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

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).


Introduction
One of the most startling results of finite-geometric approach to the field of quantum information and the so-called black-hole/qubit correspondence is, undoubtedly, a recent discovery [1,2] of the existence of a magic Veldkamp line associated with the five-dimensional binary symplectic polar space underlying geometry of the three-qubit Pauli group. Three basic constituents of this line (illustrated 1 graphically in Figure 1) host a number of extensions of generalized quadrangles with lines of size three isomorphic to affine polar spaces of rank three and order two, each having distinguished physical interpretation and in their totality offering a remarkable unifying framework for form theories of gravity and black hole entropy. The purpose of this paper is to show that this magic line has also a remarkable representation in the Veldkamp space of combinatorial Grassmannian of type G 2 (7).

Relevant Finite-Geometrical Background
To this end, we first give an overview of relevant finite geometry. We start with a finite pointline incidence structure C = (P, L, I) where P and L are, respectively, finite sets of points and lines and where incidence I ⊆ P ×L is a binary relation indicating which point-line pairs are incident (see, e. g., [3]). Here, we shall only be concerned with specific point-line incidence structures called configurations [4]. A (v r , b k )-configuration is a C where: 1) v = |P| and b = |L|, 2) every line has k points and every point is on r lines, and 3) two distinct lines intersect in at most one point and every two distinct points are joined by at most one line; a configuration where v = b and r = k is called symmetric (or balanced), and usually denoted where the symbol ∆ stands for the symmetric difference of the two geometric hyperplanes and an overbar denotes the complement of the object indicated. Our central concept is that of a combinatorial Grassmannian (see, e. g., [6,7]) G k (|X|), where k is a positive integer and X is a finite set, which is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion. It is known [6] is the Desargues (10 3 )-configuration and G 2 (6) is the Cayley-Salmon (15 4 , 20 3 )-configuration [8].
A (finite-dimensional) classical polar space (see, for example, [9,10]) describes the geometry of a d-dimensional vector space over the Galois field GF(q), V (d, q), carrying a non-degenerate reflexive sesquilinear form σ(x, y). The polar space is called symplectic, and usually denoted as W(d − 1, q), if this form is bilinear and alternating, i.e., if σ(x, x) = 0 for all x ∈ V (d, q); such a space exists only if d = 2N , where N ≥ 2 is called its rank. A subspace of V (d, q) is called totally isotropic if σ vanishes identically on it. W(2N − 1, q) can then be regarded as the space of totally isotropic subspaces of the ambient space PG(2N − 1, q), the ordinary (2N − 1)-dimensional projective space over GF(q), with respect to a symplectic form (also known as a null polarity). A quadric in PG(d, q), d ≥ 1, is the set of points whose coordinates satisfy an equation of the form d+1 i,j=1 a ij x i x j = 0, where at least one a ij = 0. Up to transformations of coordinates, there is one or two distinct kinds of non-singular quadrics in PG(d, q) according as d is even or odd, namely [9]: Q (2N, q), the parabolic quadric formed by all points of PG(2N, q) satisfying the standard equation (2N − 1, q), the elliptic quadric formed by all points of PG (2N − 1, q) satisfying the standard equation f (x 1 , x 2 )+x 3 x 4 +· · ·+x 2N −1 x 2N = 0, where f is irreducible over GF(q); and Q + (2N −1, q), the hyperbolic quadric formed by all points of PG (2N −1, q) satisfying the standard equation The number of points lying on quadrics is as follows [9]: . Given the hyperbolic quadric Q + (2N − 1, q) of PG (2N − 1, q), N ≥ 2, a set S of points such that each line joining two distinct points of S has no point in common with Q + (2N − 1, q) is called an exterior set of the quadric. It is known that |S| ≤ (q N − 1)/(q − 1); if |S| = (q N − 1)/(q − 1), then S is called a maximal exterior set. Interestingly [11], Q + (5, 2) has, up to isomorphism, a unique such set -also known, after its discoverer, as a Conwell hetpad [12].
Finally, one has to introduce a finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), which is a C satisfying the following axioms [13]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique line through x that is incident with L; from these axioms it readily follows that |P| = (s + 1)(st + 1) and |L| = (t + 1)(st + 1). In what follows we shall only be concerned with its two particular types: GQ(2, 2) ∼ = Q(4, 2) ∼ = W(3, 2) and GQ(2, 4) ∼ = Q − (5, 2).
4 Magic Three-Qubit Veldkamp Line in V(G 2 (7)) There are seven distinguished magic Veldkamp lines living in W(5, 2), one per each element of X. A representative of them, also depicted in Figure 2, is structured as follows: • Core GQ(2, 2): Its 15 points are represented by those α-points that share one digit in the second set, that is by points whose representatives are abcd:ef 7 if the common digit is '7'; its 15 lines are those of type (α, α, α) of the following particular form abcd:ef 7, abef :cd7, cdef :ab7.  (7)).

Conclusion
We have demonstrated that the Veldkamp space V(G 2 (7)) provides a rather natural environment for the magic Veldkamp line of three-qubits. Interestingly, V(G 2 (7)) was recently found to be also related to finite geometry behind the 64-dimensional real Cayley-Dickson algebra [8]. Hence, our findings seem to indicate that the nature of magic Veldkamp line may well have something to do with this particular algebra.
Research Project M1564-N27. I am grateful to my friend Petr Pracna for electronic versions of the figures.