A hyperdeterminant on Fermionic Fock Space

Twenty years ago Cayley's hyperdeterminant, the degree four invariant of the polynomial ring $\mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes \mathbb{C}^2]^{{\text{SL}_2(\mathbb{C})}^{\times 3}}$, was popularized in modern physics as separates genuine entanglement classes in the three qubit Hilbert space and is connected to entropy formulas for special solutions of black holes. In this note we compute the analogous invariant on the fermionic Fock space for $N=8$, i.e. spin particles with four different locations, and show how this invariant projects to other well-known invariants in quantum information. We also give combinatorial interpretations of these formulas.