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arXiv:1207.2967 · New Journal of Physics

General conditions for the generation of long-distance entanglement

Tomotaka Kuwahara

Published Dec 18, 2012• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/14/12/123032

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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.
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 propose two schemes to produce long-distance entanglement in a spin chain. The first is based on a controllable interaction system, one starts from an entangled kernel and adds weaken interaction spins to the boundary sites step by step, then the entanglement will be extended longer and longer and its value is equal to that of its kernel. The second is based on a uniform interaction (J) system with a bulk magnetic field (B) that is absent for the boundary qubits, as long as B|J > 5, one can obtain near perfect long distance entanglement. Ultra-low temperature is needed in both schemes.
The search for methods to create and maintain entanglement has led to the idea of environmentally induced entanglement. Roughly speaking, the usually detrimental effect of coupling a non-interacting bipartite system to an environment is turned into an advantage by using the environment to mediate an indirect interaction, which can result in entanglement of the two parts of the system under certain conditions. Of course, care has to be taken to properly evaluate the conflicting influences of the environment. Only if the indirect interaction overcompensates for the decoherence, entanglement creation can be expected. \n \nIt has been suggested that entanglement creation can be achieved in bosonic heat baths even over finite spatial separations with only a moderate polynomial decay of entanglement with distance. In this work, we look more closely at the distance dependence, for the first time employing an oscillator model that is both exactly solvable and includes dissipation. We numerically prove that entanglement creation is, in fact, extremely distance-sensitive and it is not possible to entangle objects which are further apart than approximately their own size. \n \nAdditionally, we suggest an approach how to mitigate the distance dependence. It comes at the cost of geometrically modifying the bath modes by imposing physical boundary conditions resulting in a gap in the spectrum. This is implemented by placing the system inside of an infinitely long superconducting cavity. An experimental implementation of this could be feasible.
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.
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 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.