Rocky Mountain Journal of Mathematics

A note on extremal decompositions of covariances

Zoltán Léka

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 571-580.

First available in Project Euclid: 26 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices 15B57: Hermitian, skew-Hermitian, and related matrices

Decomposition density covariance correlation extreme points


Léka, Zoltán. A note on extremal decompositions of covariances. Rocky Mountain J. Math. 46 (2016), no. 2, 571--580. doi:10.1216/RMJ-2016-46-2-571.

Export citation


  • R. Bhatia, Positive definite matrices, Princeton University Press, Oxford, 2007.
  • R. Bhatia and C. Davis, More operator versions of the Schwarz inequality, Comm. Math. Phys. 215 (2000), 239–244.
  • J.P.R. Christensen and J. Vesterstrøm, A note on extreme positive definite matrices, Math. Ann. 244 (1979), 65–68.
  • R. Grone, S. Pierce and W. Watkins, Extremal correlation matrices, Lin. Alg. Appl. 134 (1990), 63–70.
  • Z. Léka and D. Petz, Some decompositions of matrix variances, Prob. Math. Stat. 33 (2013), 191–199.
  • C-K. Li and B-S. Tam, A note on extreme correlation matrices, SIAM J. Matrix Anal. Appl. 15 (1994), 903–908.
  • K.R. Parthasarathy, An introduction to quantum stochastic calculus, Birkhäuser Verlag, Basel, 1992.
  • D. Petz and G. Tóth, Matrix variances with projections, Acta Sci. Math. 78 (2012), 683–688.
  • ––––, Extremal properties of the variance and the quantum Fisher information, Phys. Rev. 87 (2013), 032324.
  • H-J. Sommers and K. Zyckowski, Hilbert-Schmidt volume of the set of mixed quantum states, J. Phys. 36 (2003), 10115–10130.
  • S. Yu, Quantum Fisher information as the convex roof of variance, preprint, arXiv:1302.5311.