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Physical Review B

Entanglement between remote spin-qubits in one dimension by scattering in the real-space Anderson model

Matthew Habgood, J. H. Jefferson, G. A. D. Briggs

Published May 9, 2008• 14 citations
Publisher

Abstract

The possibility of generating entanglement between two solid-state electron spin-qubits by spin-dependent scattering of a third, propagating electron is attractive. A process of this kind could work well beyond the range of the direct interaction between the two spins. In this study, the binding of electrons in one-dimensional structures is theoretically modeled using a real-space analog of the Anderson model. The respective binding sites are arranged in two different geometries. The first is a straightforward series arrangement, and the second a parallel arrangement in which the binding sites are embedded in either arm of an Aharonov-Bohm style ring. For the series arrangement, the entanglement between the bound electrons is calculated and is shown to be low (concurrence $<0.2$) in all regions of significant transmission. In contrast, the parallel arrangement allows full entanglement to be generated from a ${S}_{z}=1(\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\uparrow}⟩)$ or ${S}_{z}=0(\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩)$ starting configuration with an overall probability of success up to 0.5.

DOI
10.1103/physrevb.77.195308

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A theoretical scheme is presented for the entanglement of two-electron spin qubits bound in series within a quasi-one-dimensional mesoscopic structure at a distance beyond their normal range of interaction. A third electron is scattered from them, and full entanglement is achieved upon measurement of a transmitted electron in the correct spin state. Critically, each bound electron is trapped within an individual structure that has at least two spatial states. Two simple examples of such structures are discussed here. One is a 'stub', in which a quantum dot (for example) is coupled to one side of the quasi-one-dimensional structure. The other is a pair of degenerate, coupled quantum dots, with strong interdot Coulomb repulsion, placed within the one-dimensional superstructure. Both of these are shown to allow generation of entanglement with a significant probability of success. In contrast to the results of the authors' previous works, this allows for the generation of entanglement in a series, rather than in a parallel, configuration of the bound electrons with respect to the propagating electron.
The entanglement of flying qubits with static qubits in the solid state is important in the development of quantum computing. It has recently been shown theoretically that the spin-dependent scattering of a propagating electron from a bound electron is sufficient to give full entanglement between the qubits embodied in their respective spins [J. H. Jefferson et al., Europhys. Lett. 74, 764 (2006); D. Gunlycke et al., J. Phys.: Condens. Matter 18, S851 (2006)]. In this paper a generalized, real-space Anderson model is introduced for a quasi-one-dimensional structure consisting of a binding site coupled to ideal leads. Degeneracy of both the binding site and the leads is incorporated to represent, for example, conduction band degeneracy in carbon nanotubes. The model is used to calculate the spin-dependent scattering behavior and resultant static-flying qubit entanglement created by the scattering process. Degeneracy (and more generally, multiplicity) in the binding site gives rise to inelastic scattering processes. In the elastic scattering regime, a degenerate binding site gives rise to antiresonant structures in the transmission spectrum. Additional degeneracy in the leads restricts this effect to the second set of leads, raising the possibility of spin filtration, though this is eliminated in the inelastic scattering regime. Degeneracy in the binding site also gives rise to multiple resonances in transmission that will improve the probability of obtaining entangled pairs relative to the nondegenerate case. This effect is maximized when the components of the degenerate binding site are symmetrically coupled to the leads.
We analytically study entanglement generation through an open quantum-dot system described by the two-lead Anderson model. We exactly obtain the transition rate between the nonentangled incident state in one lead and the outgoing spin-singlet state in the other lead. We find that only the spin-singlet state can transmit in the cotunneling process. To discuss such an entanglement property in the open quantum system, we construct the exact two-electron scattering state of the Anderson model. It is striking that the scattering state contains spin-singlet bound states induced by the Coulomb interaction. The bound state describes a scattering process in which the set of momenta is not conserved and hence, it is not in the form of a Bethe eigenstate.
We analytically study entanglement generation through an open quantum dot system described by the two-lead Anderson model. We exactly obtain the transition rate between the non-entangled incident state in one lead and the outgoing spin-singlet state in the other lead. In the cotunneling process, only the spin-singlet state can transmit. To discuss such an entanglement property in the open quantum system, we construct the exact two-electron scattering state of the Anderson model. It is striking that the scattering state contains spin-singlet bound states induced by the Coulomb interaction. The bound state describes the scattering process in which the set of momenta is not conserved and hence it is not in the form of a Bethe eigenstate.
We present a model where two magnetic impurities in a discrete tight-binding ring become entangled because of scattering processes associated to the injection of a conduction electron. We introduce a weak-coupling approximation that allows us to solve the problem in an analytical way and compare the theory with the exact numerical results. We obtain the generation of entanglement both in a deterministic way and in a probabilistic one. The first case is intrinsically related to the structure of the two-impurity reduced density matrix, while the second one occurs when a projection on the electron state is performed.
We theoretically study how a scattered electron can entangle molecular spin qubits (MSQs). This requires solving the inelastic transport of a single electron through a scattering region described by a tight-binding interacting Hamiltonian. We accomplish this using a Green's-function solution. We can model realistic physical implementations of MSQs by parametrizing the tight-binding Hamiltonian with first-principles descriptions of magnetic anisotropy and exchange interactions. We find that, for two-MSQ systems with inversion symmetry, projective measurement of the spin degree of freedom of the scattered electron offers probabilistic control of the degree of entanglement between the MSQs.