Skip to main content
SpringerLink
Log in

Probing tripartite entanglement and coherence dynamics in pure and mixed independent classical environments

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The type of environments to which a quantum system is exposed has a significant impact on the quantum system’s entanglement and coherence protection. In this regard, we investigate the time evolution of tripartite entanglement and coherence in GHZ-like state when subject to independent classical environments. In particular, we focus on local environments with the same and mixed disorders, resulting in various Gaussian noisy conditions, namely pure power-law noise, pure fractional Gaussian noise, power-law noise maximized, and fractional Gaussian noise maximized configurations. We show that the environments with mixed disorders are more detrimental than those having single kind of disorder for entanglement and coherence preservation using time-dependent quantum negativity, entanglement witnesses, purity, and decoherence metrics. Besides, there is no ultimate solution for avoiding the negative consequences of fractional Gaussian noise-assisted classical environments in both pure and mixed noise conditions. Not only the noise, but also the number of qubits driven by a certain noise, has been discovered to strongly influence the amount, nature of the decay, and preservation intervals. We also show that the GHZ-like states can be modelled in classical channels driven by pure power-law noise for extended quantum correlations, coherence, and quantum information preservation. In addition, we compare the entanglement measurement efficiency between entanglement witness and negativity measures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data availability

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

References

  1. Zoller, P., Beth, T., Binosi, D., Blatt, R., Briegel, H., Bruss, D., Zeilinger, A.: Quantum information processing and communication. Eur. Phys. J. D Atom. Mol. Opt. Plasma Phys. 36(2), 203–228 (2005)

    Google Scholar 

  2. Kok, P., Lovett, B.W.: Introduction to Optical Quantum Information Processing, pp. 1–488. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  3. Beth, T., Leuchs, G. (eds.): Quantum Information Processing. Wiley, New York (2006)

    Google Scholar 

  4. Monroe, C.: Quantum information processing with atoms and photons. Nature 416(6877), 238–246 (2002)

    Article  ADS  Google Scholar 

  5. Smith, A., Kim, M.S., Pollmann, F., Knolle, J.: Simulating quantum many-body dynamics on a current digital quantum computer. npj Quant. Inf. 5(1), 1–13 (2019)

    Google Scholar 

  6. Wu, L., Cai, H.J., Gong, Z.: The integer factorization algorithm with Pisano period. IEEE Access 7, 167250–167259 (2019)

    Article  Google Scholar 

  7. Jozsa, R.: Quantum factoring, discrete logarithms, and the hidden subgroup problem. Comput. Sci. Eng. 3(2), 34–43 (2001)

    Article  Google Scholar 

  8. Denchev, V.S., Pandurangan, G.: Distributed quantum computing: a new frontier in distributed systems or science fiction? ACM SIGACT News 39(3), 77–95 (2008)

    Article  Google Scholar 

  9. Morton John, J.L., et al.: Embracing the quantum limit in silicon computing. Nature 479(7373), 345–353 (2011)

    Article  ADS  Google Scholar 

  10. Lidar, D., Lian-Ao, W., Alexandre B.: Encoding and error suppression for superconducting quantum computers. U.S. Patent No. 7,307,275. 11 (2007)

  11. Preskill, J.: Simulating quantum field theory with a quantum computer. In: The 36th Annual International Symposium on Lattice Field Theory, 22–28 July, 2018. Michigan State University, East Lansing, Michigan, USA. Online at https://pos.sissa.it/334,id.24 (2018)

  12. Averin, D.V., Ruggiero, B., Silvestrini, P. (eds.): Macroscopic Quantum Coherence and Quantum Computing. Springer, New York (2012)

    Google Scholar 

  13. Suter, D., Lim, K.: Scalable architecture for spin-based quantum computers with a single type of gate. Phys. Rev. A 65(5), 052309 (2002)

    Article  ADS  Google Scholar 

  14. Anwar, M.S., et al.: Preparing high purity initial states for nuclear magnetic resonance quantum computing. Phys. Rev. Lett. 93(4), 040501 (2004)

