Three-qutrit entanglement and simple singularities

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 $P\left(H\right)=P^{26}$. Given a quantum pure state $|\phi ⟩\in P\left(H\right)$ we define the X _φ-hypersuface by cutting X with a hyperplane H _φ defined by the linear form $⟨\phi |$ (the X _φ-hypersurface of X is $X\cap H_{\phi }\subset X$). We prove that when $|\phi ⟩$ ranges over the stochastic local operation and classical communication entanglement classes, the 'worst' possible singular X _φ-hypersuface with isolated singularities, has a unique singular point of type D ₄. (paper)