Advertisement

International Journal of Theoretical Physics

, Volume 58, Issue 6, pp 2054–2067 | Cite as

Unitary and Nonunitary Evolution of Qubit States in Probability Representation of Quantum Mechanics

  • A. S. AvanesovEmail author
  • V. I. Manko
Article

Abstract

Review of the probability representation of qubit states and observables is presented as well as the picture of states of two-level systems in terms of Triada of Malevich’s squares. A new relation of introduced probability parameters is obtained. Also, it is offered a method to visualize the quantum channel’s maps of qubit states. Evolution of the two-level system is considered in terms of Triada of Malevich’s squares in case of Rabi and Demkov models.

Keywords

Open systems Quantum decoherence Tomographic probability representation of quantum mechanics Two-level quantum systems 

Notes

References

  1. 1.
    Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28(6), 1049–1070 (1926)ADSGoogle Scholar
  2. 2.
    Landau, L.D., Lifshitz, E.M.: Mechanics, vol. 3, 2nd edn. Pergamon Press, New York (1969)Google Scholar
  3. 3.
    Landau, L.D.: Das dämpfungsproblem in der Wellenmechanik. Z. Phys. 45, 430–441 (1927)ADSzbMATHGoogle Scholar
  4. 4.
    von Neumann, J.: Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Math.-Phys. Klasse 1927, 245–272 (1927)zbMATHGoogle Scholar
  5. 5.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford UK (1982)Google Scholar
  6. 6.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)ADSzbMATHGoogle Scholar
  7. 7.
    Husimi, K.: Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 22(4), 264–314 (1940)zbMATHGoogle Scholar
  8. 8.
    Kano, Y.: J. Math. Phys. 6, 12 (1965)Google Scholar
  9. 9.
    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131(6), 2766–2788 (1963)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10(7), 277–279 (1963)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Wolfgang, P.: Schleich, Quantum Optics in Phase Space. Wiley-VCH, New York (2001)Google Scholar
  12. 12.
    Zachos, C.K., Fairlie, D.B., Curtright, T.L.: Quantum Mechanics in Phase Space. World Scientific, Singapore (2005)zbMATHGoogle Scholar
  13. 13.
    Mancini, S., Manko, V.I., Tombesi, P.: Symplectic tomography as classical approach to quantum systems. Phys. Lett. A 213, 1–6 (1996)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Ibort, A., Manko, V.I., Marmo, G., Simoni, A., Ventriglia, F.: An introduction to the tomographic picture of quantum mechanics. Phys. Scr. 79(6), 065013 (2009)ADSzbMATHGoogle Scholar
  15. 15.
    Dodonov, V.V., Manko, V.I.: Positive distribution description for spin states. Phys. Lett. A 229(6), 335–339 (1997)ADSMathSciNetzbMATHGoogle Scholar
  16. 16.
    Manko, V.I., Manko, O.V.: Spin state tomography. J. Exp. Theor. Phys. 85 (3), 430–434 (1997)ADSGoogle Scholar
  17. 17.
    Korennoy, Ya.A., Man’ko, V.I.: Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles. Inter. J. Theor. Phys. 55(11), 4885–4895 (2016)zbMATHGoogle Scholar
  18. 18.
    Man’ko, V.I., Markovich, L.A.: Separability and entanglement in the hilbert space reference frames related through the generic unitary transform for four level system. Inter. J. Theor. Phys. 57(5), 1285–1303 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Manko, V.I., Marmo, G., Ventriglia, F., Vitale, P.: Metric on the space of quantum states from relative entropy. Tomographic reconstruction. J. Phys. A 50(33), 335302 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Chernega, V.N., Manko, O.V., Manko, V.I.: Triangle geometry of the qubit state in the probability representation expressed in terms of the triada of malevich’s squares. J. Russ. Las. Res. 38(2), 141–149 (2017)Google Scholar
