Low-energy-state dynamics of entanglement for spin systems

2010

We have composed the ideas of quantum renormalization group and quantum information by exploring the low energy states dynamic of entanglement resources of a system close to its quantum critical point. We demonstrate the low energy states dynamical quantities of the one dimensional magnetic systems could show the quantum phase transition point and shows the scaling behavior in the vicinity of the transition point. To present our idea, we study the evolution of two spins entanglement in the one-dimensional Ising model in the transverse field. The system is initialized as the so-called thermal ground state of the pure Ising model. We investigate evolvement of the generation of entanglement with increasing the magnetic field. We have obtained that the derivative of the time at which the entanglement reaches its maximums with respect to the transverse field, diverges at the critical point and its scaling behaviors versus the size of the system are as same as the static ground state entanglement of the the system.

Physical Review A, 2008

Two-qubit entanglement can be induced by a quantum data bus interacting with them. In this paper, with the quantum spin chain in the transverse field as an illustration of the quantum data bus, we show that such induced entanglement can be enhanced by the quantum phase transition ͑QPT͒ of the quantum data bus. We consider two external spins simultaneously coupled to a transverse field Ising chain. By adiabatically eliminating the degrees of the chain, the effective coupling between these two spins is obtained. The matrix elements of the effective Hamiltonian are expressed in terms of the dynamical structure factor ͑DSF͒ of the chain. The DSF is the Fourier transformation of the Green function of an Ising chain and can be calculated numerically by a method introduced by Derzhko and Krokhmalskii ͓Phys. Rev. B 56, 11659 ͑1997͔͒. Since all characteristics of QPT are embodied in the DSF, the dynamical evolution of the two external spins displays singularity in the vicinity of the critical point.

Physical Review A, 2004

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The "entanglement wave" evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given.

Quantum Computing in Solid State Systems, 2006

Physical Review A, 2008

The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of the entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The investigated model is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled states is considered. It was shown that, in spite of the use of a model with the direct interactions between the nearest neighbors, the entanglement between remote spins is generated.

Nature, 2002

In this Letter we discuss the entanglement near a quantum phase transition by analyzing the properties of the concurrence for a class of exactly solvable models in one dimension. We find that entanglement can be classified in the framework of scaling theory. Further, we reveal a profound difference between classical correlations and the non-local quantum correlation, entanglement: the correlation length diverges at the phase transition, whereas entanglement in general remains short ranged.

Physical Review A, 2004

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The "entanglement wave" evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given.

2007

Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time τQ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation lengthξ proportional to the square root of the quench time τQ: ξ = √ τQ. The inverse of this correlation length is known to determine average density of defects (e.g. kinks) after the transition. In this paper, we show that this sameξ controls the entropy of entanglement, e.g. entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at 1 6 log 2ξ , as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L ≈ξ, but -in the subsequent evolution in the ferromagnetic phase -a somewhat larger length scale l = √ τQ ln τQ develops as a result of a dephasing process that can be regarded as a quantum analogue of phase ordering, and the entropy saturates when L ≈ l. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation lengtĥ ξ, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined byξ. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

Physics Letters A, 2007

Non-equilibrium time evolution of entanglement is considered in a 1D critical Ising chain. At the point of the quantum phase transition, this system is maximally entangled in its thermal ground state. Using the scaling analytical expressions for the magnetization profile, the behavior of the non-equilibrium analog is studied for the single-site entanglement entropy. The latter quantity becomes the usual von Neuman entropy during the system evolution. We choose the initial finite domain configuration, which gives quite general picture of evolution processes. However the obtained results can be generalized to more complicated initial configurations, as well as to the anisotropic XY chain. It is shown that the relaxation time of single-site entanglement is strongly dependent on the characteristic size of inhomogeneity of the input state. This is an important issue for possible realization of a rescalable solid-state quantum computer.

2008

We investigate the ground state and the thermal entanglement in the two-qubit Ising model interacting with a site-dependent magnetic field. The degree of entanglement is measured by calculating the concurrence. For zero temperature and for certain direction of the applied magnetic field, the quantum phase transition observed under a uniform external magnetic field disappears once a very small non-uniformity is introduced. Furthermore, we have shown analytically and confirmed numerically that once the direction of one of the magnetic field is along the Ising axis then no entangled states can be produced, independently of the degree of non-uniformity of the magnetic fields on each site.