Advertisement

Journal of Russian Laser Research

, Volume 40, Issue 4, pp 293–312 | Cite as

Hidden Correlations and Information-Entropic Inequalities in Systems of Qudits

  • Igor Ya. Doskoch
  • Margarita A. Man’koEmail author
Article
  • 11 Downloads

Abstract

We present the results of our study of correlations in noncomposite systems like qudits (N-level atom or spin-j system). We show that they correspond to correlations in composite systems like bipartite or multipartite systems. We establish the correspondence between correlations in noncomposite systems and composite ones using the map of indices labeling the random variables in classical probability theory or matrix elements of density matrices in quantum mechanics. We obtain new information-entropic inequalities for density matrices of noncomposite systems. Also we present Bayes’ formula for noncomposite classical system with one random variable. Finally we discuss the possibility to use hidden correlations and entanglement of a single-qudit state.

Keywords

quantum suprematism triangle geometry of qubits probability distributions quantum correlations Bayes’ formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 34, 203 (2013).Google Scholar
  2. 2.
    M. A. Man’ko and V. I. Man’ko, Phys. Scr., T160, 014030 (2014).ADSGoogle Scholar
  3. 3.
    M. A. Man’ko and V. I. Man’ko, Int. J. Quantum Inf., 12, 156006 (2014).Google Scholar
  4. 4.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 35 298 (2014).Google Scholar
  5. 5.
    M. A. Man’ko and V. I. Man’ko, Entropy, 17, 2876 (2015).ADSMathSciNetGoogle Scholar
  6. 6.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 36, 301 (2015).Google Scholar
  7. 7.
    M. A. Man’ko, V. I. Man’ko, and G. Marmo, Nuovo Cimento C, 38, iss. 5, paper 167 (2015).Google Scholar
  8. 8.
    J. A. Lopez-Saldivar, O. Castaños, M. A. Man’ko, and V. I. Man’ko, Entropy, 21(8), 736 (2019).ADSGoogle Scholar
  9. 9.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 37, 1 (2016).Google Scholar
  10. 10.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 38, 211 (2017).Google Scholar
  11. 11.
    M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 39, 1 (2018).Google Scholar
  12. 12.
    M. A. Man’ko and V. I. Man’ko, Phys. Scr., 93, 084002 (2018).ADSGoogle Scholar
  13. 13.
    J. A. Lopez-Saldivar, O. Castaños, E. Nahmad-Achar, et al., Entropy, 20(9), 630 (2018).ADSGoogle Scholar
  14. 14.
    M. A. Man’ko and V. I. Man’ko, Entropy, 20(9), 692 (2018).ADSGoogle Scholar
  15. 15.
    P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, UK (1981).Google Scholar
  16. 16.
    E. Schrödinger, Ann. Phys., 384, 361; 489 (1926).ADSzbMATHGoogle Scholar
  17. 17.
    L. Landau, Z. Phys., 45, 430 (1927).ADSGoogle Scholar
  18. 18.
    J. von Neumann, Gött. Nach., 1, 245 (1927).Google Scholar
  19. 19.
    S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996).ADSMathSciNetGoogle Scholar
  20. 20.
    V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).ADSMathSciNetGoogle Scholar
  21. 21.
    M. Asorey, A. Ibort, G. Marmo, and F. Ventriglia, Phys. Scr., 90, 074031 (2015).ADSGoogle Scholar
  22. 22.
    D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244 (1993).ADSGoogle Scholar
  23. 23.
    J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).ADSMathSciNetGoogle Scholar
  24. 24.
    K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).ADSGoogle Scholar
  25. 25.
    A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys., 81, 299 (2009).ADSGoogle Scholar
  26. 26.
    R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).ADSMathSciNetGoogle Scholar
  27. 27.
    E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1963).ADSMathSciNetGoogle Scholar
  28. 28.
