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

Abstract

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of a combinatorial Grassmannian of type G 2 (7), V (G 2 (7)). The lines of the ambient symplectic polar space are those lines of V (G 2 (7)) whose cores feature an odd number of points of G 2 (7). After introducing the basic properties of three different types of points and seven distinct types of lines of 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 Q − (5, 2), a hyperbolic quadric Q + (5, 2), and a quadratic cone Q(4, 2) that are centered on the GQ(2, 2). In particular, each point of Q + (5, 2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in 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 Q(4, 2).

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

“…Among them, the unique triangle-free 15 3 -configuration (out of 245,342 ones), also known as the Cremona-Richmond configuration and in the sequel referred to as the doily, acquires a special footing. This notable role of the doily stems from the fact that it is isomorphic to three remarkable, conceptually-distinct point-line incidence structures, namely a symplectic polar space of type W(3, 2) (whose subgeometries furnish simplest observable proofs of quantum contextuality and justify the existence of the maximal sets of MUBs in the associated Hilbert space of two-qubits [3]), an orthogonal parabolic polar space of type Q(4, 2) (being the core of the magic three-qubit Veldkamp line of form theories of gravity [4][5][6]) and a generalized quadrangle of type GQ(2, 2) (being a subquadrangle of GQ(2, 4) ∼ = Q − (5, 2) that entails some important aspects of the so-called black-hole/qubit correspondence [7,8]). Employing the concept of Veldkamp space of a point-line incidence structure, this note aims at shedding some interesting light on how these three geometrical settings are interrelated.…”