New Identity for Cayley's First Hyperdeterminant with Applications to Symmetric Tensors and Entanglement

TL;DR

This paper introduces a new formula for Cayley's first hyperdeterminant using the Levi-Civita symbol, enabling efficient computation for symmetric hypermatrices and exploring applications in quantum entanglement of bosons.

Contribution

It presents a novel formula for the hyperdeterminant, along with methods for fast computation on symmetric hypermatrices and applications to quantum entanglement.

Findings

Polynomial-time computation of hyperdeterminant for symmetric hypermatrices.

Explicit formulas for hypermatrix generalizations of elimination and duplication matrices.

Application to quantum entanglement of bosons.

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.