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arXiv:0906.1082 · Annalen der Physik

Entanglement of spin chains with general boundaries and of dissipative systems

Tobias Stauber, F. Guinea

Published Jul 7, 2009• 5 citations
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Abstract

We analyze the entanglement properties of spins (qubits) close to the boundary of spin chains in the vicinity of a quantum critical point and show that the concurrence at the boundary is significantly different from the one of bulk spins. We also discuss the von Neumann entropy of dissipative environments in the vicinity of a (boundary) critical point, such as two Ising-coupled Kondo-impurities or the dissipative two-level system. Our results indicate that the entanglement (concurrence and/or von Neumann entropy) changes abruptly at the point where coherent quantum oscillations cease to exist. The phase transition modifies significantly less the entanglement if no symmetry breaking field is applied and we argue that this might be a general property of the entanglement of dissipative systems. We finally analyze the entanglement of an harmonic chain between the two ends as function of the system size.

DOI
10.1002/andp.200910357

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We analyze the entanglement properties of spins (qubits) close to the boundary of spin chains in the vicinity of a quantum critical point and show that the concurrence at the boundary is significantly different from the one of bulk spins. We also discuss the von Neumann entropy of dissipative environments in the vicinity of a (boundary) critical point, such as two Ising-coupled Kondo-impurities or the dissipative two-level system. Our results indicate that the entanglement (concurrence and/or von Neumann entropy) changes abruptly at the point where coherent quantum oscillations cease to exist. The phase transition modifies significantly less the entanglement if no symmetry breaking field is applied and we argue that this might be a general property of the entanglement of dissipative systems. We finally analyze the entanglement of an harmonic chain between the two ends as function of the system size.
Abstract We analyze the entanglement properties of spins (qubits) close to the boundary of spin chains in the vicinity of a quantum critical point and show that the concurrence at the boundary is significantly different from the one of bulk spins. We also discuss the von Neumann entropy of dissipative environments in the vicinity of a (boundary) critical point, such as two Ising‐coupled Kondo‐impurities or the dissipative two‐level system. Our results indicate that the entanglement (concurrence and/or von Neumann entropy) changes abruptly at the point where coherent quantum oscillations cease to exist. The phase transition modifies significantly less the entanglement if no symmetry breaking field is applied and we argue that this might be a general property of the entanglement of dissipative systems. We finally analyze the entanglement of an harmonic chain between the two ends as function of the system size.
We analyze the entanglement properties of spins (qubits) attached to the boundary of spin chains near quantum critical points, or to dissipative environments, near a boundary critical point, such as Kondo-like systems or the dissipative two level system. In the first case, we show that the properties of the entanglement are significantly different from those for bulk spins. The influence of the proximity to a transition is less marked at the boundary. In the second case, our results indicate that the entanglement changes abruptly at the point where coherent quantum oscillations cease to exist. The phase transition modifies the entanglement significantly less.
The von Neumann entropy of various quantum dissipative models is calculated in order to discuss the entanglement properties of these systems. First, integrable quantum dissipative models are discussed, i.e., the quantum Brownian motion and the quantum harmonic oscillator. In the case of the free particle, the related entanglement of formation shows no nonanalyticity. In the case of the dissipative harmonic oscillator, there is a nonanalyticity at the transition of underdamped to overdamped oscillations. We argue that this might be a general property of dissipative systems. We show that similar features arise in the dissipative two-level system and study different regimes using sub-Ohmic, Ohmic, and super-Ohmic baths, within a scaling approach.
We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of the critical point of the environmental system.
We compute the entropy of entanglement of two blocks of LLL spins at a distance ddd in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size L=1L=1L=1 and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both LLL and ddd. This formula is in excellent agreement with numerical results.
We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.