Properties of entanglement in a Spin Chain with impurities

Abstract

In this thesis we have studied the e®ects on ground-state entanglement and quantum-state transfer caused by the presence of one and two diagonal impu- rities in a 1D ring-shaped spin-1 2 hamiltonian with XX ferromagnetic cou- pling placed in an otherwise homogeneous transverse magnetic ¯eld. We have restricted our attention to the one-excitation sector of the Hilbert space, where the hamiltonian becomes equivalent to a tight-binding model. In both cases, SchrÄodinger equation has been solved exactly via Green's operator formalism. In the one-impurity model the presence of the defect causes a ¯rst order quantum phase transition, where the new ground state of the system becomes localized around the impurity site. In this state bipartite entanglement is di®erent from zero for spins in a region contained within a localization length, which depends on the ratio of the magnetic inhomogeneity and the exchange coupling (®). We have also analyzed the transmission of quantum information along such a chain. We have found that the presence of the defect is responsible of various phenomena: 1.) information storage: if the state is encoded initially in the impurity, it doesn't di®use away and both ¯delity and concurrence retain its initial maximum value; 2.) mirror e®ect: if the state isn't encoded initially in the defect, entanglement waves get re°ected and transmitted at the impurity site by an amount depending on re°ection and transmission coe±cients, where, they too, involve the same ratio. In the two-impurity model, a similar quantum phase transition occurs, and ground-state entanglement becomes localized too, with the same func- tional relation for the localization length as in the one-impurity model. Nev- ertheless in this case two localization points arise in the spin ring. Quantum- state transfer, in the limit of ® >> 1, exhibits : 1.) bouncing e®ect: if the initial singlet state involves one of the impurities and the external spin, en- 72 tanglement bounces between the defects with Rabi oscillations of frequency given by the energy di®erence of the two localized eigenstates; 2.) entan- glement trapping: relying on the mirror e®ect of the two impurities, the entanglement waves remain con¯ned into the region delimited by the defects sites. From an experimental point of view, we have presented a model that permits to achieve quantum information tasks by systems realizable with present-day technology and requiring minimum control operations. Further studies (inclusion of dynamical properties of the impurities, characterization of the quantum phase transition from an q-information point of view, ex- tension to higher spatial dimensionality and/or other sectors of the Hilbert space, entanglement versus disorder, etc.) should be give more insight on the fundamental physics behind entanglement's theory as well as suggest some achievable experimental protocols for quantum information development.