### Entanglement Polytope Explorer

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, it is easy to show 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 (using the connection explained in that paper). Some of us have recently studied the implications of the Borland–Dennis inequalities (which describe the first polytope) on the ground state of a harmonic system.

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