Entanglement and Entropy of a Three-Qubit System Interacting with a Quantum Spin Environment

: 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 (cid:101) 0 within the spin- qubit system and an external magnetic ﬁeld on the three-particles residual entanglement N abc , the concurrence C ( ρ ) , the information entropy H ( σ Z ) and the linear entropy P P ( t ) . The results show an inverse relationship between the entanglement and entropy, where the degree of both is controlled by controlling the temperature T and the coupling constant (cid:101) 0 . q 0 = 1, ω = 0, α = 0 and β = π 2 . We study this case at (cid:101) 0 = 2, (cid:101) 0 = 4 and (cid:101) 0 = 6. We observe the state of regularity and periodicity of all curves at any value for (cid:101) 0 . By increasing the coupling constant (cid:101) 0 , we notice that the number of oscillations of the concurrence C ( ρ ) increases signiﬁcantly; i.e., the interconnections between the atoms increase, and the numbers of oscillations of both the information entropy H ( σ Z ) and the linear entropy P P ( t ) are also signiﬁcantly reduced. The maximum values of the concurrence C ( ρ ) increase but the minimum value tends to persist. The increase in the maximum value for the information entropy H ( σ Z ) and the linear entropy P P ( t ) is minor. Additionally, the number of peaks of the linear entropy P P ( t ) in this case increases.


Introduction
Due to their significant role in quantum computation, the principles of open quantum schemes have received increasing consideration in the latest years. Usually, these theories examine the evolution concerning a quantum system in interaction with a specific nature which generally consists of bosons, fermions or localized spins [1][2][3]. The characteristics of the quantum can be obtained through the average of the degrees of freedom in the environment. Motivated by the progress in quantum information and comp, there has been a growing interest in mathematical and physical exploration of highly entangled states. This entanglement occurs when two or more particles are integrated and generated to the extent that their quantum states become inseparable and can not be depicted independently. Entangling between both near and remote pairs of qubits has yielded various applications in the quantum information processing, such as the entanglement generation in nanophotonic architectures [4][5][6][7][8][9][10]. Recently, there has been a rising curiosity in exploring the entanglement within the systems of quantum spin together with Heisenberg interactions [11][12][13][14][15][16]. Being both a simple and at the same time solid system, the Heisenberg model was used to visualize a quantum computer, and also to simulate nuclear and electronic spins [17,18], and quantum dots [19,20]. Moreover, it has demonstrated a very vital applicability in quantum state transfer [21]. However, the previous studies were not essentially discussing the relationship between the system and its surrounding environment as the pivotal point of interest, as any quantum system would naturally interact with the surrounding environment. What is the most crucial, thus, is not the interaction itself, but, practically speaking, the decoherence which that interaction leads to. Investigating the decoherence [22,23]

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: 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 ± = ∑ M r=1 S ± r , the Hamiltonian of Equations (3) and (4) can be written as follows: 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: The Hamiltonian in Equations (5) and (6) can be written as: If M −→ ∞ (when the system characterized by a finite temperature within a thermodynamic limit), H PQ and H Q can be written as follows: Here, we find the approximation that b † 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: We suppose that ρ(0) = ρ P (0) ⊗ ρ Q , i.e., ρ(0) is separable between the bath and system. The initial state of the spin system can be described by ρ P (0), and ρ Q = 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., ρ P (t) = Tr Q (ρ(t)).
To calculate exp(−iHt) |− − + , for example, we assume where U, V, X, Y, G, J, L and F are functions of operators b, b † , and time t. Here, we adopt the Schrödinger equation because it is considered to be a special case of the master equation in Lindblad form [38]. By applying the identity of Schrödinger equation and Equation (14), we get the following equations with the consideration that the initial conditions U(0) = X(0) = ......... = F(0) = 0, and V(0) = 1, from Equation (14), the pervious differential equations, are unsolvable by any traditional methods for ordinary variables, because the differential equations are composed of non-commuting operator variables. So, we can use the following transformation: Then the Equation (16) becomes, from Equation (17), as follows: with n = b † b and the initial conditions U 1 (0) = X 1 (0) = ......... = F 1 (0) = 0, and V 1 (0) = 1. Numerically we can solve the differential Equation (18). Therefore, we can obtain the variables U 1 , V 1 , X 1 , Y 1 , G 1 , J 1 , L 1 and F 1 as functions of n and t, which commute with each other. Hence, we can obtain U, V, X, Y, G, J, L and F, from Equation (17).

Results and Discussion
Since we calculated the three-particles residual entanglement N abc , the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) as applications to the previous solutions we reached, we now review the definitions and laws of these applications.

Three-Particle Residual Entanglement
The three-particle residual entanglement N abc can be defined as the following [39]: where, The term N a−bc (ρ abc ) quantifies the strengths of quantum correlations between the atom "a" and the other two atoms. The term N a−b (ρ ab ) (N a−c (ρ ac )) quantifies the pairwise entanglement between the atom "a" and "b" ("c"). And, where ρ T A is the fractional transpose of a state ρ according to subsystem A, and . 1 is the trace norm; ∑ i |µ i | is the summation of the absolute value for each eigenvalue of ρ T A .

