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Quantum correlations of tripartite entangled states under Gaussian noise

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Abstract

We investigate the preservation of quantum coherence and entanglement carried by the three non-interacting qubits during their evolution in the presence of an external fluctuating field characterized by a classical Gaussian noise. Initially, the three non-interacting qubits are considered in two different maximally entangled states, namely Greenberger–Horne–Zeilinger (\(\mathcal {X}_{G}\)) and Werner \((\mathcal {X}_{W})\) state. Besides this, we study the time evolution of the two states in three different schemes, namely: common, mixed, and independent system–environment couplings. By deploying different measures, we show that the system–environment coupling and the Gaussian noise greatly affected the quantum correlation and coherence. We also found that the non-local correlation and coherence remain more dominant and robust in common system–environment coupling for the \(\mathcal {X_G}\) state under the Gaussian noise. Hence, this kind of feature of the \(\mathcal {X}_{G}\) state is a vital resource for the transmission of quantum information with reduced loss.

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Authors and Affiliations

  1. Quantum Optics and Quantum Information Research Group, Department of Physics, University of Malakand, Chakdara Dir, Pakistan

    Atta Ur Rahman, Muhammad Noman, Muhammad Javed & Arif Ullah

  2. The School of Information Science and Technology, Southwest Jiaotong University, Chengdu, 610031, China

    Ming-Xing Luo

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Appendix

Appendix

This section describes the detailed numerical simulations that are carried out for the time evolution for the \(\mathcal {X}_{G}\) and \(\mathcal {X}_{W}\) states under \(\mathcal {G_N}\) noise for the ces, mes and ies configurations.

1.1 Appendix 1. The case of common environment setup

Using Eq. 8 for ces model, we get the final density matrix after taking the average of the time-evolved density matrix for the \(\mathcal {X_G}\) state over the noise phase, and we get:

$$\begin{aligned} \rho _{ces}^{\mathcal {X}_G }(\varphi _1, \tau )=\left[ \begin{array}{cccccccc} h_{1} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{1} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{2} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{3} &{}\quad h_{2} \\ h_{1} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{2} &{}\quad h_{1} \end{array} \right] , \end{aligned}$$
(32)

Similarly, by using the final density matrix for \(\mathcal {X}_{W}\) state given in Eq. (8) for ces configuration, we get:

$$\begin{aligned} \rho _{ces}^{\mathcal {X}_W}(\varphi _1, \tau )=\left[ \begin{array}{cccccccc} h_{4} &{}\quad 0 &{}\quad 0 &{}\quad h_{5} &{}\quad 0 &{}\quad h_{5} &{}\quad h_{5} &{}\quad 0 \\ 0 &{}\quad h_{6} &{}\quad h_{6} &{}\quad 0 &{}\quad h_{6} &{}\quad 0 &{}\quad 0 &{}\quad h_{7} \\ 0 &{}\quad h_{6} &{}\quad h_{6} &{}\quad 0 &{}\quad h_{6} &{}\quad 0 &{}\quad 0 &{}\quad h_{7} \\ h_{5} &{}\quad 0 &{}\quad 0 &{}\quad h_{8} &{}\quad 0 &{}\quad h_{8} &{}\quad h_{8} &{}\quad 0 \\ 0 &{}\quad h_{6} &{}\quad h_{6} &{}\quad 0 &{}\quad h_{6} &{}\quad 0 &{}\quad 0 &{}\quad h_{7} \\ h_{5} &{}\quad 0 &{}\quad 0 &{}\quad h_{8} &{}\quad 0 &{}\quad h_{8} &{}\quad h_{8} &{}\quad 0 \\ h_{5} &{}\quad 0 &{}\quad 0 &{}\quad h_{8} &{}\quad 0 &{}\quad h_{8} &{}\quad h_{8} &{}\quad 0 \\ 0 &{}\quad h_{7} &{}\quad h_{7} &{}\quad 0 &{}\quad h_{7} &{}\quad 0 &{}\quad 0 &{}\quad h_{9} \end{array} \right] . \end{aligned}$$
(33)

