Singularity of type $D_4$ arising from four qubit systems

Abstract

An intriguing correspondence between four-qubit systems and simple singularity of type $D_4$ 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 $X\cap H\subset X$, has an isolated singularity of type $D_4$; 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 $2\times 2\times 2\times 2$, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the $D_4$-singularity.

20 pages, 5 tables

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Cited by in corpus (7)

• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Three-qutrit entanglement and simple singularities
• Learning Algebraic Models of Quantum Entanglement
• Multi-qubits and Polyvalent Singularity in Type II Supestring Theory
• Maximally entangled real states and SLOCC invariants: the 3-qutrit case
• Toward Jordan Decompositions of Tensors