Maximally informative ensembles for SIC-POVMs in dimension 3

Abstract

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 of 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 three-dimensional group-covariant (Weyl–Heisenberg) symmetric informationally complete positive operator valued measures (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 a certain Weyl matrix (algebraically).