Geometry of entanglement in the Bloch sphere

TL;DR

This paper offers a comprehensive geometric framework for understanding entanglement in rank-2 quantum states using Bloch spheres, classifying states into five distinct entanglement classes.

Contribution

It introduces a novel geometric classification of rank-2 quantum states into five entanglement classes based on their Bloch sphere representations.

Findings

Five classes of entanglement and separability identified

Complete analysis for bipartite states, partial for multipartite states

Unique Bloch sphere for each rank-2 mixed state

Abstract

Entanglement is an important concept in quantum information, quantum communication, and quantum computing. We provide a geometrical analysis of entanglement and separability for all the rank-2 quantum mixed states: complete analysis for the bipartite states, and partial analysis for the multipartite states. For each rank-2 mixed state, we define its unique Bloch sphere, that is spanned by the eigenstates of its density matrix. We characterize those Bloch spheres into exactly five classes of entanglement and separability, give examples for each class, and prove that those are the only classes.