Entanglement and the geometry of two qubits

TL;DR

This paper provides a geometric framework for understanding entanglement in two-qubit systems, including separability, entanglement measures, and key quantum information protocols, with new formulas and proofs.

Contribution

It introduces a three- and four-dimensional geometric description of two-qubit states, including a formula for Bell states and geometric proofs of key criteria.

Findings

Geometric description of separability and entanglement in 3D and 4D

A new formula for Bell states facilitating teleportation proof

Geometric proof of the Peres-Horodecki separability criterion

Abstract

Two qubits is the simplest system where the notions of separable and entangled states and entanglement witnesses first appear. We give a three dimensional geometric description of these notions. This description however carries no quantitative information on the measure of entanglement. A four dimensional description captures also the entanglement measure. We give a neat formula for the Bell states which leads to a slick proof of the fundamental teleportation identity. We describe optimal distillation of two qubits geometrically and present a simple geometric proof of the Peres-Horodecki separability criterion.