Charting the real four-qubit Pauli group via ovoids of a hyperbolic quadric of PG(7, 2)

Abstract

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7, 2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7, 2), the seven-dimensional projective space of order 2. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circumstances, the third element on the line defined by any two points of an ovoid is skew-symmetric, as is the nucleus of the conic defined by any three points of an ovoid. Each ovoid thus yields 36/84 elements of the former/latter type, accounting for all 120 skew-symmetric elements of the group. There are a number of notable types of ovoid-associated subgeometries of the group, of which we mention the following: a subset of 12 skew-symmetric elements lying on four mutually skew lines that span the whole ambient space; a subset of 15 symmetric elements that corresponds to two ovoids sharing three points; a subset of 19 symmetric elements generated by two ovoids on a common point; a subset of 27 symmetric elements that can be partitioned into three ovoids in two unique ways; a subset of 27 skew-symmetric elements that exhibits a 15 + 2 × 6 split similar to that exhibited by an elliptic quadric of PG(5, 2); and a subset of seven skew-symmetric elements formed by the nuclei of seven conics having two points in common, which is an analog of a Conwell heptad of PG(5, 2). The strategy we employed is completely novel and unique in its nature, mainly due to the fact that hyperbolic quadrics in binary projective spaces of higher dimensions have no ovoids. Such a detailed dissection of the geometry of the group in question may, for example, be crucial in getting further insights into the still-puzzling black-hole-qubit correspondence/analogy.