Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map, and let $J(\Phi)$ be its Choi representation. As is well known, any such map can be written in a Kraus representation of the form $$\Phi(X)=\sum_a p_a A_a X A_a^\dagger,\tag A$$ where $p_a\ge0$ and $\operatorname{Tr}(A_a^\dagger A_b)=\delta_{ab}$ (as mentioned for example in this other question of mine). The positive numbers $p_a$ appearing here can be seen to be the eigenvalues of $J(\Phi)$.
Is there a way to directly relate the eigenvalues $p_a$ to some properties of the channel, or to how the channel acts on states?