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?
In your latest comments you talked about matrices (channels) that meet some commutation relations. This is best tackled with the representation matrix (also known as "natural representation") with respect to vectorization---as linked in one of the comments, cf. also this answer---which is the unique matrix $\hat\Phi$ such that ${\rm vec}(\Phi(\rho))=\hat\Phi{\rm vec}(\rho)$ for all $\rho$. Given any $\Phi:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$ the explicit expression for $\hat\Phi$ reads $$ \hat\Phi=\sum_{i,j,k,l=1}^n\;\langle l|\Phi(|i\rangle\langle j|)|k\rangle\;|k\rangle\langle j|\otimes|l\rangle\langle i| $$ as is readily verified. The key property here is that $\widehat{(\cdot)}$ is a (linear) homomorphism, that is, $\widehat{\Phi_1\Phi_2}=\widehat{\Phi_1}\widehat{\Phi_2}$ for all $\Phi_1,\Phi_2$; thus commutators of linear maps are in one-to-one correspondence with commutators of their representation matrices. Then in order to check complete positivity of $\Phi$ you have to check the Choi matrix $\mathsf C(\Phi)$ which is just a re-shuffling of $\hat\Phi$. More precisely, it is easy to see that $$ \mathsf C(\Phi)=\sum_{i,j,k,l}\;\langle l|\Phi(|i\rangle\langle j|)|k\rangle\;\color{red}{|i\rangle}\langle j|\otimes|l\rangle\color{red}{\langle k|} $$ so the linear (and even unitary) map $S:\mathbb C^{n^2\times n^2}\to \mathbb C^{n^2\times n^2}$ uniquely defined via $S(|k\otimes l\rangle\langle j\otimes i|):=|i\otimes l\rangle\langle j\otimes k|$ satisfies $S(\hat\Phi)=\mathsf C(\Phi)$ (and even $S(\mathsf C(\Phi))=\hat\Phi$ because $S^2={\rm id}$). From the Choi matrix complete positivity and Kraus operators are obtained as usual.
Finally, because you explicitly mentioned it let me say that as per this result a unitary representation matrix $\hat\Phi$ corresponds to a channel if and only if $\hat\Phi=\overline{U}\otimes U$ for some $U\in\mathsf U(n)$. An easy way to check this would be to look at the partial traces of $\hat\Phi$ with respect to the first and second tensor factor and to check that they are both unitary and coincide up to complex conjugation $\overline{(\cdot)}$.
