I am interested in characterizing the space of all collections of $d^2$ linearly independent projectors on the Hilbert space $\mathbb{C}^d$. The linear independence I desire is in the vector space of $B(\mathbb{C}^d)$ of bounded operators on the Hilbert space.
I'm interested in this space because such sets of projectors correspond to minimal informationally complete rank-1 POVMs in quantum measurement theory (this shouldn't be necessary information for addressing my problem, but I include it for context). This has a connection to frame theory (a minimal informationally complete rank-1 POVM is a tight frame), but literature I've found on the subject seems to be missing the mark so far.
For instance, Manifold structure of spaces of spherical tight frames appears to be a promising article, but spherical tight frames are both too general and too restrictive for my interests (too general since I would like to fix the number of frame elements to $d^2$ and require the operators $\vert v\rangle\langle v\vert$ made from the frame elements $\vert v\rangle$ to be linearly independent operators; too restrictive since I would like to allow my frame elements to have different norms unlike the spherical case).
Does anyone know of literature addressing this structure?