# Check that a channel implements a particular unitary

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.

The most straightforward way to see this is by realising that the Choi operator of a unitary channel is rank $$1$$, and that therefore the rank of the Choi operator of the channel $$C$$ should also be rank $$1$$. The eigenvectors of the Choi operator are in direct one-to-one correspondence with the Kraus operators (by the vectorisation mapping). Therefore, the only way for the two Choi operators to coincide is when the Kraus operators are either all zero except for one, or if they are scalar multiples of each other.