How can one check if a given quantum channel is unitary?

Question

A unitary channel is a channel $\mathcal{U}$ of the following form: $\mathcal{U}(\rho) = U\rho U^{\dagger}$.

A mixed unitary channel is a channel $\mathcal{U}_m$ of the form: $\mathcal{U}_m(\rho) = \sum_{k=1}^n p_kU_k\rho U_k^{\dagger}$, where each $U_k$ is a unitary matrix.

So given a general quantum channel, determining whether or not it is mixed unitary is an NP-hard problem. Is there any easy test to check if it is unitary?

Answer 1

Compute the rank of the Choi matrix $C_\Phi$. For any quantum channel $\Phi$, the Choi matrix will be rank 1 if and only if $\Phi$ can be written in the form $U\rho U^\dagger$ for a unitary $U$.