Consider a channel $C$ with Kraus operators $\{K_k\}$ and a unitary U. How can I check that $C$ implements $U$ ?
One can write that their Choi matrices are equal i.e:
\begin{equation} \sum_{i,j}|i\rangle\langle j|\otimes \sum_{k}K_k|i\rangle\langle j|K_k^\dagger = \sum_{i,j}|i\rangle\langle j|\otimes U|i\rangle\langle j|U^\dagger \end{equation} but I'm not sure if this can be simplified to obtain a condition on the $K_k$'s.