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arXiv:1204.2337 · Journal of Physics A Mathematical and Theoretical

Role of external fields in enhancing long-distance entanglement at finite temperatures

Tomotaka Kuwahara

Published Mar 28, 2013• 1 citation
arXivHTMLPDF

Abstract

We investigate the end-to-end entanglement of a general XYZ -spin chain at the non-zero temperatures. The entanglement usually vanishes at a certain critical temperature T c , but external fields can make T c higher. We obtain a general statement on the increase of the critical temperature T c by the external fields. We prove that if the two end spins are separated by two spins or more, then the critical temperature cannot be higher than a certain finite temperature ( ), that is, the entanglement must vanish above the temperature for any values of the external fields. On the other hand, if the two end spins are separated by one spin, then the entanglement maximized by the external fields exhibits a power-law decay of the temperature, being finite at any temperatures. In order to demonstrate the former case, we numerically calculate the temperature in XX and XY four-spin chains. We find that the temperature shows qualitatively different behavior, depending on the conservation of the angular momentum in the z direction.

DOI
10.1088/1751-8113/46/15/155302

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We investigate the end-to-end entanglement of a general XYZ -spin chain at the non-zero temperatures. The entanglement usually vanishes at a certain critical temperature T c , but external fields can make T c higher. We obtain a general statement on the increase of the critical temperature T c by the external fields. We prove that if the two end spins are separated by two spins or more, then the critical temperature cannot be higher than a certain finite temperature ( ), that is, the entanglement must vanish above the temperature for any values of the external fields. On the other hand, if the two end spins are separated by one spin, then the entanglement maximized by the external fields exhibits a power-law decay of the temperature, being finite at any temperatures. In order to demonstrate the former case, we numerically calculate the temperature in XX and XY four-spin chains. We find that the temperature shows qualitatively different behavior, depending on the conservation of the angular momentum in the z direction.
In the present thesis, we report several general properties of the enhancement of the entanglement by external fields. First, we investigate the thermal entanglement of interacting two qubits. We maximize it by tuning a local Hamiltonian under a given interaction Hamiltonian. We prove that the optimizing local Hamiltonian takes a simple form which dose not depend on the temperature and that the corresponding optimized thermal entanglement decays as 1/(T log T ) at high temperatures. We also find that at low temperatures the thermal entanglement is maximum without any local Hamiltonians and that the second derivative of the maximized thermal entanglement changes discontinuously at the boundary between the highand low-temperature phases. Second, we investigate the maximized entanglement of indirectly interacting two spins, that is, through other spins. We present a necessary condition for the indirect interaction to give a non-zero maximized entanglement between the focused spins. We also prove that if the focused spins are separated by two spins, there is a critical temperature above which the maximized entanglement between the focused spins vanishes. Then, we numerically calculate the maximized entanglement between the end spins of three-spin chains and four-spin chains. We discover that the maximizing local fields on the end spins have asymmetric forms. In the three-spin chains, we attribute the entanglement enhancement to the asymmetry of the local fields qualitatively and quantitatively in terms of the magnons. In XX and XY four-spin chains, we find that the critical temperature shows qualitatively different behavior depending on the conservation of the angular momentum in the z direction.
We consider an infinite one-dimensional anisotropic XY\mathit{XY}XY spin chain with a nearest-neighbor time-dependent Heisenberg coupling J(t)J(t)J(t) between the spins in presence of a time-dependent magnetic field h(t)h(t)h(t). We discuss a general solution for the system and present an exact solution for particular choice of JJJ and hhh of practical interest. We investigate the dynamics of entanglement for different degrees of anisotropy of the system and at both zero and finite temperatures. We find that the time evolution of entanglement in the system shows nonergodic and critical behavior at zero and finite temperatures and different degrees of anisotropy. The asymptotic behavior of entanglement at the infinite time limit at zero temperature and constant JJJ and hhh depends only the parameter $\ensuremath{\lambda}=J/h$ rather than the individual values of JJJ and hhh for all degrees of anisotropy but changes for nonzero temperature. Furthermore, the asymptotic behavior is very sensitive to the initial values of JJJ and hhh and for particular choices we may create finite asymptotic entanglement regardless of the final values of JJJ and hhh. The persistence of quantum effects in the system as it evolves and as the temperature is raised is studied by monitoring the entanglement. We find that the quantum effects dominate within certain regions of the kT\mathit{kT}kT-$\ensuremath{\lambda}$ space that vary significantly depending on the degree of the anisotropy of the system. Particularly, the quantum effects in the Ising model case persist in the vicinity of both its critical phase transition point and zero temperature as it evolves in time. Moreover, the interplay between the different system parameters to tune and control the entanglement evolution is explored.
In this article, we investigate thermal entanglements of the two-site, three-site and four-site mixed spin (1/2,1) XYsystems. The entanglement versus temperature and external magnetic field is discussed. It is found that the entanglements decrease monotonically as temperature increases in the presence and absence of a weak external magnetic field. For the two-site and four-site XY systems, thermal entanglements disappear at the same temperature which is called critical temperature no matter in the ferromagnetic case or antiferromagnetic. It also shows that the critical temperature is independent of external magnetic field. For the three-site system, the corresponding critical temperature is also irrelevant to external magnetic field, while the critical temperature for the ferromagnetic case is higher than that for the antiferromagnetic case. The entanglement of XY systems can develop a few stable platform in an environment of low temperature, but the entanglement vanishes when external magnetic field exceeds some critical value. In this article, we also analyze the difference in thermal entanglement between mixed-spin system and single-spin system, and find that there exists multi-level level crossing in the mixed-spin system.
In this paper, the entanglement in a spin-1 Heisenberg XXZ chain with uniform and nonuniform external magnetic field is investigated in the terms of the measure of entanglement called 'negativity'. The effects of the external magnetic field, the anisotropy parameter and the temperature on the negativity are discussed respectively. It can be seen that the entanglement in this spin chain for any temperature is symmetric with respect to zero magnetic field. We calculate the partially transposed density matrix and give qualitatively results at different temperatures. In order to show the results clearly, the behavior of negativity is presented for five different cases. It can be found that the temperature and the external magnetic field can effect the feature of the thermal entanglement significantly. In addition, under certain condition, the negativity may be enhanced under a nonuniform magnetic field.
We study and compare the persistence of bipartite entanglement (BE) and multipartite entanglement (ME) in one-dimensional and two-dimensional spin XY models in an external transverse magnetic field under the effect of thermal excitations. We compare the threshold temperature at which the entanglement vanishes in both types of entanglement. We use the entanglement of formation as a measure of the BE and the geometric measure to evaluate the ME of the system. We have found that in both dimensions in the anisotropic and partially anisotropic spin systems at zero temperatures, all types of entanglement decay as the magnetic field increases but are sustained with very small magnitudes at high field values. Also we found that for the same systems, the threshold temperatures of the nearest neighbour (nn) BEs are higher than both of the next-to-nearest neighbour BEs and MEs and the three of them increase monotonically with the magnetic field strength. Thus, as the temperature increases, the ME and the far parts BE of the system become more fragile to thermal excitations compared to the nn BE. For the isotropic system, all types of entanglement and threshold temperatures vanish at the same exact small value of the magnetic field. We emphasise the major role played by both the properties of the ground state of the system and the energy gap in controlling the characteristics of the entanglement and threshold temperatures. In addition, we have shown how an inserted magnetic impurity can be used to preserve all types of entanglement and enhance their threshold temperatures. Furthermore, we found that the quantum effects in the spin systems can be maintained at high temperatures, as the different types of entanglements in the spin lattices are sustained at high temperatures by applying sufficiently high magnetic fields.