## 2 The Physical Model and Its Master Equation
In ESR method, we used the canonical transform: \(W(t)=e^{i\hat{H}t}\hat{\rho}(t)e^{-i\hat{H}t}\), the master equation of Equation (5) becomes  
\[\dot{W} = \frac{\gamma}{2}\sum_{i=A,B}e^{i\hat{H}t}[2|1_{i}\rangle\langle 1_{i }|\hat{\rho}(t)|1_{i}\rangle\langle 1_{i}|-|1_{i}\rangle\langle 1_{i}|\hat{ \rho}(t)-\hat{\rho}(t)|1_{i}\rangle\langle 1_{i}|]e^{-i\hat{H}t}. \tag{6}\]  
Then, we write the two-qubit operators in Equation (6) in terms ESR. Consequently, the Equation (6) can be written as  
\[\frac{dW}{dt} = \frac{\gamma}{2}\bigg{\{}|\Psi_{1}\rangle\langle\Psi_{1}|W|\Psi_ {1}\rangle\langle\Psi_{1}|+|\Psi_{1}\rangle\langle\Psi_{4}|W|\Psi_{4}\rangle \langle\Psi_{1}|+|\Psi_{4}\rangle\langle\Psi_{1}|W|\Psi_{1}\rangle\langle\Psi_ {4}| \tag{7}\] \[+|\Psi_{4}\rangle\langle\Psi_{4}|W|\Psi_{4}\rangle\langle\Psi_{4 }|+|\Psi_{2}\rangle\langle\Psi_{2}|W|\Psi_{2}\rangle\langle\Psi_{2}|+|\Psi_{3 }\rangle\langle\Psi_{3}|W|\Psi_{3}\rangle\langle\Psi_{3}|]\] \[+|\Psi_{3}\rangle\langle\Psi_{2}|W|\Psi_{2}\rangle\langle\Psi_{3 }|+|\Psi_{2}\rangle\langle\Psi_{3}|W|\Psi_{3}\rangle\langle\Psi_{2}|\] \[-\sum_{n=1}^{4}[W|\Psi_{n}\rangle\langle\Psi_{n}|+|\Psi_{n} \rangle\langle\Psi_{n}|W\bigg{\}}.\]  
We assume that the two qubits are initially started with the important class of mixed Werner state:  
\[\rho^{AB}(0) = p|\varphi\rangle\langle\varphi|+(1-p)|1_{A}1_{B}\rangle\langle 1_{A }1_{B}|,\] \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]0 \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]1where \(0\leq p\leq 1\) is the purity indicator of the initial two-qubit states. The state \(|\varphi\rangle\) is taken to describes the maximally correlated non-symmetric Bell state: \(|\varphi\rangle=\frac{1}{\sqrt{2}}[|10\rangle+|01\rangle]\), that is not the eigenstate Hamiltonian of Equation (1). With the chosen class of the Werner state of Equation (13), we can study the generation of the NLCs when the system starts with uncorrected states, and investigate the robustness of the initial NLCs against the interactions. The Werner states have manageable features that they are robust against nonlocal loss [33]. The generation of Werner states and maximally entangled mixed states are introduced by several universal sources [34; 35]  
The elements of the density matrix \(W\) of the case of \(i=j\), \(W_{i}(t)=\langle\Psi_{i}|W(t)|\Psi_{i}\rangle\), are calculated by solving the simple differential equations:  
\[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]2  
whereas in the the case of \(i\
eq j\), \(W_{ij}(t)=\langle\Psi_{i}|W(t)|\Psi_{j}\rangle\), are given by  
\[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]3  
By using Equations (13)-(15), the density matrix of the two qubits \(\rho^{AB}(t)\) in the basis two-qubit states: \(\{|\varpi_{i}\rangle\}\), is given by  
\[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]4  
the non-vanishing elements of \(\rho_{ij}(t)\) are  
\[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]5 \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]6 \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]7 \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]8 \[E_{3} = J_{z}-\sqrt{4\chi_{z}^{2}+S^{2}},\quad E_{4}=-J_{x}+J_{y}+J_{z},\]9 \[\frac{d\hat{\rho}(t)}{dt} = -i[\hat{H},\hat{\rho}(t)]\]0  
\(\beta=2\sqrt{4\chi_{x}^{2}+S^{2}}\), and the difference \(\Delta=J_{x}-J_{y}\) stands for the anisotropy parameter. The analytical solution of the density matrix of the two qubits will be used to investigate the different NLCs.