Tripartite quantum correlations in XXZ Heisenberg spin chain with Dzyaloshinskii–Moriya interaction

Abstract

Quantum correlations as the information resources play an important role in quantum computing and information processing. Most of the previous studies on quantum correlation dynamics are limited to bipartite systems, but only partial solutions are known to detect and quantify of multipartite systems. In this paper, we exploit the notions of tripartite quantum discord ${\mathcal {D}}^{(3)}$ and tripartite negativity $\tau _{ABC}$ as a measure of quantum correlations in a model of a three-qubit anisotropic Heisenberg XXZ chain in the presence of an external magnetic field and Dzyaloshinskii–Moriya (DM) interaction. We compare the dynamics of thermal ${\mathcal {D}}^{(3)}$ with that of thermal correlation quantified by $\tau _{ABC}$ in thermal equilibrium with external magnetic field and DM interaction. Our results show that the magnetic field, the anisotropic coupling coefficient, and the DM interaction parameter are all efficient control parameters for quantum correlation creation and enhancement. Taking the effect of the intrinsic decoherence into account, the dynamics of ${\mathcal {D}}^{(3)}$ and the tripartite negativity $\tau _{ABC}$ are strongly affected by the input configuration of initial states. We find that the tripartite quantum correlations dynamics for the initial GHZ-type states are only related to the external magnetic field parameter. Conversely, for the initial W-type states, the tripartite quantum correlations dynamics is independent of the external magnetic field parameter but depended on the rest parameters of the system. Furthermore, the results also show that the tripartite quantum correlations quantified by ${\mathcal {D}}^{(3)}$ exhibit more robust against the intrinsic decoherence than that of tripartite negativity $\tau _{ABC}$.

Appendix A

In this section, we give the explicit expressions of the $\tau _{ABC}(\rho )$ and ${\mathcal {D}}^{3}(\rho )$ for thermal equilibrium state given by Eq. (13). It is easy to conclude that the eigenvalues of the partial transpose of the total density matrix given by Eq. (13) have the same form. Hence, the analytical form of $\tau _{ABC}(\rho )$ reduces to

$$\begin{aligned} \tau _{ABC}[\rho (T)]=\Sigma _{i=1}^{8}|\lambda _{i}|-1 \end{aligned}$$

(A1)

where $\lambda _{1}=\frac{\Sigma _{i=2,4,6}m_{i}}{3M}$, $\lambda _{2}=\frac{\Sigma _{i=1,3,5}m_{i}}{3M}$, $\lambda _{i}(i=3,4,5)$, and $\lambda _{j}(j=6,7,8)$ are the root of

$$\begin{aligned}{} & {} \frac{1}{27}\left[ 3M\alpha \alpha ^{*}e^{-\frac{E_{7}}{KT}}-3M (\Sigma _{i=2,4,6}m_{i})^2 e^{-\frac{E_{7}}{KT}}\right. \nonumber \\{} & {} \quad \left. +2M^3\beta \beta ^{*}(\Sigma _{i=2,4,6}m_{i})-M^3(\alpha \beta ^2+\alpha ^{*}\beta ^{*2})\right] \nonumber \\{} & {} \quad +\frac{1}{9}\left[ 6(\Sigma _{i=2,4,6}m_{i}) e^{-\frac{E_{7}}{KT}}+M^2(\Sigma _{i=2,4,6}m_{i})^2\right. \nonumber \\{} & {} \quad \left. -M^2(\alpha \alpha ^{*}+2\beta \beta ^{*})\right] \sharp 1-\frac{1}{3M}\left[ 3e^{\frac{-E_{7}}{KT}}+2M^2(\Sigma _{i=2,4,6}m_{i})\right] \sharp 1^{2}+\sharp 1^{3}, \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} \frac{1}{27}\left[ 3M\beta \beta ^{*}e^{-\frac{E_{8}}{KT}}-3M (\Sigma _{i=1,3,4}m_{i})^2 e^{-\frac{E_{8}}{KT}}\right. \nonumber \\{} & {} \quad \left. +2M^3\alpha \alpha ^{*}(\Sigma _{i=1,3,5}m_{i})-M^3(\beta \alpha ^2+\beta ^{*}\alpha ^{*2})\right] \nonumber \\{} & {} \quad +\frac{1}{9}\left[ 6(\Sigma _{i=1,3,5}m_{i}) e^{-\frac{E_{8}}{KT}}+M^2(\Sigma _{i=1,3,5}m_{i})^2\right. \nonumber \\{} & {} \left. \quad -M^2(2\alpha \alpha ^{*}+\beta \beta ^{*})\right] \sharp 1-\frac{1}{3M}\left[ 3e^{\frac{-E_{8}}{KT}}+2M^2(\Sigma _{i=1,3,5}m_{i})\right] \sharp 1^{2}+\sharp 1^{3}, \end{aligned}$$