The Information Entropy and the Linear Entropy
The information entropy H(σ Z ) of the atomic operator σ z can be written as follows [40,41]: where the probability distribution P j (σ Z ) for ε probable outcomes of measurements for a random quantum state of an atomic operator σ Z is where ρ represents the density matrix of the total quantum system and Φ Zj eigenvector of the atomic operator σ Z : where υ Zj is the eigenvalue of the atomic operator σ β shown in Equation (27). For a two-qubit, = 4, but in the case of the three-qubit ε = 8. The evolution of the linear entropy P P (t) is given by where ρ P (t) = Tr Q ρ(t) denotes the reduced density matrix for bipartite system.

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: , an arrangement in descending order. Furthermore, the transposition of density matrix ρ is denoted by ρ . 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(σ Z ) and the linear entropy P P (t) in case of 0 = 2, q = q 0 = 1, ω = 0, α = 0 and β = π 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(σ 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(σ 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 ρ P , (ρ ij (t) = Tr k (ρ P (t)); i, j and k are equal; a, b, c, i = j = k, ρ P = ρ abc ). In Figure 2, we study the influence of the temperature T on the behavior of the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) in case of 0 = 2, q = q 0 = 1, ω = 0, α = 0 and β = π 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(ρ) are almost regular, but at T = 2, the regularity of curves is slightly decreased. The maximum values of the concurrence C(ρ) decrease from C(ρ) = 0.7 to C(ρ) = 0.3 and the minimum value keeps its stability. For the information entropy H(σ 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 0 between the spin-qubit system, and an external magnetic field along the Z direction on the behavior of the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) in case of T = 1, q = q 0 = 1, ω = 0, α = 0 and β = π 2 . We study this case at 0 = 2, 0 = 4 and 0 = 6. We observe the state of regularity and periodicity of all curves at any value for 0 . By increasing the coupling constant 0 , we notice that the number of oscillations of the concurrence C(ρ) increases significantly; i.e., the interconnections between the atoms increase, and the numbers of oscillations of both the information entropy H(σ Z ) and the linear entropy P P (t) are also significantly reduced. The maximum values of the concurrence C(ρ) increase but the minimum value tends to persist. The increase in the maximum value for the information entropy H(σ Z ) and the linear entropy P P (t) is minor. Additionally, the number of peaks of the linear entropy P P (t) in this case increases. Linear entropy Figure 1. The time evolution of the entanglement N abc , the information entropy H(σ Z ) and the linear entropy P P (t) in the case of three-qubit for parameters 0 = 2, ω = 0, q = q 0 = 1, α = 0 and β = π 2 . Where solid green, red dots and blue curves correspond, respectively, to T = 0.1, 1 and 3. Linear entropy Figure 2. The time evolution of the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) in the case of two-qubit for parameters 0 = 2, ω = 0, q = q 0 = 1, α = 0 and β = π 2 . Solid green, red dots and blue curves correspond, respectively, to T = 0.1, 1 and 2.  . The time evolution of the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) in the case of two-qubit for parameters T = 2, ω = 0, q = q 0 = 1, α = 0 and β = π 2 , where solid green, red dots and blue curves correspond, respectively, to 0 = 2, 4 and 6.
From the above-mentioned discussion, we can observe that oscillations in the results always take indefinite forms. That is because we have chosen an initial entangled state, and this, as we observed, reduces the effect of the surrounding environment, in addition to changing the values of parameters which we control. Thus, the oscillations would continue to take the indefinite appearance, rather than a steady state, in the figures.

Conclusions
We analytically solved a quantum system composed of three qubits characterized by a finite temperature within a thermodynamic limit influenced by a quantum spin environment. The case of the two-qubit was obtained by deriving the trace for the density matrix. We studied the effect of the temperature T and the coupling constant 0 between the spin-qubit system and an external magnetic field on the three-particle residual entanglement N abc , the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t). Our work is considered to be an extension of a previous study [32], yet with much larger calculations to reach an exact solution of a non-Markovian case of three-qubit quantum system by using a novel operator technique. We observed that all curves, whether in the case of three-qubit or two-qubit, are regular and periodic at low temperatures T, but at high temperatures T tend to be irregular. We also noticed a significant change in the number of oscillations, increasing or decreasing, for the concurrence C(ρ), the information entropy H(σ Z ) and the linear entropy P P (t) when the coupling constant 0 changed. Also note the strong relationship between the entanglement, N abc and C(ρ), and the entropy, H(σ Z ) and P P (t). When we saw an increase in the entanglement between the atoms, there was a marked decrease in entropy and vice versa. From there, we proved an inverse relationship between the entanglement and the entropy, that we controlled the degree of entanglement, and subsequently, the degree of entropy by controlling the temperature T and the coupling constant 0 .