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Entanglement witness and linear entropy in an open system influenced by FG noise

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Abstract

We investigate the behavior of tripartite entanglement and entropy disorder in three non-interacting qubits that are first prepared in a mixture state containing Greenberger–Horne–Zeilinger (GHZ) and Werner (W) states, and then exposed to classical transmitting channels influenced by fractional Gaussian (FG) noise. To investigate the coupling of the three non-interacting qubits, single, double, and triple local channel configurations are used. We address the question of how GHZ and W states can have different capacities to protect entanglement while avoiding entropy disorder in different qubit-channel couplings. Using entanglement witness and linear entropy functions, we find that GHZ state is more resourceful when coupled with a single channel, whereas W state remains more resourceful when exposed to more than one local channel. The statistical ensemble states that are initially designed, as well as the designs of the medium to which they are exposed, have a strong influence on the initial state entanglement retention in quantum systems. Moreover, we realize that FG noise is more harmful than frequently found local noises, resulting in faster entropy disorder generation and, as a result, the destruction of tripartite quantum correlations. However, the tripartite correlations can be preserved by increasing the FG noise parameter known as the Hurst exponent.

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Authors and Affiliations

  1. School of Physics, University of Chinese Academy of Science, Yuquan Road 19A, Beijing 100049, China

    Atta Ur Rahman

  2. Faculty of Physics, Semnan University, P.O. Box 35195-363, Semnan, Iran

    Saeed Haddadi

  3. Saeed’s Quantum Information Group, P.O. Box 19395-0560, Tehran, Iran

    Saeed Haddadi

  4. Quantum Optics and Information Research Team, Department of Physics, University of Malakand, Dir(L) Chakdara, KPK, Pakistan

    Muhammad Javed & Arif Ullah

  5. Research Unit of Condensed Matter, Electronic and Signal Processing, Department of Physics, Dschang School of Sciences and Technology, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Lionel Tenemeza Kenfack

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A Appendix

A Appendix

Here, we present the explicit form of statistical ensembles of final density matrices of SQC, DQC and TQC configurations. For three non-interacting qubits initially prepared in the state \(\rho (0)\) given in Eq. (3) when considered in SQC configuration under FG noise, Eq. (11), the corresponding density matrix takes the form

$$\begin{aligned} \rho _{\textrm{SQC}}(t)=\left[ \begin{array}{cccccccc} \mathcal {A}_{1} &{} \mathcal {A}_{2} &{} \mathcal {A}_{2} &{} \mathcal {A}_{3} &{} \mathcal {A}_{2} &{} \mathcal {A}_{3} &{} \mathcal {A}_{3} &{} \mathcal {A}_{4} \\ \mathcal {A}_{2} &{} \mathcal {A}_{5} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{7} \\ \mathcal {A}_{2} &{} \mathcal {A}_{5} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{7} \\ \mathcal {A}_{3} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{8} &{} \mathcal {A}_{2} \\ \mathcal {A}_{2} &{} \mathcal {A}_{5} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{5} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{7} \\ \mathcal {A}_{3} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{8} &{} \mathcal {A}_{2} \\ \mathcal {A}_{3} &{} \mathcal {A}_{6} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{6} &{} \mathcal {A}_{8} &{} \mathcal {A}_{8} &{} \mathcal {A}_{2} \\ \mathcal {A}_{4} &{} \mathcal {A}_{7} &{} \mathcal {A}_{7} &{} \mathcal {A}_{2} &{} \mathcal {A}_{7} &{} \mathcal {A}_{2} &{} \mathcal {A}_{2} &{} \mathcal {A}_{9} \\ \end{array} \right] \end{aligned}$$
(A.1)

