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arXiv:0906.0801 · Physical Review A

Entanglement between distant qubits in cyclic XX chains

N. Canosa, R. Rossignoli

Published Mar 30, 2007• 17 citations
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Abstract

We evaluate the exact concurrence between any two spins in a cyclic $XX$ chain of $n$ spins placed in a uniform transverse magnetic field, at both zero and finite temperature, by means of the Jordan-Wigner transformation plus a number-parity-projected statistics. It is shown that, while at $T=0$ there is always entanglement between any two spins in a narrow field interval before the transition to the aligned state, at low but nonzero temperatures the entanglement remains nonzero for arbitrarily high fields, for any pair separation $L$, although its magnitude decreases exponentially with increasing field. It is also demonstrated that the associated limit temperatures approach a constant nonzero value in this limit, which decreases as ${L}^{\ensuremath{-}2}$ for $L⪡n$, but exhibit special finite-size effects for distant qubits $(L\ensuremath{\approx}n∕2)$. Related aspects such as the different behavior of even and odd antiferromagnetic chains, the existence of $n$ ground-state transitions, and the thermodynamic limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ are also discussed.

DOI
10.1103/physreva.75.032350

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We examine the entanglement of cyclic spin-1/2 chains with anisotropic XYZXYZXYZ Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually nonzero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization jump. Illustrative exact results are shown for chains with (I) full range and (II) nearest-neighbor couplings. Global entanglement properties at such points are also discussed.
We examine the maximum entanglement reached by an initially fully aligned state evolving in an XYXYXY Heisenberg spin chain placed in a uniform transverse magnetic field. Both the global entanglement between one qubit and the rest of the chain and the pairwise entanglement between adjacent qubits is analyzed. It is shown that in both cases the maximum is not a monotonous decreasing function of the aligning field, exhibiting instead a resonant behavior for low anisotropies, with pronounced peaks (a total of [n∕2][n∕2][n∕2] peaks in the global entanglement for an nnn-spin chain), whose width is proportional to the anisotropy and whose height remains finite in the limit of small anisotropy. It is also seen that the maximum pairwise entanglement is not a smooth function of the field even in small finite chains, where it may exhibit narrow peaks above strict plateaus. Explicit analytical results for small chains, as well as general exact results for finite nnn-spin chains obtained through the Jordan-Wigner mapping, are discussed.
Abstract We investigate entanglement and coherence in an XXZ spin s pair immersed in a non-uniform transverse magnetic field. The ground state and thermal entanglement phase diagrams are analyzed in detail in both the ferromagnetic and antiferromagnetic cases. It is shown that a non-uniform field can control the energy levels and the entanglement of the corresponding eigenstates, making it possible to entangle the system for any value of the exchange couplings, both at zero and finite temperatures. Moreover, the limit temperature for entanglement is shown to depend only on the difference ∣ h 1 − h 2 ∣ between the fields applied at each spin, leading for T > 0 to a separability stripe in the ( h 1 , h 2 ) field plane such that the system becomes entangled above a threshold value of ∣ h 1 − h 2 ∣ . These results are demonstrated to be rigorously valid for any spin s . On the other hand, the relative entropy of coherence in the standard basis, which coincides with the ground state entanglement entropy at T = 0 for any s , becomes non-zero for any value of the fields at T > 0 , decreasing uniformly for sufficiently high T . A special critical point arising at T = 0 for non-uniform fields in the ferromagnetic case is also discussed.
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The concurrence of two alternate qubits in a four-qubit Heisenberg XXXXXX chain is investigated when a uniform magnetic field BBB is included. It is found that there is no thermal entanglement between alternate qubits if BBB is close to zero. A magnetic field can induce entanglement in a certain range both for the antiferromagnetic and ferromagnetic cases. Near zero temperature, the entanglement undergoes two sudden changes with increasing value of the magnetic field BBB. This is due to the changes in the ground state. This novel property may be used as a quantum entanglement switch. The anisotropy in the system can also induce the entanglement between two alternate qubits.
The thermal pairwise entanglement (TE) of the S=1/2S=1/2S=1/2 XYXYXY chain in a transverse magnetic field is exactly resolved by means of the Jordan-Wigner transformation in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that the TE vanishes at a fixed point with temperature ${T}_{c}\ensuremath{\simeq}0.484\text{ }3J$, which is independent of the magnetic field. A thermal quantity is proposed to witness the entangled state. Furthermore, the TE of the S=1/2S=1/2S=1/2 antiferromagnetic-ferromagnetic (AF-F) Heisenberg chain is studied by the transfer-matrix renormalization-group method. The TEs of the spins coupled by AF and F interactions are found to behave distinctively. The vanishing temperature of the field-induced TE of the spins coupled by F interactions is observed dependent on the magnetic field. The results are further confirmed and analyzed within a mean-field framework.