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Triangle Geometry of Spin States and Nonlinear Superposition of Probabilities Describing These States†

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Journal of Russian Laser Research Aims and scope

Abstract

On the spin-1/2 example, we demonstrate that quantum states can be described by standard probability distributions, which contain the same information that the wave function and the density matrix do. Within the framework of this approach, called for spin-1/2 the quantum suprematism representation, the probability distributions are illustrated by simplex or triangle geometry or by the Triada of Malevich’s Squares (black, red, and white) associated with the triangles, and new quantum relations for areas of the squares are obtained. The superposition principle for spin states and quantum interference phenomenon are expressed as an explicit new nonlinear addition rule for the probability distributions describing the quantum states and illustrated as the addition of two Triadas of Malevich’s squares. We discuss some analogy of the triangle geometry of spin-1/2 states related to the O(3) symmetry group and the pyramide geometry related to the hydrogen-atom dynamical symmetry O(2, 4).

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References

  1. V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 141 (2017).

    Article  Google Scholar 

  2. V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 324 (2017).

    Article  Google Scholar 

  3. V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 416 (2017).

    Article  Google Scholar 

  4. V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Phys. Conf. Ser., 1071, 012008 (2018).

    Article  Google Scholar 

  5. V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 39, 128 (2018).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys., 81, 299 (2009).

    Article  ADS  Google Scholar 

  10. M. Asorey, A. Ibort, G. Marmo, and F. Ventriglia, Phys. Scr., 90, 074031 (2015).

    Article  ADS  Google Scholar 

  11. V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 34, 204 (2013).

    Google Scholar 

  14. M. A. Man’ko and V. I. Man’ko, Entropy, 17, 2876 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  15. M. A. Man’ko, V. I. Man’ko, and G. Marmo, Nuovo Cimento C, 38, iss. 5, paper 167 (2015).

    ADS  Google Scholar 

  16. M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 38, 211 (2017).

    Article  Google Scholar 

  17. M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 39, 1 (2018).

    Article  Google Scholar 

  18. J. A. Lopez-Saldivar, O. Castaños, M. A. Man’ko, and V. I. Man’ko, Phys. Rev. E, 97, 022128 (2018).

  19. M. A. Man’ko, J. Phys.: Conf. Ser., 1071, 012015 (2018).

    Google Scholar 

  20. J. A. Lopez-Saldivar, O. Castaños, M. A. Man’ko, and V. I. Man’ko, Physica A, 491, 64 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  21. M. A. Man’ko and V. I. Man’ko, Phys. Scr., 93, 084002 (2018).

    Article  ADS  Google Scholar 

  22. M. A. Man’ko and V. I. Man’ko, Entropy, 20, 692 (2018).

    Article  ADS  Google Scholar 

  23. J. A. Lopez-Saldivar, O. Castaños, E. Nahmad-Achar, et al., Entropy, 20, 630 (2018).

    Article  ADS  Google Scholar 

  24. L. Landau, Z. Phys., 45, 430 (1927).

    Article  ADS  Google Scholar 

  25. J. von Neumann, Gött. Nach., 1, 245 (1927).

    Google Scholar 

  26. E. C. G. Sudarshan, J. Russ. Laser Res., 24, 195 (2003).

    Article  Google Scholar 

  27. V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 20, 421 (1999).

    Article  Google Scholar 

  28. V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 21, 305 (2000).

    Article  Google Scholar 

  29. V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 24, 507 (2003).

    Article  Google Scholar 

  30. V. I. Man’ ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Phys. A: Math. Gen., 35, 7137 (2002).

    Article  ADS  Google Scholar 

  31. V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Lett. A, 327, 353 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  32. I. A. Malkin and V. I. Man’ ko, JETP Lett., 2, 230 (1965).

    MathSciNet  Google Scholar 

  33. Yu. B. Rumer and A. I. Fet, Theor. Math. Phys., 103, 1221 (2004).

    Google Scholar 

  34. B. G. Konopelchenko and Yu. B. Rumer, Uspekhi Fiz. Nauk, 129, 339 (1979) [Sov. Phys. Usp, 22, 837 (1979)].

  35. B. G. Konopelchenko, “Nonlinear integrable equations” in: Lecture Notes in Physics, Springer (1987), Vol. 270, p. 371.

  36. M. Kibler, J. Mol. Struct. (Theochem.), 187, 83 (1989).

    Article  Google Scholar 

  37. A. Shatskikh, Black Square: Malevich and the Origin of Suprematism, Yale University Press, New Haven, CT, USA (2012).

    Book  Google Scholar 

  38. V. I. Man’ko, G. Marmo, F. Ventriglia, and P. Vitale, J. Phys. A: Math. Theor., 50, 335302 (2017).

    Article  Google Scholar 

  39. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys., 17, 821 (1976).

    Article  ADS  Google Scholar 

  40. G. Lindblad, Commun. Math. Phys., 48, 119 (1976).

    Article  ADS  Google Scholar 

  41. A. Khrennikov, Probability and Randomness. Quantum versus Classical, World Scientific, Singapore (2016).

    Book  MATH  Google Scholar 

  42. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North Holland, Amsterdam (1982).

    MATH  Google Scholar 

  43. 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.

    Article  ADS  Google Scholar 

Download references

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Authors and Affiliations

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

    Margarita A. Man’ko & Vladimir I. Man’ko

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  1. Margarita A. Man’ko
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  2. Vladimir I. Man’ko
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Correspondence to Margarita A. Man’ko.

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Man’ko, M.A., Man’ko, V.I. Triangle Geometry of Spin States and Nonlinear Superposition of Probabilities Describing These States†. J Russ Laser Res 40, 6–18 (2019). https://doi.org/10.1007/s10946-019-09762-y

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