The Veldkamp Space of Two-Qubits

Abstract

Given a remarkable representation of the generalized Pauli operators of two-qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements - the so-called Veldkamp space of W(2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the "classical" subsets answering to geometric hyperplanes of W(2).

Figures (3)

• Figure 1: The three kinds of geometric hyperplanes of $W(2)$. The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each picture -- except that in the bottom right-hand corner -- stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.
• Figure 2: Left: -- The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -- A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order ($1,2$)).
• Figure 3: The five different kinds of the lines of $\mathcal{V}(W(2))$, each being uniquely determined by the properties of its core-set (black bullets). Note that the "yellow" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of $W(2)$.