On exceptional Lie geometries

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

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 Q + ( 5 , 2 ) , an elliptic quadric Q − ( 5 , 2 ) and a quadratic cone Q ^ ( 4 , 2 ) that share a parabolic quadric Q ( 4 , 2 ) , the doily, is shown to provide an interesting model for the Veldkamp space of the doily. The model is based on the facts that: (a) the 20 off-doily points of Q + ( 5 , 2 ) form ten complementary pairs, each corresponding to a unique grid of the doily; (b) the 12 off-doily points of Q − ( 5 , 2 ) form six complementary pairs, each corresponding to a unique ovoid of the doily; and (c) the 15 off-doily points of Q ^ ( 4 , 2 ) , disregarding the nucleus of 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.