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

A magic three-qubit Veldkamp line of W ( 5 , 2 ) , i.e., the line comprising a hyperbolic quadric Q + ( 5 , 2 ) , an elliptic quadric Q − ( 5 , 2 ) and a quadratic cone Q ^ ( 4 , 2 ) that share a parabolic quadric Q ( 4 , 2 ) , the doily, is shown to provide an interesting model for the Veldkamp space of the doily. The model is based on the facts that: (a) the 20 off-doily points of Q + ( 5 , 2 ) form ten complementary pairs, each corresponding to a unique grid of the doily; (b) the 12 off-doily points of Q − ( 5 , 2 ) form six complementary pairs, each corresponding to a unique ovoid of the doily; and (c) the 15 off-doily points of Q ^ ( 4 , 2 ) , disregarding the nucleus of 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.

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THE VELDKAMP SPACE OF GQ(2, 4)

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887810004762 ◽

2010 ◽

Vol 07 (07) ◽

pp. 1133-1145 ◽

Cited By ~ 4

Author(s):

M. SANIGA ◽

R. M. GREEN ◽

P. LÉVAY ◽

P. VRANA ◽

P. PRACNA

Keyword(s):

Information Theory ◽

Black Hole ◽

Quantum Information ◽

Quantum Information Theory ◽

Generalized Quadrangle ◽

Combinatorial Properties ◽

Elliptic Quadric

It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic to PG(5, 2). Since the GQ(2, 4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2, 2)s, the 63 points of PG(5, 2) split into two families; 27 being represented by perp-sets and 36 by GQ(2, 2)s. The 651 lines of PG(5, 2) are found to fall into four distinct classes: in particular, 45 of them feature only perp-sets, 216 comprise two perp-sets and one GQ(2, 2), 270 consist of one perp-set and two GQ(2, 2)s and the remaining 120 are composed solely of GQ(2, 2)s, according to the intersection of two distinct hyperplanes determining the (Veldkamp) line is, respectively, a line, an ovoid, a perp-set and a grid (i.e. GQ(2, 1)) of a copy of GQ(2, 2). A direct "by-hand" derivation of the above-listed properties is followed by their heuristic justification based on the properties of an elliptic quadric of PG(5, 2) and complemented by a proof employing combinatorial properties of a 2-(28, 12, 11)-design and associated Steiner complexes. Surmised relevance of these findings for quantum (information) theory and the so-called black hole analogy is also outlined.

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The Valuations of the Near Octagon ${\Bbb I}_4$

The Electronic Journal of Combinatorics ◽

10.37236/1102 ◽

2006 ◽

Vol 13 (1) ◽

Cited By ~ 1

Author(s):

Bart De Bruyn ◽

Pieter Vandecasteele

Keyword(s):

Polar Space ◽

Polar Spaces ◽

Near Polygon ◽

Hyperbolic Quadric ◽

Dual Polar Space ◽

Full Embedding ◽

Parabolic Quadric ◽

Dual Polar Spaces

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric $Q(2n,2)$, $n \geq 2$, which are not contained in a given hyperbolic quadric $Q^+(2n-1,2) \subset Q(2n,2)$ define a sub near polygon ${\Bbb I}_n$ of the dual polar space $DQ(2n,2)$. It is known that every valuation of $DQ(2n,2)$ induces a valuation of ${\Bbb I}_n$. In this paper, we classify all valuations of the near octagon ${\Bbb I}_4$ and show that they are all induced by a valuation of $DQ(8,2)$. We use this classification to show that there exists up to isomorphism a unique isometric full embedding of ${\Bbb I}_n$ into each of the dual polar spaces $DQ(2n,2)$ and $DH(2n-1,4)$.

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