In the past 20 years classical concepts of algebraic geometry such as the notion of dual varieties have been introduced in the quantum information literature to distinguish different classes of entanglement, a quantum property recognized as a resource in quantum information processing. In this talk, after introducing the connection between the basics of quantum information and the geometry of tensors, I will explain how new equations of the duals of homogeneous varieties can be obtained from graded simple Lie algebras. I will in particular focus on the dual of the spinor varieties $\mathbb{S}_{16}$, the projectivization of the highest weight orbit of the 128 dimensional spin module and its connection with what is known in physics as Fermionic Fock spaces.