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
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