### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 096, 15 pages      arXiv:0903.5418      https://doi.org/10.3842/SIGMA.2009.096

### Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Hans Havlicek a, c, Boris Odehnal a and Metod Saniga b, c
a) Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040 Wien, Austria
b) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
c) Center for Interdisciplineary Research (ZiF), University of Bielefeld, D-33615 Bielefeld, Germany

Received August 19, 2009, in final form October 02, 2009; Published online October 13, 2009

Abstract
Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute. Restricting to p = 2, we search for ''refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ''condensation'' of several distinct elements of G. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.

Key words: groups; symplectic and orthogonal polar spaces; geometry of generalised Pauli groups.

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