I see, thanks for the information

It seems like before the components part the important thing was to find $\mathcal{I}^{-1}$. What is the significance of that? Find the inverse in the first place?

So why is it justified to use $A$ instead of $\alpha$ in what I wrote? Is it just abuse of notation?

So according to Wikipedia: "The rank of [a matrix] equals the number of non-zero singular values". Since the density matrices are 4x4, a rank of 2 would mean that two of the eigenvalues would be zero, making the determinant zero. Right?

I actually don't know what either of those mean for this, but I'll try to figure it out.