A hyperdeterminant on fermionic Fock space

Abstract

Twenty years ago, Cayley's hyperdeterminant, the degree four invariant of the polynomial ring $\mathbb{C}[{\mathbb{C}}^2\otimes {\mathbb{C}}^2\otimes {\mathbb{C}}^2{]}^{{{\mathrm{SL}}_2(\mathbb{C})}^{\times 3}}$, was popularized in modern physics as it 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.