A note on extremal decomposition of covariances

TL;DR

This paper introduces an elementary method for extremal decomposition of quantum covariance matrices, offering a new proof of existing results and sharp estimates on the ranks of densities involved.

Contribution

It provides a simplified proof and improved rank estimates for extremal decompositions of quantum covariance matrices.

Findings

New elementary proof of extremal decomposition results

Sharp rank estimates for densities in the decomposition

Enhanced understanding of quantum covariance matrix structure

Abstract

We shall present an elementary approach to extremal decompositions of (quantum) covariance matrices determined by densities. We give a new proof on former results and provide a sharp estimate of the ranks of the densities that appear in the decomposition theorem.