Maximal entanglement of two delocalized spin-$\frac{1}{2}$ particles

TL;DR

This paper analyzes the entanglement properties of two indistinguishable delocalized spin-1/2 particles across three spatial modes, identifying conditions for maximal entanglement and characterizing different entanglement types.

Contribution

It provides the first detailed description of maximally entangled states in such a delocalized particle system and derives conditions for maximal entanglement in arbitrary configurations.

Findings

Maximally entangled states exist only in three-mode systems.

Two entanglement invariants fully characterize the entanglement.

A necessary and sufficient condition for maximal entanglement in general systems is established.

Abstract

We describe the entanglement of two indistinguishable delocalized spin-$\frac{1}{2}$ particles in the simplest spatial configuration of three spatial modes with the constraint that at most one particle occupy each mode. It is show that this is the only number of modes for which maximally entangled states exist in such a system. The set of entangled states, including the set of maximally entangled states, is described and different types of entanglement in terms of Bell-nonlocal correlations for different partitions of the system are identified. In particular we focus on the entangled states that are Bell-local for a tri-partition of the system and cannot be described as a superposition of bi-partite entangled pairs of localized particles. Two entanglement invariants are constructed and it is shown that all entanglement monotones are functions of these. Furthermore, the system has a generic non-trivial local unitary symmetry with a corresponding $2\pi/3$ fractional topological phase. In addition to this a necessary and sufficient condition for the existence of maximally entangled states in systems of arbitrary numbers of delocalized particles with arbitrary spin, where at most one particle can occupy each mode, is derived.