In order to find out for which initial states of the system the uncertainty
of the measurement outcomes will be minimal, one can look for the minimizers of
the Shannon entropy of the measurement. In case of group covariant measurements
this question becomes closely related to the problem how informative the
measurement is in the sense of its informational power. Namely, the orbit under
group action of the entropy minimizer corresponds to a maximally informative
ensemble of equiprobable elements. We give a characterization of such ensembles
for 3-dimensional group covariant (Weyl-Heisenberg) SIC-POVMs in both geometric
and algebraic terms. It turns out that a maximally informative ensemble arises
from the input state orthogonal to a subspace spanned by three linearly
dependent vectors defining a SIC-POVM (geometrically) or from an eigenstate of
certain Weyl's matrix (algebraically).Comment: 11 page
