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

Abstract

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of combinatorial Grassmannian of type $G_2(7)$, $\mathcal{V}(G_2(7))$. The lines of the ambient symplectic polar space are those lines of $\mathcal{V}(G_2(7))$ whose cores feature an odd number of points of $G_2(7)$. After introducing basic properties of three different types of points and six distinct types of lines of $\mathcal{V}(G_2(7))$, 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 $\hat{\mathcal{Q}}$(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 $\hat{\mathcal{Q}}$(4,2).

6 pages, 2 figures

References in corpus (3)

• Black Hole Entropy and Finite Geometry
• Mermin pentagrams arising from Veldkamp lines for three qubits
• The Complement of Binary Klein Quadric as a Combinatorial Grassmannian