Where

$$\begin{aligned} h_{1}=&\frac{1}{16}(5+3 e^{-8 \beta }),&h_{2}=&\frac{1}{16}t(-1+e^{-8 \beta }),\\ h_{3}=&\frac{1}{16}(1-e^{-8 \beta }),&h_{4}=&\frac{1}{32} e^{-18 \beta }(-3-6 e^{10 \beta }+3 e^{16 \beta }+6 e^{18 \beta },\\ h_{5}=&\frac{1}{32} e^{-18 \beta } (-3-2 e^{10 \beta }+3 e^{16 \beta }+2 e^{18 \beta }),&h_{6}=&\frac{1}{96} (10+9 e^{-18 \beta }+6 e^{-8 \beta }+7 e^{-2 \beta }),\\ h_{7}=&\frac{1}{32} e^{-18 \beta }(3-2 e^{10 \beta }-3 e^{16 \beta }+2 e^{18 \beta }),&h_{8}=&\frac{1}{96}(10-9 e^{-18 \beta }+6 e^{-8 \beta }-7 e^{-2 \beta }),\\ h_{9}=&\frac{1}{32} e^{-18 \beta }(3-6 e^{10 \beta }-3 e^{16 \beta }+6 e^{18 \beta }). \end{aligned}$$

1.2 Appendix 2. The case of mixed environment setup

By using Eq. (9) for mes, the average of the final density matrix for \(\mathcal {X}_{G}\) state in Eq. (3) is taken over the noise phase and we get;

$$\begin{aligned} \rho _{mes}^{\mathcal {X}_G}(\varphi _1,\varphi _2, \tau )=\left[ \begin{array}{cccccccc} h_{10} &{}\quad 0 &{}\quad h_{11} &{}\quad 0 &{}\quad 0 &{}\quad h_{11} &{}\quad 0 &{}\quad h_{10} \\ 0 &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad 0 \\ h_{11} &{}\quad 0 &{}\quad h_{13} &{}\quad 0 &{}\quad 0 &{}\quad h_{13} &{}\quad 0 &{}\quad h_{11} \\ 0 &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad 0 \\ 0 &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad 0 \\ h_{11} &{}\quad 0 &{}\quad h_{13} &{}\quad 0 &{}\quad 0 &{}\quad h_{13} &{}\quad 0 &{}\quad h_{11} \\ 0 &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad h_{12} &{}\quad 0 &{}\quad h_{12} &{}\quad 0 \\ h_{10} &{}\quad 0 &{}\quad h_{11} &{}\quad 0 &{}\quad 0 &{}\quad h_{11} &{}\quad 0 &{}\quad h_{10} \end{array} \right] . \end{aligned}$$
(34)

By utilizing the final density matrix Eq. (9) for the \(\mathcal {X}_{W}\) state in mes situation, we obtain:

$$\begin{aligned} \rho _{mes}^{\mathcal {X}_W}(\varphi _1,\varphi _2, \tau )=\left[ \begin{array}{cccccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{14} &{}\quad 0 &{}\quad 0 &{}\quad h_{14} &{}\quad 0 \\ 0 &{}\quad h_{15} &{}\quad h_{16} &{}\quad 0 &{}\quad h_{15} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad h_{16} &{}\quad h_{17} &{}\quad 0 &{}\quad h_{16} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ h_{14} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{18} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad h_{15} &{}\quad h_{16} &{}\quad 0 &{}\quad h_{15} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{18} &{}\quad 0 &{}\quad 0 &{}\quad h_{18} &{}\quad 0 \\ h_{14} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{18} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{array} \right] \end{aligned}$$
(35)

where

$$\begin{aligned} h_{10}=&\frac{1}{16} e^{-4 \beta } (4+e^{2 \beta }+3 e^{4 \beta }),&h_{11}=&\frac{1}{16} (-1+e^{-2 \beta }),\\ h_{12}=&-h_{11},&h_{13}=&\frac{1}{16} e^{-4 \beta } (-4+e^{2 \beta }+3 e^{4 \beta }),\\ h_{14}=&\frac{1}{12} e^{-10 \beta } (-1-e^{6 \beta }+e^{8 \beta }+e^{10 \beta }),&h_{15}=&\frac{e^{-5 \beta }}{3},\\ h_{16}=&\frac{1}{12} e^{-10 \beta } (1+e^{6 \beta }+e^{8 \beta }+e^{10 \beta }),&h_{17}=&\frac{e^{-3 \beta }}{3},\\ h_{18}=&\frac{1}{12} e^{-10 \beta } (-1+e^{6 \beta }-e^{8 \beta }+e^{10 \beta }). \end{aligned}$$