where

$$\begin{aligned} \mathcal {A}_{1}&=\frac{1}{32} e^{-18 \beta (t) } \left[ -3 e^{16 \beta (t) } (r-1)+6 e^{10 \beta (t) } (2 r-1)+e^{18 \beta (t) } (4 r+6)+3 (r-1)\right] ,\\ \mathcal {A}_{2}&=\frac{1}{16} \left( e^{-8 \beta (t) }-1\right) r,\\ \mathcal {A}_{3}&=-\frac{1}{32} e^{-18 \beta (t) } \left( e^{8 \beta (t) }-1\right) \left[ 3 e^{8 \beta (t) } (r-1)+e^{10 \beta (t) } (4 r-2)+3 (r-1)\right] ,\\ \mathcal {A}_4&=\frac{1}{16} \left( 3 e^{-8 \beta (t) }+5\right) r,\\ \mathcal {A}_{5}&=\frac{1}{96} e^{-18 \beta (t) } \left[ -7 e^{16 \beta (t) } (r-1)+e^{18 \beta (t) } (10-4 r)+e^{10 \beta (t) } (6-12 r)-9(r-1)\right] ,\\ \mathcal {A}_{6}&=\frac{1}{16} \left( 1-e^{-8 \beta (t) }\right) r,\\ \mathcal {A}_{7}&=-\frac{1}{32} e^{-18 \beta (t) } \left( e^{8 \beta (t) }-1\right) \left[ -3 e^{8 \beta (t) } (r-1)+e^{10 \beta (t) } (4 r-2)-3(r-1)\right] ,\\ \mathcal {A}_{8}&=\frac{1}{96} e^{-18 \beta (t) } \left[ 7 e^{16 \beta (t) } (r-1)+e^{18 \beta (t) } (10-4 r)+e^{10 \beta (t) } (6-12 r)+9 (r-1)\right] ,\\ \mathcal {A}_{9}&=\frac{1}{32} e^{-18 \beta (t) } \left[ 3 e^{16 \beta (t) } (r-1)+6 e^{10 \beta (t) } (2 r-1)+e^{18 \beta (t) } (4 r+6)-3(r-1)\right] .&\end{aligned}$$

The final density matrix for the case of DQC configuration driven by FG noise, Eq. (12), has the following form

$$\begin{aligned} \rho _{\textrm{DQC}}(t)=\left[ \begin{array}{cccccccc} \mathcal {B}_{1} &{}\mathcal {B}_{2} &{}\mathcal {B}_{3} &{}\mathcal {B}_{4} &{}\mathcal {B}_{3} &{}\mathcal {B}_{4} &{} \mathcal {B}_{5} &{} \mathcal {B}_{6} \\ \mathcal {B}_{7} &{} \mathcal {B}_{8} &{} \mathcal {B}_{9} &{} \mathcal {B}_{10} &{} \mathcal {B}_{9} &{} \mathcal {B}_{10} &{} \mathcal {B}_{11} &{} \mathcal {B}_{12} \\ \mathcal {B}_{10} &{} \mathcal {B}_{9} &{} \mathcal {B}_{13} &{} \mathcal {B}_{14} &{} \mathcal {B}_{13} &{} \mathcal {B}_{14} &{} \mathcal {B}_{10} &{} \mathcal {B}_{15} \\ \mathcal {B}_{4} &{}\mathcal {B}_{3} &{} \mathcal {B}_{16} &{} \mathcal {B}_{17} &{} \mathcal {B}_{16} &{} \mathcal {B}_{17} &{} \mathcal {B}_{18} &{}\mathcal {B}_{3} \\ \mathcal {B}_{10} &{} \mathcal {B}_{9} &{} \mathcal {B}_{13} &{} \mathcal {B}_{14} &{} \mathcal {B}_{13} &{} \mathcal {B}_{14} &{} \mathcal {B}_{10} &{} \mathcal {B}_{15} \\ \mathcal {B}_{4} &{}\mathcal {B}_{3} &{} \mathcal {B}_{16} &{} \mathcal {B}_{17} &{} \mathcal {B}_{16} &{} \mathcal {B}_{17} &{} \mathcal {B}_{18} &{}\mathcal {B}_{3} \\ \mathcal {B}_{5} &{} \mathcal {B}_{19} &{}\mathcal {B}_{3} &{} \mathcal {B}_{18} &{}\mathcal {B}_{3} &{} \mathcal {B}_{18} &{} \mathcal {B}_{20} &{}\mathcal {B}_{2} \\ \mathcal {B}_{21}&{} \mathcal {B}_{12} &{} \mathcal {B}_{15} &{} \mathcal {B}_{10} &{} \mathcal {B}_{15} &{} \mathcal {B}_{10} &{} \mathcal {B}_{7} &{} \mathcal {B}_{22} \\ \end{array} \right] \end{aligned}$$
(A.2)

