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Entanglement Dynamics and Symmetry Breaking in Symmetric four Qubits GHZ- and W-type States Coupled to Classical Random Telegraph Noise in Mixed Environments

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Abstract

This paper investigates the time behavior of entanglement (quantified by negativity) and the symmetry breaking in a symmetric four-qubit system (initialized either in the GHZ-type or W-type states), interacting with a Markov and non-Markov random telegraph noise (RTN). Two different qubit-noise coupling configurations, namely C1 and C2 are investigated. In the first one (C1) it is assumed that, three of the qubits interact with the noise in a common environment (CE) and the remaining qubit in its own local environment while in the second one (C2) it is rather assumed that two of them interact in a CE and the rest also in their own CE. Using the entanglement between the different non-equivalent bipartitions of the qubits (obtained by dividing the system into two arbitrary blocks) as probe, it is demonstrated that the breaking of symmetry in the initialized state of the qubits occurs due to bipartitions of system in which none of the qubit(s) in the right block/partition share the same environment with the remaining qubit(s) in the left block/partition. On the other hand, it is shown that the CE induces an indirect interaction between the qubits which plays a constructive role in reducing the decay rate of entanglement. As a matter of fact, it is shown that the higher the number of qubit interacting in a CE, the more protected the entanglement of the overall system, demonstrating that the C1 scheme is more efficient for shield the system from the detrimental impacts induced by the RTN than the C2 one. Finally, it is shown that strong qubit-environment coupling strength also favors the exchanges of information between the qubits and the external environment.

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References

  1. Benenti, G.S.G., Casati, G.: Principles of Quantum Computation and Information: Basic Tools and Special Topics. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  2. Amico, L., Fazio, R., Osterloh, A., Vedral, V.: . Rev. Mod. Phys. 80, 517 (2008)

    ADS  Google Scholar 

  3. Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  4. Bennett, C.H.: . Phys. Rev. Lett. 68, 3121 (1992)

    ADS  MathSciNet  Google Scholar 

  5. Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.: . Phys. Rev. Lett. 70, 1895 (1993)

    ADS  MathSciNet  Google Scholar 

  6. Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: . Rev. Mod. Phys. 74, 145 (2002)

    ADS  Google Scholar 

  7. Bennett, C., Brenstein, H., Popescu, S., Schumacher, B.: . Phys. Rev. A 53, 2046 (1996)

    ADS  Google Scholar 

  8. Leggio, B., Franco, R.L., Soares-Pinto, D.O., Horodecki, P., Compagno, G.: . Phys. Rev. A 92, 032311 (2015)

    ADS  Google Scholar 

  9. DArrigo, A., et al.: . Int. J. Quant. Inf. 12, 1461005 (2014)

    Google Scholar 

  10. Xu, J.-S., Sun, K., Li, C.-F., et al.: . Nat. Commun. 4, 2851 (2013)

    ADS  Google Scholar 

  11. Orieux, A., D’Arrigo, A., Ferranti, G., Franco, R.L., et al.: . Sci. Rep. 5, 8575 (2015)

    Google Scholar 

  12. Mu, Q., Zhao, X.: . Int. J. Theor. Phys. 55, 2711 (2016)

    Google Scholar 

  13. Guo, J., Zhang, X.: . Sci. Rep. 6, 32634 (2016)

    ADS  Google Scholar 

  14. Cui, W., Xi, Z., Pan, Y.: . J. Phys. A: Math. Theor. 42, 155303 (2009)

    ADS  Google Scholar 

  15. Szańkowski, P., Trippenbach, M., Cywiński, L., Band, Y.: . Quantum Inf. Process 14, 3367 (2015)

    ADS  MathSciNet  Google Scholar 

  16. Mortezapour, A., Borji, M.A., Franco, R.: . Laser Phys. Lett. 14, 055201 (2017)

    ADS  Google Scholar 

  17. González-Gutiérrez, C., Román-Ancheyta, R., Espitia, D., Franco, R.: . Int. J. Quantum Inform. 14, 1650031 (2016)

    ADS  Google Scholar 

  18. Espoukeh, P., Rahimi, R., Salimi, S., Pedram, P.: . Int. J. Quantum Inform. 13, 1550044 (2015)

    ADS  Google Scholar 

  19. Espoukeh, P., Pedram, P.: . Int. J. Quantum Inform. 13, 1550004 (2015)

    ADS  Google Scholar 

  20. Ali, M.: . Chin. Phys. B 24, 120303 (2015)

    Google Scholar 

  21. Ali, M.: . Open Systems and Information Dynamics 21, 1450008 (2014)

    MathSciNet  Google Scholar 

  22. Siomau, M.: . Phys. Rev. A 82, 062327 (2010)

    ADS  Google Scholar 

  23. Majtey, A., et al.: . Int. J. Quant. Inf. 8, 505 (2010)

    Google Scholar 

  24. Borras, A., et al.: . Phys. Rev. A 79, 022108 (2009)

    ADS  Google Scholar 

  25. Aolita, L., Chaves, R., Cavalcanti, D., Acín, A., Davidovich, L.: . Phys. Rev. Lett. 100, 080501 (2008)

    ADS  Google Scholar 

  26. Man, Z.-X., Xia, Y.-J., An, N.: . New J. Phys. 12, 033020 (2010)

    ADS  MathSciNet  Google Scholar 

  27. Feng, L.-J., Zhang, Y.-J., Zhang, L., Xia, Y.-J.: . Chin. Phys. B 24, 110305 (2015)

    ADS  Google Scholar 

  28. Ma, X., Wang, A., Yang, X., You, H.: . J. Phys. A: Math. Gen. 38, 2761 (2005)

    ADS  Google Scholar 

  29. Man, Z.-X., Xia, Y.-J., Franco, R.: . Sci. Rep. 5, 13843 (2015)

    ADS  Google Scholar 

  30. Bellomo, B., Franco, R., Andersson, E., Cresser, J., Compagno, G.: . Phys. Scr. T 147, 014004 (2012)

    ADS  Google Scholar 

  31. D’Arrigo, A., Franco, R., Benenti, G., Paladino, E., Falci, G.: . Ann. Phys. 35, 0211 (2014)

    ADS  Google Scholar 

  32. Helm, J., Strunz, W.T.: . Phys. Rev. A 80, 042108 (2009)

    ADS  Google Scholar 

  33. Helm, J., Strunz, W.T., Rietzler, S., Wuringer, L.E.: . Phys. Rev. A 83, 042103 (2011)

    ADS  Google Scholar 

  34. Witzel, W.M., Young, K., Sarma, S.D.: . Phys. Rev. B 90, 115431 (2014)

    ADS  Google Scholar 

  35. Strunz, W.T., Diosi, L., Gisin, N.: . Phys. Rev. Lett. 82, 1801 (1999)

    ADS  MathSciNet  Google Scholar 

  36. Stockburger, J., Grabert, H.: . Phys. Rev. Lett. 88, 170407 (2002)

    ADS  Google Scholar 

  37. Crow, D., Joynt, R.: . Phys. Rev. A 89, 042123 (2014)

    ADS  Google Scholar 

  38. Rossi, M., Benedetti, C., Matteo, G.: . Int. J. Quantum Inform. 12, 1560003 (2014)

    ADS  Google Scholar 

  39. Benedetti, C., Buscemi, F., Bordone, P., Paris, M.G.A.: . Phys. Rev. A 87, 052328 (2013)

    ADS  Google Scholar 

  40. Benedetti, C., Paris, M.G.A., Buscemi, F., Bordone, P.: Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise. Proceedings of the 22nd International Conference on Noise and Fluctuations (ICNF) 323, 6578952 (2013). https://doi.org/10.1109/ICNF

