Singularity of type D4D_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 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.Comment: 20 pages, 5 table

Two new non-equivalent three-qubit CHSH games

In this paper, we generalize to three players the well-known CHSH quantum game. To do so, we consider all possible 3 variables Boolean functions and search among them which ones correspond to a game scenario with a quantum advantage (for a given entangled state). In particular we provide two new three players quantum games where, in one case, the best quantum strategy is obtained when the players share a $GHZ$ state, while in the other one the players have a better advantage when they use a $W$ state as their quantum resource. To illustrate our findings we implement our game scenarios on an online quantum computer and prove experimentally the advantage of the corresponding quantum resource for each game.Comment: 19 pages, 3 figure