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.
"A note on extremal decompositions of covariances." Rocky Mountain J. Math. 46 (2) 571 - 580, 2016. https://doi.org/10.1216/RMJ-2016-46-2-571