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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?

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    $\begingroup$ Can you specify exactly what you mean by an 'isometric' quantum channel? $\endgroup$
    – JSdJ
    Nov 11, 2021 at 12:52
  • $\begingroup$ I have mentioned it in the edits. Thanks for pointing it out. $\endgroup$ Nov 11, 2021 at 13:20
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    $\begingroup$ 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? $\endgroup$
    – Condo
    Nov 11, 2021 at 14:59
  • $\begingroup$ 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). $\endgroup$ Nov 11, 2021 at 15:11
  • $\begingroup$ 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) $\endgroup$
    – glS
    Nov 11, 2021 at 15:45

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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 $U$.

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  • $\begingroup$ Thanks, this clarifies my doubt. Could you refer any material so that I can read more on this? $\endgroup$ Nov 11, 2021 at 15:26
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    $\begingroup$ @HarikrishnanSV Watrous' book, cs.uwaterloo.ca/~watrous/TQI, is always a good reference $\endgroup$
    – glS
    Nov 11, 2021 at 15:41
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    $\begingroup$ @HarikrishnanSV as pointed to by glS in the comment above, Watrous' book is an excellent reference and should provide all the relevant information and background. I believe chapters 2 and 4 discuss quantum channels and mixed-unitary channels. $\endgroup$
    – Condo
    Nov 11, 2021 at 16:47
  • $\begingroup$ Thanks. I will check it out. $\endgroup$ Nov 11, 2021 at 17:11

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