Singularity of type D-4 arising from four-qubit systems

An intriguing correspondence between four-qubit systems and simple singularity of type D-4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space P(H) = P-15. Then, cutting X with a specific hyperplane H, we prove that the X-hypersurface, defined from the section X boolean AND H subset of X, has an isolated singularity of type D-4; 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 x 2 x 2 x 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 D-4-singularity. As a consequence of this correspondence one obtains a finer- grained classification of entanglement classes of four-qubit systems.