}K_{a,j}^\dagger K_{a,j}
=\sum_a\sqrt{E_a}\left(\sum_j|j\rangle\langle j|\right)\sqrt{E_a}
=\sum_aE_a=I.
$$
Therefore $\Phi$ is a <completely positive map> and is trace preserving: it is a <quantum channel> representing a deterministic <measurement channel>.
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