We derive some entanglement properties of the ground states for two
classes of quantum spin chains described by the Fredkin model, for
half-integer spins, and the Motzkin model, for integer ones. Since the
ground states of the two models are known analytically, we can calculate
exactly the entanglement entropy, the negativity and the quantum mutual
information. We show, in particular, that these systems exhibit
long-distance entanglement, namely two disjoint regions of the chains
remain entangled even when their separation is sent to infinity, i.e.
these systems are not affected by decoherence. This strongly entangled
behavior, Finally, we show that this behavior involves disjoint segments
located both at the edges and in the bulk of the chains.
Entanglement versus mutual information in quantum spin chains
