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

## TL;DR

This paper explores the structure of the Veldkamp space of the doily using a specific three-qubit Veldkamp line, revealing new geometric correspondences and suggesting potential relevance for quantum information theory.

## Contribution

It introduces a novel model of the doily's Veldkamp space based on a three-qubit Veldkamp line, linking geometric configurations to quantum information concepts.

## Key findings

- The 20 off-doily points of $ ext{Q}^+(5,2)$ form ten pairs linked to doily grids.
- The 12 off-doily points of $ ext{Q}^-(5,2)$ form six pairs linked to doily ovoids.
- The 15 off-doily points of $ ext{Q}(4,2)$ correspond to the doily's perp-sets.

## Abstract

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.