# Entanglement and Entropy of a Three-Qubit System Interacting with a Quantum Spin Environment
## 2 The Model  
We study a quantum system composed of three qubits characterized by a finite temperature within a thermodynamic limit influenced by a quantum spin environment. The Hamiltonian we are studying is the extension of the Hamiltonian studied in the reference [32]. Here, we consider that the anisotropy and the magnetic field are not homogeneous, yet the spin environment remains without change. The Hamiltonian system can be written as:\[H = H_{P}+H_{PQ}+H_{Q}, \tag{1}\] \[H_{P} = \epsilon_{0}(S_{01}^{z}+S_{02}^{z}+S_{03}^{z})+\omega(S_{01}^{+}S_{0 2}^{-}+S_{01}^{-}S_{02}^{+}+S_{02}^{+}S_{03}^{-}+S_{02}^{-}S_{03}^{+}\] \[+S_{01}^{+}S_{03}^{-}+S_{01}^{-}S_{03}^{+}),\] \[H_{Q} = \frac{q}{M}\sum_{r\
eq j}^{M}(S_{r}^{+}S_{j}^{-}+S_{r}^{-}S_{j}^{+}),\] (3) \[H_{PQ} = \frac{q_{0}}{\sqrt{M}}[(S_{01}^{+}+S_{02}^{+}+S_{03}^{+})\sum_{r= 1}^{M}S_{r}^{-}]+\frac{q_{0}}{\sqrt{M}}[(S_{01}^{-}+S_{02}^{-}+S_{03}^{-})\sum_ {r=1}^{M}S_{r}^{+}], \tag{4}\]  
where \(H_{P}\), \(H_{Q}\) and \(H_{PQ}\) are the system Hamiltonian, bath and system-bath interaction. \(S_{0r}^{+}\) and \(S_{0r}^{-}\) (\(r=1,2,3\)) denote the spin system operators' [33; 34; 35; 36]. Through the substitution of the operator of the collective angular momentum, \(J_{\pm}=\sum_{r=1}^{M}S_{r}^{\pm}\), the Hamiltonian of Equations (3) and (4) can be written as follows:  
\[H_{PQ}=\frac{q_{0}}{\sqrt{M}}[(S_{01}^{+}+S_{02}^{+}+S_{03}^{+})J_{-}]+\frac{q _{0}}{\sqrt{M}}[(S_{01}^{-}+S_{02}^{-}+S_{03}^{-})J_{+}], \tag{5}\]  
\[H_{Q}=\frac{q}{M}\sum_{r\
eq j}^{M}(J_{+}J_{-}+J_{-}J_{+})-q. \tag{6}\]  
We apply the Holstein-Primakoff transformation in order to transform the infinite-dimensional Fock space of boson creation and annihilation operators to finite-dimensional subspaces of the spin operators [37]. The transformation of Holstein-Primakoff can be expressed as:  
\[J_{+}=b^{\dagger}(\sqrt{M-b^{\dagger}b}),\;J_{-}=(\sqrt{M-b^{\dagger}b})b,\; \text{with}\;\left[b^{\dagger},b\right]=1. \tag{7}\]  
The Hamiltonian in Equations (5) and (6) can be written as:  
\[H_{PQ}=q_{0}[(S_{01}^{+}+S_{02}^{+}+S_{03}^{+})(\sqrt{1-\frac{b^{\dagger}b}{M }})b]+q_{0}[(S_{01}^{-}+S_{02}^{-}+S_{03}^{-})b^{\dagger}(\sqrt{1-\frac{b^{ \dagger}b}{M}})], \tag{8}\]  
\[H_{Q}=q[b^{\dagger}(1-\frac{b^{\dagger}b}{M})b+(\sqrt{1-\frac{b^{\dagger}b}{M} })bb^{\dagger}(\sqrt{1-\frac{b^{\dagger}b}{M}})]-q. \tag{9}\]  
If \(M\longrightarrow\infty\) (when the system characterized by a finite temperature within a thermodynamic limit), \(H_{PQ}\) and \(H_{Q}\) can be written as follows:  
\[H_{P} = \epsilon_{0}(S_{01}^{z}+S_{02}^{z}+S_{03}^{z})+\omega(S_{01}^{+}S_{0 2}^{-}+S_{01}^{-}S_{02}^{+}+S_{02}^{+}S_{03}^{-}+S_{02}^{-}S_{03}^{+}\]0 \[H_{P} = \epsilon_{0}(S_{01}^{z}+S_{02}^{z}+S_{03}^{z})+\omega(S_{01}^{+}S_{0 2}^{-}+S_{01}^{-}S_{02}^{+}+S_{02}^{+}S_{03}^{-}+S_{02}^{-}S_{03}^{+}\]1  
Here, we find the approximation that \(\frac{b^{\dagger}b}{M}\) approaches zero, due to the deficient energy of the elementary excitations according to the interconnection between the system and the bath. Accordingly, we deduce the time evolution of the density operator of the spin quantum system. Taking into consideration that the Hamiltonian is time-independent, we can present the density matrix as follows:  
\[H_{P} = \epsilon_{0}(S_{01}^{z}+S_{02}^{z}+S_{03}^{z})+\omega(S_{01}^{+}S_{0 2}^{-}+S_{01}^{-}S_{02}^{+}+S_{02}^{+}S_{03}^{-}+S_{02}^{-}S_{03}^{+}\]2We suppose that \(\rho(0)=\rho_{P}(0)\otimes\rho_{Q}\), i.e., \(\rho(0)\) is separable between the bath and system. The initial state of the spin system can be described by \(\rho_{P}(0)\), and \(\rho_{Q}=\frac{\exp(\frac{-H_{Q}}{RT})}{R}\) defines the spin environment, where \(R=\frac{1}{1-\exp(\frac{-2t}{T})}\) is the separation function, \(T\) is the temperature and \(K\) is the Boltzmann constant (\(K=1\)). The reduced density matrix of scheme is attained by deriving the trace over the spin system; i.e., \(\rho_{P}(t)=Tr_{Q}(\rho(t))\).