    Article  Google Scholar 

  41. Benedetti, C., Buscemi, F., Bordone, P., Paris, M.G.A.: . Int. J. Quant. Inf. 10, 1241005 (2012)

    Google Scholar 

  42. Javed, M., Khan, S., Ullah, S.: . J. Russ. Laser Res. 37, 562 (2016)

    Google Scholar 

  43. Bordone, P., Buscemi, F., Benedetti, C.: . Fluct. Noise Lett. 11, 1242003 (2012)

    Google Scholar 

  44. Shamirzaie, M., Khan, S., Ullah, S.: . Fluct. Noise Lett. 17, 1850023 (2018)

    ADS  Google Scholar 

  45. Kenfack, L.T., Tchoffo, M., Jipdi, M.N., Fuoukeng, J.C., Fai, L.C.: . J. Phys. Commun. 2, 055011 (2018)

    Google Scholar 

  46. Buscemi, F., Bordone, P.: . Phys. Rev. A 87, 042310 (2013)

    ADS  Google Scholar 

  47. Kenfack, L.T., Tchoffo, M., Fai, L.C.: . Eur. Phys. J. Plus. 132, 91 (2017)

    Google Scholar 

  48. Tchoffo, M., Kenfack, L.T., Fouokeng, G.C., Fai, L.C.: . Eur. Phys. J. Plus. 131, 380 (2016)

    Google Scholar 

  49. Lionel, T.K., Martin, T., Collince, F.G., Fai, L.C.: . Int. J. Mod. Phys. B 30, 1750046 (2016)

    Google Scholar 

  50. Kenfack, L.T., Tchoffo, M., Fouokeng, G.C, Fai, L.C.: . Quantum Inf. Process 17, 76 (2018)

    ADS  Google Scholar 

  51. Kenfack, L.T., Tchoffo, L.C., Fai, G.C.: . Physica B 511, 123 (2017)

    ADS  Google Scholar 

  52. Kenfack, L.T., Tchoffo, M., Fouokeng, G.C., Fai, L.C.: . Int. J. Quant. Inf. 15, 1750038 (2017)

    Google Scholar 

  53. Rossi, M., Paris, M.: . J. Chem. Phys. 144, 024113 (2016)

    ADS  Google Scholar 

  54. Paris, M.: . Physica A. 413, 256 (2014)

    ADS  MathSciNet  Google Scholar 

  55. Hamadou-Ibrahim, A., Plastino, A., Zander, C.: . J. Phys. A: Math. Theor. 43, 055305 (2010)

    ADS  Google Scholar 

  56. Kenfack, L.T., Tchoffo, M., Fai, L.C.: . Phys. Lett. A 382, 2818 (2018)

    Google Scholar 

  57. Man, Z.-X., Tavakoli, A., Brask, J., Xia, Y.-J.: . Phys. Scr. 94, 075101 (2019)

    ADS  Google Scholar 

  58. Man, Z.-X., Zhang, Q., Xia, Y.-J.: . Quant. Inf. Process. 18, 157 (2019)

    ADS  Google Scholar 

Download references

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

  1. Research Unit of Condensed Matter, Electronic and Signal Processing, Department of Physics, Dschang school of Sciences and Technology, University of Dschang, PO Box: 67, Dschang, Cameroon

    Lionel Tenemeza Kenfack, Martin Tchoffo & Lukong Cornelius Fai

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  1. Lionel Tenemeza Kenfack
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  2. Martin Tchoffo
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  3. Lukong Cornelius Fai
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Correspondence to Lionel Tenemeza Kenfack.

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Appendices

Appendix

Here we write explicitly the forms of the system’s density matrices obtained after the evaluation of the expectation value over all the possible temporal sequences of the noise phases, (5). Note that more detailed about the calculation processes can be found in Ref. [45] and references therein.

Appendix A: The case of GHZ-type states

As a result, we have, for the first qubit-noise configuration (C1),

$$ { \rho }_{{ GHZ}}^{{ C1}}(t) =\left[ \begin{array}{cccccccccccccccc} \varphi(t) & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & \phi(t) & \phi(t) & {{ 0}} & {{ 0}} & \phi(t) & \phi(t) & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & \omega(t) \\ {{ 0}} & \chi(t) & \psi(t) & {{ 0}} & \psi(t) & {{ 0}} & {{ 0}} & \psi(t) & \psi(t) & {{ 0}} & {{ 0}} & \psi(t) & {{ 0}} & \psi(t) & \upsilon(t) & {{ 0}} \\ {{ 0}} & \psi(t) & \kappa(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & {{ 0}} & \alpha(t) & \psi(t) & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \lambda(t) & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & \delta(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & \psi(t) & \alpha(t) & {{ 0}} & \kappa(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & {{ 0}} & \alpha(t) & \psi(t) & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \lambda(t) & \delta(t) & {{ 0}} & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & \delta(t) & {{ 0}} & {{ 0}} & \phi(t) \\ \phi(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \delta(t) & \lambda(t) & {{ 0}} & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & \delta(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & \psi(t) & \alpha(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \kappa(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & {{ 0}} & \alpha(t) & \psi(t) & {{ 0}} \\ {{ 0}} & \psi(t) & \alpha(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \kappa(t) & {{ 0}} & {{ 0}} & \kappa(t) & {{ 0}} & \kappa(t) & \psi(t) & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & {{ 0}} & \lambda(t) & \delta(t) & {{ 0}} & \delta(t) & {{ 0}} & {{ 0}} & \phi(t) \\ \phi(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & {{ 0}} & \delta(t) & \lambda(t) & {{ 0}} & \delta(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & \psi(t) & \alpha(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \kappa(t) & {{ 0}} & \alpha(t) & \psi(t) & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & {{ 0}} & \delta(t) & \delta(t) & {{ 0}} & \lambda(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & \psi(t) & \alpha(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & {{ 0}} & \kappa(t) & \psi(t) & {{ 0}} \\ {{ 0}} & \upsilon(t) & \psi(t) & {{ 0}} & \psi(t) & {{ 0}} & {{ 0}} & \psi(t) & \psi(t) & {{ 0}} & {{ 0}} & \psi(t) & {{ 0}} & \psi(t) & \chi(t) & {{ 0}} \\ \omega(t) & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & \phi(t) & \phi(t) & {{ 0}} & {{ 0}} & \phi(t) & \phi(t) & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & \varphi(t) \end{array} \right], $$
(10)

