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

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 $\hat{\mathcal{Q}}(4,2)$ 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 $\hat{\mathcal{Q}}(4,2)$ -- 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.

7 pages, 3 figures

References in corpus (4)

• Black Hole Entropy and Finite Geometry
• Mermin pentagrams arising from Veldkamp lines for three qubits
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

Cited by in corpus (1)

• $X$-States From a Finite Geometric Perspective