  An unexpected connection exists between compatibility criteria for quantum
states and symmetric informationally complete POVMs. Beginning with Caves,
Fuchs and Schack's "Conditions for compatibility of quantum state assignments"
[Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in
other contexts enjoys additional interesting properties. Compatibility criteria
provide a new way to understand the relationship between SIC-POVMs and mutually
unbiased bases, as calculations in the SIC representation of quantum states
make clear. This, in turn, illuminates the resources necessary for magic-state
quantum computation, and why hidden-variable models fail to capture the
vitality of quantum mechanics.