with

$$ \begin{array}{@{}rcl@{}} \varphi(t) &=&\frac{1}{16} \left( 1+\frac{3q}{2}\right)+\frac{15q}{64} \beta_{2}(t) \eta_{2}(t) +\frac{3q}{32} \eta_{4}(t) +\frac{q}{64} \eta_{6}(t) \beta_{2}(t), \phi(t)\\&=&\frac{q}{64} \left( -2+2\eta_{4}(t) -\eta_{2}(t) \beta_{2}(t) +\eta_{6}(t) \beta_{2}(t) \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \omega(t) &&=\frac{q}{64} \left( 15\eta_{2}(t) \beta_{2}(t) +6\eta_{4}(t) +10+\eta_{6}(t) \beta_{2}(t) \right), \chi(t) =\frac{1}{16} \left( 1+\frac{3q}{2}\right)\\&&-\frac{15q}{64}\eta_{2}(t) \beta_{2}(t)+\frac{3q}{32} \eta_{4}(t) -\frac{q}{64} \eta_{6}(t) \beta_{2}(t) , \end{array} $$
$$ \begin{array}{@{}rcl@{}} \psi(t) &=&-\frac{q}{64} \left( 2-2\eta_{4}(t) -\eta_{2}(t) \beta_{2}(t) +\eta_{6}(t) \beta_{2}(t) \right), \upsilon(t) \\&=&-\frac{r}{64} \left( 15\eta_{2}(t) \beta_{2}(t) +\eta_{6}(t) \beta_{2}(t) -10-6\eta_{4}(t) \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \kappa(t) &=&\frac{1}{16} \left( 1-\frac{q}{2}\right)+\frac{q}{64} \eta_{2}(t) \beta_{2}(t) -\frac{q}{32} \eta_{4}(t) -\frac{q}{64} \eta_{6}(t) \beta_{2}(t), \alpha(t) \\&=&-\frac{q}{64} \left( -\eta_{2}(t) \beta_{2}(t) +\eta_{6}(t) \beta_{2}(t) -2+2\eta_{4}(t) \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \lambda(t) &=&\frac{1}{16} \left( 1-\frac{q}{2} \right)-\frac{q}{64} \eta_{2}(t) \beta_{2}(t) +\frac{q}{64} \eta_{6}(t) \beta_{2}(t) -\frac{q}{32} \eta_{4}(t) , \delta(t) \\&=&\frac{q}{64} \left( 2-2\eta_{4}(t) +\eta_{6}(t) \beta_{2}(t) -\eta_{2}(t) \beta_{2}(t) \right). \end{array} $$

while for the second configuration (C2) we have

$$ { \rho }_{{ GHZ}}^{{ C2}} \left( { t}\right)=\left[\begin{array}{cccccccccccccccc} \varphi(t) & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & \lambda(t) \\ {{ 0}} & \chi(t) & \omega(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \omega(t) & \omega(t) & {{ 0}} \\ {{ 0}} & \omega(t) & \chi(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \omega(t) & \omega(t) & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \tau(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \kappa(t) & {{ 0}} & {{ 0}} & \upsilon(t) & \upsilon(t) & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \pi(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \alpha(t) & \pi(t) & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & \kappa(t) & \upsilon(t) & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & \upsilon(t) & \kappa(t) & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \pi(t) & \alpha(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \alpha(t) & \alpha(t) & {{ 0}} & {{ 0}} & \alpha(t) & \pi(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & {{ 0}} & \upsilon(t) & \upsilon(t) & {{ 0}} & {{ 0}} & \kappa(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ \phi(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \tau(t) & {{ 0}} & {{ 0}} & \phi(t) \\ {{ 0}} & \omega(t) & \omega(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \chi(t) & \omega(t) & {{ 0}} \\ {{ 0}} & \omega(t) & \omega(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \omega(t) & \chi(t) & {{ 0}} \\ \lambda(t) & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & \varphi(t) \end{array}\right], $$
(11)

with

$$ \begin{array}{@{}rcl@{}} \varphi(t) &=&\frac{1}{16} \left( 1+\frac{3}{2} q\right)+\frac{q}{32} \left( \beta_{4}(t))+\eta_{4}(t))\right)+\frac{q}{4} \eta_{2}(t) \beta_{2}(t) +\frac{q}{32} \eta_{4}(t) \beta_{4}(t), \phi(t) \\&=&\frac{q}{64} \left( -6+2\eta_{4}(t) +2\eta_{4}(t) \beta_{4}(t) +2\beta_{4}(t) \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \lambda(t) &=&\frac{q}{64} \left( 16\eta_{2}(t) \beta_{2}(t) +10+2\eta_{4}(t) +2\eta_{4}(t) \beta_{4}(t) +2\beta_{4}(t) \right), \chi(t) \\&=&\frac{1}{16} \left( 1-\frac{q}{2} \right)-\frac{q}{32} \left( \beta_{4}(t) -\eta_{4}(t) \right)-\frac{q}{32} \eta_{4}(t) \beta_{4}(t), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \omega(t)&&=-\frac{q}{64} \left( -2-2\eta_{4}(t) +2\eta_{4}(t) \beta_{4}(t) +2\beta_{4}(t) \right), \tau(t)=\frac{1}{16} \left( 1+\frac{3q}{2} \right)\\&&\quad+\frac{q}{32} \eta_{4}(t) \beta_{4}(t) +\frac{q}{32} \left( \eta_{4}(t) +\beta_{4}(t) \right)-\frac{r}{4} \eta_{2}(t) \beta_{2}(t), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \mu(t) &&=\frac{q}{64} \left( 10+2\beta_{4}(t) +2\eta_{4}(t) -16\eta_{2}(t) \beta_{2}(t) +2\eta_{4}(t) \beta_{4}(t) \right), \kappa(t)=\frac{1}{16} \left( 1-\frac{q}{2} \right)\\&&\quad-\frac{q}{32} \eta_{4}(t) \beta_{4}(t) +\frac{q}{32} \left( \beta_{4}(t) -\eta_{4}(t) \right), \end{array} $$
$$ \alpha(t) =\frac{q}{64} \left( 2-2\eta_{4}(t) -2\beta_{4}(t) +2\eta_{4}(t) \beta_{4}(t) \right), \upsilon(t) =-\frac{q}{64} \left( -2-2\beta_{4}(t) +2\eta_{4}(t) +2\eta_{4}(t) \beta_{4}(t) \right), $$
$$ \pi(t)=\frac{1}{16} \left( 1-\frac{q}{2} \right)+\frac{q}{32} \eta_{4}(t) \beta_{4}(t) -\frac{q}{32} \left( \eta_{4}(t) +\beta_{4}(t) \right). $$

