Multi-distributed entanglement in finitely correlated chains

¹ Dipartimento di Fisica Teorica, Università di Trieste Strada Costiera 11, 34014 Trieste, Italy
² Istituto Nazionale di Fisica Nucleare, Sezione di Trieste - 34100 Trieste, Italy
³ Institut für Theoretische Physik - Boltzmanngasse 5, A-1090 Vienna, Austria

Received: 14 April 2005
Accepted: 2 August 2005

Abstract

The entanglement-sharing properties of an infinite spin-chain are studied when the state of the chain is a pure, translation-invariant state with a matrix-product structure (Klümper A., Schadschneider A. and Zittartz J., J. Phys. A, 24 (1991) L955; Z. Phys. B, 87 (1992) 281; Europhys. Lett., 24 (1993) 293). We study the entanglement properties of such states by means of their finitely correlated structure (Fannes M., Nachtergaele B. and Werner R. F., Comm. Math. Phys., 144 (1992) 443; Europhys. Lett., 10 (1989) 633; J. Phys. A, 24 (1991) L185). These states are recursively constructed by means of an auxiliary density matrix ρ on a matrix algebra and a completely positive map , where is the spin matrix algebra. General structural results for the infinite chain are therefore obtained by explicit calculations in (finite) matrix algebras. In particular, we study not only the entanglement shared by nearest-neighbours, but also, differently from previous works (Wootters W. K., Contemp. Math., 305 (2002) 299) the entanglement shared between connected regions of the spin-chain. This range of possible applications is illustrated and the maximal concurrence (Coffman V., Kundu J. and Wootters W. K., Phys. Rev. A, 61 (2000) 052306) for the entanglement of connected regions can actually be reached.