Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2) - Archive ouverte HAL Access content directly
Journal Articles Journal of Physics A: Mathematical and Theoretical Year : 2012

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 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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Dates and versions

hal-00669929 , version 1 (14-02-2012)

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Metod Saniga, Peter Levay, Petr Pracna. Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2). Journal of Physics A: Mathematical and Theoretical, 2012, 45, pp.295304. ⟨10.1088/1751-8113/45/29/295304⟩. ⟨hal-00669929⟩

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