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).
Highlights
One of the most startling results of the finite-geometric approach to the field of quantum information and the so-called black-hole/qubit correspondence is undoubtedly the recent discovery [1,2] of the existence of a magic Veldkamp line associated with the five-dimensional binary symplectic polar space W (5, 2) underlying the geometry of the three-qubit Pauli group
The purpose of this paper is to show that this magic line has a remarkable representation in the Veldkamp space of a combinatorial Grassmannian of type G2 (7)
We have demonstrated that the Veldkamp space V ( G2 (7)) provides a rather natural environment for the magic Veldkamp line of three-qubits
Summary
One of the most startling results of the finite-geometric approach to the field of quantum information and the so-called black-hole/qubit correspondence is undoubtedly the recent discovery [1,2] of the existence of a magic Veldkamp line associated with the five-dimensional binary symplectic polar space W (5, 2) ( the finite-geometrical concepts, symbols, and notation are explained ) underlying the geometry of the three-qubit Pauli group. The one we are interested in features an elliptic quadric, a hyperbolic quadric, and a quadratic cone over a parabolic quadric Q(4, 2), the three objects having the latter quadric in common and no other pairwise intersection. The three basic constituents of this line ( illustrated 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 main reason why this particular Veldkamp line is referred to as “magic” is the fact that it features a remarkable 20-point extension of the generalized quadrangle of type GQ(2, 1) that hosts 12 interwoven copies of a so-called magic Mermin pentagram;. The numbers inside the triangles indicate the number of points in the complement of GQ(2, 2) of the geometrical object in question
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