  We consider the implementation of a symmetric informationally complete
probability-operator measurement (SIC POM) in the Hilbert space of a d-level
system by a two-step measurement process: a diagonal-operator measurement with
high-rank outcomes, followed by a rank-1 measurement in a basis chosen in
accordance with the result of the first measurement. We find that any
Heisenberg-Weyl group-covariant SIC POM can be realized by such a sequence
where the second measurement is simply a measurement in the Fourier basis,
independent of the result of the first measurement. Furthermore, at least for
the particular cases studied, of dimension 2, 3, 4, and 8, this scheme reveals
an unexpected operational relation between mutually unbiased bases and SIC
POMs; the former are used to construct the latter. As a laboratory application
of the two-step measurement process, we propose feasible optical experiments
that would realize SIC POMs in various dimensions.
