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On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems

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A mapping between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich–Weyl kernel. It is shown that the moduli space of Stratonovich–Weyl kernels is given by the intersection of the coadjoint orbit space of the group SU(N) and a unit (N − 2)-dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems.

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References

  1. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Verlag von Julius Springer, Berlin (1932).

  2. H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel-Verlag, Leipzig (1928).

  3. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev., 40, 749–759 (1932).

    Article  MATH  Google Scholar 

  4. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Japan, 22, 264–314 (1940).

    MATH  Google Scholar 

  5. H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica, 12, 405–460 (1946).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. E. Moyal, “Quantum mechanics as a statistical theory,” Math. Proc. Cambridge Philos. Soc., 45, 99–124 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Dito and D. Sternheimer, “Deformation quantization: genesis, developments and metamorphoses,” arXiv:math/0201168 (2002).

  8. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett., 10, 277–279 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  9. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev., 131, 2766–2788 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Phys. Rep., 106, 121–167 (1984).

    Article  MathSciNet  Google Scholar 

  11. D. J. Rowe, B. C. Sanders, and H. de Guise, “Representations of the Weyl group and Wigner functions for SU(3),” J. Math. Phys., 40, 3604–3615 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Chumakov, A. Klimov, and K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101 (2000).

    Article  Google Scholar 

  13. M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces,” J. Math. Phys., 43, 5857–5871 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys., 81, 299–332 (2009).

    Article  Google Scholar 

  15. A. B. Klimov and H. de Guise, “General approach to GU(n) quasi-distribution functions,” J. Phys. A, 43, 402001 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Rigas, L. L. S´anchez-Soto, A. B. Klimov, J. Rehacek, and Z. Hradil, “Orbital angular momentum in phase space,” Ann. Phys., 326, 426–439 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, “Wigner functions for arbitrary quantum systems,” Phys. Rev. Lett., 117, 180401 (2016).

    Article  Google Scholar 

  18. A. B. Klimov, J. L. Romero, and H. de Guise, “Generalized SU(2) covariant Wigner functions and some of their applications,” J. Phys. A, 50, 323001 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge Univ. Press, Cambridge (2006).

  20. I. Bengtsson and K. E. Życzkowski, Geometry of Quantum States. An Introduction to Quantum Entanglement, 2nd edition, Cambridge Univ. Press, Cambridge (2017).

    Book  MATH  Google Scholar 

  21. R. P. Feynman, “Negative probability,” in: B. J. Hiley and D. Peat (eds.), Quantum Implications: Essays in Honour of David Bohm, Routledge (1987), pp. 235–248.

  22. R. L. Stratonovich, “On distributions in representation space,” Sov. Phys. JETP, 4, No. 6, 891–898 (1957).

    MathSciNet  MATH  Google Scholar 

  23. C. Briff and A. Mann, “Phase space formulation of quantum mechanics and quantumstate reconstruction for physical systems with Lie-group symmetries,” Phys. Rev., 59, 971 (1999).

    Article  MathSciNet  Google Scholar 

  24. A. Khvedelidze and V. Abgaryan, “On the family of Wigner functions for N-level quantum system,” arXiv:1708.05981 (2018).

  25. A. A. Kirillov, Lectures on the Orbit Method, Amer. Math. Soc. (2004).

  26. D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank,” J. Math. Phys., 19, 999–1001 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Collins, “Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability,” Int. Math. Res. Not., 17, 953–982 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Collins and P. Sniady, “Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,” Comm. Math. Phys., 264, No. 3, 773–795 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Luis, “A SU(3) Wigner function for three-dimensional systems,” J. Phys. A, 41, 495302 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. I. Arnold, “Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect,” Selecta Math., New Series, 1, No. 1, 1–19 (1995).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Russia

    V. Abgaryan, A. Khvedelidze & A. Torosyan

  2. A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University and Institute of Quantum Physics and Engineering Technologies, Georgian Technical University, Tbilisi, Georgia

    A. Khvedelidze

Authors
  1. V. Abgaryan
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  2. A. Khvedelidze
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  3. A. Torosyan
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Correspondence to V. Abgaryan.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 468, 2018, pp. 177–201.

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Abgaryan, V., Khvedelidze, A. & Torosyan, A. On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems. J Math Sci 240, 617–633 (2019). https://doi.org/10.1007/s10958-019-04379-7

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  • DOI: https://doi.org/10.1007/s10958-019-04379-7

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