Appendix B: The case of W-type states

As a result, we have, for the first configuration,

$$ { \rho }_{{ W}}^{{ C1}} \left( {t}\right)=\left[\begin{array}{cccccccccccccccc} \omega(t) & {{ 0}} & {{ 0}} & \sigma(t) & {{ 0}} & \sigma(t) & \nu(t) & {{ 0}} & {{ 0}} & \sigma(t) & \nu(t) & {{ 0}} & \nu(t) & {{ 0}} & {{ 0}} & \omega1(t) \\ {{ 0}} & \psi(t) & \varrho(t) & {{ 0}} & \varrho(t) & {{ 0}} & {{ 0}} & \mu(t) & \varrho(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & \mu(t) & \psi1(t) & {{ 0}} \\ {{ 0}} & \varrho(t) & \phi(t) & {{ 0}} & \zeta(t) & {{ 0}} & {{ 0}} & \chi1(t) & \zeta(t) & {{ 0}} & {{ 0}} & \chi1(t) & {{ 0}} & \chi1(t) & \lambda(t) & {{ 0}} \\ \sigma(t) & {{ 0}} & {{ 0}} & \tau(t) & {{ 0}} & \varepsilon(t) & \varphi1(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & \varphi1(t) & {{ 0}} & \varphi1(t) & {{ 0}} & {{ 0}} & \iota(t) \\ {{ 0}} & \varrho(t) & \zeta(t) & {{ 0}} & \phi(t) & {{ 0}} & {{ 0}} & \phi1(t) & \zeta(t) & {{ 0}} & {{ 0}} & \phi1(t) & {{ 0}} & \phi1(t) & \kappa(t) & {{ 0}} \\ \sigma(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & {{ 0}} & \tau(t) & \upsilon1(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & \upsilon1(t) & {{ 0}} & \upsilon1(t) & {{ 0}} & {{ 0}} & \theta(t) \\ \nu(t) & {{ 0}} & {{ 0}} & \varphi1(t) & {{ 0}} & \upsilon1(t) & \varsigma(t) & {{ 0}} & {{ 0}} & \upsilon1(t) & \alpha(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \varpi(t) \\ {{ 0}} & \mu(t) & \chi1(t) & {{ 0}} & \phi1(t) & {{ 0}} & {{ 0}} & \upsilon(t) & \chi1(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \delta(t) & \pi(t) & {{ 0}} \\ {{ 0}} & \varrho(t) & \zeta(t) & {{ 0}} & \zeta(t) & {{ 0}} & {{ 0}} & \chi1(t) & \phi(t) & {{ 0}} & {{ 0}} & \phi1(t) & {{ 0}} & \phi1(t) & \kappa(t) & {{ 0}} \\ \sigma(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & {{ 0}} & \varepsilon(t) & \upsilon1(t) & {{ 0}} & {{ 0}} & \tau(t) & \upsilon1(t) & {{ 0}} & \upsilon1(t) & {{ 0}} & {{ 0}} & \theta(t) \\ \nu(t) & {{ 0}} & {{ 0}} & \varphi1(t) & {{ 0}} & \upsilon1(t) & \alpha(t) & {{ 0}} & {{ 0}} & \upsilon1(t) & \varsigma(t) & {{ 0}} & \alpha(t) & {{ 0}} & {{ 0}} & \varpi(t) \\ {{ 0}} & \mu(t) & \chi1(t) & {{ 0}} & \phi1(t) & {{ 0}} & {{ 0}} & \delta(t) & \phi1(t) & {{ 0}} & {{ 0}} & \upsilon(t) & {{ 0}} & \delta(t) & \pi(t) & {{ 0}} \\ \nu(t) & {{ 0}} & {{ 0}} & \varphi1(t) & {{ 0}} & \upsilon1(t) & \alpha(t) & {{ 0}} & {{ 0}} & \upsilon1(t) & \alpha(t) & {{ 0}} & \varsigma(t) & {{ 0}} & {{ 0}} & \xi(t) \\ {{ 0}} & \mu(t) & \chi1(t) & {{ 0}} & \phi1(t) & {{ 0}} & {{ 0}} & \delta(t) & \phi1(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \upsilon(t) & \pi(t) & {{ 0}} \\ {{ 0}} & \psi1(t) & \lambda(t) & {{ 0}} & \kappa(t) & {{ 0}} & {{ 0}} & \pi(t) & \kappa(t) & {{ 0}} & {{ 0}} & \delta(t) & {{ 0}} & \pi(t) & \chi(t) & {{ 0}} \\ \omega1(t) & {{ 0}} & {{ 0}} & \iota(t) & {{ 0}} & \theta(t) & \varpi(t) & {{ 0}} & {{ 0}} & \theta(t) & \varpi(t) & {{ 0}} & \xi(t) & {{ 0}} & {{ 0}} & \varphi(t) \end{array}\right] $$
(12)

