# Entanglement and Entropy of a Three-Qubit System Interacting with a Quantum Spin Environment
###### Abstract  
Throughout this paper, we describe an analytical solution for a three-qubit system characterized by a finite temperature within a thermodynamic limit influenced by a quantum spin environment. As applications to the presented solution, we investigate the effect of the temperature, the coupling constant \(\epsilon_{0}\) within the spin- qubit system and an external magnetic field on the three-particles residual entanglement \(N_{abc}\), the concurrence \(C(\rho)\), the information entropy \(H(\sigma_{Z})  
process is undoubtedly fundamental for the theoretical basics. However, penetrating the process of decoherence within the quantum system is more important for building strategies of error correction, by which the collapse of quantum computers [24] can be detained. When interacting with an environment, the components of a composite quantum system, such as a multi-qubit quantum computer, are disentangled under the influence of the decoherence process. Qubits of a quantum computer may sometimes interact with substantially independent environments. Yet, in some cases, those environments are not completely independent, as the correlations within them are effectively significant.  
Increasingly applied in quantum information theory and quantum computation [25], the dynamic behavior of an individual spin or multi-spin system's interaction with a spin bath has recently gained much attention [26; 27; 28; 29]. A study of three-qubit quantum state has special significance because a three-qubit quantum state is the real start towards investigating multipartite entanglement [30]. Three-qubit quantum entanglement can also exhibit more complex entanglement structure than the two-qubit quantum entanglement [31]. However, in most cases of the non-Markovian process, it is quite tricky to get a specific solution to the time evolution of the reduced density matrix which is traced over by the surrounding environment. This casts more importance on our endeavor to study the case of the non-Markovian three-qubit quantum system by employing an innovative operator procedure to reach a specific solution to our system.  
The problem we are introducing in this paper is a quantum system composed of three qubits characterized by a finite temperature within a thermodynamic limit influenced by a quantum spin environment. We obtained the situation of a two-qubit system through the tracing of a three-qubit case. The article [32] was studying the two-qubit state, and our paper has studied a three-qubit, so our paper is the extension of [32]. Through our case, we can reach the state of the two-qubit, and by changing some parameters we get the results obtained in [32]; i.e., the results are consistent with the results of reference [32]. Additionally, if we follow the same treatment, we can solve more atoms. We have calculated the entanglement and entropy as applications on our problem. We calculated the three-particle residual entanglement \(N_{abc}\) as a kind of entanglement on the case of a three-qubit system, while we calculated the concurrence \(C(\rho)\) on a two-qubit quantum system. We calculated the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\) in both cases, two-qubit and three-qubit. The aim of these calculations was to establish a relationship between the entanglement and entropy, determined by the change of parameters.  
This paper is ordered according to the following arrangement: The Hamiltonian system, along with the analytical solution of a three-qubit system characterized by a finite temperature within a thermodynamic limit influenced by a quantum spin environment, is discussed in Section 2. In Section 3, we discuss the three-particles residual entanglement \(N_{abc}\), the concurrence \(C(\rho)\), the information entropy \(H(\sigma_{Z})\) and the linear entropy \(P_{P}(t)\). Finally, Section 4 presents the conclusion.