  21. 21.
    Chernega, V.N., Manko, O.V., Manko, V.I.: Probability representation of quantum observables and quantum states. J. Russ. Las. Res. 38(4), 324–333 (2017)Google Scholar
  22. 22.
    Chernega, V.N., Manko, O.V., Manko, V.I.: Triangle geometry for qutrit states in the probability representation. J. Russ. Las. Res. 38(5), 416–425 (2017)Google Scholar
  23. 23.
    Chernega, V.N., Manko, O.V., Manko, V.I.: Quantum suprematism picture of Triada of Malevich’s squares for spin states and the parametric oscillator evolution in the probability representation of quantum mechanics. J. Phys.: Conf. Ser. 1071(1), 012008 (2017)Google Scholar
  24. 24.
    López-Saldívar, J.A., Castaños, O., Nahmad-Achar, E., López-Peña, R., Man’ko, M.A., Man’ko, V.I.: Geometry and entanglement of Two-Qubit states in the quantum probabilistic representation. Entropy 20(9), 630–646 (2018)ADSMathSciNetGoogle Scholar
  25. 25.
    Manko, M.A., Manko, V.I.: New entropic inequalities and hidden correlations in quantum suprematism picture of qudit states. Entropy 20(9), 692–708 (2018)ADSMathSciNetGoogle Scholar
  26. 26.
    Shatskikh, A., Square, B.: Malevich and the Origin of Suprematism. Yale University Press, New Haven (2012)Google Scholar
  27. 27.
    Khrennikov, A., Alodjants, A.: Classical (Local and contextual) probability model for Bohm–Bell type experiments. No-Signaling as Independence of Random Variables. Entropy 21, 157 (2019)ADSGoogle Scholar
  28. 28.
    Khrennikov, A.: Prequantum classical statistical field theory: Schrödinger dynamics of entangled systems as a classical stochastic process. Found. Phys. 41, 317–329 (2011)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Mancini, S., Manko, O.V., Manko, V.I., Tombesi, P.: The Pauli equation for probability distributions. J. Phys. A: Math. Gen. 34, 3461 (2001)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Rabi, I.I.: Space quantization in a gyrating magnetic field. Phys. Rev. 51, 652–654 (1937)ADSzbMATHGoogle Scholar
  31. 31.
    Demkov, Y.N.: Charge transfer at small resonance defects. Sov. Phys. JETP 18 (1), 138–142 (1964)MathSciNetGoogle Scholar
  32. 32.
    Zlatanov, K.N., Vasilev, G.S., Ivanov, P.A., Vitanov, N.V.: Exact solution of the Bloch equations for the nonresonant exponential model in the presence of dephasing. Phys. Rev. A 92, 043404 (2015)ADSGoogle Scholar
  33. 33.
    Bloch, F.: Nuclear Induction. Phys. Rev. 70, 460–474 (1946)ADSGoogle Scholar
  34. 34.
    Sudarshan, E.C.G., Mathews, P.M., Rau, Jayaseetha: Stochastic dynamics of Quantum-Mechanical system. Phys. Rev. 121(3), 920–924 (1961)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Stinespring, W.F.: Positive Functions on C*-algebras. Proc. Amer. Math. Soc., 211–216 (1955)Google Scholar
  36. 36.
    Amosov, G.G., Mancini, S., Manko, V.I.: Tomographic Portrait of Quantum Channels. Rep. Math. Phys. 81, 165 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kraus, K.: States, Efects and Operations: Fundamental Notions of Quantum Theory. Springer, Berlin (1983)Google Scholar
  38. 38.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  39. 39.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)ADSMathSciNetzbMATHGoogle Scholar
  40. 40.
    Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821 (1976)ADSMathSciNetGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of General and Applied PhysicsMoscow Institute of Physics and Technology (State University) Institutskii per. 9DolgoprudnyiRussia
  2. 2.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Department of PhysicsTomsk State UniversityTomskRussia

Personalised recommendations