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 tensors in polynomial time with respect to their order (assuming fixed side length). Applications to quantifying the entanglement of states of bosonic quantum systems are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric tensors, generalized elimination and duplication matrices are defined, and their explicit formulas are derived.
Fields of Study & Subjects
Algebraic and Geometric Analysis
Quantum Mechanics and Applications
Quantum Information and Cryptography
Publication details
- Journal / source
- Electronic Journal of Linear Algebra
- Publisher
- University of Wyoming Libraries
- Volume
- 42
- Pages
- 399–410
- Publication date
- 10 May 2026