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I am aware that the rank of the Choi state is a test for whether or not a channel is unitary.

Are there any other equivalent tests that we can do to test if a channel is unitary without involving a reference register? I am assuming I have the channel available to me as a black box and can send arbitrary inputs and measure the outputs (even tomography on the outputs).

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    $\begingroup$ Tomography? ____ $\endgroup$
    – Norbert Schuch
    Commented Aug 4 at 12:44
  • $\begingroup$ what kind of context do you have in mind? What kind of information is given? Do you assume to know a full description of the channel itself? Or you have physically so you can probe it with quantum states and measure the result? Or you know some general property of it? $\endgroup$
    – glS ♦
    Commented Aug 4 at 14:25
  • $\begingroup$ @gIS I would prefer to probe it with states i.e. it is a blackbox to me. I added this to the question now, thanks for the comment $\endgroup$
    – asdfghj
    Commented Aug 4 at 14:29

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A naive approach is to perform quantum process tomography and characterize the process matrix $\chi$. However, there is a body of work on learning quantum dynamics, some of which explicitly focuses on learning the unitarity, which is a measure of how unitary the channel is. Here are a couple of possibly helpful papers.

[1] https://arxiv.org/abs/1503.07865

[2] https://arxiv.org/abs/2212.09319

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