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General theory of entanglement enhancement by external fields in spin chains

Tomotaka Kuwahara

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

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.

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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 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.
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 does not depend on the temperature and that the corresponding optimized thermal entanglement decays as 1/(TlnT)1/(T\mathrm{ln}T)1/(TlnT) 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 high- and low-temperature phases.
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 investigate the mixed-state entanglement between two spins embedded in the XXZ Heisenberg chain under thermal equilibrium. By deriving an analytical expression for the entanglement of two-spin thermal states and extending this analysis to larger spin chains, we demonstrate that mixed-state entanglement is profoundly shaped by both disorder and temperature. Our results reveal a sharp distinction between many-body localized (MBL) and ergodic phases, with entanglement vanishing above different finite temperature thresholds. Furthermore, by analyzing non-adjacent spins, we uncover an approximate exponential decay of entanglement with separation. This work advances the understanding of the quantum-to-classical transition by linking the entanglement properties of small subsystems to the broader thermal environment, offering a explanation for the absence of entanglement in macroscopic systems. These findings provide critical insights into quantum many-body physics, bridging concepts from thermalization, localization, and quantum information theory.
The thermal entanglement in the Heisenberg XYZ\mathrm{XYZ}XYZ chain is investigated in the presence of an external magnetic field B. In the two-qubit system, the critical magnetic field Bc{B}_{c}Bc​ is increased by introducing the interaction of the z component of two neighboring spins Jz.{J}_{z}.Jz​. This interaction not only improves the critical temperature Tc,{T}_{c},Tc​, but also enhances the entanglement for a particular fixed B. We also analyze the pairwise entanglement between nearest neighbors in three qubits. The pairwise entanglement, for a fixed T, can become strong by controlling B and Jz.{J}_{z}.Jz​.