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

TL;DR

This paper reveals that the magic three-qubit Veldkamp line can be naturally represented within the combinatorial Grassmannian of type G_2(7), connecting quantum information structures with combinatorial and geometric configurations.

Contribution

It provides a novel combinatorial Grassmannian framework for understanding the magic three-qubit Veldkamp line, detailing its composition with geometric and combinatorial configurations.

Findings

Veldkamp line occurs within the Grassmannian of type G_2(7)

Explicit combinatorial representation of core and additional points and lines

Connections between quantum structures and geometric configurations

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 $\widehat{\mathcal{Q}}$(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 (Schl\"afli) 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).