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Chinese Physics Letters

Entanglement of Two-Qubit Quantum Heisenberg XYZ Chain

Xiao-Qiang Xi, Hao San-Ru, Chen Wen-xue, Rui‐Hong Yue

Published Jul 30, 2002• 15 citations
Publisher

Abstract

We derive the analytic expression of the concurrence in the quantum Heisenberg XYZ model and discuss the influence of parameters J , Δ and Γ on the concurrence. By choosing different values of Γ and Δ, we obtain the XX , XY , XXX and XXZ chains. The concurrence decreases with increasing temperature. When T →0, the concurrence reaches its maximum value 1, i.e. the entangled state, |Ψ⟩ = (((2) 1/2 )/2)(|01⟩-|10⟩), is maximum entanglement. For the XXZ chain, when Γ→∞, the concurrence will meet its maximum value 𝒞 max = (sinh (1/ T ))/(cosh (1/ T )).

DOI
10.1088/0256-307x/19/8/305

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We derive the analytic expression of the concurrence in the quantum Heisenberg XYZ model and discuss the influence of parameters J , Δ and Γ on the concurrence. By choosing different values of Γ and Δ, we obtain the XX , XY , XXX and XXZ chains. The concurrence decreases with increasing temperature. When T →0, the concurrence reaches its maximum value 1, i.e. the entangled state, |Ψ⟩ = (((2) 1/2 )/2)(|01⟩-|10⟩), is maximum entanglement. For the XXZ chain, when Γ→∞, the concurrence will meet its maximum value 𝒞 max = (sinh (1/ T ))/(cosh (1/ T )).
We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C.It is found that for impurity entanglement,C appears only when J1 J for J 0,and J1 0 for J 0,and in these two regions C increases with the increase of J1,so is the critical temperature Tc.When J1 |J|,C reaches its maximum value 0.5 and Tc reaches the asymptotic value Tc = 3.41448J1.For entanglement between the normal lattices,C appears only when J 0 and 2J J1 J,and initially increases with the increase of J1 and arrives at the maximum value Cmax =(e4JIT-3)/(e4JIT+3)before it decays to zero gradually,so is the critical temperature Tc with,however,the maximum value Tcmax = 4J/ln3.
By using the concept of concurrence, we study pairwise entanglement between the two end spins in the open-ended Heisenberg XXX and XY chains up to ten spins. The results show that by introducing two boundary impurities, one can obtain maximum entanglement at the limit of the impurity parameter |J1| ≪ J for the even-number qubits. When |J1/J| > 0, the entanglement always decreases with the increase in the absolute value of J1/J, and for the Heisenberg XXX chain, C disappears when J1/J exceeds a certain critical point Jic, and attains an asymptotic value C0 when |J1| ≫ J(J1 < 0), while for the Heisenberg XY chain, C always disappears when |J1/J| exceeds a certain critical point Jic. Both C0 and Jic decrease with the increase of the length of the chain.
We study the entanglement of a two-qubit one-dimensional XYZ Heisenberg chain in thermal equilibrium at temperature T. We obtain an analytical expression for the concurrence of this system in terms of the parameters of the Hamiltonian and T. We show that depending on the relation among the coupling constants, it is possible to increase the amount of entanglement of the system by increasing its anisotropy. We also show numerically that for all sets of the coupling constants entanglement is a monotonically decreasing function of the temperature T, proving that we must have at least an external magnetic field in the z-direction to obtain a behavior where entanglement increases with T.
The pairwise entanglement in the four-qubit Heisenberg XX open chain with external magnetic fields is calculated,and the analytical expressions for the measures of entanglement Concurrence are obtained.It is found that the entanglement between the boundary qubits can reach the value of 0.28 in the system with uniform external magnetic fields.In addition,in the Heisenberg XX model with external magnetic fields on the two boundary qubits,the two non-boundary qubits will be in the maximal entanglement by increasing the magnetic field at low temperature.