A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of combinatorial Grassmannian of type G₂(7), mathcal{V}\(G₂(7\)). The lines of the ambient symplectic polar space are those lines of mathcal{V}\(G₂(7\)) whose cores feature an odd number of points of G₂(7). After introducing basic properties of three different types of points and six distinct types of lines of mathcal{V}\(G₂(7\)), we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core generalized quadrangle GQ(2,2), and then additional points and lines of a specific elliptic quadric mathcal{Q}^-(5,2), a hyperbolic quadric mathcal{Q}⁺(5,2) and a quadratic cone widehat{mathcal{Q}}(4,2) that are centered on the GQ(2,2). In particular, each point of mathcal{Q}⁺(5,2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in mathcal{Q}^-(5,2) comprise six Cayley-Salmon configurations and six Desargues configurations with their complementary points, and the remaining Cayley-Salmon configuration stands for the vertex of widehat{mathcal{Q}}(4,2).