with

$$\omega(t)=\frac{1}{16} \left( 1+\frac{3q}{4} \right)+\frac{q}{32} \left( -\frac{5}{4}\eta_{6} \beta_{2}(t)-\frac{5}{4}\eta_{2}(t) \beta_{2}(t)-\eta_{6} -\frac{3}{2} \eta_{4}(t) -3\eta_{4}(t) \beta_{2}(t) +3\eta_{2}(t) +\beta_{2}(t) \right)$$
$$\psi(t)=\frac{1}{16} \left( 1+\frac{3q}{4} \right)-\frac{q}{32} \left( -\frac{5}{4}\eta_{6} \beta_{2}(t)-\frac{5}{4}\eta_{2}(t) \beta_{2}(t) +\eta_{6} +\frac{3}{2} \eta_{4}(t) -3\eta_{4}(t) \beta_{2}(t) -3\eta_{2}(t) +\beta_{2}(t) \right)$$
$$\chi(t)=\frac{1}{16} \left( 1+\frac{3q}{4} \right)+\frac{q}{32} \left( \frac{5}{4}\eta_{6} \beta_{2}(t)+\frac{5}{4}\eta_{2}(t) \beta_{2}(t)+\eta_{6} -\frac{3}{2} \eta_{4}(t) -3\eta_{4}(t) \beta_{2}(t) -3\eta_{2}(t) +\beta_{2}(t) \right)$$
$$\varphi(t)=\frac{1}{16} \left( 1+\frac{3q}{4} \right)+\frac{q}{32} \left( -\frac{5}{4}\eta_{6} \beta_{2}(t) -\frac{5}{4}\eta_{2}(t) \beta_{2}(t)+\eta_{6} -\frac{3}{2} \eta_{4}(t) +3\eta_{4}(t) \beta_{2}(t) -3\eta_{2}(t) -\beta_{2}(t) \right)$$
$$\phi(t)=\frac{1}{16} \left( 1-\frac{q}{4} \right)+\frac{q}{32} \left( \frac{5}{4}\eta_{6} \beta_{2}(t) +\frac{3}{4}\eta_{2}(t) \beta_{2}(t)+\eta_{6} +\frac{1}{2} \eta_{4}(t) +\eta_{4}(t) \beta_{2}(t) +\eta_{2}(t) +\beta_{2}(t) \right)$$
$$\upsilon(t) =\frac{1}{16} \left( 1-\frac{q}{4} \right)+\frac{q}{32} \left( \frac{5}{4}\eta_{6} \beta_{2}(t) +\frac{3}{4}\eta_{2}(t) \beta_{2}(t) -\eta_{6} +\frac{1}{2} \eta_{4}(t) -\eta_{4}(t) \beta_{2}(t) -\eta_{2}(t) -\beta_{2}(t) \right)$$
$$\tau(t)=\frac{1}{16} \left( 1-\frac{q}{4} \right)+\frac{q}{32} \left( -\frac{5}{4}\eta_{6} \beta_{2}(t) -\frac{3}{4}\eta_{2}(t) \beta_{2}(t)+\eta_{6} +\frac{1}{2} \eta_{4}(t) -\eta_{4}(t) \beta_{2}(t) +\eta_{2}(t) -\beta_{2}(t) \right)$$
$$\varsigma(t)=\frac{1}{16} \left( 1-\frac{q}{4} \right)+\frac{q}{32} \left( -\frac{5}{4}\eta_{6} \beta_{2}(t) -\frac{3}{4}\eta_{2}(t) \beta_{2}(t)-\eta_{6} +\frac{1}{2} \eta_{4}(t) +\eta_{4}(t) \beta_{2}(t) -\eta_{2}(t) +\beta_{2}(t) \right)$$
$$\omega1(t)=\frac{q}{128} \left( 3\eta_{2}(t) \beta_{2}(t) -3\eta_{6} \beta_{2}(t) +6\eta_{4}(t) -6\right),\psi1(t) =\frac{q}{128} \left( -3\eta_{2}(t) \beta_{2}(t) +3\eta_{6} \beta_{2}(t) -6+6\eta_{4}(t) \right),$$
$$\chi1(t)=\frac{q}{128} \left( -3\eta_{2}(t) \beta_{2}(t) +3\eta_{6} \beta_{2}(t) +2-2\eta_{4}(t) \right),\varphi1(t) =\frac{q}{128} \left( 3\eta_{2}(t) \beta_{2}(t) -3\eta_{6} \beta_{2}(t) +2-2\eta_{4}(t) \right),$$
$$\phi1(t)=\frac{q}{128} \left( -3\eta_{2}(t) \beta_{2}(t) +3\eta_{6} \beta_{2}(t) +2-2\eta_{4}(t) \right),\upsilon1(t) =\frac{q}{128} \left( 3\eta_{2}(t) \beta_{2}(t) +2-2\eta_{4}(t) -3\eta_{6} \beta_{2}(t) \right),$$
$$\delta(t)=\frac{q}{128} \left( 3\eta_{2}(t) \beta_{2}(t) +6+2\eta_{4}(t) -4\eta_{4}(t) \beta_{2}(t) -4\eta_{2}(t) -4\eta_{6} -4\beta_{2}(t) +5\eta_{6} \beta_{2}(t) \right),$$
$$\alpha(t)=\frac{q}{128} \left( -3\eta_{2}(t) \beta_{2}(t) +6+2\eta_{4}(t) +4\eta_{4}(t) \beta_{2}(t) -4\eta_{2}(t) -4\eta_{6} +4\beta_{2}(t) -5\eta_{6} \beta_{2}(t) \right)$$
$$\mu(t)=\frac{q}{128} \left( -5\eta_{2}(t) \beta_{2}(t) +5\eta_{6} \beta_{2}(t) +2-2\eta_{4}(t) -4\beta_{2}(t) +4\eta_{2}(t) +4\eta_{4}(t) \beta_{2}(t) -4\eta_{6} \right)$$
$$\lambda(t)=\frac{q}{128} \left( -4\eta_{2}(t) +4\beta_{2}(t) +4\eta_{6} -5\eta_{2}(t) \beta_{2}(t) +5\eta_{6} \beta_{2}(t) +2-2\eta_{4}(t) -4\eta_{4}(t) \beta_{2}(t) \right),$$
$$\kappa(t)=\frac{q}{128} \left( 4\eta_{6} -5\eta_{2}(t) \beta_{2}(t) +5\eta_{6} \beta_{2}(t) + 4\beta_{2}(t) +2\!-2\eta_{4}(t) \!-4\eta_{2}(t) -4\eta_{4}(t) \beta_{2}(t) \right)$$
$$\iota(t)=\frac{q}{128} \left( 4\eta_{6} +5\eta_{2}(t) \beta_{2}(t) -5\eta_{6} \beta_{2}(t) +2\!-2\eta_{4}(t) \!-4\beta_{2}(t) + 4\eta_{4}(t) \beta_{2}(t) \!-4\eta_{2}(t) \right),$$
$$\theta(t) =\frac{q}{128} \left( 2-2\eta_{4}(t) -4\eta_{2}(t) +5\eta_{2}(t) \beta_{2}(t) \!+4\eta_{4}(t) \beta_{2}(t) + 4\eta_{6} -4\beta_{2}(t) -5\eta_{6} \beta_{2}(t) \right)$$
$$\zeta(t)=\frac{q}{128} \left( 4\beta_{2}(t) +4\eta_{6} +4\eta_{2}(t) +3\eta_{2}(t) \beta_{2}(t) +5\eta_{6} \beta_{2}(t) + 6 + 2\eta_{4}(t) + 4\eta_{4}(t) \beta_{2}(t) \right),$$
$$\varepsilon(t)=\frac{q}{128} \left( 4\eta_{6} +4\eta_{2}(t) -3\eta_{2}(t) \beta_{2}(t) -5\eta_{6} \beta_{2}(t) +6 + 2\eta_{4}(t) \!-4\eta_{4}(t) \beta_{2}(t) - 4\beta_{2}(t) \right)$$
$$\nu(t)=\frac{q}{128} \left( 5\eta_{2}(t) \beta_{2}(t) + 4\beta_{2}(t) -4\eta_{6} -4\eta_{4}(t) \beta_{2}(t) -5\eta_{6} \beta_{2}(t) + 2-2\eta_{4}(t) + 4\eta_{2}(t) \right),$$
$$\sigma(t)=\frac{q}{128} \left( 8\eta_{2}(t) -8\eta_{4}(t) \beta_{2}(t) -5\eta_{2}(t) \beta_{2}(t) +2\eta_{4}(t) -3\eta_{6} \beta_{2}(t) +6\right),$$
$$\varrho(t)=\frac{q}{128} \left( 5\eta_{2}(t) \beta_{2}(t) +3\eta_{6} \beta_{2}(t) +6+8\eta_{2}(t) +2\eta_{4}(t) +8\eta_{4}(t) \beta_{2}(t) \right),$$
$$\varpi(t)=\frac{q}{128} \left( -5\eta_{2}(t) \beta_{2}(t) -8\eta_{2}(t) +6+2\eta_{4}(t) +8\eta_{4}(t) \beta_{2}(t) -3\eta_{6} \beta_{2}(t) \right)$$
$$\pi(t)=\frac{q}{128} \left( 6+2\eta_{4}(t) +5\eta_{2}(t) \beta_{2}(t) -8\eta_{2}(t) -8\eta_{4}(t) \beta_{2}(t) +3\eta_{6} \beta_{2}(t) \right),$$
$$\xi(t)=\frac{q}{128} \left( 6+2\eta_{4}(t) -8\eta_{2}(t) -5\eta_{2}(t) \beta_{2}(t) -3\eta_{6} \beta_{2}(t) +8\eta_{4}(t) \beta_{2}(t) \right),$$

