Entanglement Classification by Algebraic Geometry

Abstract

The role of entanglement is to substantially enhance the speed of computations by a process that classical computers could not achieve. Indeed, entanglement is something mysterious from quantum mechanics that has no equivalent in classical mechanics. An enormous quantity of work has been carried in the past decades to answer some natural questions about entanglement. How to know if a state is entangled or not (Separability problem)? If a state is entangled, how much is it, and how far from a separable state is it (Entanglement measure)? Can an entangled state perform the same tasks as another entangled state (Entanglement classification)? These questions are increasingly mastered over time and are the keystone of progress in quantum computation, in conjunction with technical progress. Entanglement classification fails to be finite when we consider something greater than a 4-qubit system. This issue has been analysed and solved by Masoud Gharahi, Stefano Mancini and Giorgio Ottaviani in their paper with help of algebraic geometry. The purpose of this thesis is to make a detailed overview of the notions this article needs to be understood.