where

$$\begin{aligned} \mathcal {B}_{1}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -20 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)+4 e^{10 \beta (t) } (2 r+7)\right. \\&\left. \quad +\, e^{6 \beta (t) } (65 r-17)+19 (r-1)\right] ,\\ \mathcal {B}_{2}&=\frac{1}{192} e^{-10 \beta (t) } \left[ e^{6 \beta (t) } (r-1)+12 e^{2 \beta (t) } r-12 e^{10 \beta (t) } r-r+1\right] ,\\ \mathcal {B}_{3}&=-\frac{1}{192} e^{-10 \beta (t) } \left( e^{6 \beta (t) }-1\right) (r-1),\\ \mathcal {B}_{4}&=-\frac{1}{192} e^{-10 \beta (t) } \left[ -15 e^{6 \beta (t) }+16 e^{8 \beta (t) }+16 e^{10 \beta (t) }-17\right] (r-1),\\ \mathcal {B}_{5}&=\frac{1}{192} e^{-10 \beta (t) } \left[ e^{6 \beta (t) } (r-1)-20 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)\right. \\&\left. \quad +\, e^{10 \beta (t) } (12-24 r)+19 (r-1)\right] ,\\ \mathcal {B}_{6}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 12 e^{2 \beta (t) } r+36 e^{10 \beta (t) } r+e^{6 \beta (t) } (49 r-1)-r+1\right] ,\\ \mathcal {B}_{7}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -e^{6 \beta (t) } (r-1)+12 e^{2 \beta (t) } r-12 e^{10 \beta (t) } r+r-1\right] ,\\ \mathcal {B}_{8}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -12 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)+4 e^{10 \beta (t) } (2 r+7)\right. \\&\left. \quad +\, e^{6 \beta (t) } (17-65 r)-19 (r-1)\right] ,\\ \mathcal {B}_{9}&=-\frac{1}{192} e^{-10 \beta (t) } \left[ 15 e^{6 \beta (t) }+16 e^{8 \beta (t) }+16 e^{10 \beta (t) }+17\right] (r-1),\\ \mathcal {B}_{10}&=\frac{1}{192} e^{-10 \beta (t) } \left( e^{6 \beta (t) }-1\right) (r-1),\\ \mathcal {B}_{11}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 12 e^{2 \beta (t) } r+36 e^{10 \beta (t) } r+e^{6 \beta (t) } (1-49 r)+r-1\right] ,\\ \mathcal {B}_{12}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -e^{6 \beta (t) } (r-1)+20 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)\right. \\&\left. \quad +\, e^{10 \beta (t) } (12-24 r)-19 (r-1)\right] ,\\ \mathcal {B}_{13}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -e^{6 \beta (t) } (r-1)-12 e^{8 \beta (t) } (r-1)+e^{10 \beta (t) } (20-8 r)\right. \\&\left. \quad +\, e^{2 \beta (t) } (12-24 r)-19 (r-1)\right] ,\\ \mathcal {B}_{14}&=\frac{1}{192} e^{-10 \beta (t) } \left[ -e^{6 \beta (t) } (r-1)-12 e^{2 \beta (t) } r+12 e^{10 \beta (t) } r+r-1\right] ,\\ \mathcal {B}_{15}&=-\frac{1}{192} e^{-10 \beta (t) } \left[ -17 e^{6 \beta (t) }-16 e^{8 \beta (t) }+16 e^{10 \beta (t) }+17\right] (r-1),\\ \mathcal {B}_{16}&=\frac{1}{192} e^{-10 \beta (t) } \left[ e^{6 \beta (t) } (r-1)-12 e^{2 \beta (t) } r+12 e^{10 \beta (t) } r-r+1\right] ,\\ \mathcal {B}_{17}&=\frac{1}{192} e^{-10 \beta (t) } \left[ e^{6 \beta (t) } (r-1)+12 e^{8 \beta (t) } (r-1)+e^{10 \beta (t) } (20-8 r)\right. \\&\left. \quad + \, e^{2 \beta (t) } (12-24 r)+19 (r-1)\right] ,\\ \mathcal {B}_{18}&=-\frac{1}{192} e^{-10 \beta (t) } \left[ 17 e^{6 \beta (t) }-16 e^{8 \beta (t) }+16 e^{10 \beta (t) }-17\right] (r-1),\\ \mathcal {B}_{19}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 12 e^{2 \beta (t) } r+36 e^{10 \beta (t) } r-e^{6 \beta (t) } (47 r+1)-r+1\right] ,\\ \mathcal {B}_{20}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 12 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)+4 e^{10 \beta (t) } (2 r+7)\right. \\&\left. \quad +\, e^{6 \beta (t) } (15-63 r)+19 (r-1)\right] ,\\ \mathcal {B}_{21}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 12 e^{2 \beta (t) } r+36 e^{10 \beta (t) } r+e^{6 \beta (t) } (47 r+1)+r-1\right] ,\\ \mathcal {B}_{22}&=\frac{1}{192} e^{-10 \beta (t) } \left[ 20 e^{8 \beta (t) } (r-1)+12 e^{2 \beta (t) } (2 r-1)+4 e^{10 \beta (t) } (2 r+7)\right. \\&\left. \quad +\, 3 e^{6 \beta (t) } (21 r-5)-19 (r-1)\right] . \end{aligned}$$