while for the second configuration,

$$ { \rho }_{{ W}}^{{ C2}} \left( { t}\right)=\left[\begin{array}{cccccccccccccccc} {\omega } & {{ 0}} & {{ 0}} & \psi(t) & {{ 0}} & \nu(t) & \nu(t) & {{ 0}} & {{ 0}} & \nu(t) & \nu(t) & {{ 0}} & \varpi(t) & {{ 0}} & {{ 0}} & \varphi1(t) \\ {{ 0}} & \upsilon(t) & \delta(t) & {{ 0}} & \iota(t) & {{ 0}} & {{ 0}} & \theta(t) & \iota(t) & {{ 0}} & {{ 0}} & \theta(t) & {{ 0}} & \upsilon1(t) & \upsilon1(t) & {{ 0}} \\ {{ 0}} & \delta(t) & \upsilon(t) & {{ 0}} & \iota(t) & {{ 0}} & {{ 0}} & \theta(t) & \iota(t) & {{ 0}} & {{ 0}} & \theta(t) & {{ 0}} & \upsilon1(t) & \upsilon1(t) & {{ 0}} \\ \psi(t) & {{ 0}} & {{ 0}} & \chi(t) & {{ 0}} & \mu(t) & \mu(t) & {{ 0}} & {{ 0}} & \mu(t) & \mu(t) & {{ 0}} & \chi1(t) & {{ 0}} & {{ 0}} & \xi(t) \\ {{ 0}} & \iota(t) & \iota(t) & {{ 0}} & \tau(t) & {{ 0}} & {{ 0}} & \phi1 & \omega1 & {{ 0}} & {{ 0}} & \phi1(t) & {{ 0}} & \zeta(t) & \zeta(t) & {{ 0}} \\ \nu(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & \varrho(t) & \tau1(t) & {{ 0}} & {{ 0}} & \tau1(t) & \tau1(t) & {{ 0}} & \lambda(t) & {{ 0}} & {{ 0}} & \kappa(t) \\ \nu(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & \tau1(t) & \varrho(t) & {{ 0}} & {{ 0}} & \tau1(t) & \tau1(t) & {{ 0}} & \lambda(t) & {{ 0}} & {{ 0}} & \kappa(t) \\ {{ 0}} & \theta(t) & \theta(t) & {{ 0}} & \phi1(t) & {{ 0}} & {{ 0}} & \varsigma(t) & \phi1(t) & {{ 0}} & {{ 0}} & \psi1(t) & {{ 0}} & \varepsilon(t) & \varepsilon(t) & {{ 0}} \\ {{ 0}} & \iota(t) & \iota(t) & {{ 0}} & \omega1(t) & {{ 0}} & {{ 0}} & \phi1(t) & \tau(t) & {{ 0}} & {{ 0}} & \phi1(t) & {{ 0}} & \zeta(t) & \zeta(t) & {{ 0}} \\ \nu(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & \tau1(t) & \tau1(t) & {{ 0}} & {{ 0}} & \varrho(t) & \tau1(t) & {{ 0}} & \lambda(t) & {{ 0}} & {{ 0}} & \kappa(t) \\ \nu(t) & {{ 0}} & {{ 0}} & \mu(t) & {{ 0}} & \tau1(t) & \tau1(t) & {{ 0}} & {{ 0}} & \tau1(t) & \varrho(t) & {{ 0}} & \lambda(t) & {{ 0}} & {{ 0}} & \kappa(t) \\ {{ 0}} & \theta(t) & \theta(t) & {{ 0}} & \phi1(t) & {{ 0}} & {{ 0}} & \psi1(t) & \phi1(t) & {{ 0}} & {{ 0}} & \varsigma(t) & {{ 0}} & \varepsilon(t) & \varepsilon(t) & {{ 0}} \\ \varpi(t) & {{ 0}} & {{ 0}} & \chi1(t) & {{ 0}} & \lambda(t) & \lambda(t) & {{ 0}} & {{ 0}} & \lambda(t) & \lambda(t) & {{ 0}} & \varphi(t) & {{ 0}} & {{ 0}} & \pi(t) \\ {{ 0}} & \upsilon1(t) & \upsilon1(t) & {{ 0}} & \zeta(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & \zeta(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & {{ 0}} & \sigma(t) & \alpha(t) & {{ 0}} \\ {{ 0}} & \upsilon1(t) & \upsilon1(t) & {{ 0}} & \zeta(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & \zeta(t) & {{ 0}} & {{ 0}} & \varepsilon(t) & {{ 0}} & \alpha(t) & \sigma(t) & {{ 0}} \\ \varphi1(t) & {{ 0}} & {{ 0}} & \xi(t) & {{ 0}} & \kappa(t) & \kappa(t) & {{ 0}} & {{ 0}} & \kappa(t) & \kappa(t) & {{ 0}} & \pi(t) & {{ 0}} & {{ 0}} & \phi(t) \end{array}\right] $$
(13)

