Borland and Dennis studied a system of three fermions with local rank six. That is, the wave function is an element of the Hilbert space $$\mathcal H = \bigwedge^3 \mathbb C^6$$. Using the results of arXiv:0806.4076, one finds that there are four entanglement classes, with entanglement polytopes as displayed below. Intriguingly, this can be understood as a “symmetrization” of the three-qubit system.

No. Class Description Vertices Faces
1 $$\operatorname{GEN}_1$$ Genuinely entangled three-fermion state (analogue of GHZ state) 4 4
2 $$\operatorname{GEN}_2$$ Genuinely entangled three-fermion state (analogue of W state) 4 4
3 $$\operatorname{BISEP}$$ Bi-separable three-fermion state 2 2
4 $$\operatorname{SEP}$$ A single Slater determinant 1 1