Skip to main content
Log in

Geometry and Structure of Quantum Phase Space

  • Published:
Foundations of Physics Aims and scope Submit manuscript


The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an important role in foundations of quantum mechanics and quantum information. In this work we discuss a geometric framework for mixed quantum states represented by density matrices, where the quantum phase space of density matrices is equipped with a symplectic structure, an almost complex structure, and a compatible Riemannian metric. This compatible triple allow us to investigate arbitrary quantum systems. We will also discuss some applications of the geometric framework.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Günther, C.: Prequantum bundles and projective Hilbert geometries. Int. J. Theoret. Phys. 16, 447–464 (1977)

    Article  MATH  Google Scholar 

  2. Kibble, T.W.B.: Geometrization of quantum mechanics. Commun. Math. Phys. 65, 189–201 (1979)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Ashtekar, A., Schilling, T.A.: Geometrical formulation of quantum mechanics. In: Harvey, A. (ed.) On Einstein’s Path, pp. 23–65. Springer, New York (1998)

  4. Brody, D.C., Hughston, L.P.: Geometrization of statistical mechanics. Proc. Math. Phys. Eng. Sci. 455(1985), 1683–1715 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Andersson, O., Heydari, H.: Operational geometric phase for mixed quantum states. New J. Phys. 15(5), 053006 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  6. Andersson, O., Heydari, H.: Motion in bundles of purifications over spaces of isospectral density matrices. AIP Conf. Proc. 1508(1), 350–353 (2012). GP, MB. DD, GQE, GUR, QSL

    ADS  Google Scholar 

  7. Andersson, O., Heydari, H.: Dynamic distance measure on spaces of isospectral mixed quantum states. Entropy 15(9), 3688–3697 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  8. Andersson, O., Heydari, H.: Geometry of quantum evolution for mixed quantum states. Phys. Scripta 2014(T160), 014004 (2014)

    Article  Google Scholar 

  9. Andersson, O., Heydari, H.: Geometric uncertainty relation for mixed quantum states. J. Math. Phys. 55(4), 042110 (2014)

    Article  ADS  Google Scholar 

  10. Andersson, O., Heydari, H.: Quantum speed limits and optimal hamiltonians for driven systems in mixed states. J. Phys. A 47(21), 215301 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  11. Heydari, H.: A geometric framework for mixed quantum states based on a Kahler structure. arXiv:1409.6891

  12. Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)

    Article  ADS  Google Scholar 

Download references


The author acknowledges useful comments and discussions with Ole Andersson at Stockholm University and also discussion with Faisal Shah Khan at Khalifa University. The author also acknowledges the financial support from the Swedish Research Council (VR).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hoshang Heydari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heydari, H. Geometry and Structure of Quantum Phase Space. Found Phys 45, 851–857 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: