Singularity of type D₄ arising from four qubit systems

An intriguing correspondence between four-qubit systems and simple singularity of type D₄ is established. We first consider an algebraic variety X of separable states within the projective Hilbert space mathbb{P}\(mathcal{H}\)=mathbb{P}¹⁵. Then, cutting X with a specific hyperplane H, we prove that the X-hypersurface, defined from the section Xcap Hsubset X, has an isolated singularity of type D₄; it is also shown that this is the "worst-possible" isolated singularity 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 2times 2times 2times 2, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the D₄-singularity.