(A3)

respectively. On the other hand, according to Eq. (9), the expression of ${\mathcal {D}}^{3}(\rho )$ is calculated as follows:

$$\begin{aligned} {\mathcal {D}}^{3}[\rho (T)]= & {} \frac{M}{3}\left( \Sigma _{i=2,4,6}m_{i}+\alpha +\alpha ^{*}\right) \log _{2}\left[ \frac{M}{3}(\Sigma _{i=2,4,6}m_{i}+\alpha +\alpha ^{*})\right] \nonumber \\{} & {} -M(\Sigma _{i=2,4,6}m_{i})\log _{2}\left[ \frac{\Sigma _{i=2,4,6}m_{i}}{3}\right] \nonumber \\{} & {} +\frac{M}{3}(\Sigma _{i=1,3,5}m_{i}+\beta +\beta ^{*})\log _{2}\left[ \frac{M}{3}(\Sigma _{i=1,3,5}m_{i}+\beta +\beta ^{*})\right] \nonumber \\{} & {} -M(\Sigma _{i=1,3,5}m_{i})\log _{2}\left[ \frac{\Sigma _{i=1,3,5}m_{i}}{3}\right] \nonumber \\{} & {} +\frac{1}{6}[M(2\Sigma _{i=2,4,6}m_{i}-\alpha -\alpha ^{*})\nonumber \\{} & {} -i\sqrt{3}M(\alpha -\alpha ^{*})]\log _{2}\frac{1}{6}\left[ M(2\Sigma _{i=2,4,6}m_{i}-\alpha -\alpha ^{*})-i\sqrt{3}M(\alpha -\alpha ^{*})\right] \nonumber \\{} & {} +\frac{1}{6}\left[ M\left( 2\Sigma _{i=2,4,6}m_{i}-\alpha -\alpha ^{*}\right) \right. \nonumber \\{} & {} \left. +i\sqrt{3}M(\alpha -\alpha ^{*})\right] \log _{2}\frac{1}{6}\left[ M(2\Sigma _{i=2,4,6}m_{i}-\alpha -\alpha ^{*})+i\sqrt{3}M(\alpha -\alpha ^{*})\right] \nonumber \\{} & {} +\frac{1}{6}[M(2\Sigma _{i=1,3,5}m_{i}-\beta -\beta ^{*})\nonumber \\{} & {} -i\sqrt{3}M(\beta -\beta ^{*})]\log _{2}\frac{1}{6}\left[ M(2\Sigma _{i=1,3,5}m_{i}-\beta -\beta ^{*})-i\sqrt{3}M(\beta -\beta ^{*})\right] \nonumber \\{} & {} +\frac{1}{6}\left[ M(2\Sigma _{i=1,3,5}m_{i}-\beta -\beta ^{*})\right. \nonumber \\{} & {} \left. +i\sqrt{3}M(\beta -\beta ^{*})\right] \log _{2}\frac{1}{6}\left[ M(2\Sigma _{i=1,3,5}m_{i}\right. \nonumber \\{} & {} \left. -\beta -\beta ^{*})+i\sqrt{3}M(\beta -\beta ^{*})\right] \end{aligned}$$

(A4)

Following, we give the explicit time-evolved density matrix of the system when the initial state is prepared in W-type state. According to Eq. (17), the time-evolved density matrix is given

$$\begin{aligned} \rho _{W}(t)= \frac{1}{8}\left( \begin{array}{ c c c c c c c c cl r } 1-r &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} \mu (t) &{} \Delta (t) &{} 0 &{} \delta (t) &{} 0 &{}0 &{}0\\ 0 &{} \Delta ^{*}(t) &{} \Xi (t) &{} 0 &{} \nu (t) &{} 0 &{}0 &{}0\\ 0 &{} 0 &{} 0 &{} 1-r &{} 0 &{}0 &{}0 &{}0\\ 0 &{} \delta ^{*}(t) &{} \nu ^{*}(t) &{} 0 &{} \kappa (t) &{} 0 &{}0 &{}0\\ 0 &{} 0 &{}0 &{} 0 &{} 0 &{} 1-r &{}0 &{}0\\ 0 &{} 0 &{}0 &{}0 &{} 0 &{} 0 &{}1-r &{}0\\ 0 &{} 0 &{}0 &{} 0 &{} 0 &{} 0 &{}0 &{}1-r\\ \end{array} \right) , \end{aligned}$$

(A5)

where

$$\begin{aligned} \mu (t)= & {} \frac{1}{72} \{\left( 12-8 \sqrt{2}\right) \cos \left( 2 \sqrt{3} D_{z}t\right) e^{-6 D_{z}^2 \text { t }\Gamma }\nonumber \\{} & {} +\,4 \sqrt{2}\cos \left[ \left( 3 J-\sqrt{3} D_{z}\right) t\right] e^{-\frac{1}{2} \text { t }\Gamma } \left( 3 J-\sqrt{3} D_{z}\right) ^2\nonumber \\{} & {} +\,4 \sqrt{2} \cos \left[ \left( \sqrt{3} D_{z}+3 J\right) t\right] e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3} D_{z}+3 J\right) ^2}+15 r+9\} \end{aligned}$$

