Geometric constructions over $\mathbb{C}$ and $\mathbb{F}_2$ for Quantum Information

TL;DR

This review explores two geometric constructions over complex numbers and finite fields that connect Lie group representations to quantum entanglement and contextuality, providing a unified classification framework.

Contribution

It introduces two geometric models linked to Lie groups that unify the classification of quantum entanglement and contextuality resources.

Findings

Unified classification of entanglement classes for tripartite systems.

Representation theory links to quantum resource geometry.

Weight diagrams encode commutation relations of generalized Pauli groups.

Abstract

In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers $\mathbb{C}$ and the other one is over the two elements field $\mathbb{F}_2$. Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems ($3$ qubits, $3$ fermions, $3$ bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized $N$-Pauli group.