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(H) = P-26. Given a quantum pure state vertical bar phi > is an element of P(H) we define the X-phi-hypersuface by cutting X with a hyperplane H-phi defined by the linear form ranges over the stochastic local operation and classical communication entanglement classes, the `worst' possible singular X-phi-hypersuface with isolated singularities, has a unique singular point of type D-4.