Abstract
An intriguing correspondence between four-qubit systems and simple singularity of type D4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space . Then, cutting X with a specific hyperplane H, we prove that the X-hypersurface, defined from the section X∩H⊂X, has an isolated singularity of type D4; it is also shown that this is the ‘worst-possible’ isolated singularity that one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version, by proving that the equation of the dual variety of X, which is nothing but the Cayley hyperdeterminant of type 2 × 2 × 2 × 2, can be expressed in terms of the stochastic local operation and classical communication-invariant polynomials as the discriminant of the miniversal deformation of the D4-singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems.
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