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Journal of Russian Laser Research

, Volume 39, Issue 4, pp 360–375 | Cite as

Symbols of Multiqubit States Admitting a Physical Interpretation*

  • Peter Adam
  • Vladimir A. Andreev
  • Margarita A. Man’ko
  • Vladimir I. Man’ko
Article
  • 16 Downloads

Abstract

We study the multiqubit states and the corresponding minimal sets of dequantizers and quantizers, such that symbols of the operators (observables) obtained with the help of the dequantizers admit a physical interpretation; more precisely, the symbols can be measured in the experiments. We consider two types of such quantities: (i) the probabilities of spin projections onto certain directions and (ii) the mean values of these projections. We provide an explicit description of the systems of dequantizers and quantizers for both types of these quantities corresponding to N-qubit states. We show that, in view of such symbols, it is possible to represent the density matrices of the N-qubit states in the form of series expansion in terms of quantizers with the coefficients as measurable observables.

Keywords

density matrix symbol quantizer dequantizer spin projection probability mean value 

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References

  1. 1.
    E. P. Wigner, Phys.Rev., 40, 749 (1932).ADSCrossRefGoogle Scholar
  2. 2.
    K. Husimi, Proc. Phys. Math. Soc. Jpn., 22, 264 (1940).Google Scholar
  3. 3.
    Y. J. Kano, J. Math. Phys., 6, 1913 (1965).ADSCrossRefGoogle Scholar
  4. 4.
    R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1963).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    S. L. Stratonovich, Sov. Phys. JETP, 4, 891 (1957).MathSciNetGoogle Scholar
  7. 7.
    J. Schwinger, Proc. Nat. Acad. Sci. U.S.A., 46, 1401 (1960).ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    W. K. Wootters, IBM J. Res. Dev., 48, 99 (2004).CrossRefGoogle Scholar
  9. 9.
    W. K. Wootters, Ann. Phys., 176, 1 (1987).ADSCrossRefGoogle Scholar
  10. 10.
    U. Leonhardt, Phys. Rev. Lett., 74, 4101 (1995).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    U. Leonhardt, Phys. Rev. A, 53, 2998 (1996).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Asplund and G. Björk, Phys. Rev. A, 64, (2001) 012106.ADSCrossRefGoogle Scholar
  13. 13.
    K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A, 70, 062101 (2004).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, J. Phys. A: Math. Gen., 40, 3987 (2007).ADSCrossRefGoogle Scholar
  15. 15.
    G. Björk, J. L. Romero, A. B. Klimov, and L. L. Sánchez-Soto, arXiv:quant-ph/0608173v2 23 Aug 2006.Google Scholar
  16. 16.
    A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, arXiv:quant-ph/0806.072v1 17 Nov 2008.Google Scholar
  17. 17.
    F. Lizzi and P. Vitale, SIGMA, 10, 086 (2014).Google Scholar
  18. 18.
    A. Vourdas, Acta Appl. Math., 93, 197 (2006).MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Stóvicék and J. Tolar, Rep. Math. Phys., 20, 157 (1984).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Chaturvedi, E. Ercolessi, G. Marmo, et al., J. Phys. A: Math. Gen., 39, 1405 (2006).ADSCrossRefGoogle Scholar
  21. 21.
    S. N. Filippov and V. I. Man’ko, Phys. Scripta, T143, 014010 (2011).ADSCrossRefGoogle Scholar
  22. 22.
    S. N. Filippov and V. I. Man’ko, Phys. Scripta, 83, 058101 (2011).ADSCrossRefGoogle Scholar
  23. 23.
    S. N. Filippov and V. I. Man’ko, J. Phys. A: Math. Gen., 32, 56 (2011).Google Scholar
  24. 24.
    F. Bayen, M. Flato, C. Fronsdal, et al., Lett. Math. Phys., 1, 521 (1977).ADSCrossRefGoogle Scholar
  25. 25.
    O. V. Man’ko, V. I. Man’ko, and G. Marmo, J. Phys. A: Math. Gen., 35, 699 (2002).ADSCrossRefGoogle Scholar
