On symmetric decompositions of positive operators

TL;DR

This paper studies symmetric decompositions of positive operators in finite-dimensional Hilbert spaces, characterizing their structure and comparing with quantum measurement frameworks, while also extending Welch-type inequalities.

Contribution

It provides a complete characterization of symmetric decompositions of positive operators and relates them to SIC-POVMs and Welch-type inequalities in quantum information theory.

Findings

Characterization of all symmetric decompositions of positive operators.

Comparison with SIC-POVMs in quantum information.

Generalization of Welch-type inequalities.

Abstract

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.