On symmetric decompositions of positive operators

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.