An Algebraic-Geometric Characterization of Tripartite Entanglement

Abstract

To characterize entanglement of tripartite \(\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d\) systems, we employ algebraic-geometric tools that are invariants under Stochastic Local Operation and Classical Communication (SLOCC), namely \(k\)-secants and one-multilinear ranks. Indeed, by means of them, we present a classification of tripartite pure states in terms of a finite number of families and subfamilies. At the core of it stands out a fine-structure grouping of three-qutrit entanglement.