Wigner function of noisy accelerated two-qubit system

  • M. Y. Abd-Rabbou
  • N. MetwallyEmail author
  • M. M. A. Ahmed
  • A.-S. F. Obada


In this manuscript, the behavior of the Wigner function of accelerated and non-accelerated two-qubit system passing through different noisy channels is discussed. The decoherence of the initial quantum correlation due to the noisy channels and the acceleration process is investigated by means of Wigner function. The negative (positive) behavior of the Wigner function predicts the gain of the quantum (classical) correlations. Based on the upper and lower bounds of the Wigner function, the entangled initial state loses its quantum correlation due to the acceleration process and the strengths of the noisy channels. However, by controlling the distribution angles, the decoherence of these quantum correlations may be suppressed. For the accelerated state, the robustness of the quantum correlations that contained in the initial state appears in different ranges of the distribution angles depending on the noisy type. For the bit-phase flip and the phase flip channels, the robustness of the quantum correlations are shown at any acceleration and large range of distribution angles. However, the fragility of the quantum correlation is depicted for large values for strength of the bit flip channel. Different profiles of the Wigner function are exhibited for the quantum and classical correlations, cup, lune and hemisphere. The amount of quantum correlation is quantified by using the quantum discord, where its maximum/minimum bounds are consistence with that depicted by the Wigner function. It is shown that the degree of withstanding against the decoherence due to the acceleration is depicted for the amplitude damping channel.


Wigner function Qubits Non-inertial frame Noisy channels 



We would like to thank the referees for their important remarks which helped us to improve our manuscript and go deeply through our results.


  1. 1.
    Moya-Cessa, H., Knight, P.: Series representation of quantum-field quasiprobabilities. Phys. Rev. A 48, 2479–2481 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Deleglise, S., Dotsenko, I., Sayrin, C., Bernu, J., Brune, M., Raimond, J.M., Haroche, S.: Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature 455(7212), 510 (2008)ADSCrossRefGoogle Scholar
  3. 3.
    McConnell, R., Zhang, H., Hu, J., Ćuk, S., Vuletić, V.: Entanglement with negative Wigner function of almost 3000 atoms heralded by one photon. Nature 519(7544), 439 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    Mohamed, A.-B.A., Metwally, N.: Nonclassical features of two SC-qubit system interacting with a coherent SC-cavity. Phys. E 102, 1–7 (2018)CrossRefGoogle Scholar
  5. 5.
    Giraud, R., Braun, P., Braun, D.: Classicality of spin states. Phys. Rev. A 78, 4702 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Obada, A.-S.F., Hessian, H.A., Mohamed, A.-B.A., Hashem, M.: Wigner function and phase properties for a two-qubit field system under pure phase noise. J. Russ. Laser Res. 33(4), 369–378 (2012)CrossRefGoogle Scholar
  7. 7.
    Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics: Theory and Application. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  8. 8.
    Klimov, A.B., Romero, J., Guise, H.: Generalized SU(2) covariant Wigner functions and some of their applications. J. Phys. A 50(32), 323001 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Husimi, K.: Some formal properties of the density matrix. Proc. Phys.-Math. Soc 22, 264–314 (1940)zbMATHGoogle Scholar
  10. 10.
    Agarwal, G.S.: State reconstruction for a collection of two-level systems. Phys. Rev. A 57, 671–673 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    Várilly, J.C., Gracia-Bondía, J.: The Moyal representation for spin. Ann. phys. 190(1), 107–148 (1989)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Klimov, A.B., Chumakov, S.M.: On the SU (2) Wigner function dynamics. Revista mexicana defísica 48(4), 317–324 (2002)ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gibbons, K.S., Hoffman, M.J., Wootters, W.K.: Discrete phase space based on finite fields. Phys. Rev. A 70, 062101 (2004)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Reboiro, M., Civitarese, O., Tielas, D.: Use of discrete Wigner functions in the study of decoherence of a system of superconducting flux-qubits. Phys. Scr. 90(7), 074028 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    Ciampini, M.A., Tilma, T., Everitt, M.J., Munro, W.J., Mataloni, P., Nemoto, K., Barbieri, M.: Wigner function reconstruction of experimental three-qubit GHZ and W states (2017). arXiv preprint arXiv:1710.02460
  17. 17.
    Tilma, T., Everitt, M.J., Samson, J.H., Munro, W.J., Nemoto, K.: Wigner functions for arbitrary quantum systems. Phys. Rev. Lett. 117, 180401 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Koczor, B., Zeier, R., Glaser, S.J.: Time evolution of coupled spin systems in a generalized Wigner representation. Ann. Phys. 408, 1–50 (2019)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Franco, R., Penna, V.: Discrete Wigner distribution for two qubits: a characterization of entanglement properties. J. Phys. A Math. Gen. 39, 5907–5919 (2006)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Agarwal, G.S.: Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions. Phys. Rev. A 24, 2889–2896 (1981)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Metwally, N., Rabbou, M.Y., Ahmed, M.M.A., Obada, A.-S.F.: Wigner function of accelerated and non-accelerated Greenberger Horne Zeilinger state (2019). arXiv preprint arXiv:1901.08828
  22. 22.
    Bruschi, D.E., Louko, J., Martín-Martínez, E., Dragan, A., Fuentes, I.: Unruh effect in quantum information beyond the single mode approximation. Phys. Rev. A 82, 042332 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Martín-Martínez, E., Fuentes, I.: Redistribution of particle and antiparticle entanglement in non-inertial frames. Phys. Rev. A 83, 052306 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Metwally, N.: Estimation of teleported and gained parameters in a non-inertial frame. Laser Phys. Lett. 14(4), 045202 (2017)ADSCrossRefGoogle Scholar
  25. 25.
    Salles, A., de Melo, F., Almeida, M.P., Hor-Meyll, M., Walborn, S.P., Souto Ribeiro, P.H., Davidovich, L.: Experimental investigation of the dynamics of entanglement: sudden death, complementarity, and continuous monitoring of the environment. Phys. Rev. A 78, 022322 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    Horst, B., Bartkiewicz, K., Miranowicz, A.: Two-qubit mixed states more entangled than pure states: comparison of the relative entropy of entanglement for a given nonlocality. Phys. Rev. A 87, 042108 (2013)ADSCrossRefGoogle Scholar
  27. 27.
    Mari, A., Eisert, J.: Positive Wigner functions render classical simulation of quantum computation efficient. Phys. Rev. Lett. 109, 230503 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  29. 29.
    Fanchini, F.F., Werlang, T., Brasil, C.A., Arruda, L.G.E., Caldeira, A.O.: Non-Markovian dynamics of quantum discord. Phys. Rev. A 81, 052107 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Ding, B.-F., Wang, X.-Y., Zhao, H.-P.: Quantum and classical correlations for a two-qubit X structure density matrix. Chin. Phys. B 20, 100302 (2011)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematics Department, Faculty of ScienceAl-Azhar UniversityNasr CityEgypt
  2. 2.Mathematics Department, College of ScienceUniversity of BahrainZallaqBahrain
  3. 3.Department of MathematicsAswan UniversityAswanEgypt

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