  Since Renes et al. [J. Math. Phys. 45, 2171 (2004)], there has been much
effort in the quantum information community to prove (or disprove) the
existence of symmetric informationally complete (SIC) sets of quantum states in
arbitrary finite dimension. This paper strengthens the urgency of this question
by showing that if SIC-sets exist: 1) by a natural measure of orthonormality,
they are as close to being an orthonormal basis for the space of density
operators as possible, and 2) in prime dimensions, the standard construction
for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant
SIC-sets are intimately related: The latter represent minimum uncertainty
states for the former in the sense of Wootters and Sussman. Finally, we
contribute to the question of existence by conjecturing a quadratic redundancy
in the equations for Weyl-Heisenberg SIC-sets.
