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

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 to use the Pl\"ucker relations as a sufficient and necessary condition of separability.