Geometry of entanglement and separability in Hilbert subspaces of dimension up to three

TL;DR

This paper classifies the geometric structures of entangled and separable states in three-dimensional Hilbert subspaces, revealing 14 possible shapes and their implications for quantum entanglement.

Contribution

It provides a complete classification of the geometric forms of entangled and separable states in three-dimensional Hilbert subspaces, including new classes unique to these dimensions.

Findings

Identified 14 geometric classes of separable states in 3D subspaces

Characterized the geometry of entangled and separable states in bipartite and multipartite systems

Illustrated the shapes and structures of these state sets

Abstract

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer, Liss and Mor [Phys. Rev. A 95:032308, 2017], and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.