Singularity of typeD₄arising 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 . 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 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 × 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 D 4 -singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems.