# Long-distance entanglement in Motzkin and Fredkin spin chains

## TL;DR

This paper analytically investigates the entanglement properties of ground states in Fredkin and Motzkin spin chains, revealing persistent long-distance entanglement across the chains, including at edges and in the bulk.

## Contribution

It provides exact calculations of entanglement measures in these models and demonstrates the presence of long-distance entanglement, a novel feature for these systems.

## Key findings

- Long-distance entanglement persists even at infinite separation.
- Entanglement involves both edge and bulk segments.
- Models violate the cluster decomposition property.

## Abstract

We derive some entanglement properties of the ground states of 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 the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior, occurring both for colorful versions of the models (with spin larger than 1/2 or 1, respectively) and for colorless cases (spin 1/2 and 1), is consistent with the violation of the cluster decomposition property. Moreover we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains.