A hyperdeterminant on Fermionic Fock Space

TL;DR

This work extends Cayley’s hyperdeterminant to the fermionic Fock space with $N=8$, using the Spin$(16,\mathbb{C})$ action arising from the $\mathbb{Z}_2$-graded Lie algebra $\mathfrak{e}_8=\mathfrak{so}(16,\mathbb{C})\oplus\mathcal{F}_8^+$. The authors compute the semi-simple restriction of the invariant, obtaining $\operatorname{HDet}_{\operatorname{Spin}(16,\mathbb{C})|\bigwedge^4\mathbb{C}^8}(\Psi)=Q^{2}T^{4}$, where $Q$ encodes a piece related to $\Delta_{E_7|\bigwedge^4\mathbb{C}^8}$ and $T$ is a separate $2\times2\times2\times2$ hyperdeterminant factor. A rich combinatorial picture ties the invariant to planes in a cube over $\mathbb{Z}_2$ and to the Fano plane, and the construction reveals connections to the four-fermion and four-qubit entanglement invariants. The paper also discusses how the Spin$(16)$ invariant ring is generated by fundamental invariants of degrees $2,8,12,14,18,20,24,30$, and comments on the challenge of expressing the hyperdeterminant in terms of these generators.

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.

Figures (4)

• Figure 1: Monomials of Cayley's Hyperdeterminant from a combinatorial perspective: The four diagonals (red) of the cube provide the first four monomials of degree $4$ of type $a_{ijk}a_{\overline{ijk}}$ (where bar denotes the bit-complement), the six parallelograms (green) provide the six monomials of type $a_{ijk}a_{\overline{ijk}}a_{i'j'k'}a_{\overline{i'j'k'}}$ (where $ijk$ and $i'j'k'$ have one bit difference), and the two tetrahedra (blue) give the two monomials of type $a_{ijk}a_{i\overline{jk}}a_{\overline{i}j\overline{k}}a_{\overline{ij}k}$.
• Figure 2: Planes of the cube in $(\mathbb{Z}_2)^3$: $6$ planes as the "regular" faces, $6$ planes through the "opposite edges" and $2$ "tetrahedron"-planes.
• Figure 3: The Fano plane obtained by considering the projective space associated to the cube $(\mathbb{Z}_2)^3$ when the point corresponding to variable $y_8$ is chosen as the center. The planes of $(\mathbb{Z}_2)^3$ passing through $y_8$ are sent to lines and the planes that do not go through $y_8$ are mapped to affine plane of the Fano plane.
• Figure 4: The affine plane obtained by removing the projective line $y_5y_6y_7$.

Theorems & Definitions (1)

• Remark 3.1