with

$$\delta(t)=\frac{q}{16} \left( \frac{1}{2} \beta_{4}(t) \left( \eta_{4}(t) +1\right)+\eta_{2}(t) \left( 1+\beta_{4}(t) \right)+1\right),\alpha(t)=\frac{q}{16} \left( \frac{1}{2} \beta_{4}(t) \left( \eta_{4}(t) +1\right)-\eta_{2}(t) \left( \beta_{4}(t) +1\right)+1\right)$$
$$\omega1(t)=\frac{q}{16} \left( \frac{1}{2} \eta_{4}(t) \left( \beta_{4}(t) +1\right)+\beta_{2}(t) \left( \eta_{4}(t) +1\right)+1\right),\psi1(t) =\frac{q}{16} \left( \frac{1}{2} \eta_{4}(t) \left( \beta_{4}(t) +1\right)-\beta_{2}(t) \left( \eta_{4}(t) +1\right)+1\right),$$
$$ \begin{array}{@{}rcl@{}} \phi1(t)&=&\frac{q}{32} \eta_{4}(t) \left( \beta_{4}(t) -1\right),\upsilon1(t)=\frac{q}{32} \beta_{4}(t) \left( \eta_{4}(t) -1\right),\tau1(t)\\&=&\frac{q}{32} \left( 1-\eta_{4}(t) \beta_{4}(t) \right),\sigma(t)=\frac{{ 1}}{{ 16}} \left( { 1}-\frac{{ q}}{{ 2}} \right)-\frac{{ q}}{{ 32}} { \eta }_{{ 4}}(t) { \beta }_{{ 4}}(t).\end{array} $$
$$ \begin{array}{@{}rcl@{}}\phi(t)&=&\frac{{ 1}}{{ 16}} \left( { 1}+{q}\left( \frac{{ 1}}{{ 2}} { \beta }_{{ 4}}(t) \left( { 1}+{ \eta }_{{ 4}}(t) \right)+{ \eta }_{{ 2}}(t) \left( { \beta }_{{ 4}}(t) +{ 1}\right)\right)\right),\upsilon(t) \\&=&\frac{{ 1}}{{ 16}} \left( { 1}+{q}\left( \frac{{ 1}}{{ 2}} { \eta }_{{ 4}}(t) \left( { 1}+{ \beta }_{{ 4}}(t) \right)+{ \beta }_{{ 2}}(t) \left( { \eta }_{{ 4}}(t) +{ 1}\right)\right)\right), \end{array} $$
$$ \begin{array}{@{}rcl@{}}\tau(t) &=&\frac{{ 1}}{{ 16}} \left( { 1}+{q}\left( \frac{{ 1}}{{ 2}} { \eta }_{{ 4}}(t) \left( { 1}+{ \beta }_{{ 4}}(t) \right)-{ \beta }_{{ 2}}(t) \left( { \eta }_{{ 4}}(t) +{ 1}\right)\right)\right), \varsigma(t) \\&=&\frac{{ 1}}{{ 16}} \left( { 1}+{ q}\left( \frac{{ 1}}{{ 2}} { \beta }_{{ 4}}(t) \left( { 1}+{ \eta }_{{ 4}}(t) \right)-{ \eta }_{{ 2}}(t) \left( { 1}+{ \beta }_{{ 4}}(t) \right)\right)\right), \end{array} $$
$$\omega(t) =\frac{{ 1}}{{ 16}} \left( { q}\left( -\frac{{ 1}}{{ 2}} \left( { \eta }_{{ 4}}(t) { \beta }_{{ 4}}(t) +{ \eta }_{{ 4}}(t) +{ \beta }_{{ 4}}(t) -{ 1}\right)-{ \beta }_{{ 4}}(t) { \eta }_{{ 2}}(t) -{ \eta }_{{ 4}}(t) { \beta }_{{ 2}}(t) +{ \eta }_{{ 2}}(t) +{ \beta }_{{ 2}}(t) \right)+{ 1}\right),$$
$$\psi(t) =\frac{{ 1}}{{ 16}} \left( { 1}+{ q}\left( -\frac{{ 1}}{{ 2}} \left( { \eta }_{{ 4}}(t) +{ \beta }_{{ 4}}(t) +{ \eta }_{{ 4}}(t) { \beta }_{{ 4}}(t) -{ 1}\right)-{ \beta }_{{ 2}}(t) +{ \eta }_{{ 2}}(t) +{ \eta }_{{ 4}}(t) { \beta }_{{ 2}}(t) -{ \eta }_{{ 2}}(t) { \beta }_{{ 4}}(t) \right)\right), $$
$$\chi(t)=\frac{{ 1}}{{ 16}} \left( { 1}+{q}\left( -\frac{{ 1}}{{ 2}} \left( { \eta }_{{ 4}}(t) +{ \beta }_{{ 4}}(t) +{ \eta }_{{ 4}}(t) { \beta }_{{ 4}}(t) -{ 1}\right)-{ \eta }_{{ 4}}(t) { \beta }_{{ 2}}(t) +{ \beta }_{{ 2}}(t) -{ \eta }_{{ 2}}(t) +{ \eta }_{{ 2}}(t) { \beta }_{{ 4}}(t) \right)\right),$$
$$\varphi(t)=\frac{{ 1}}{{ 16}} \left( { 1}+{q}\left( -\frac{{ 1}}{{ 2}} \left( { \eta }_{{ 4}}(t) +{ \beta }_{{ 4}}(t) +{ \eta }_{{ 4}}(t) { \beta }_{{ 4}}(t) -{ 1}\right)+{ \eta }_{{ 4}}(t) { \beta }_{{ 2}}(t) -{ \beta }_{{ 2}}(t) -{ \eta }_{{ 2}}(t) +{ \eta }_{{ 2}}(t) { \beta }_{{ 4}}(t) \right)\right),$$
$$\chi1(t)=\frac{q}{32} \left( \eta_{4}(t) +\beta_{4}(t) \left( 1-\eta_{4}(t) \right)-1\right),\varphi1(t)=\frac{q}{32} \left( \beta_{4}(t) +\eta_{4}(t) \left( 1-\beta_{4}(t) \right)-1\right)$$
$$\zeta(t) =\frac{q}{32} \left( \eta_{4}(t) \left( \beta_{4}(t) +\beta_{2}(t) \right)-2\eta_{2}(t) \left( \beta_{2}(t) +\frac{1}{2} \beta_{4}(t) \right)-\eta_{2}(t) +\beta_{2}(t) +1\right),$$
$$\varepsilon(t) =\frac{q}{32} \left( \eta_{4}(t) \left( \beta_{4}(t) -\beta_{2}(t) \right)+2\eta_{2}(t) \left( \beta_{2}(t) -\frac{1}{2} \beta_{4}(t) \right)-\eta_{2}(t) -\beta_{2}(t) +1\right),$$
$$\theta(t)=\frac{q}{32} \left( \eta_{4}(t) \left( \beta_{4}(t) -\beta_{2}(t) \right)-2\eta_{2}(t) \left( \beta_{2}(t) -\frac{1}{2} \beta_{4}(t) \right)+\eta_{2}(t) -\beta_{2}(t) +1\right),$$
$$\iota(t)=\frac{q}{32} \left( \eta_{4}(t) \left( \beta_{4}(t) +\beta_{2}(t) \right)+2\eta_{2}(t) \left( \beta_{2}(t) +\frac{1}{2} \beta_{4}(t) \right)+\eta_{2}(t) +\beta_{2}(t) +1\right),$$
$$\lambda(t)=\frac{q}{32} \left( -\eta_{4}(t) \left( \beta_{2}(t) +\beta_{4}(t) \right)+\eta_{2}(t) \left( \beta_{4}(t) -1\right)+\beta_{2}(t) +1\right),$$
$$\kappa(t)=\frac{q}{32} \left( -\eta_{4}(t) \left( \beta_{4}(t) -\beta_{2}(t) \right)+\eta_{2}(t) \left( \beta_{4}(t) -1\right)-\beta_{2}(t) +1\right),$$
$$\pi(t)=\frac{q}{16} \left( \frac{1}{2} \left( \eta_{4}(t) -\beta_{4}(t) \left( 1+\eta_{4}(t) \right)+1\right)+\eta_{2}(t) \left( \beta_{4}(t) -1\right)\right),$$
$$\xi(t)=\frac{q}{16} \left( \frac{1}{2} \left( -\eta_{4}(t) +\beta_{4}(t) \left( 1-\eta_{4}(t) \right)+1\right)+\beta_{2}(t) \left( \eta_{4}(t) -1\right)\right)$$
$$\varrho(t) =\frac{{q}}{{ 16}} \left( \frac{{ 1}}{{ 2}} \left( { \eta }_{{ 4}}(t) -{ \beta }_{{ 4}}(t) \left( { 1}+{ \eta }_{{ 4}}(t) \right)+{ 1}\right)-{ \eta }_{{ 2}}(t) \left( { \beta }_{{ 4}}(t) -{ 1}\right)\right),$$
$$\varpi(t)=\frac{q}{16} \left( \frac{1}{2} \left( -\eta_{4}(t) +\beta_{4}(t) \left( 1-\eta_{4}(t) \right)+1\right)-\beta_{2}(t) \left( \eta_{4}(t) -1\right)\right)$$
$$\nu(t)=\frac{q}{32} \left( -\eta_{4}(t) \left( \beta_{4}(t) +\beta_{2}(t) \right)-\eta_{2}(t) \left( \beta_{4}(t) -1\right)+\beta_{2}(t) +1\right),$$
$$ \mu(t)=\frac{q}{32} \left( -\eta_{4}(t) \left( \beta_{4}(t) -\beta_{2}(t) \right)-\eta_{2}(t) \left( \beta_{4}(t) -1\right)-\beta_{2}(t) +1\right), $$

In above density matrices, the time-dependent functions ηn(t) and βn(t) with n = 2, 4, 6 and t ≡ νt are decoherence factors coming from the coupling of the qubits to the first and the second reservoir respectively. Since both reservoirs are sources of RTN, we have ηn(t) ≡ βn(t) which is given by:

$$ \beta_{n}(t)=\left\lbrace\begin{array}{ll} &\displaystyle e^{\displaystyle-\gamma t}\left[ \cosh(\theta t)+\frac{\gamma}{\Gamma}\sinh(\theta t)\right] \rightarrow \gamma>n\nu, \theta=\sqrt{\gamma^{2}-(n\nu)^{2}} \\\\& \displaystyle e^{\displaystyle-\gamma t}\left[ \cos(\theta t)+\frac{\gamma}{\Gamma}\sin(\theta t)\right] \rightarrow \gamma<n\nu, \theta=\sqrt{(n\nu)^{2}-\gamma^{2}} \end{array}, \right. $$
(14)

Therefore, this research could be easily extended to the situation where the two reservoirs are sources of different kinds of classical noise. In such a situation the decoherence factors ηn(t) and βn(t) should be different from one another.

