Coherent states, entanglement, and geometric invariant theory

TL;DR

This paper explores the deep mathematical connection between quantum entanglement and Geometric Invariant Theory, revealing how entangled states relate to minimal vectors and orbit structures within this geometric framework.

Contribution

It establishes a theoretical link between entanglement properties and Geometric Invariant Theory, providing a new mathematical perspective on quantum states.

Findings

Maximal total variance characterizes completely entangled states

Completely entangled states are minimal vectors in their orbits

Noncompletely entangled states correspond to GIT semistable vectors

Abstract

The main objective of the paper is to unveil an adequate mathematics hidden behind entanglement, that is Geometric Invariant Theory. More specifically relation between these two subjects can be described by the following theses. (i) Total variance of completely entangled state is maximal. (ii) This distinguishes the state as a minimal vector in its orbit under action of complexified dynamic group. (iii) An ultimate aim of Geometric Invariant Theory is a description of complex orbits and their minimal vectors. It suggests that noncompletely entangled states are just GIT semistable vectors.