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Geometry and Structure of Quantum Phase Space


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.

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

    MATH  Article  Google Scholar 

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

    MATH  MathSciNet  ADS  Article  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)

    MATH  MathSciNet  Article  Google Scholar 

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

    MathSciNet  ADS  Article  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)

    MathSciNet  ADS  Article  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)

    ADS  Article  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)

    MathSciNet  ADS  Article  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)

    ADS  Article  Google Scholar 

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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).

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Correspondence to Hoshang Heydari.

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

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  • Quantum mixed states
  • Density operators
  • Quantum phase space
  • Uncertainty relation
  • Geometry