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I am working on a project and I expect to have expressions of a bunch of quantum channels of interest. The quantum channels will be in matrix form. For example for a 2 qubit system, the quantum channel will be given by a 16 by 16 matrix.

Is there any systematic way to use the matrix representations of quantum channels and find the corresponding Kraus operators etc? Find which channels correspond to unitary channels etc?

user22511
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  • This might be helpful: https://quantumcomputing.stackexchange.com/a/5816/13968 – narip Nov 21 '22 at 02:10
  • There are multiple representations of channels using a single, necessarily square matrix, most notably the Choi and $\chi$ matrix. See e.g. this previous answer of mine. For both these we have: 1) the eigenvectors correspond to the Kraus operators and 2) if the matrix is single rank it corresponds to a unitary. If you let us know what representation you will work with I can write up a more detailed answer. – JSdJ Nov 21 '22 at 08:28
  • you can also find a bunch of explicit examples of matrix representations (with a focus on Chois and Stinespring isometries) in https://quantumcomputing.stackexchange.com/q/24511/55. Does any of these linked posts answer your question? If not, could you further clarify what you are asking and how it differs from the other related discussions? – glS Nov 21 '22 at 09:32
  • I have yet to perform the computation but I expect to have a bunch of matrices in the usual sense. I'll have to throw away the matrices that don't correspond to CPTP maps. I'll then like to identify the remaining matrices/CPTP maps with their Kraus and/or Choi decomposition. – user22511 Nov 21 '22 at 16:12
  • @user22511 that still doesn't clarify the problem. There are multiple possible matrix representations of a channel. You need to specify which one you are talking about. – glS Nov 21 '22 at 18:13
  • Well I am first trying to classify using representation theory certain matrices that meet some commutation relations. For a 2 qubit system, I'll have 16 by 16 matrices representing some linear maps taking 4 by 4 matrices to 4 by 4 matrices. I'll then have to throw those away that are not representing CPTP maps. – user22511 Nov 21 '22 at 19:35
  • Does this help? It's just linear algebra at this point. After this point, I need to start interpreting everything in terms of quantum channels so I'd like to convert the CPTP matrices to their Kraus/Choi representations. – user22511 Nov 21 '22 at 19:36

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