    Article  ADS  Google Scholar 

  15. Rohde, P.P.: Optical quantum computing with photons of arbitrarily low fidelity and purity. Phys. Rev. A 86(5), 052321 (2012)

    Article  ADS  Google Scholar 

  16. Häffner, H., Roos, C.F., Blatt, R.: Quantum computing with trapped ions. Phys. Rep. 469(4), 155–203 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  17. Herrera-Martí, D.A., Fowler, A.G., Jennings, D., Rudolph, T.: Photonic implementation for the topological cluster-state quantum computer. Phys. Rev. A 82(3), 032332 (2010)

    Article  ADS  Google Scholar 

  18. Peng, X., Jiangfeng, D., Suter, D.: Quantum phase transition of ground-state entanglement in a Heisenberg spin chain simulated in an NMR quantum computer. Phys. Rev. A 71(1), 012307 (2005)

    Article  ADS  Google Scholar 

  19. Riebe, M., et al.: Deterministic entanglement swapping with an ion-trap quantum computer. Nat. Phys. 4(11), 839–842 (2008)

    Article  Google Scholar 

  20. Franco, R.L., Compagno, G.: Quantum entanglement of identical particles by standard information-theoretic notions. Sci. Rep. 6, 20603 (2016)

    Article  Google Scholar 

  21. Horodecki, R., et al.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Ghisi, F., Ulyanov, S.V.: The information role of entanglement and interference operators in Shor quantum algorithm gate dynamics. J. Mod. Opt. 47(12), 2079–2090 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Behera, B.K., et al.: Demonstration of entanglement purification and swapping protocol to design quantum repeater in IBM quantum computer. Quant. Inf. Process. 18(4), 108 (2019)

    Article  ADS  MATH  Google Scholar 

  24. Peng, J.Y., Bai, M.Q., Mo, Z.W.: Remote information concentration via \(W\) W state: reverse of ancilla-free phase-covariant telecloning. Quant. Inf. Process. 12(11), 3511–3525 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Xu, J.S., et al.: Experimental investigation of classical and quantum correlations under decoherence. Nat. Commun. 1(1), 1–6 (2010)

    Article  ADS  Google Scholar 

  26. Girolami, D., Adesso, G.: Interplay between computable measures of entanglement and other quantum correlations. Phys. Rev. A 84(5), 052110 (2011)

    Article  ADS  Google Scholar 

  27. Bhaskar, M.K., et al.: Experimental demonstration of memory-enhanced quantum communication. Nature 5, 1–5 (2020)

    Google Scholar 

  28. Dong, X. Dong, B., Wang, X.: Quantum attacks on some Feistel block ciphers. Des. Codes Cryptogr. 1–25 (2020)

  29. Ji, Z.X., et al.: Several two-party protocols for quantum private comparison using entanglement and dense coding. Opt. Commun. 459, 124911 (2020)

    Article  Google Scholar 

  30. Meher, N.: Scheme for realizing quantum dense coding via entanglement swapping. J. Phys. B Atom. Mol. Opt. Phys. (2020)

  31. Gu, J., Hwang, T., Tsai, C.-W.: On the controlled cyclic quantum teleportation of an arbitrary two-qubit entangled state by using a ten-qubit entangled state. Int. J. Theor. Phys. 59(1), 200–205 (2020)

    Article  MATH  Google Scholar 

  32. Zhu, F., et al.: Experimental long-distance quantum secure direct communication. Sci. Bull. 62(22), 1519–1524 (2017)

    Article  Google Scholar 

  33. Derkach, I., Usenko, V.C., Filip, R.: Squeezing-enhanced quantum key distribution over atmospheric channels. N. J. Phys. (2020)

  34. Soe, M.T., et al.: Stable Polarization Entanglement based quantum key distribution over metropolitan fibre network. Bull. Am. Phys. Soc. (2020)

