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

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.