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 . Given a quantum pure state we define the Xφ-hypersuface by cutting X with a hyperplane Hφ defined by the linear form (the Xφ-hypersurface of X is ). We prove that when ranges over the stochastic local operation and classical communication entanglement classes, the ‘worst’ possible singular Xφ-hypersuface with isolated singularities, has a unique singular point of type D4.
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