  35. Karimi, E., Soljanin, E., Whiting, P.: Increasing the raw key rate in energy-time entanglement based quantum key distribution. arXiv preprint arXiv:2001.09049 (2020)

  36. Bennett, C.H., et al.: Concentrating partial entanglement by local operations. Phys. Rev. A 53(4), 2046 (1996)

    Article  ADS  Google Scholar 

  37. Moehring, D.L., et al.: Entanglement of single-atom quantum bits at a distance. Nature 449(7158), 68–71 (2007)

    Article  ADS  Google Scholar 

  38. Preskill, J.: Fault-tolerant quantum computation. In: Introduction to Quantum Computation and Information, pp. 213–269 (1998)

  39. Yu, T., Eberly, J.H.: Qubit disentanglement and decoherence via dephasing. Phys. Rev. B 68(16), 165322 (2003)

    Article  ADS  Google Scholar 

  40. Dodd, P.J., Halliwell, J.J.: Disentanglement and decoherence by open system dynamics. Phys. Rev. A 69(5), 052105 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  41. Barreiro, J.T., et al.: An open-system quantum simulator with trapped ions. Nature 470(7335), 486–491 (2011)

    Article  ADS  Google Scholar 

  42. Sun, Z., Wang, X., Sun, C.P.: Disentanglement in a quantum-critical environment. Phys. Rev. A 75(6), 062312 (2007)

    Article  ADS  Google Scholar 

  43. Rossini, D., et al.: Decoherence induced by interacting quantum spin baths. Phys. Rev. A 75(3), 032333 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  44. Hu, M.-L.: Environment-induced decay of teleportation fidelity of the one-qubit state. Phys. Lett. A 375(21), 2140–2143 (2011)

    Article  ADS  Google Scholar 

  45. Tahira, R., et al.: Entanglement dynamics of high-dimensional bipartite field states inside the cavities in dissipative environments. J. Phys. B Atom. Mol. Opt. Phys. 43(3), 035502 (2010)

    Article  ADS  Google Scholar 

  46. Paz, J.P., Roncaglia, A.J.: Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment. Phys. Rev. A 79(3), 032102 (2009)

    Article  ADS  Google Scholar 

  47. Benedetti, C., et al.: Effects of classical environmental noise on entanglement and quantum discord dynamics. Int. J. Quant. Inf. 10(08), 1241005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Arthur, T.T., Martin, T., Fai, L.C.: Quantum correlations and coherence dynamics in qutrit-qutrit systems under mixed classical environmental noises. Int. J. Quant. Inf. 15(06), 1750047 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Szańkowski, P., Trippenbach, M., Cywiński, Ł: Spectroscopy of cross correlations of environmental noises with two qubits. Phys. Rev. A 94(1), 012109 (2016)

    Article  ADS  Google Scholar 

  50. Suter, D., Álvarez, G.A.: Colloquium: protecting quantum information against environmental noise. Rev. Mod. Phys. 88(4), 041001 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  51. Rossi, M., Matteo, G.A.: Entangled quantum probes for dynamical environmental noise. Phys. Rev. A 92(1), 010302 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  52. Astafiev, O., et al.: Quantum noise in the Josephson charge qubit. Phys. Rev. Lett. 93(26), 267007 (2004)

    Article  ADS  Google Scholar 

  53. Benedetti, C., Matteo, G.A.: Characterization of classical Gaussian processes using quantum probes. Phys. Lett. A 378(34), 2495–2500 (2014)

    Article  ADS  MATH  Google Scholar 

  54. Tchoffo, M., et al.: Quantum correlations dynamics and decoherence of a three-qubit system subject to classical environmental noise. Eur. Phys. J. Plus 131(10), 380 (2016)