  26. 26.
    P. Adam, V. A. Andreev, I. Ghiu, et al., J. Russ. Laser Res., 35, 3 (2014).CrossRefGoogle Scholar
  27. 27.
    P. Adam, V. A. Andreev, I. Ghiu, et al., J. Russ. Laser Res., 35, 427 (2014).CrossRefGoogle Scholar
  28. 28.
    P. Adam, V. A. Andreev, A. Isar, et al., Theor. Math. Phys., 186, 346 (2016).CrossRefGoogle Scholar
  29. 29.
    P. Adam, V. A. Andreev, A. Isar, et al., J. Russ. Laser Res., 37, 544 (2016).CrossRefGoogle Scholar
  30. 30.
    P. Adam, V. A. Andreev, A. Isar, et al., Phys. Lett. A, 381, 2778 (2017).ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    P. Adam, V. A. Andreev, M. A. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 491 (2017).CrossRefGoogle Scholar
  32. 32.
    A. Weyl, Acta Math., 111, 143 (1964).MathSciNetCrossRefGoogle Scholar
  33. 33.
    G. Folland, Harmonic Analysis in Phase Space, Princeton University Press (1989).Google Scholar
  34. 34.
    J. O’Neill, P. Flandrin, and W. Williams, IEEE Signal Process. Lett., 6, 304 (1999).ADSCrossRefGoogle Scholar
  35. 35.
    B. Boashash (Ed.), Time Frequency Signal Analysis and Processing. A Comprehensive Reference, Elsevier, Amsterdam (2003).Google Scholar
  36. 36.
    M. Asorey, P. Facchi, V. I. Man’ko, et al., Phys. Scr., 90, 065101 (2015).ADSCrossRefGoogle Scholar
  37. 37.
    T Opartny, V. Bu\( \tilde{\mathrm{z}} \)ek, J. Bajer, and G. Drobny, Phys. Rev. A, 52, 2419 (1995).Google Scholar
  38. 38.
    M. A. Marchiolli, E. C. Silva, and D. Galetti, Phys. Rev. A, 79, 022114 (2009).ADSCrossRefGoogle Scholar
  39. 39.
    E. C. Silva and D. Galetti, J. Phys. A: Math. Theor., 13, 022114 (2009).Google Scholar
  40. 40.
    S. Mancini, V. I. Man’ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615 (1995).ADSCrossRefGoogle Scholar
  41. 41.
    S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996).ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    S. Mancini, V. I. Man’ko, and P. Tombesi, Found. Phys., 27, 801 (1997).ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 79, 065013 (2009).ADSCrossRefGoogle Scholar
  44. 44.
    V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    V. I. Man’ko and O. V. Man’ko, J. Exp. Theor. Phys., 85, 430 (1997).ADSCrossRefGoogle Scholar
  46. 46.
    V. A. Andreev and V. I. Man’ko, J. Exp. Theor. Phys., 87, 239 (1998).ADSCrossRefGoogle Scholar
  47. 47.
    V. A. Andreev, O. V. Man’ko, V. I. Man’ko, and S. S. Safonov, J. Russ. Laser Res., 19, 340 (1998).CrossRefGoogle Scholar
  48. 48.
    V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, Phys. Lett. A, 372, 6490 (2008).ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    G. M. D’Ariano, L. Maccone, and M. Paini, J. Opt. B: Quantum Semiclass. Opt., 5, 77 (2003).ADSCrossRefGoogle Scholar
  50. 50.
    S. Weigert, Int. J. Mod. Phys., 20, 1942 (2006) .ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    J. P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 32, 2777 (1999).ADSCrossRefGoogle Scholar
  52. 52.
    V. A. Andreev and V. I. Man’ko, JETP Lett., 72, 93 (2000).ADSCrossRefGoogle Scholar
  53. 53.
    V. A. Andreev and V. I. Man’ko, Phys. Lett. A, 254, 272 (2001).Google Scholar
  54. 54.
    V. A. Andreev, Theor. Math. Phys., 152, 1286 (2007).CrossRefGoogle Scholar
  55. 55.
    V. A. Andreev, Theor. Math. Phys., 158, 196 (2009).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Peter Adam
    • 1
  • Vladimir A. Andreev
    • 2
  • Margarita A. Man’ko
    • 2
  • Vladimir I. Man’ko
    • 2
  1. 1.Institute for Solid State Physics and Optics, Wigner Research Centre for PhysicsHungarian Academy of SciencesBudapestHungary
  2. 2.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia

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