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Communications in Theoretical Physics

Energy as Entanglement Witness in Bilinear-Biquadratic Spin-1 Chain

Feng Xu, Wang An-Min, Zhao Ning-Bo, Xiao-Qiang Su, Ren-Gui Zhu

Published Oct 1, 2006• 0 citations
Publisher

Abstract

Energy is introduced as an entanglement witness to describe the entanglement property of a quantum system. The thermal equilibrium system is guaranteed to be entangled when system is cooled down below the entanglement temperature T E . By virtue of this concept we exploit the minimum separable state energy and entanglement temperature T E of the bilinear-biquadratic antiferromagnetic spin-1 chain model. We numerically calculate T E for arbitrary values of the strength of biquadratic exchange interaction Q up to N = 7. We find T E decreases with Q for fixed N when Q is between -3 and 1/3 ( J = 1). In this regime T E also decreases with N for fixed Q and varies slowly for large N . While the thermal system is always entangled when Q is smaller than -3.

DOI
10.1088/0253-6102/46/4/004

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Energy is introduced as an entanglement witness to describe the entanglement property of a quantum system. The thermal equilibrium system is guaranteed to be entangled when system is cooled down below the entanglement temperature T E . By virtue of this concept we exploit the minimum separable state energy and entanglement temperature T E of the bilinear-biquadratic antiferromagnetic spin-1 chain model. We numerically calculate T E for arbitrary values of the strength of biquadratic exchange interaction Q up to N = 7. We find T E decreases with Q for fixed N when Q is between -3 and 1/3 ( J = 1). In this regime T E also decreases with N for fixed Q and varies slowly for large N . While the thermal system is always entangled when Q is smaller than -3.
The entanglement in a general Heisenberg antiferromagnetic chain of arbitrary spin-sss is investigated. The entanglement is witnessed by the thermal energy which equals the minimum energy of any separable state. There is a characteristic temperature below that an entangled thermal state exists. The characteristic temperature for thermal entanglement is increased with spin sss. When the total number of lattice is increased, the characteristic temperature decreases and then approaches a constant. This effect shows that the thermal entanglement can be detected in a real solid state system of larger number of lattices for finite temperature. The comparison of negativity and entanglement witness is obtained from the separability of the unentangled states. It is found that the thermal energy provides a sufficient condition for the existence of the thermal entanglement in a spin-sss antiferromagnetic Heisenberg chain.
We investigate quantum many-body systems where all low-energy states are entangled. As a tool for quantifying such systems, we introduce the concept of the entanglement gap, which is the difference in energy between the ground-state energy and the minimum energy that a separable (unentangled) state may attain. If the energy of the system lies within the entanglement gap, the state of the system is guaranteed to be entangled. We find Hamiltonians that have the largest possible entanglement gap; for a system consisting of two interacting spin-1∕21∕21∕2 subsystems, the Heisenberg antiferromagnet is one such example. We also introduce a related concept, the entanglement-gap temperature: the temperature below which the thermal state is certainly entangled, as witnessed by its energy. We give an example of a bipartite Hamiltonian with an arbitrarily high entanglement-gap temperature for fixed total energy range. For bipartite spin lattices we prove a theorem demonstrating that the entanglement gap necessarily decreases as the coordination number is increased. We investigate frustrated lattices and quantum phase transitions as physical phenomena that affect the entanglement gap.
By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled.
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We use entanglement witnesses to detect entanglement in the XYXYXY chain in thermal equilibrium and determine the temperature bound below which the state is detected as entangled. We consider the entanglement witness based on the Hamiltonian. Such a witness detects a state as entangled if its energy is smaller than the energy of separable states. We also consider a family of entanglement witnesses related to the entanglement negativity of the state. We test the witnesses in infinite and finite systems. We study how the temperature bounds obtained are influenced by a quantum phase transition or a disorder line in the ground state. Very strong finite-size corrections are observed in the ordered phase due to the presence of a quasidegenerate excitation. We also study the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. In the case of the Ising model, we find that the mixed postquench state is detected as entangled by the two methods if the parameters of the Hamiltonian before and after the quench are close to each other. We find that the two witnesses give qualitatively similar results, showing that energy-based entanglement witnesses are efficient in detecting the nearest-neighbor entanglement in spin chains in various circumstances. For other XY models, we find that the negativity-based witnesses also detect states in some parameter regions where the energy-based witness does not, in particular, if the quench is performed from the paramagnetic phase to the ferromagnetic phase and vice versa. The domains in parameter space corresponding to postquench states detected as entangled by the energy-based witness have been determined analytically, which stresses further the utility of our method.
We construct entanglement witnesses using fundamental quantum operators of spin models which contain two-particle interactions and have a certain symmetry. By choosing the Hamiltonian as such an operator, our method can be used for detecting entanglement by energy measurement. We apply this method to the Heisenberg model in a cubic lattice with a magnetic field, the XYXYXY model, and other familiar spin systems. Our method provides a temperature bound for separable states for systems in thermal equilibrium. We also study the Bose-Hubbard model and relate its energy minimum for separable states to the minimum obtained from the Gutzwiller ansatz.