Appendix C: General partial transposes

The time-evolved state of the system initially prepared either in GHZ-type states or W-type states is a 16 × 16 matrix. This matrix can be written as a n × n block matrix as follows:

$$ \rho(t) =\left( \begin{array}{cccc} {\left[\rho_{1,1} \right]_{m\times m} } & {\left[\rho_{1,2} \right]_{m\times m} } & {{\cdots} } & {\left[\rho_{1,n} \right]_{m\times m} } \\ {\left[\rho_{2,1} \right]_{m\times m} } & {\left[\rho_{2,2} \right]_{m\times m} } & {{\cdots} } & {\left[\rho_{2,n} \right]_{m\times m} } \\ {{\vdots} } & {{\cdots} } & {{\ddots} } & {{\vdots} } \\ {\left[\rho_{n,1} \right]_{m\times m} } & {\left[\rho_{n,2} \right]_{m\times m} } & {{\cdots} } & {\left[\rho_{n,n} \right]_{m\times m} } \end{array}\right), $$
(15)

with m = 2, 4, 8 and n = 16/m. \( {\left [\rho _{\imath ,\jmath } \right ]_{m\times m} } \) is a m × m matrix defined as:

$$ \left[\rho_{\imath\jmath} \right]_{m\times m} =\left( \begin{array}{cccc} {\rho_{m\left( \imath-1\right)+1,m\left( \jmath-1\right)+1} } & {\rho_{m\left( \imath-1\right)+1,m\left( \jmath-1\right)+2} } & {{\cdots} } & {\rho_{m\left( \imath-1\right)+1,m\left( \jmath-1\right)+m} } \\ {\rho_{m\left( \imath-1\right)+2,m\left( \jmath-1\right)+1} } & {\rho_{m\left( \imath-1\right)+2,m\left( \jmath-1\right)+2} } & {{\cdots} } & {\rho_{m\left( \imath-1\right)+2,m\left( \jmath-1\right)+m} } \\ {{\vdots} } & {{\cdots} } & {{\ddots} } & {{\vdots} } \\ {\rho_{m\left( \imath-1\right)+m,m\left( \jmath-1\right)+1} } & {\rho_{m\left( \imath-1\right)+m,m\left( \jmath-1\right)+2} } & {{\cdots} } & {\rho_{m\left( \imath-1\right)+m,m\left( \jmath-1\right)+m} } \end{array}\right) $$
(16)

With the above notations, the different partial transposes of the time-evolved state of system are obtained as:

$$ \rho^{T_{1}}(t)=\left( \begin{array}{cc} {\left[\rho_{1,1} \right]_{8\times 8} } & {\left[\rho_{2,1} \right]_{8\times 8} } \\\\ {\left[\rho_{1,2} \right]_{8\times 8} } & {\left[\rho_{2,2} \right]_{8\times 8} } \end{array}\right) $$
(17)

where \(\rho ^{T_{1}} \) is the partial transpose of the time-evolved state of the system with respect to the first qubit.

$$ \rho^{T_{2}}(t)=\left( \begin{array}{cc} {B_{11} } & {B_{12} } \\ {B_{21} } & {B_{22} } \end{array}\right), \text{and } \rho^{T_{12}}(t)=\left( \begin{array}{cc} {B_{11} } & {B_{21} } \\ {B_{12} } & {B_{22} } \end{array}\right) $$
(18)

where \(\rho ^{T_{2}}(t) \) and \(\rho ^{T_{12}}(t) \) are respectively the partial transposes of the final state of the system with respect to qubit 1 and both qubits 1 and 2 and

$$ B_{\ell k} =\left( \begin{array}{cc} {\left[\rho_{2\left( \ell-1\right)+1,2\left( k-1\right)+1} \right]_{4\times 4} } & {\left[\rho_{2\left( \ell-1\right)+2,2\left( k-1\right)+1} \right]_{4\times 4} } \\\\ {\left[\rho_{2\left( \ell-1\right)+1,2\left( k-1\right)+2} \right]_{4\times 4} } & {\left[\rho_{2\left( \ell-1\right)+2,2\left( k-1\right)+2} \right]_{4\times 4} } \end{array}\right) $$

On the other hand we have

$$ \rho^{T_{3}}(t)=\left( \begin{array}{cccc} {A_{11} } & {A_{12} } & {A_{13} } & {A_{14} } \\ {A_{21} } & {A_{22} } & {A_{23} } & {A_{24} } \\ {A_{31} } & {A_{32} } & {A_{33} } & {A_{34} } \\ {A_{41} } & {A_{42} } & {A_{43} } & {A_{44} } \end{array}\right) \text{and } \rho^{T_{13}}(t)=\left( \begin{array}{cccc} {A_{11} } & {A_{12} } & {A_{31} } & {A_{32} } \\ {A_{21} } & {A_{22} } & {A_{41} } & {A_{42} } \\ {A_{13} } & {A_{14} } & {A_{33} } & {A_{34} } \\ {A_{23} } & {A_{24} } & {A_{43} } & {A_{44} } \end{array}\right) $$
(19)

where

$$ A_{\ell k} =\left( \begin{array}{cc} {\left[\rho_{2\left( \ell-1\right)+1,2\left( k-1\right)+1} \right]_{2\times 2} } & {\left[\rho_{2\left( \ell-1\right)+2,2\left( k-1\right)+1} \right]_{2\times 2} } \\\\ {\left[\rho_{2\left( \ell-1\right)+1,2\left( k-1\right)+2} \right]_{2\times 2} } & {\left[\rho_{2\left( \ell-1\right)+2,2\left( k-1\right)+2} \right]_{2\times 2} } \end{array}\right) $$

Finally, the partial transposes of the final state of the system with respect to qubit 4 and both qubits 1 and 4 can be written as:

$$ \rho^{T_{4}}(t)=\left( \begin{array}{cccc} {C_{11} } & {C_{12} } & {C_{13} } & {C_{14} } \\ {C_{21} } & {C_{22} } & {C_{23} } & {C_{24} } \\ {C_{31} } & {C_{32} } & {C_{33} } & {C_{34} } \\ {C_{41} } & {C_{42} } & {C_{43} } & {C_{44} } \end{array}\right) \text{and } \rho^{T_{14}}(t)=\left( \begin{array}{cccc} {C_{11} } & {C_{12} } & {C_{31} } & {C_{32} } \\ {C_{21} } & {C_{22} } & {C_{41} } & {C_{42} } \\ {C_{13} } & {C_{14} } & {C_{33} } & {C_{34} } \\ {C_{23} } & {C_{24} } & {C_{43} } & {C_{44} } \end{array}\right) $$
(20)

with

$$ C_{\ell k} =\left( \begin{array}{cccc} {\rho_{4\left( \ell-1\right)+1,4\left( k-1\right)+1} } & {\rho_{4\left( \ell-1\right)+2,4\left( k-1\right)+1} } & {\rho_{4\left( \ell-1\right)+1,4\left( k-1\right)+3} } & {\rho_{4\left( \ell-1\right)+2,4\left( k-1\right)+3} } \\ {\rho_{4\left( \ell-1\right)+1,4\left( k-1\right)+2} } & {\rho_{4\left( \ell-1\right)+2,4\left( k-1\right)+2} } & {\rho_{4\left( \ell-1\right)+1,4\left( k-1\right)+4} } & {\rho_{4\left( \ell-1\right)+2,4\left( k-1\right)+4} } \\ {\rho_{4\left( \ell-1\right)+3,4\left( k-1\right)+1} } & {\rho_{4\left( \ell-1\right)+4,4\left( k-1\right)+1} } & {\rho_{4\left( \ell-1\right)+3,4\left( k-1\right)+3} } & {\rho_{4\left( \ell-1\right)+4,4\left( k-1\right)+3} } \\ {\rho_{4\left( \ell-1\right)+3,4\left( k-1\right)+2} } & {\rho_{4\left( \ell-1\right)+2,4\left( k-1\right)+4} } & {\rho_{4\left( \ell-1\right)+3,4\left( k-1\right)+4} } & {\rho_{4\left( \ell-1\right)+4,4\left( k-1\right)+4} } \end{array}\right) $$

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Kenfack, L.T., Tchoffo, M. & Fai, L.C. Entanglement Dynamics and Symmetry Breaking in Symmetric four Qubits GHZ- and W-type States Coupled to Classical Random Telegraph Noise in Mixed Environments. Int J Theor Phys 58, 4258–4277 (2019). https://doi.org/10.1007/s10773-019-04299-x

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