An Algebraic-Geometric Characterization of Tripartite Entanglement

TL;DR

This paper introduces an algebraic-geometric framework using invariants like secant varieties and multilinear ranks to classify tripartite entanglement in quantum systems, providing a detailed taxonomy including a refined classification of three-qutrit states.

Contribution

It develops a novel algebraic-geometric approach for classifying tripartite entanglement, offering a finite classification scheme based on SLOCC invariants and detailed structures for three-qutrit systems.

Findings

Classification of tripartite pure states into finite families and subfamilies.

Introduction of algebraic-geometric tools as invariants under SLOCC.

Detailed grouping of three-qutrit entanglement structures.

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 $k$-secant varieties 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.