Computing Entanglement Polytopes

TL;DR

This paper explores new computational methods, including geometric, algebraic, and numerical techniques, to determine entanglement polytopes in multipartite quantum systems, extending previous approaches and classifying specific system types.

Contribution

It introduces alternative methods for computing entanglement polytopes, including inequalities and a numerical approach for classifying entanglement in complex quantum systems.

Findings

Presented geometric and algebraic tools for outer approximation of entanglement polytopes.

Developed a numerical method to compute entanglement polytopes for SLOCC classes.

Classified entanglement polytopes for 2 x 3 x N quantum systems.

Abstract

In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the entanglement polytopes. Furthermore we present a numerical method which, in theory, can compute the entanglement polytope of any given SLOCC class given a representative. Using it we classify the entanglement polytopes of $2 \times 3 \times N$ systems.