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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 $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|φ\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_φ$-hypersuface by cutting $X$ with a hyperplane $H_φ$ defined by the linear form $\langleφ|$ (the $X_φ$-hypersurface of $X$ is $X\cap H_φ\subset X$). We prove that when $|φ\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_φ$-hypersuface with isolated singularities, has a unique singular point of type $D_4$.