  We consider sets of trace-normalized non-negative operators in
Hilbert-Schmidt balls that maximize their mutual Hilbert-Schmidt distance;
these are optimal arrangements in the sets of purity-limited classical or
quantum states on a finite-dimensional Hilbert space. Classical states are
understood to be represented by diagonal matrices, with the diagonal entries
forming a probability vector. We also introduce the concept of spectrahedron
arrangements which provides a unified framework for classical and quantum
arrangements and the flexibility to define new types of optimal packings.
Continuing a prior work, we combine combinatorial structures and line packings
associated with frames to arrive at optimal arrangements of higher-rank quantum
states. One new construction that is presented involves generating an optimal
arrangement we call a Gabor-Steiner equiangular tight frame as the orbit of a
projective representation of the Weyl-Heisenberg group over any finite abelian
group. The minimal sets of linearly dependent vectors, the so-called binder, of
the Gabor-Steiner equiangular tight frames are then characterized; under
certain conditions these form combinatorial block designs and in one case
generate a new class of block designs. The projections onto the span of minimal
linearly dependent sets in the Gabor-Steiner equiangular tight frame are then
used to generate further optimal spectrahedron arrangements.
