I have two Choi matrix I suspect be equivalent. Can I manipulate them?

Question

I am performing a process tomography over a protocol I suspect to be equivalent to the $CS$ gate.

To compare basic operators I usually compute the Choi matrix of the target gate -- which in this case should be $|CS\rangle\rangle\langle\langle CS|$ -- and check if it is (almost) equivalent to the experimental Choi matrix.

Now, when I try to get the experimental matrix of the circuit I wrote, this has some differences with $|CS\rangle\rangle\langle\langle CS|$.

Is it possible that they are still equivalent, what kind of transformation I am allowed to do to the Choi matrix without changing the channel it is representing?

Answer 1

There is a one-to-one relation between Choi states$^1$ and quantum channels (Choi-Jamiołkowski isomorphism). So, you cannot transform the Choi matrix without changing the channel it represents. If the two Choi matrices are different, then they are different quantum channels.

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^{1: Choi states are just normalized Choi matrices. If your map is a valid quantum map, then normalized Choi matrix, i.e. Choi state is a valid quantum state, i.e., the matrix is positive-semidefinite & has unit trace.}