  The generalized Bloch decomposition of a bipartite quantum state gives rise
to a correlation matrix whose singular values provide rich information about
non-local properties of the state, such as the dimensionality of entanglement.
While some entanglement criteria based on the singular values exist, a complete
understanding of the geometry of admissible correlation matrices is lacking. We
provide a deeper insight into the geometry of the singular values of the
correlation matrices of limited Schmidt number. First, we provide a link to the
framework of randomized measurements and show how to obtain knowledge about the
singular values in this framework by constructing observables that yield the
same moments as one obtains from orthogonal averages over the Bloch sphere. We
then focus on the case of separable states and characterize the boundary of the
set of the first two non-vanishing moments by giving explicit constructions for
some of the faces and extremal points. These constructions yield a connection
between the geometry of the correlation matrices and the existence problems of
maximal sets of mutually unbiased bases, as well as SIC-POVMs.
