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

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?

• Can you specify exactly what you mean by an 'isometric' quantum channel?
– JSdJ
Nov 11, 2021 at 12:52
• I have mentioned it in the edits. Thanks for pointing it out. Nov 11, 2021 at 13:20
• In the title, you ask "how one can check if a given isometric channel is unitary" but then in the question you state that "an isometric channel is a unitary channel". Are you just wondering when a mixed unitary channel has a single term in the sum? Nov 11, 2021 at 14:59
• Thanks for pointing out the confusion. I have edited it and hope it is clear now. Given a quantum channel, I was wondering how we can check if it is unitary with only one term in the sum (not mixed unitary). Nov 11, 2021 at 15:11
• which problem exactly are you saying is NP-hard? Checking whether a given channel is isometric surely isn't no? Just check the rank of the Choi, as per the answer. And an isometric channel is a unitary channel iff the input and output dimensions are the same (I'm assuming the $A$ in your post is also an isometry, otherwise the map you get is not a (CPTP) channel)
– glS
Nov 11, 2021 at 15:45

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 $$U$$.