New Identity for Cayley's First Hyperdeterminant with Applications to Symmetric Tensors and Entanglement
Isaac Dobes
TL;DR
The paper derives a new Levi-Civita-based formula for Cayley’s first hyperdeterminant, connecting $\mathrm{hdet}(\mathcal{A})$ to a multilinear contraction with $d$-dimensional Levi-Civita symbols and the canonical hypermatrix vectorization $\mathrm{hvec}(\mathcal{A})$. It then specializes to symmetric hypermatrices, introducing the $1/N$-hypermatrix vectorization $\mathrm{hvec}_{1/N}$ and generalized duplication matrices $D_d^{(N)}$ to achieve a polynomial-time algorithm in the order $N$ (for fixed side length $d$) for computing $\mathrm{hdet}$, via an efficiently reusable precomputed tensor $\mathcal{E}_d^{(N)}$. The approach yields a concrete pathway to quantify $2n$-way entanglement for $2n$-qudit bosons through $\mathrm{hdet}$, linking to broader entanglement measures such as concurrence. The work also lays out directions for non-cubical and group-symmetric generalizations and provides appendices with explicit construction proofs for generalized elimination and duplication matrices. Overall, the results offer a new, scalable method for hyperdeterminants with direct implications for quantum entanglement in symmetric bosonic systems.
Abstract
In this article, a new formula for computing Cayley's first hyperdeterminant in terms of the Levi-Civita symbol is given. It is then shown that this formula can be used to compute the hyperdeterminant of symmetric hypermatrices in polynomial time with respect to their order (assuming fixed side length). Applications to the quantum entanglement of bosons are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric hypermatrices, hypermatrix generalizations of elimination and duplication matrices are defined, and explicit formulas for them are derived in the appendix of this article.