  Coherence, treated as a resource in quantum information theory, is a
basis-dependent quantity. Looking for states that have constant coherence under
canonical changes of basis yields highly symmetric structures in state space.
For the case of a qubit, we find an easy construction of qubit SICs (Symmetric
Informationally Complete POVMs). SICs in dimension 3 and 8 are also shown to be
equicoherent.