1.3 Appendix 3. The case of independent environment setup

Here, the final matrix for ies scheme is obtained from Eq. (10) by taking the average of the time-evolved density matrix for \(\mathcal {X}_{G}\) state given in Eq. (3) over the \(\mathcal {G_N}\) noise phase from Eq. (7). We followed it as:

$$\begin{aligned} \rho _{ies}^{\mathcal {X}_G}(\varphi _1,\varphi _2,\varphi _3, \tau )=\left[ \begin{array}{cccccccc} h_{19} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{19} \\ 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad h_{20} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{20} &{}\quad 0 \\ h_{19} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad h_{19} \end{array} \right] . \end{aligned}$$
(36)

For \(\mathcal {X}_{W}\) state, we use the density matrix given in Eq. (10) for the ies model and we get:

$$\begin{aligned} \rho _{ies}^{\mathcal {X}_W}(\varphi _1,\varphi _2,\varphi _3, \tau )=\left[ \begin{array}{cccccccc} h_{21} &{}\quad 0 &{}\quad 0 &{}\quad h_{22} &{}\quad 0 &{}\quad h_{22} &{}\quad h_{22} &{}\quad 0 \\ 0 &{}\quad h_{23} &{}\quad h_{24} &{}\quad 0 &{}\quad h_{24} &{}\quad 0 &{}\quad 0 &{}\quad h_{25} \\ 0 &{}\quad h_{24} &{}\quad h_{23} &{}\quad 0 &{}\quad h_{24} &{}\quad 0 &{}\quad 0 &{}\quad h_{25} \\ h_{22} &{}\quad 0 &{}\quad 0 &{}\quad h_{26} &{}\quad 0 &{}\quad h_{27} &{}\quad h_{27} &{}\quad 0 \\ 0 &{}\quad h_{24} &{}\quad h_{24} &{}\quad 0 &{}\quad h_{23} &{}\quad 0 &{}\quad 0 &{}\quad h_{25} \\ h_{22} &{}\quad 0 &{}\quad 0 &{}\quad h_{27} &{}\quad 0 &{}\quad h_{26} &{}\quad h_{27} &{}\quad 0 \\ h_{22} &{}\quad 0 &{}\quad 0 &{}\quad h_{27} &{}\quad 0 &{}\quad h_{27} &{}\quad h_{26} &{}\quad 0 \\ 0 &{}\quad h_{25} &{}\quad h_{25} &{}\quad 0 &{}\quad h_{25} &{}\quad 0 &{}\quad 0 &{}\quad h_{28} \end{array} \right] . \end{aligned}$$
(37)

Where

$$\begin{aligned} h_{19}=&\frac{1}{8}+\frac{3 e^{-4 \beta }}{8},&h_{20}=&\frac{1}{8}-\frac{e^{-4 \beta }}{8},\\ h_{21}=&\frac{1}{8} e^{-6 \beta } (-1+e^{2 \beta }) (1+e^{2 \beta })^2,&h_{22}=&\frac{1}{12} e^{-6 \beta } (-1+e^{2 \beta }) (1+e^{2 \beta })^2,\\\ h_{23}=&\frac{1}{24} e^{-6 \beta } (3+e^{2 \beta }+e^{4 \beta }+3 e^{6 \beta }),&h_{24}=&\frac{1}{12} e^{-6 \beta } (1+e^{2 \beta }+e^{4 \beta }+e^{6 \beta }),\\ h_{25}=&\frac{1}{12} e^{-6 \beta } (-1+e^{2 \beta })^2 (1+e^{2 \beta }),&h_{26}=&\frac{1}{24} e^{-6 \beta } (-3+e^{2 \beta }-e^{4 \beta }+3 e^{6 \beta }),\\ h_{27}=&\frac{1}{12} e^{-6 \beta } (-1+e^{2 \beta }-e^{4 \beta }+e^{6 \beta }),&h_{28}=&\frac{1}{8} e^{-6 \beta } (-1+e^{2 \beta })^2 (1+e^{2 \beta }). \end{aligned}$$

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Rahman, A.U., Noman, M., Javed, M. et al. Quantum correlations of tripartite entangled states under Gaussian noise. Quantum Inf Process 20, 290 (2021). https://doi.org/10.1007/s11128-021-03231-9

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