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arXiv:0708.0320 · Physical Review A

Analytic results on long-distance entanglement mediated by gapped spin chains

Aires Ferreira, J. M. B. Lopes dos Santos

Published Mar 11, 2008• 15 citations
arXivHTMLPDF

Abstract

Recent numerical results showed that spin chains are able to produce long-distance entanglement (LDE). We develop a formalism that allows the computation of LDE for weakly interacting probes with gapped many-body systems. At zero temperature, a dc response function determines the ability of the physical system to generate genuine quantum correlations between the probes. We show that the biquadratic Heisenberg spin-1 chain is able to produce LDE in the thermodynamic limit and that the finite antiferromagnetic Heisenberg chain maximally entangles two spin-$1/2$ probes very far apart. These results support the current perspective of using quantum spin chains as entanglers or quantum channels in quantum information devices.

DOI
10.1103/physreva.77.034301

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Recent numerical results showed that spin chains are able to produce long-distance entanglement (LDE). We develop a formalism that allows the computation of LDE for weakly interacting probes with gapped many-body systems. At zero temperature, a dc response function determines the ability of the physical system to generate genuine quantum correlations between the probes. We show that the biquadratic Heisenberg spin-1 chain is able to produce LDE in the thermodynamic limit and that the finite antiferromagnetic Heisenberg chain maximally entangles two spin-1/21/21/2 probes very far apart. These results support the current perspective of using quantum spin chains as entanglers or quantum channels in quantum information devices.
We discuss the phenomenon of long-distance entanglement (LDE) in the ground state of quantum spin models, its use in high-fidelity and robust quantum communication, and its realization in many-body systems of ultracold atoms in optical lattices and in arrays of coupled optical cavities. We investigate XX quantum spin models on one-dimensional lattices with open ends and different patterns of site-dependent interaction couplings, singling out two general settings: patterns that allow for perfect LDE in the ground state of the system, namely such that the end-to-end entanglement remains finite in the thermodynamic limit, and patterns of quasi-long-distance entanglement (QLDE) in the ground state of the system, namely such that the end-to-end entanglement vanishes with a very slow power-law decay as the length of the spin chain is increased. We discuss physical realizations of these models in ensembles of ultracold bosonic atoms loaded in optical lattices. We show how, using either suitably engineered super-lattice structures or exploiting the presence of edge impurities in lattices with single periodicity, it is possible to realize models endowed with nonvanishing LDE or QLDE. We then study how to realize models that optimize the robustness of QLDE at finite temperature and in the presence of imperfections using suitably engineered arrays of coupled optical cavities. For both cases the numerical estimates of the end-to-end entanglement in the actual physical systems are thoroughly compared with the analytical results obtained for the spin model systems. We finally introduce LDE-based schemes of long-distance quantum teleportation in linear arrays of coupled cavities, and show that they allow for high-fidelity and high success rates even at moderately high temperatures.
Several recent experiments have shown that long-range exchange interactions can determine collective magnetic ground states of nanostructures in bulk and on surfaces. The ability to generate and control entanglement in a system with long-range interaction will be of great importance for future quantum technology. An important step forward to reach this goal is the creation of entangled states for spins of distant magnetic atoms. Herein, the generation of long-distance entanglement between remote spins at large separations in bulk and on surface is studied theoretically, based on a quantum spin Hamiltonian and time-dependent Schrödinger equation for experimentally realized conditions. We demonstrate that long-distance entanglement can be generated between remote spins by using an appropriate quantum spin chain (a quantum mediator), composed by sets of antiferromagnetically coupled spin dimers. Ground state properties and quantum spin dynamics of entangled atoms are studied. We demonstrate that one can increase or suppress entanglement by adding a single spin in the mediator. The obtained result is explained by monogamy property of entanglement distribution inside a quantum spin system. We present a novel approach for non-local sensing of remote magnetic adatoms via spin entanglement.
Isotropic XXXXXX models of one-dimensional spin-1/2 chains are investigated with the aim to elucidate the formal structure and the physical properties that allow these systems to act as channels for long-distance, high-fidelity quantum teleportation. We introduce two types of models: (i) open, dimerized XXXXXX chains, and (ii) open XXXXXX chains with small end bonds. For both models we obtain the exact expressions for the end-to-end correlations and the scaling of the energy gap with the length of the chain. We determine the end-to-end concurrence and show that model (i) supports true long-distance entanglement at zero temperature, while model (ii) supports ``quasi-long-distance'' entanglement that slowly falls off with the size of the chain. Due to the different scalings of the gaps, respectively exponential for model (i) and algebraic in model (ii), we demonstrate that the latter allows for efficient qubit teleportation with high fidelity in sufficiently long chains even at moderately low temperatures.
We generally investigate necessary conditions for the generation of long-distance entanglement. We consider a quantum system in which a system mediates the indirect interaction between two spins, which we refer to as probe spins. Firstly, we weaken the coupling between each probe spin and the mediator system to the infinitesimal strength in order to generate the long-distance entanglement. We give two necessary conditions for the mediator system to generate the long-distance entanglement. We prove that indirect interaction cannot generate the entanglement if it is 'classical'. We also give a necessary condition for the effective fields on the probe spins to satisfy. Secondly, we generate the long-distance entanglement by the use of only external fields. We show that external fields on the adjacent spins to the probes are necessary in addition to external fields on the probe spins. Finally, we consider the cases where the coupling strength between each probe spin and the mediator system is finite. In particular, we show two examples where the external fields on the mediator system highly enhance the long-distance entanglement.
A strongly coupled Heisenberg chain provides an important channel for quantum communication through its many-body ground state. Yet, the nature of the effective interactions and the ability to mediate long-range entanglement differs significantly for chains of opposite parity. Here, we contrast the characters of even- and odd-size chains when they are coupled to external qubits. Additional parity effects emerge in both cases, depending on the positions of the attached qubits. Some striking results include (i) the emergence of maximal entanglement and (ii) spin-chain Ruderman-Kittel-Kasuya-Yosida interactions for qubits attached to an even chain, and (iii) the ability of chains of either parity to mediate qubit entanglement that is undiminished by distance.
We investigate entanglement generation in one-dimensional quantum spin systems with the sinusoidal deformation. In the system, the energy scale of each local term in the Hamiltonian is modified according to a position-dependent function ${\mathrm{sin}}^{\ensuremath{\alpha}}[\frac{\ensuremath{\pi}}{N}(x\ensuremath{-}\frac{1}{2})]$, where xxx is the position of the local term and NNN is the length of the system. We show that at zero temperature the system with $\ensuremath{\alpha}\ensuremath{\ge}2$ is able to generate a sizable entanglement between two spins at open edges, even when the two spins are infinitely far apart. This long-distance entanglement is rather robust against thermal fluctuations and survives up to a temperature that decays with the system size slowly, in an algebraic form.