Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily

A magic three-qubit Veldkamp line of $W(5,2)$, i.\,e. the line comprising a hyperbolic quadric $\mathcal{Q}^+(5,2)$, an elliptic quadric $\mathcal{Q}^-(5,2)$ and a quadratic cone $\widehat{\mathcal{Q}}(4,2)$ that share a parabolic quadric $\mathcal{Q}(4,2)$, the doily, is shown to provide an interesting model for the Veldkamp space of the latter. The model is based on the facts that: a) the 20 off-doily points of $\mathcal{Q}^+(5,2)$ form ten complementary pairs, each corresponding to a unique grid of the doily; b) the 12 off-doily points of $\mathcal{Q}^-(5,2)$ form six complementary pairs, each corresponding to a unique ovoid of the doily; and c) the 15 off-doily points of $\widehat{\mathcal{Q}}(4,2)$ -- disregarding the nucleus of $\mathcal{Q}(4,2)$ -- are in bijection with the 15 perp-sets of the doily. These findings lead to a conjecture that also parapolar spaces can be relevant for quantum information.