Finally, using Eq. (13), the density matrix for TQC configuration of three qubits under FG noise can be written as

$$\begin{aligned} \rho _{\textrm{TQC}}(t)=\left[ \begin{array}{cccccccc} \mathcal {C}_{1} &{} 0 &{} 0 &{}\mathcal {C}_{2} &{} 0 &{}\mathcal {C}_{2} &{}\mathcal {C}_{2} &{}\mathcal {C}_{3} \\ 0 &{}\mathcal {C}_{4} &{}\mathcal {C}_{5} &{} 0 &{}\mathcal {C}_{5} &{} 0 &{}\mathcal {C}_{6} &{}\mathcal {C}_{7} \\ 0 &{}\mathcal {C}_{5} &{}\mathcal {C}_{4} &{} 0 &{}\mathcal {C}_{5} &{}\mathcal {C}_{6} &{} 0 &{}\mathcal {C}_{7} \\ \mathcal {C}_{2} &{} 0 &{} 0 &{}\mathcal {C}_{8} &{}\mathcal {C}_{6} &{}\mathcal {C}_{9} &{}\mathcal {C}_{9} &{} 0 \\ 0 &{}\mathcal {C}_{5} &{}\mathcal {C}_{5} &{}\mathcal {C}_{6} &{}\mathcal {C}_{4} &{} 0 &{} 0 &{}\mathcal {C}_{7} \\ \mathcal {C}_{2} &{} 0 &{}\mathcal {C}_{6} &{}\mathcal {C}_{9} &{} 0 &{}\mathcal {C}_{8} &{}\mathcal {C}_{9} &{} 0 \\ \mathcal {C}_{2} &{}\mathcal {C}_{6} &{} 0 &{}\mathcal {C}_{9} &{} 0 &{}\mathcal {C}_{9} &{}\mathcal {C}_{8} &{} 0 \\ \mathcal {C}_{3} &{}\mathcal {C}_{7} &{}\mathcal {C}_{7} &{} 0 &{}\mathcal {C}_{7} &{} 0 &{} 0 &{} \mathcal {C}_{10} \\ \end{array} \right] \end{aligned}$$
(A.3)

where

$$\begin{aligned} \mathcal {C}_{1}&=\frac{1}{8} e^{-6 \beta (t) } \left[ e^{6 \beta (t) }-e^{4 \beta (t) } (r-1)+e^{2 \beta (t) } (4 r-1)+r-1\right] ,\\ \mathcal {C}_{2}&=-\frac{1}{12} e^{-6 \beta (t) } \left( e^{2 \beta (t) }-1\right) \left( e^{2 \beta (t) }+1\right) ^2 (r-1),\\ \mathcal {C}_{3}&=\frac{1}{8} \left( 3 e^{-4 \beta (t) }+1\right) r,\\ \mathcal {C}_{4}&=\frac{1}{24} e^{-6 \beta (t) } \left( e^{2 \beta (t) }+1\right) \left[ 3 e^{4 \beta (t) }-e^{2 \beta (t) } (r+2)-3 r+3\right] ,\\ \mathcal {C}_{5}&=-\frac{1}{12} e^{-6 \beta (t) } \left[ e^{2 \beta (t) }+e^{4 \beta (t) }+e^{6 \beta (t) }+1\right] (r-1),\\ \mathcal {C}_{6}&=\frac{1}{8} \left( 1-e^{-4 \beta (t) }\right) ,\\ \mathcal {C}_{7}&=-\frac{1}{12} e^{-6 \beta (t) } \left( e^{2 \beta (t) }-1\right) ^2 \left( e^{2 \beta (t) }+1\right) (r-1),\\ \mathcal {C}_{8}&=\frac{1}{24} e^{-6 \beta (t) } \left( e^{2 \beta (t) }-1\right) \left[ 3 e^{4 \beta (t) }+e^{2 \beta (t) } (r+2)-3 r+3\right] ,\\ \mathcal {C}_{9}&=-\frac{1}{12} e^{-6 \beta (t) } \left[ e^{2 \beta (t) }-e^{4 \beta (t) }+e^{6 \beta (t) }-1\right] (r-1),\\ \mathcal {C}_{10}&=\frac{1}{8} e^{-6 \beta (t) } \left[ e^{6 \beta (t) }+e^{4 \beta (t) } (r-1)+e^{2 \beta (t) } (4 r-1)-r+1\right] . \end{aligned}$$

