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

2012, Journal of Physics A: Mathematical and Theoretical

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 two. 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 reminding 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 analogue of a Conwell heptad of PG(5, 2).