    Article  Google Scholar 

  55. Tsokeng, A.T., Tchoffo, M., Fai, L.C.: Quantum correlations and decoherence dynamics for a qutrit-qutrit system under random telegraph noise. Quant. Inf. Process. 16(8), 191 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Benedetti, C., et al.: Time-evolution of entanglement and quantum discord of bipartite systems subject to \(\dfrac{1}{f^{\alpha }}\) noise. In: 2013 22nd International Conference on Noise and Fluctuations (ICNF). IEEE (2013)

  57. Scala, M., Migliore, R., Messina, A.: Dissipation and entanglement dynamics for two interacting qubits coupled to independent reservoirs. J. Phys. A Math. Theor. 41(43), 435304 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Cui, W., Xi, Z., Pan, Yu.: The entanglement dynamics of the bipartite quantum system: toward entanglement sudden death. J. Phys. A Math. Theor. 42(2), 025303 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Kim, K.-I., Li, H.-M., Zhao, B.-K.: Genuine tripartite entanglement dynamics and transfer in a triple Jaynes–Cummings model. Int. J. Theor. Phys. 55(1), 241–254 (2016)

    Article  MATH  Google Scholar 

  60. Leggio, B., et al.: Distributed correlations and information flows within a hybrid multipartite quantum-classical system. Phys. Rev. A 92(3), 032311 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  61. Kenfack, L.T., et al.: Dynamics of tripartite quantum correlations in mixed classical environments: the joint effects of the random telegraph and static noises. Int. J. Quant. Inf. 15(05), 1750038 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kenfack, L.T., et al.: Decoherence and tripartite entanglement dynamics in the presence of Gaussian and non-Gaussian classical noise. Physica B Condens. Matter 511, 123–133 (2017)

    Article  ADS  Google Scholar 

  63. Rossi, M.A., Benedetti, C., Paris, M.G.: Engineering decoherence for two-qubit systems interacting with a classical environment. Int. J. Quant. Inf. 12(07–08), 1560003 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  64. Pezzè, L., Li, Y., Li, W., Smerzi, A.: Witnessing entanglement without entanglement witness operators. Proc. Natl. Acad. Sci. 113(41), 11459–11464 (2016)

    Article  ADS  Google Scholar 

  65. Barbieri, M., De Martini, F., Di Nepi, G., Mataloni, P., D’Ariano, G.M., Macchiavello, C.: Detection of entanglement with polarized photons: experimental realization of an entanglement witness. Phys. Rev. Lett. 91(22), 227901 (2003)

  66. Li, Z.D., Zhao, Q., Zhang, R., Liu, L.Z., Yin, X.F., Zhang, X., Pan, J.W.: Measurement-device-independent entanglement witness of tripartite entangled states and its applications. Phys. Rev. Lett. 124(16), 160503 (2020)

    Article  ADS  Google Scholar 

  67. Brandao, F.G., Vianna, R.O.: Witnessed entanglement. Int. J. Quant. Inf. 4(02), 331–340 (2006)

    Article  MATH  Google Scholar 

  68. Brandao, F.: Quantifying entanglement with witness operators. Phys. Rev. A 72(2), 022310 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  69. Tchoffo, M., Kenfack, L.T., Fouokeng, G.C., Fai, L.C.: Quantum correlations dynamics and decoherence of a three-qubit system subject to classical environmental noise. Eur. Phys. J. Plus 131(10), 1–18 (2016)

    Article  Google Scholar 

  70. Zhang, Y.J., Man, Z.X., Zou, X.B., Xia, Y.J., Guo, G.C.: Dynamics of multipartite entanglement in the non-Markovian environments. J. Phys. B At. Mol. Opt. Phys. 43(4), 045502 (2010)

    Article  ADS  Google Scholar 

  71. Dutta, P., Horn, P.M.: Low-frequency fluctuations in solids: 1 f noise. Rev. Mod. Phys. 53(3), 497 (1981)

    Article  ADS  Google Scholar 

  72. Postma, H.W.C., Teepen, T., Yao, Z., Grifoni, M., Dekker, C.: Carbon nanotube single-electron transistors at room temperature. Science 293(5527), 76–79 (2001)

