Classification of multipartite systems featuring only $|W\rangle$ and $|GHZ\rangle$ genuine entangled states

TL;DR

This paper classifies multipartite quantum systems with only |W> and |GHZ> genuine entangled states using geometric and algebraic methods, linking quantum entanglement classes to Lie algebra structures.

Contribution

It provides a geometric and algebraic classification of multipartite systems with specific entanglement types, connecting quantum information theory to algebraic geometry and Lie algebras.

Findings

Classification of systems based on entanglement types

Connection between quantum states and algebraic geometric structures

Identification of fundamental subvarieties related to entanglement

Abstract

In this paper we present several multipartite quantum systems featuring the same type of genuine (tripartite) entanglement. Based on a geometric interpretation of the so-called $|W\rangle$ and $|GHZ\rangle$ states we show that the classification of all multipartite systems featuring those and only those two classes of genuine entanglement can be deduced from earlier work of algebraic geometers. This classification corresponds in fact to classification of fundamental subadjoint varieties and establish a connection between those systems, well known in Quantum Information Theory and fundamental simple Lie algebras.