On the Purity and Entropy of Mixed Gaussian States

  • Maurice de GossonEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The notions of purity and entropy play a fundamental role in the theory of density operators. These are nonnegative trace class operators with unit trace. We review and complement some results from a rigorous point of view.



Maurice de Gosson has been financed by the Grant P27773-N23 of the Austrian Research Foundation FWF.


  1. 1.
    G. Adesso, S. Ragy, and A. R. Lee. Continuous variable quantum information: Gaussian states and beyond. Open Systems & Information Dynamics 21, 01n02, 1440001 (2014)Google Scholar
  2. 2.
    G. S. Agarwal, Entropy, the Wigner distribution function, and the approach to equilibrium of a system of coupled harmonic oscillators. Phys. Rev. A3(2), 828–831 (1971)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. de Gosson and F. Luef, Symplectic Capacities and the Geometry of Uncertainty: the Irruption of Symplectic Topology in Classical and Quantum Mechanics. Phys. Reps. 484 (2009)Google Scholar
  4. 4.
    M. de Gosson, Quantum blobs, Found. Phys. 43(4), 440–457 (2013)Google Scholar
  5. 5.
    M. de Gosson, Symplectic geometry and quantum mechanics. Vol. 166. Springer Science & Business Media (2006)Google Scholar
  6. 6.
    M. de Gosson, Symplectic methods in harmonic analysis and in mathematical physics. Vol. 7. Springer Science & Business Media (2011)Google Scholar
  7. 7.
    M. de Gosson, The symplectic camel and phase space quantization. J. Phys. A: Math. Gen. 34(47), 10085 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. de Gosson, The symplectic camel and the uncertainty principle: the tip of an iceberg? Found. Phys. 99 (2009)Google Scholar
  9. 9.
    B. Dutta, N. Mukunda, and R. Simon, The real symplectic groups in quantum mechanics and optics, Pramana 45(6), 471–497 (1995)CrossRefGoogle Scholar
  10. 10.
    G. B. Folland, Harmonic Analysis in Phase space, Annals of Mathematics studies, Princeton University Press, Princeton, N.J. (1989)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics (NuHAG)University of ViennaViennaAustria

Personalised recommendations