# Entanglement and Entropy of a Three-Qubit System Interacting with a Quantum Spin Environment
## 6 Concurrence  
In the case of the proposed system, which consists of two quantum bits, the concurrence of the system can be calculated as proposed in [42] to calculate the quantum correlation. Firstly, in the pure state case, we can write the concurrence is as follows:\(C(\rho)=\max\{0,\sqrt{\xi_{1}}-\sqrt{\xi_{2}}-\sqrt{\xi_{3}}-\sqrt{\xi_{4}}\}\), where \(\xi_{k}\) are the eigenvalues of the matrix \(\varrho=\rho\bar{\rho}=\rho(\sigma_{y}^{1}\otimes\sigma_{y}^{2})\rho^{\intercal }(\sigma_{y}^{1}\otimes\sigma_{y}^{2})\), an arrangement in descending order. Furthermore, the transposition of density matrix \(\rho\) is denoted by \(\rho^{\intercal}\). It is obvious that the value of concurrence is bounded between zero and one. We are now explaining the results we have been able to deduce through the figures we have made. We discuss the effect of changing parameters on the applications we are studying. In Figure 1, we discuss the case of a system that consists of three quantum bits under the influence of a quantum spin environment. We study the effects of the temperature \(T\) on the behavior of the three-particle residual entanglement \(N_{abc}\), the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\) in case of \(\epsilon_{0}=2\), \(q=q_{0}=1\), \(\omega=0\), \(\alpha=0\) and \(\beta=\frac{\pi}{2}\). We study this case at \(T=0.1\), \(T=1\) and \(T=3\). The curves of entanglement \(N_{abc}\) at \(T=0.1\) and \(T=1\) have regular and periodic oscillations while at \(T=3\), they lose their regularity. The maximum values of the entanglement decrease with increasing temperature \(T\), while the minimum value remains unchanged. In the case of the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\), we also notice the regularity of the curves and their periodicity, and then they are irregular at \(T=3\). But it is clear here that both the maximum and the minimum value of the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\) increases with increasing temperature \(T\). That is, we note that by increasing the temperature the purity of the atoms and the degree of entanglement between the atoms and some decrease. So it is normal and expected that the increase will occur for the entropy. In Figures 2 and 3, we discuss the case of a two-qubit system, where we get to the case of the two atoms by taking the trace of the density matrix operator \(\rho_{P}\), \((\rho_{ij}(t)=Tr_{k}(\rho_{P}(t));i,j\) and \(k\) are equal; \(a,b,c\), \(i\
eq j\
eq k\), \(\rho_{P}=\rho_{abc})\). In Figure 2, we study the influence of the temperature \(T\) on the behavior of the concurrence \(C(\rho)\), the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\) in case of \(\epsilon_{0}=2\), \(q=q_{0}=1\), \(\omega=0\), \(\alpha=0\) and \(\beta=\frac{\pi}{2}\). We study this case at \(T=0.1\), \(T=1\) and \(T=2\). At \(T=0.1\) and \(T=1\), the curves of the concurrence \(C(\rho)\) are almost regular, but at \(T=2\), the regularity of curves is slightly decreased. The maximum values of the concurrence \(C(\rho)\) decrease from \(C(\rho)=0.7\) to \(C(\rho)=0.3\) and the minimum value keeps its stability. For the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\), at \(T=0.1\) the curves are regular, and then the regularity of the curves gradually decreases by increasing the temperature \(T\) to \(T=1\) and then to \(T=2\). Both the maximum and minimum values increase significantly with increasing temperature \(T\). We also notice an increase in the number of peaks for the linear entropy \(P_{P}(t)\). In Figure 3, we study the influence of the coupling constant \(\epsilon_{0}\) between the spin- qubit system, and an external magnetic field along the \(Z\) direction on the behavior of the concurrence \(C(\rho)\), the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\) in case of \(T=1\), \(q=q_{0}=1\), \(\omega=0\), \(\alpha=0\) and \(\beta=\frac{\pi}{2}\).