  Inspired by some problems in Quantum Information Theory, we present some
results concerning decompositions of positive operators acting on finite
dimensional Hilbert spaces. We focus on decompositions by families having
geometrical symmetry with respect to the Euclidean scalar product and we
characterize all such decompositions, comparing our results with the case of
SIC--POVMs from Quantum Information Theory. We also generalize some Welch--type
inequalities from the literature.