    E. C. G. Sudarshan, P. M. Mathews, and Jayaseetha Rau, Phys. Rev., 121, 920 (1961).ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    K. Kraus, States, Effects and Operations, Springer, Berlin (1988).Google Scholar
  30. 30.
    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys., 17, 821 (1976).ADSCrossRefGoogle Scholar
  31. 31.
    G. Lindblad, Commun. Math. Phys., 48, 119 (1976).ADSCrossRefGoogle Scholar
  32. 32.
    E. Wigner, Phys. Rev., 40, 749 (1932).ADSGoogle Scholar
  33. 33.
    K. Husimi, Proc. Phys. Math. Soc. Jpn., 22, 264 (1940).Google Scholar
  34. 34.
    Y. J. Kano, J. Math. Phys., 6, 1913 (1965).ADSGoogle Scholar
  35. 35.
    R. Glauber and V. I. Man’ko, “Damping and fluctuations in the systems of two entangled quantum oscillators,” in: Group Theory, Gravitation, and Physics of Elementary Particles, Proceedings of the Lebedev Physical Institute [in Russian], Nauka, Moscow (1986), Vol. 167 [English translation by Nova Science, Commack, New York (1987), Vol. 167].Google Scholar
  36. 36.
    R. Glauber and V. I. Man’ko, Zh. Éksp. Teor. Fiz., 87, 790 (1984) [Sov. Phys. JETP, 60, 450 (1984)].Google Scholar
  37. 37.
    G. Schrade, V. I. Man’ko, W. P. Schleich, and R. Glauber, Quantum Semiclass. Opt., 7, 307 (1995).ADSGoogle Scholar
  38. 38.
    V. I. Man’ko and O. V. Man’ko, J. Exp. Theor. Phys., 85, 430 (1997).ADSGoogle Scholar
  39. 39.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 35, 278 (2014).Google Scholar
  40. 40.
    E. C. G. Sudarshan, J. Russ. Laser Res., 24, 195 (2003).Google Scholar
  41. 41.
    V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 20, 421 (1999).Google Scholar
  42. 42.
    V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 21, 305 (2000).Google Scholar
  43. 43.
    V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 24, 507 (2003).Google Scholar
  44. 44.
    J. Klauder and E. C. G. Sudarshan, Foundation of Quantum Optics, Russin translation, Mir, Moscow (1970).Google Scholar
  45. 45.
    M. A. Man’ko, V. I. Man’ko, G. Marmo, et al., “Dichotomic probability representation of quantum states,” arXiv:1905.10561 quant-ph (submitted to J. Phys. A: Math. Theor., 2019).Google Scholar
  46. 46.
    A. N. Kolmogorov, Foundation of the Theory of Probability, Chelsea, New York (1956).Google Scholar
  47. 47.
    A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North Holland, Amsterdam (1982).Google Scholar
  48. 48.
    Albert N. Shiryaev, Probability-1, Springer, Berlin, Germany (2016).Google Scholar
  49. 49.
    A. Khrennikov, Probability and Randomness. Quantum versus Classical, World Scientific, Singapore (2016).Google Scholar
  50. 50.
    V. I. Man’ko and Zh. Zeilov, J. Russ. Laser Res., 38 50 (2017).Google Scholar
  51. 51.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 141 (2017).Google Scholar
  52. 52.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 324 (2017).Google Scholar
  53. 53.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 416 (2017).Google Scholar
  54. 54.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Phys. Conf. Ser., 1071, 012008 (2018).Google Scholar
  55. 55.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 39, 128 (2018).Google Scholar
  56. 56.
    M. A. Man’ko, J. Phys.: Conf. Ser., 1071, 012015 (2018).Google Scholar
  57. 57.
    J. A. Lopez-Saldivar, O. Casta˜nos, M. A. Man’ko, and V. I. Man’ko, Physica A, 491, 64 (2018).Google Scholar
  58. 58.
    J. A. Lopez-Saldivar, O. Castaños, M. A. Man’ko, and V. I. Man’ko, Phys. Rev. E, 97, 022128 (2018).Google Scholar
  59. 59.
    V. I. Man’ko, G. Marmo, F. Ventriglia, and P. Vitale, J. Phys. A: Math. Theor., 50, 335302 (2017).Google Scholar
  60. 60.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, Eur. Phys. J. D, 73, 10 (2019); DOI:  https://doi.org/10.1140/epjd/e2018-90487-9.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations