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Journal Articles Symmetry, Integrability and Geometry : Methods and Applications Year : 2009

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

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 $\vG$, we first construct vector spaces over $\GF(p)$, $p$ a prime, by factorising $\vG$ over appropriate normal subgroups. Then, by expressing $\GF(p)$ in terms of the commutator subgroup of $\vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $\vG$ 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 $\vG$ is $\leq 2$. 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 $\vG$. 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.
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Dates and versions

hal-00372071 , version 1 (31-03-2009)
hal-00372071 , version 2 (18-08-2009)
hal-00372071 , version 3 (13-10-2009)

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Hans Havlicek, Boris Odehnal, Metod Saniga. Factor-Group-Generated Polar Spaces and (Multi-)Qudits. Symmetry, Integrability and Geometry : Methods and Applications, 2009, 5, 096 (15 p.). ⟨10.3842/SIGMA.2009.096⟩. ⟨hal-00372071v3⟩

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