A hyperdeterminant on Fermionic Fock Space

TL;DR

This paper computes a hyperdeterminant invariant on fermionic Fock space for eight particles, linking it to quantum entanglement classifications and black hole entropy, with combinatorial insights.

Contribution

It introduces a new hyperdeterminant invariant for fermionic Fock space and connects it to existing quantum information invariants.

Findings

Computed the hyperdeterminant invariant for N=8 fermionic particles.

Connected the invariant to known quantum information invariants.

Provided combinatorial interpretations of the formulas.

Abstract

Twenty years ago Cayley's hyperdeterminant, the degree four invariant of the polynomial ring $\mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes \mathbb{C}^2]^{{\text{SL}_2(\mathbb{C})}^{\times 3}}$, 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.