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Quantum correlations and tomographic representation

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Abstract

We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Rényi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.

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References

  1. S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A 213, 1 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res. 18, 407 (1997).

    Article  Google Scholar 

  3. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Fizmatlit, Moscow, 1963, 4th ed.; Pergamon, New York, 1977, 3rd ed.).

    Google Scholar 

  4. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932).

    MATH  Google Scholar 

  5. E. Wigner, Phys. Rev. 40, 749 (1932).

    Article  ADS  Google Scholar 

  6. J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. J. Radon, Ber. Sachs. Akad. Wiss. (Leipzig) 69, 262 (1917).

    Google Scholar 

  8. A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr. 79, 065013 (2009).

    Article  ADS  Google Scholar 

  9. V. V. Dodonov and V. I. Man’ko, Phys. Lett. A 239, 335 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  10. V. I. Man’ko and O. V. Man’ko, J. Exp. Theor. Phys. 85, 430 (1997).

    Article  ADS  Google Scholar 

  11. O. V. Man’ko, V. I. Man’ko, and G. Marmo, J. Phys. A 35, 699 (2002).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. S. L. Stratonovich, Sov. Phys. JETP 4, 891 (1957).

    MathSciNet  Google Scholar 

  13. H. Weyl, Zeitschr. Phys. 46, 1 (1928).

    Article  ADS  Google Scholar 

  14. H. Groenewold, Physica 12, 405 (1946).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. O. V. Man’ko, V. I. Man’ko, G. Marmo, and P. Vitale, Phys. Lett. A 360, 522 (2007).

    Article  MathSciNet  ADS  Google Scholar 

  16. S. Mancini, V. I. Man’ko, and P. Tombesi, Europhys. Lett. 37, 79 (1997).

    Article  ADS  Google Scholar 

  17. S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).

    Article  ADS  Google Scholar 

  18. K. Banaszek and K. Wodkiewicz, Phys. Rev. Lett. 76, 4344 (1996).

    Article  ADS  Google Scholar 

  19. A. S. Arkhipov and V. I. Man’ko, Phys. Rev. A 71, 012101 (2005).

    Article  ADS  Google Scholar 

  20. O. V. Man’ko, AIP Conf. Proc. 1424, 221 (2012).

    Article  ADS  Google Scholar 

  21. O. V. Man’ko, Phys. Scr. T 151, 6 (2013).

    Google Scholar 

  22. V. V. Dodonov, V. I. Man’ko, and O. V. Man’ko, in Compressed and Correlated States of Quantum Systems, Works of Phys. Inst. Acad. Sci., Vol. 200 (Nauka, Moscow, 1991), p. 155 [in Russian].

    Google Scholar 

  23. V. V. Dodonov and A. V. Dodonov, J. Russ. Laser Res. 26, 445 (2005).

    Article  Google Scholar 

  24. V. V. Dodonov, O. V. Man’ko, and V. I. Man’ko, J. Sov. Laser Res. 10, 413 (1989).

    Article  Google Scholar 

  25. O. V. Man’ko, J. Korean Phys. Soc. 27, 1 (1994).

    MathSciNet  Google Scholar 

  26. K. Takashima, N. Hatakenaka, S. Kurihara, and A. Zeilinger, J. Phys. A: Math. Theor. 41, 164036 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  27. C. M. Wilson, G. Johansson, A. Pourkabirian, et al., Nature 479, 376 (2011).

    Article  ADS  Google Scholar 

  28. S. V. Kuznetsov, A. V. Kusev, O. V. Man’ko, and N. V. Tcherniega, J. Opt. B: Quantum Semiclass. Phys. 5, S503 (2003).

    Article  ADS  Google Scholar 

  29. S. V. Kuznetsov, A. V. Kusev, O. V. Man’ko, and N. V. Tcherniega, Acta Phys. Hung., Quantum Electron. Ser. 20(1), 11 (2004).

    ADS  Google Scholar 

  30. O. V. Man’ko, and N. V. Tcherniega, Proc. SPIE 6256, 62560W–1 (2006).

    Article  Google Scholar 

  31. O. V. Man’ko and V. I. Man’ko, Laser Phys. 19, 1804 (2009).

    Article  ADS  Google Scholar 

  32. O. V. Man’ko and V. I. Man’ko, Phys. Scr. T 140, 014028 (2010).

    Article  ADS  Google Scholar 

  33. O. V. Man’ko, AIP Conf. Proc. 1508, 275 (2012).

    Article  ADS  Google Scholar 

  34. O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res. 31, 608 (2010).

    Article  Google Scholar 

  35. O. V. Man’ko, AIP Conf. Proc. 1327, 166 (2011).

    Article  ADS  Google Scholar 

  36. M. Bellini, A. S. Coelho, S. N. Filippov, et al., Phys. Rev. A 85, 052129 (2012).

    Article  ADS  Google Scholar 

  37. V. N. Chernega and V. I. Man’ko, Int. J. Quantum Inf. 10(8) (2012); arXiv:1210.0464.

    Google Scholar 

  38. V. N. Chernega, Phys. Scr. T 147, 014006 (2012).

    Article  ADS  Google Scholar 

  39. V. N. Chernega, J. Russ. Las. Res. 34(2) (2013, in press); arXiv:1303.5238.

    Google Scholar 

  40. V. I. Man’ko, M. Moshinsky, and A. Sharma, Phys. Rev. A 59, 1809 (1999).

    Article  ADS  Google Scholar 

  41. K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).

    Article  ADS  Google Scholar 

  42. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).

    Article  ADS  Google Scholar 

  43. O. V. Man’ko and V. I. Man’ko, Fortschr. Phys. 57, 1064 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  44. V. V. Dodonov and V. I. Man’ko, Invariants and Evolution of Non-Stationary Quantum Systems, Works of Phys. Inst. Acad. Sci., Vol. 183 (Nauka, Moscow, 1987), p. 286 [in Russian].

    Google Scholar 

  45. V. V. Dodonov, A. B. Klimov, and V. I. Man’ko, Phys. Lett. A 134, 211 (1989).

    Article  ADS  Google Scholar 

  46. V. V. Dodonov, A. V. Klimov, and V. I. Man’ko, in Compressed and Correlatedd States of Quantum Systems, Works of Phys. Inst. Acad. Sci., Vol. 200 (Nauka, Moscow, 1991), p. 56 [in Russian].

    Google Scholar 

  47. V. I. Vysotskii, M. V. Vysotskii, and S. V. Adamenko, J. Exp. Theor. Phys. 115, 551 (2012).

    Article  ADS  Google Scholar 

  48. M. A. Man’ko and V. I. Man’ko, Found. Phys. 41, 330 (2011).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. V. I. Man’ko and S. N. Filippov, J. Rus. Laser Res. 34, 1 (2013).

    Article  Google Scholar 

  50. M. A. Man’ko and V. I. Man’ko, AIP Conf. Proc. 1488, 110 (2012).

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119991, Russia

    O. V. Man’ko & V. N. Chernega

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  1. O. V. Man’ko
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  2. V. N. Chernega
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Correspondence to O. V. Man’ko.

Additional information

Original Russian Text © O.V. Man’ko, V.N. Chernega, 2013, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 97, No. 9, pp. 642–648.

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Man’ko, O.V., Chernega, V.N. Quantum correlations and tomographic representation. Jetp Lett. 97, 557–563 (2013). https://doi.org/10.1134/S0021364013090099

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