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

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.