Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model

Abstract

We study the threshold temperature for pairwise thermal entanglement in the spin-1/2 isotropic Heisenberg model up to 11 spins and find that the threshold temperature for odd and even number of qubits approaches the thermal dynamical limit from below and above, respectively. The threshold temperature in the thermodynamical limit is estimated. We investigate the many-particle entanglement in both ground states and thermal states of the system, and find that the thermal state in the four-qubit model is four-particle entangled before a threshold temperature.PACS numbers: 03.65. Ud, 75.10.Jm Entanglement is a nonlocal quantum correlation and entangled states constitute indeed a valuable resource in quantum information processing [1]. Entanglement in systems of interacting spins [2-6] as well as in systems of indistinguishable particles [7][8][9][10][11] has been investigated. In particular entanglement in both the ground state [2,3] and thermal state [4-6] of a spin-1/2 Heisenberg spin chain have been analyzed in the literature. The intriguing issue of the relation between entanglement and quantum phase transition [12] have been addressed in a few quite recent papers [13,14].The state we will consider is the thermal equilibrium state in a canonical ensemble. The system state at finite temperature is given by the Gibb's density operator ρ T = exp (−H/kT ) /Z, where Z =tr[exp (−H/kT )] is the partition function, H the system Hamiltonian, k is Boltzmann's constant which we henceforth will take equal to 1, and T the temperature. As ρ T represents a thermal state, the entanglement in the state is called thermal entanglement [4]. It is important to stress that, although the central object of statistical physics, the partition function, is determined by the eigenvalues of H only, thermal entanglement properties generally require in addition the knowledge of the energy eigenstates. In some models such as the isotropic Heisenberg qubit rings the pairwise thermal entanglement can be completely determined by the partition function [15].Let us briefly review the main results of Ref.[15]. We consider a physical model of a ring of N qubits interacting via the isotropic Heisenberg Hamiltonianwhere S j,j+1 = 1 2 (1 + σ i · σ i+1 ) is the swap operator between qubit i and j, σ i = (σ ix , σ iy , σ iz ) is the vector of Pauli matrices, and J is the exchange constant. In Ref.[15] we proved that there is no thermal entanglement between two nearest-neighbour qubits in the ferromagnetic (J < 0) isotropic Heisenberg rings at any temperature. So we only consider the antiferromagnetic case and the exchange constant J is then assumed to be 1. A direct relation is established between the concurrence [16] quantifying two-qubit entanglement and a macroscopic thermodynamical function, the internal energy U [15]:where β = 1/T and C(N ) refers to the concurrence for two nearest-neighbor qubits of the N -qubit ring. The entanglement is uniquely determined by the partition function of the system. Note that there is a slight difference between the...

Entanglement in thermal equilibrium states

“…At present there are finite algorithms deciding whether a given state is entangled or not only for two qubits (N = 2 with d 1 = d 2 = 2); and for N = 2 with d 1 = 2 and d 2 = 3. The entanglement studies presented in the literature have, perforce, been restricted to investigations of the entanglement of the Gibbs-state reduced to the possible twocomponent subsystems, or to the study of particular so called entanglement witnesses or other entanglement monotone functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In section II, we present some features of the entanglement issue for thermal equilibrium states for the bipartite systems mentioned.…”

Scaling of Tripartite Entanglement at Impurity Quantum Phase Transitions

“…This leads to the conjecture that it is the multipartite entanglement that is maximal at criticality. However, verification of this conjecture faces a big obstacle as quantification of multipartite entanglement is still a challenging problem and can only be evidenced via appropriate witness operators [6][7][8] or for the case of pure states through either multipartite generalized global entanglement [9] or fidelity approaches [10,11]. Some of these methods have also been used in spin chains [12][13][14][15].…”

Detection of multipartite entanglement in spin rings by use of exchange energy

“…In addition to their fundamental interest, multipartite entangled states have important applications in quantum computation [10,11] and in quantum metrology, where they allow one to reach high sensitivities, beyond the standard quantum limit [12][13][14][15]. More generally, multipartite entanglement can also be found in the ground and thermal states of prototypical many-body systems, such as spin systems with isotropic, XY , and noncollinear Ising interactions [16][17][18][19][20].…”