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International Journal of Quantum Information

EFFECTS OF ANISOTROPY ON DYNAMICS OF PAIRWISE ENTANGLEMENT IN THE FOUR-QUBIT HEISENBERG XXZ MODEL

Jin‐Liang Guo, He‐Shan Song

Published Oct 1, 2009• 0 citations
Publisher

Abstract

We investigate the effects of anisotropy on the dynamic evolution of pairwise entanglement in the four-qubit Heisenberg XXZ model. The entanglement between the nearest-neighbor and the next nearest-neighbor qubits are calculated by making use of concurrence. Our results show that the sudden death effect can be weakened when introducing the anisotropy Δ. In addition, we also find that the anisotropy Δ can protect the entanglement between the nearest-neighbor qubits from being completely destroyed by the intrinsic decoherence, but it cannot do this for the entanglement between the next nearest-neighbor qubits.

DOI
10.1142/s0219749909005900

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We discuss the dynamic evolution of pairwise entanglement in a four-qubit Heisenberg XX spin chain with the periodic boundary condition and the external magnetic field considered. The results show that the so-called entanglement sudden death (ESD) effect can be weakened when introducing the periodic boundary condition and the magnetic impurity. In addition, we also find that the ESD affects the dynamic evolution of entanglement between weakly interacting alternate qubits stronger than that for strongly interacting nearest qubits.
The pair-wise thermal entanglement in a four-qubit Heisenberg XXZ chain is investigated to study the role of anisotropy when an external magnetic field is included. It is found that pair-wise entanglement is absent between nearest- and next-nearest neighbouring qubits with anisotropic parameter Δ⩽−1. For two nearest-neighbouring qubits, increasing the parameter can not only induce the entanglement, but also extend the entanglement region in terms of magnetic field B and temperature T . For two next-nearest-neighbouring qubits, increasing anisotropic parameter can shift the location of the entanglement and control the extent of the entanglement in terms of magnetic field at a finite temperature.
The pairwise entanglement of the Heisenberg XX chain with next-nearest-neighbor (NNN) interactions was investigated by using the concurrence measure. The results show that for the nearest-neighbor sites, the entanglement may be improved or suppressed depending on the magnitudes of the NNN coupling constant J, while for the next-nearest-neighbor sites, it always increases with the increase of |J|. The critical temperature Tc decreases with the increase of J for the nearest-neighbor entanglement and increases with the increase of |J| for the next-nearest-neighbor entanglement, respectively. We also show that the general Heisenberg XX model still can be used to create the entangled W states of three and four qubits, and that the presence of NNN coupling has no effect on the creation of four-qubit W states, while it shifts the instant of time at which the three-qubit W states are created.
We investigate the effect of the Dzyaloshinskii–Moriya anisotropic antisymmetric interaction on entanglement in a three-qubit Ising model with intrinsic decoherence. The entanglement of the nearest and the next-nearest neighbor qubit is calculated. The results show that, taking into account the intrinsic decoherence, when the qubits are initially in the maximal entangled state, the concurrence of the system decreases with the increasing Dzyaloshinskii–Moriya interaction following the evolution of the time t. When the qubits are initially in the disentangled state, the destructive effect of intrinsic decoherence on entanglement can be moderated by the Dzyaloshinskii–Moriya interaction, the concurrence increases with the increasing Dzyaloshinskii–Moriya interaction following the evolution of the time t.
For a three-qubit Heisenberg XX spin chain with the uniform magnetic field and magnetic impurity, dynamic evolution of pairwise entanglement are studied. We show that the phenomenon of entanglement sudden death (ESD) occurs in the evolution of entanglement for some initial states. We also find that magnetic impurity cannot only eliminate the phenomenon of ESD but also induce the ESD effects, which is different from the case of uniform magnetic field where the ESD can be depressed but cannot be eliminated by introducing the uniform magnetic field.
In this paper, we study the property of thermal entanglement in four-qubit Heisenberg model, where Dzyaloshinskii-Moriya (DM) interaction is considered, and investigate the pairwise concurrences of two nearest-neighbor qubits and two next-neighbor qubits to study this entanglement property. The result shows that for the two-next-neighbor-qubit case, there exists pairwise concurrence neigher in ferromagnetic model nor in antiferromagnetic model; but for the two-nearest-neighbor-qubits case, the DM interaction and the parameter of the anisotropy exchange coupling have a significant influence on the pairwise entanglement and critical temperature Tc. Moreover, the pairwise concurrence will decrease with the increase of temperature. When the temperature execeds its critical value, the pairwise concurrence disappears. Therefore, the pairwise entanglement can be controlled and enhanced by choosing the appropriate parameters of the DM interaction and the anisotropy exchange coupling.