On the Dynamics of Correlations in 2 ⊗ 3 Heisenberg Chains with Inhomogeneous Magnetic Field

Abstract

The majority of entanglement measurement research concentrates on systems that can be represented by two interacting qubits. This is because there aren’t many researches that show analytical calculations for entanglement in systems with spin s > 1/2. In ref. (S.L.L. Silva Int. J. Theor. Phys. 60, 3861, 2021) the author used distance between states to study thermal entanglement in 2 ⊗ 3 Heisenberg chain. In this work various dynamic correlations as well as quantum Fisher information phenomena are studied in a 2 ⊗ 3 Heisenberg chain. First, we introduce our model for spin-$\frac {1}{2}$ and spin-1 located in inhomogeneous magnetic field, then we obtain the eigenvalues and eigenvectors of Hamiltonian to deduce the thermal state matrix. On the other hand, the dynamic behavior of the system is studied by considering the system initially prepared in the thermal state. Also, the influence of both nonuniform and uniform external magnetic field on the degree of entanglement between the spin-$\frac {1}{2}$ and spin-1 are discussed through some correlation functions, namely, concurrence, von Neumann entropy and geometric quantum discord. Finally we study the quantum Fisher information phenomena.

Appendices

Appendix A: Concurrence Vector

First, as mentioned previously, by using the basis of spin-1 (eigenvectors of S_z)

$$ |1\rangle=\left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right), |0\rangle=\left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right) |-1\rangle=\left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) $$

(A1)

and, by using (3.9) together with (3.31), we get

$$ \hat{R}_{1}=\left( \begin{array}{cccccc} \rho_{11} \rho_{55}^{*} & 0 & 0 & 0 & 0 & 0 \\ 0 & |\rho_{24}|^{2}+\rho_{22} \rho_{44}^{*} & 0 & \rho_{24} \rho_{22}^{*}+\rho_{22} \rho_{42}^{*} & 0 & 0 \\ 0 & 0 & 0 & 0 & \rho_{11} \rho_{35} & 0 \\ 0 & \rho_{44} \rho_{24}^{*}+\rho_{42} \rho_{44}^{*} & 0 & \rho_{44} \rho_{22}^{*}+|\rho_{42}|^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \rho_{11} \rho_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right) $$

(A2)

with nonzero eigenvalues

$$ \begin{array}{@{}rcl@{}} {\mu_{1}^{1}}& = &({\mu_{1}^{2}})^{*}=\rho_{11} \rho_{55}\\ \mu_{1}^{3,4}& = & \frac{\rho_{44} \rho_{22}^{*} + \rho_{22} \rho_{44}^{*} + |\rho_{24}|^{2} + |\rho_{42}|^{2}\!\pm\!\sqrt{(\rho_{44} \rho_{22}^{*} + \rho_{22} \rho_{44}^{*} + |\rho_{24}|^{2} + |\rho_{42}|^{2})^{2} - 4|\rho_{24}\rho_{42} - \rho_{22} \rho_{44}|^{2}}}{2} \end{array} $$

(A3)

Also,

$$ \hat{R}_{2}=\left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \rho_{22} \rho_{66} & 0 & 0 & 0 & 0 \\ 0 & 0 & |\rho_{35}|^{2}+\rho_{33} \rho_{55}^{*} & 0 & \rho_{35} \rho_{33}^{*}+\rho_{33} \rho_{53}^{*} & 0 \\ 0 & \rho_{42} \rho_{66} & 0 & 0 & 0 & 0 \\ 0 & 0 & \rho_{55} \rho_{35}^{*}+\rho_{53} \rho_{55}^{*} & 0 & \rho_{55} \rho_{33}^{*}+|\rho_{53}|^{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \rho_{66} \rho_{22}^{*} \end{array} \right) $$

(A4)

with nonzero eigenvalues

$$ \begin{array}{@{}rcl@{}} {\mu_{2}^{1}}& = &({\mu_{2}^{2}})^{*}=\rho_{22} \rho_{66}\\ \mu_{2}^{3,4}& = & \frac{\rho_{55} \rho_{33}^{*} + \rho_{33} \rho_{55}^{*} + |\rho_{35}|^{2} + |\rho_{53}|^{2}\!\pm\!\sqrt{(\rho_{55} \rho_{33}^{*} + \rho_{33}\rho_{55}^{*} + |\rho_{35}|^{2} + |\rho_{53}|^{2})^{2} - 4|\rho_{35}\rho_{53} - \rho_{33} \rho_{55}|^{2}}}{2} \end{array} $$

(A5)

and,

$$ \hat{R}_{3}=\left( \begin{array}{cccccc} \rho_{11} \rho_{66} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \rho_{24} \rho_{33}^{*} & 0 & 0 \\ 0 & 0 & \rho_{33} \rho_{44}^{*} & 0 & 0 & 0 \\ 0 & 0 & 0 & \rho_{44} \rho_{33}^{*} & 0 & 0 \\ 0 & 0 & \rho_{53} \rho_{44}^{*} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \rho_{11} \rho_{66} \end{array}\right) $$

(A6)

with nonzero eigenvalues,

$$ \begin{array}{@{}rcl@{}} {\mu_{3}^{1}}&=&{\mu_{3}^{2}}=\rho_{11} \rho_{66}\\ {\mu_{3}^{3}}&=&\left( {\mu_{3}^{4}}\right)^{*}=\rho_{33} \rho_{44}^{*} \end{array} $$

(A7)

Appendix B: Geometric Quantum Discord

By using (3.9) and, we recall that

$$ \begin{array}{@{}rcl@{}} x_{1}&=&x_{2}=0, \qquad x_{3}=\rho_{11}+\rho_{22}+\rho_{33}-\rho_{44}-\rho_{55}-\rho_{66} \end{array} $$

(B1)

$$ \begin{array}{@{}rcl@{}} r_{11}&=&r_{22}=\frac{1}{\sqrt{2}}\left( \rho_{24}+\rho_{42}+\rho_{35}+\rho_{53}\right)\\ r_{21}&=&r_{12}^{*}=-\frac{i}{\sqrt{2}}\left( \rho_{24}+\rho_{35}-\rho_{42}-\rho_{53}\right)\\ r_{33}&=&\rho_{11}+\rho_{66}-\rho_{33}-\rho_{44} \end{array} $$

(B2)

then, we get

$$ \hat{X}=\left( \begin{array}{cccccc} 0 \\ 0\\ x_{3} \end{array}\right) $$

(B3)

and

$$ \hat{R}=\left( \begin{array}{cccccc} r_{11} & r_{12} & 0 \\ r_{12}^{*} & r_{22} & 0 \\ 0 & 0 &r_{33} \end{array} \right) $$

(B4)

So, we obtain

$$ \hat{K}=\left( \begin{array}{cccccc} k_{1} & 0 & 0 \\ 0 & k_{1} & 0 \\ 0 & 0 &k_{2} \end{array} \right)\\ $$

(B5)

where, $k_{1}=\frac {1}{2}(\rho _{24}+\rho _{35})(\rho _{42}+\rho _{53})$ and $k_{2}=K_{\max \limits }=\frac {1}{6}(2(\rho _{11}+\rho _{22}+\rho _{33})-1)^{2}+\frac {1}{4}(\rho _{11}+\rho _{66}-\rho _{33}-\rho _{44})$ From all of the above we can deduce that

$$ \begin{array}{@{}rcl@{}} D_{G}(\rho)=2 k_{1}=.(\rho_{24}+\rho_{35})(\rho_{42}+\rho_{53}). \end{array} $$

(B6)