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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

Effects of impurity on the entanglement of the three-qubit Heisenberg XXX spin chain

Hu Ming

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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.
We investigate the impurity entanglement of 3-qutrit XXX-type Heisenberg chains in the presence of a uniform magnetic field along z axis by means of negativity. The entanglement is calculated as a function of the coupling constants J, impurity parameter J1, magnetic fields B and temperature T. Through calculating the negativity between sites 1 and (2,3) N1-23 and sites 3 and (1,2) N12-3, we show that the critical temperature Tc above which the entanglement vanishes increases with the increase of J1. The existence of magnetic fields B can obviously reduce the entanglement, and it is found that the limiting temperature Tc depends on the impurity parameter J1 but not on the magnetic field.
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.
In this paper,we investigated the influence of impurity on the entanglement between the normal lattice in a three-qubit Ising chain.The results show that the impurity parameter has great effects on the pairwise entanglement,it can make the normal lattice disentangled and sometimes at certain regions of magnetic field B and temperature T,it can even make the concurrence between the normal lattice greater than the general pairwise entanglement of three qubits(J1=J) or two qubits(J1=0).When T=0,for the case of antiferromagnet with the conditions of θ=π/2,B→0 or θ→0,B2J,the concurrence shows a square-wave structure with the variation of the impurity parameter J1.
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 )).
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