Wednesday, February 15, 2012

1202.2973 (Metod Saniga et al.)

Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic
Quadric of PG(7,2)

Metod Saniga, Peter Levay, Petr Pracna
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 x 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).
The strategy we employed is completely novel and unique in its nature, as are
the results obtained. 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.
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