(A6)

$$\begin{aligned} \Xi (t)= & {} \frac{1}{72} \{\left( 12-8 \sqrt{2}\right) r e^{-6 D_{z}^2 \text { t }\Gamma } \sin \left( 2 \sqrt{3} D_{z}t-\frac{\pi }{6}\right) \nonumber \\{} & {} -\,4 \sqrt{2} r e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3} D_{z}+3 J\right) ^2} \sin \left[ t \left( \sqrt{3} D_{z}+3 J\right) +\frac{\pi }{6}\right] \nonumber \\{} & {} +\,4 \sqrt{2} r e^{-\frac{1}{2} \text { t }\Gamma \left( 3 J-\sqrt{3} D_{z}\right) ^2} \sin \left[ t \left( 3 J-\sqrt{3} D_{z}\right) -\frac{\pi }{6}\right] +15 r+9\} \end{aligned}$$

(A7)

$$\begin{aligned} \kappa (t)= & {} \frac{1}{72} \{\left( 8 \sqrt{2}-12\right) r e^{-6 D_{z}^2 \text { t }\Gamma } \sin \left( 2 \sqrt{3} D_{z} t+\frac{\pi }{6}\right) \nonumber \\{} & {} -\,4 \sqrt{2} r e^{-\frac{1}{2} \text { t }\Gamma \left( 3 J-\sqrt{3} D_{z}\right) ^2} \sin \left[ t \left( 3 J-\sqrt{3} D_{z}\right) +\frac{\pi }{6}\right] \nonumber \\{} & {} +\,4 \sqrt{2} r e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3} D_{z}+3 J\right) ^2} \sin \left[ t \left( \sqrt{3} D_{z}+3 J\right) -\frac{\pi }{6}\right] +15 r+9\} \end{aligned}$$

(A8)

$$\begin{aligned} \Delta (t)= & {} \frac{1}{36} r \{\left( 4 \sqrt{2}-6\right) e^{-6 D_{z}^2 \text { t }\Gamma } \sin \left( 2 \sqrt{3} D_{z}t+\frac{\pi }{6}\right) \nonumber \\{} & {} +\,i \sqrt{2} e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3} D_{z}+3 J\right) ^2} \left[ 2 \sin \left( \sqrt{3} D_{z}+3 J\right) t-\sqrt{3} e^{ -i\left( \sqrt{3} D_{z}+3 J\right) t}\right] \nonumber \\{} & {} +\,i \sqrt{2} e^{-\frac{1}{2} \text { t }\Gamma \left( 3 J-\sqrt{3} D_{z}\right) ^2} \left[ 2 \sin \left( 3 J-\sqrt{3} D_{z}\right) t\right. \nonumber \\{} & {} \left. +\,\sqrt{3} e^{ -i\left( 3 J-\sqrt{3} D_{z}\right) t}\right] +6 \sqrt{2}+3\} \end{aligned}$$

(A9)

$$\begin{aligned} \delta (t)= & {} \frac{1}{36} r \{ \left( 6-4 \sqrt{2}\right) e^{-6 D_{z}^2 \text { t }\Gamma } \sin \left( 2 \sqrt{3} D_{z} t-\frac{\pi }{6}\right) \nonumber \\{} & {} +\,\sqrt{2} \left( \sqrt{3} i-e^{-\frac{1}{2} \text { t }\Gamma \left( 3 J-\sqrt{3} D_{z}\right) ^2}\right) \sin \left[ t \left( 3 J-\sqrt{3} D_{z}\right) -\frac{\pi }{6}\right] \nonumber \\{} & {} +\,\sqrt{2} \left( i \sqrt{3}+e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3} D_{z}+3 J \right) ^2}\right) \sin \left[ t \left( \sqrt{3} D_{z}+3 J\right) +\frac{\pi }{6}\right] \nonumber \\{} & {} +\,6 \sqrt{2}+3\} \end{aligned}$$

(A10)

$$\begin{aligned} \nu (t)= & {} \frac{1}{36} r \{\left( 6-4 \sqrt{2}\right) e^{-6 D_{z}^2 \text { t }\Gamma } \cos \left( 2 \sqrt{3} D_{z} t\right) \nonumber \\{} & {} -\,\sqrt{2}\left( 1+i \sqrt{3}\right) e^{-\frac{1}{2} \text { t }\Gamma \left( 3 J-\sqrt{3} D_{z}\right) ^2} \cos \left[ t \left( 3 J-\sqrt{3} D_{z}\right) \right] \nonumber \\{} & {} -\,\sqrt{2}\left( 1-\sqrt{3} i\right) e^{-\frac{1}{2} \text { t }\Gamma \left( \sqrt{3}D_{z}+3 J\right) ^2} \cos \left[ t \left( \sqrt{3} D_{z}+3 J\right) \right] \nonumber \\{} & {} +\,6 \sqrt{2}+3\} \end{aligned}$$

(A11)