Three fermions with six single particle states can be entangled in two inequivalent ways

TL;DR

This paper classifies entanglement types in three-fermion systems with six single-particle states using a generalized hyperdeterminant, revealing two inequivalent entanglement classes and connecting to three-qubit systems.

Contribution

It introduces a new measure of tripartite fermionic entanglement and provides a SLOCC classification for three-fermion systems with six single-particle states.

Findings

Identifies two inequivalent entanglement classes for three fermions.

Shows a subclass of these systems shares properties with three-qubit entanglement.

Proposes Plücker relations as criteria for separability in general fermionic systems.

Abstract

Using a generalization of Cayley's hyperdeterminant as a new measure of tripartite fermionic entanglement we obtain the SLOCC classification of three-fermion systems with six single particle states. A special subclass of such three-fermion systems is shown to have the same properties as the well-known three-qubit ones. Our results can be presented in a unified way using Freudenthal triple systems based on cubic Jordan algebras. For systems with an arbitrary number of fermions and single particle states we propose the Pl\"ucker relations as a sufficient and necessary condition of separability.