    Article  ADS  Google Scholar 

  73. Kasdin, N.J.: Discrete simulation of colored noise and stochastic processes and 1/f/sup/spl alpha//power law noise generation. Proc. IEEE 83(5), 802–827 (1995)

    Article  Google Scholar 

  74. Kasdin, N.J., Walter, T.: Discrete simulation of power law noise (for oscillator stability evaluation). In: Proceedings of the 1992 IEEE Frequency Control Symposium, pp. 274–283. IEEE (1992)

  75. Lakhmanskiy, K., Holz, P.C., Schärtl, D., Ames, B., Assouly, R., Monz, T., Blatt, R.: Observation of superconductivity and surface noise using a single trapped ion as a field probe. Phys. Rev. A 99(2), 023405 (2019)

    Article  ADS  Google Scholar 

  76. Rilling, G., Flandrin, P., Gonçalves, P.: Empirical mode decomposition, fractional Gaussian noise and Hurst exponent estimation. In: Proceedings (ICASSP’05). IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 4. IEEE (2005)

  77. Lima, F.W.S.: Simulation of majority rule disturbed by power-law noise on directed and undirected Barabási-Albert networks. Int. J. Mod. Phys. C 19(07), 1063–1067 (2008)

    Article  ADS  MATH  Google Scholar 

  78. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Kaplan, L.M., Kuo, C.C.: Extending self-similarity for fractional Brownian motion. DEEE Trans. Signal Process. 42(12), 3526–3530 (1994)

    Article  ADS  Google Scholar 

  80. Abry, P., Flandrin, P., Taqqu, M.S., Veitch, D.: Self-similarity and long-range dependence through the wavelet lens. In: Theory and Applications of Long-Range Dependence, pp. 527–556 (2003)

  81. Pentland, A.P.: Fractal-based description of natural scenes. DEEE Trans. Pattern Anal. Mach. Intell. 6, 661–674 (1984)

    Article  Google Scholar 

  82. Zunino, L., Pérez, D.G., Kowalski, A., Martín, M.T., Garavaglia, M., Plastino, A., Rosso, O.A.: Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy. Physica A 387(24), 6057–6068 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. Rilling, G., Flandrin, P., Gonçalves P.: Empirical mode decomposition, fractional Gaussian noise and Hurst exponent estimation. In: DEEE International Conference on Acoustics, Speech, and Signal Processing, Proceedings (ICASSP’05), vol. 4, pp. iv–489. DEEE (2005)

  84. LeDEsma, S., Liu, D.: Synthesis of fractional Gaussian noise using linear approximation for generating self-similar network traffic. ACM SIGCOMM Comput. Commun. Rev. 30(2), 4–17 (2000)

    Article  Google Scholar 

  85. Koutsoyiannis, D.: The Hurst phenomenon and fractional Gaussian noise made easy. Hydrol. Scdences J. 47(4), 573–595 (2002)

    Article  Google Scholar 

  86. Koutsoyiannis, D.: The Hurst phenomenon and fractional Gaussian noise made easy. Hydrol. Sci. J. 47(4), 573–595 (2002)

    Article  Google Scholar 

  87. Fabrice, K.F., Arthur, T.T., Pernel, N.N., Martin, T., Fai, L.C.: Tripartite quantum discord dynamics in qubits driven by the joint influence of distinct classical noises. Quant. Inf. Process. 20(1), 1–15 (2021)

    Article  MathSciNet  Google Scholar 

  88. Benedetti, C., Paris, M.G.: Characterization of classical Gaussian processes using quantum probes. Phys. Lett. A 378(34), 2495–2500 (2014)

    Article  ADS  MATH  Google Scholar 

  89. Buscemi, F., Bordone, P.: Time evolution of tripartite quantum discord and entanglement under local and nonlocal random telegraph noise. Phys. Rev. A 87(4), 042310 (2013)

    Article  ADS  Google Scholar 

  90. Javed, M., Khan, S., Ullah, S.A.: The dynamics of quantum correlations in mixed classical environments. J. Russ. Laser Res. 37(6), 562–571 (2016)

    Article  Google Scholar 

  91. Lionel, T.K., et al.: Effects of static noise on the dynamics of quantum correlations for a system of three qubits. Int. J. Mod. Phys. B 31(8), 1750046 (2017)

    Article  ADS  MATH  Google Scholar 

  92. Benedetti, C., et al.: Dynamics of quantum correlations in colored-noise environments. Phys. Rev. A 87(5), 052328 (2013)

    Article  ADS  Google Scholar 

  93. Rahman, A.U., Javed, M., Noman, M., Ullah, A., Dir, C., Luo, M.X.: Effects of classical fluctuating environments on decoherence and bipartite quantum correlations dynamics. arXiv preprint arXiv:2107.11241 (2021)

  94. Rahman, A.U., Noman, M., Javed, M., Ullah, A.: Dynamics of bipartite quantum correlations and coherence in classical environments described by pure and mixed Gaussian noises. Eur. Phys. J. Plus 136(8), 1–19 (2021)

    Article  Google Scholar 

  95. Rahman, A.U., Javed, M., Ullah, A.: Probing multipartite entanglement, coherence and quantum information preservation under classical Ornstein-Uhlenbeck noise. arXiv preprint arXiv:2107.11251 (2021)

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

    Atta Ur Rahman, Muhammad Javed & Arif Ullah

  2. Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, Wuhan, 430072, China

    Zhaoxu Ji

Authors
  1. Atta Ur Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Javed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arif Ullah
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhaoxu Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atta Ur Rahman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

In this section, we give the details of the final density matrices obtained for the time evolution of the three qubits initially prepared in the state \(\rho _{\mathrm{GHZ}}(0)\) under the effects of pure and mixed noisy configurations. Using Eq. (13), we get the final density matrix under pure Gaussian noise case as:

$$\begin{aligned} \rho _p(t)\!=\!\frac{1}{8}\left[ \begin{array}{cccccccc} 1+3 \mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1+3 \mathcal {X}_1 \\ 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1-\mathcal {X}_1 &{}\quad 0 \\ 1+3 \mathcal {X}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1+3 \mathcal {X}_1 \end{array} \right] . \end{aligned}$$
(28)

Next, using Eq. (15), we obtained the final density matrix ensemble state as:

$$\begin{aligned} \rho _{XYZ}(t)=\frac{1}{8}\left[ \begin{array}{cccccccc} \mathcal {H}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_1 \\ 0 &{}\quad \mathcal {H}_2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_2 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_3 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \mathcal {H}_2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_2 &{}\quad 0 \\ \mathcal {H}_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \mathcal {H}_1 \end{array} \right] \end{aligned}$$
(29)

where \(\rho _{XYZ}(t) \in \{ \rho _{\mathrm{PLM}}(t), \rho _{\mathrm{FGM}}(t)\}\), \(\mathcal {X}_i=\exp [-n\beta _{\mathcal {A_B}}(t)]\) with \(\beta _{\mathcal {A_B}}(t) \in \beta _{\mathcal {P_L}}(t), \beta _{\mathcal {F_G}}(t)\) and \(\mathcal {Y}=\exp [-n\beta _{mix}(t)]\) with \(\beta _{mix}(t)=\beta _{\mathcal {P_L}}(t)+\beta _{\mathcal {F_G}}(t)\), \(\mathcal {H}_1=1+\mathcal {X}_2+2 \mathcal {Y}\), \(\mathcal {H}_2=1+\mathcal {X}_2-2 \mathcal {Y}\), \(\mathcal {H}_3=1-\mathcal {X}_2\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rahman, A.U., Javed, M., Ullah, A. et al. Probing tripartite entanglement and coherence dynamics in pure and mixed independent classical environments. Quantum Inf Process 20, 321 (2021). https://doi.org/10.1007/s11128-021-03257-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-03257-z

Keywords

Search

Navigation