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

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2009
• Hauptverfasser: Havlicek, H., Odehnal, B., Saniga, M.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2009
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149117
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Factor-Group-Generated Polar Spaces and (Multi-)Qudits / H. Havlicek, B. Odehnal, M. Saniga // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 32 назв. — англ.

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.