Now, we provide the analytical results for entanglement witness and linear entropy for the dynamics of the physical model described in the above density matrices. For three non-interacting qubits initially created in a mixture state and assumed in SQC, DQC, and TQC couplings given in Eqs. (A.1), (A.2), and (A.3) under the impact of FG noise, we use Eqs. (1) and (2) to analyze entanglement witness and linear entropy. The analytical expressions are given as

$$\begin{aligned} E[\rho _{\textrm{SQC}}(t)]&=\frac{1}{32} e^{-18 \beta (t) } \left[ 6 e^{10 \beta (t) } \left( 5 r^2-4 r+1\right) \right. \nonumber \\&\left. \quad +\, 2 e^{18 \beta (t) } \left( 9 r^2-4 r-3\right) +7 e^{16 \beta (t) } (r-1)^2+9 (r-1)^2\right] , \end{aligned}$$
(A.4)
$$\begin{aligned} L[\rho _{\textrm{SQC}}(t)]&=1-\left\{ \frac{1}{32} e^{-36 \beta (t) } \left[ 7 e^{32 \beta (t) } (r-1)^2+6 e^{20 \beta (t) } (r (5 r-4)+1)\right. \right. \nonumber \\&\left. \left. \quad +\, 2 e^{36 \beta (t) } (r (9 r-4)+5)+9 (r-1)^2\right] \right\} , \end{aligned}$$
(A.5)

For the DQC configurations, the entanglement witness and linear entropy results are

$$\begin{aligned} E[\rho _{\textrm{DQC}}(t)]&=\frac{1}{576} e^{-10 \beta (t) } \left[ a+b+36 e^{2 \beta (t) } \left( 5 r^2-4 r+1\right) \right. \nonumber \\&\left. \quad +\, 124 e^{8 \beta (t) } (r-1)^2+163 (r-1)^2\right] , \end{aligned}$$
(A.6)
$$\begin{aligned} L[\rho _{\textrm{DQC}}(t)]&=1-\left\{ \frac{1}{576} e^{-20 \beta (t)} \left[ 124 e^{16 \beta (t) } (r-1)^2\right. \right. \nonumber \\&\left. \left. \quad +\, 36 e^{4 \beta (t) } (r (5 r-4)+1)+163 (r-1)^2+c + d\right] \right\} , \end{aligned}$$
(A.7)

where

$$\begin{aligned} a&=4 e^{10 \beta (t) } (55 r^2-44 r-29),&b&=3 e^{6 \beta (t) } (155 r^2-86 r+27),\\ c&=4 e^{20 \beta (t) } (55 r^2-44 r+43),&d&=3 e^{12 \beta (t) } (155 r^2-86 r+27). \end{aligned}$$

Similarly, for the TQC configurations, the analytical expressions of entanglement witness and linear entropy are as follows

$$\begin{aligned} E[\rho _{\textrm{TQC}}(t)]&=\frac{1}{24} e^{-6 \beta (t) } \left[ 5 e^{4 \beta (t) } (r-1)^2+e^{6 \beta (t) } (r (7 r-8)-5)\right. \nonumber \\&\left. \quad +\, e^{2\beta (t) } (r (29 r-16)+5)+7 (r-1)^2\right] , \end{aligned}$$
(A.8)
$$\begin{aligned} L[\rho _{\textrm{TQC}}(t)]&=1-\left\{ \frac{1}{24} e^{-12 \beta (t) } \left[ 5 e^{8 \beta (t) } (r-1)^2+e^{12 \beta (t) } (r (7 r-8)+7)\right. \right. \nonumber \\&\left. \left. \quad +\, e^{4 \beta (t) } (r (29 r-16)+5)+7 (r-1)^2\right] \right\} . \end{aligned}$$
(A.9)

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Rahman, A.U., Haddadi, S., Javed, M. et al. Entanglement witness and linear entropy in an open system influenced by FG noise. Quantum Inf Process 21, 368 (2022). https://doi.org/10.1007/s11128-022-03720-5

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