Consider two channels, $\Phi,\Psi\in\mathrm C(\mathcal X)$ acting on some space $\mathcal X$, and suppose they commute, that is, $$\Phi(\Psi(\rho))=\Psi(\Phi(\rho))$$ for all states $\rho$. Can anything be said about the structure, e.g. in terms of their Kraus operators, that this implies? For example, does there have to be a specific relation between their Kraus operators?
Suppose $\Phi(X)=\sum_a A_a X A_a^\dagger$ and $\Psi(X)=\sum_b B_b X B_b^\dagger$. Then the question relates to the channels with Kraus operators $\{A_a B_b\}_{ab}$ and $\{B_b A_a\}_{ab}$. There are a few cases in which the solution is obvious, e.g. if the Kraus operators commute with each others, $[A_a,B_b]=0$, but I'm not sure how to proceed in the more general case. Is there anything general that can be said about the problem?