A geometric approach to quantum entanglement classification

Abstract

[en] Quantum entanglement represents one of the fundamental differences between classical and quantum physics, with crucial roles in quantum information theory, super dense coding and quantum teleportation among others. A particularly simple description of entanglement of quantum states arises in the setting of complex algebraic geometry, via the Segre embedding. This is a map of algebraic varieties that serves as a tensor product and allows detecting separable (non-entangled states). In this thesis, we review the main features of the geometric approach to entanglement. We focus on SLOCC equivalence, which is defined as the set of possible states that a quantum state may transform into. We construct generalizations of previous results for concrete instances, giving a classification formula for all states. Some applications concerning quantum information are also given.