Three-qutrit entanglement and simple singularities

TL;DR

This paper applies singularity theory to analyze the entanglement structure of pure three-qutrit quantum states, revealing that the most complex entanglement class corresponds to a hypersurface with a unique D4 singularity.

Contribution

It introduces a novel geometric approach using singularity theory to classify three-qutrit entanglement and identifies the D4 singularity as the most complex case.

Findings

The worst-case entanglement hypersurface has a D4 singularity.

Singularity types classify different entanglement classes.

The approach links algebraic geometry with quantum entanglement analysis.

Abstract

In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is $X\cap H_\varphi \subset X$). We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated singularities, has a unique singular point of type $D_4$.