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Communications in Theoretical Physics

Bipartite and Tripartite Entanglement in a Three-Qubit Heisenberg Model

Jie Ren, Zhu Shi-Qun

Published Dec 1, 2006• 3 citations
Publisher

Abstract

The bipartite and tripartite entanglement in a three-qubit Heisenberg XY model with a nonuniform magnetic field is studied. There are two or four peaks in the concurrence of the bipartite entanglement when the amplitudes of the magnetic fields are differently distributed between the three qubits. It is very interesting to note that there is no tangle of tripartite entanglement between the three qubits when the amplitudes of the magnetic fields are varied. However, the variation of the magnetic field direction can induce the tangle. The tangle is periodic about the angle between the magnetic field and the z axis of the spin.

DOI
10.1088/0253-6102/46/6/003

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The bipartite and tripartite entanglement in a three-qubit Heisenberg XY model with a nonuniform magnetic field is studied. There are two or four peaks in the concurrence of the bipartite entanglement when the amplitudes of the magnetic fields are differently distributed between the three qubits. It is very interesting to note that there is no tangle of tripartite entanglement between the three qubits when the amplitudes of the magnetic fields are varied. However, the variation of the magnetic field direction can induce the tangle. The tangle is periodic about the angle between the magnetic field and the z axis of the spin.
We investigate the thermal entanglement of two-qubit anisotropic Heisenberg XY model in the presence of an external nonuniform magnetic field B along the z-axis. Concurrence, the measure of entanglement is calculated and its property is studied in different cases. Two best models, Ising model under a uniform magnetic and isotropic model in a nonuniform magnetic field, are discovered. In the two models, the critical temperature Tc (above which there is no entanglement) can be enhanced and its concurrence is maximal.
We investigate the thermal entanglement in the two-qubit Heisenberg XY model with a nonuniform magnetic field. Concurrence, the measurement of entanglement, is calculated. The behavior of concurrence is present at three different cases. Contrary to the uniform magnetic field case, we find that the entanglement and the critical temperature TC{T}_{C}TC​ may be enhanced under a nonuniform magnetic field.
We study the bipartite entanglement dynamics in an anisotropic two-qubit Heisenberg XYZXYZXYZ model under the influence of population relaxation and in the presence of various types of magnetic fields. We find that the maximal value of the concurrence has an intrinsic rigidity for the different external magnetic fields. Moreover, our results demonstrate that it is possible to produce entangled states and to control or to modulate the concurrence within the intrinsic maximal value with the help of external time-varying fields despite the existence of dissipation.
The effects of spin-spin interaction on thermal entanglement of a two-qubit Heisenberg XYZ model with different inhomogeneous magnetic fields are investigated. It is shown that the entanglement is dependent on the spin-spin interaction and the inhomogeneous magnetic fields. The larger the J i (i-axis spin-spin interaction), the higher critical value the B i (i-axis uniform magnetic field) has. Moreover, in the weak-field regime, the larger J i corresponds to more entanglement, while in the strong-field regime, different J i correspond to the same entanglement. In addition, it is found that with the increase of J i , the concurrence can approach the maximum value more rapidly for the smaller B i , and can reach a larger value for the smaller b i (i-axis nonuniform magnetic field). So we can get more entanglement by increasing the spin-spin interaction J i , or by decreasing the uniform magnetic field B i and the nonuniform magnetic field b i .
Three-dimensional Heisenberg model in the form of a tetrahedron lattice is investigated. The concurrence and multipartite entanglement are calculated through 2-concurrence C and 4-concurrence C 4 . The concurrence C and multipartite entanglement C 4 depend on different coupling strengths J i and are decreased when the temperature T is increased. For a symmetric tetrahedron lattice, the concurrence C is symmetric about J 1 when J 2 is negative while the multipartite entanglement C 4 is symmetric about J 1 when J 2 <2. For a regular tetrahedron lattice, the concurrence C of ground state is 1/3 for ferromagnetic case while C = 0 for antiferromagnetic case. However, there is no multipartite entanglement since C 4 = 0 in a regular tetrahedron lattice. The external magnetic field B can increase the maximum value of the concurrence C B and induce two or three peaks in C B . There is a peak in the multipartite entanglement C 4 B when C 4 B is varied as a function of the temperature T . This peak is mainly induced by the magnetic field B .
This paper investigates bipartite entanglement of a two-qubit system with anisotropic couplings under an inhomogeneous magnetic field. This work is mainly to investigate the characteristics of a Heisenberg XYZ chain and obtains some meaningful results. By the concept of negativity, it finds that the inhomogeneity of magnetic field may induce entanglement and the critical magnetic field is independent of J z . The inhomogeneous magnetic field can increase the value of critical magnetic field B c . It also finds that the magnetic field not only suppresses the entanglement but also can induce it to revival for some time.