Three-qutrit entanglement and simple singularities

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 $|\phi ⟩\in \mathbb{P}(\mathcal{H})$ we define the $X_{\phi }$-hypersuface by cutting $X$ with a hyperplane $H_{\phi }$ defined by the linear form $⟨\phi |$ (the $X_{\phi }$-hypersurface of $X$ is $X\cap H_{\phi }\subset X$). We prove that when $|\phi ⟩$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_{\phi }$-hypersuface with isolated singularities, has a unique singular point of type $D_4$.

6 pages, 1 figure, 2 tables

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• The 3 x 3 x 3 hyperdeterminant as a polynomial in the fundamental invariants for SL(3,C) x SL(3,C) x SL(3,C)

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• Maximally entangled real states and SLOCC invariants: the 3-qutrit case
• Persistent Tensors and Multiqudit Entanglement Transformation