Is there a physical interpretation for the diagonal terms of a $chi$ matrix?

I assume $mathcal{E} in mathcal{L}(mathcal{L}(H))$ is a CPTP map. I call ${B_i}$ an orthonormal basis for Hilbert-Schmidt scalar product of $mathcal{L}(H)$

This quantum map can be decomposed as:

$$mathcal{E}(rho)=sum_{ij} chi_{ij} B_i rho B_j^{dagger}$$

The matrix having for element $chi_{ij}$ is called the chi-matrix of the process.

My question are:

Is there a physical interpretation for the diagonal coefficients of this chi matrix in a general case (i.e interpretation valid for any basis). I know that in quantum error correction theory they might have an importance, but even without Q.E.C can we relate them to some physical property ?

Same question in the particular case the chi-matrix is actually diagonal, is there a general interpretation of the $chi_{ii}$ ?