  Surprisingly often decoherence is due to classical fluctuations of ambient
fields and may thus be described in terms of random unitary (RU) dynamics.
However, there are decoherence channels where such a representation cannot
exist. Based on a simple and intuitive geometric measure for the distance of an
extremal channel to the convex set of RU channels we are able to characterize
the set of true quantum phase-damping channels. Remarkably, using the
Caley-Menger determinant, our measure may be assessed directly from the matrix
representation of the channel. We find that the channel of maximum quantumness
is closely related to a symmetric, informationally-complete positive
operator-valued measure (SIC-POVM) on the environment. Our findings are in line
with numerical results based on the entanglement of assistance.
