Factor-Group-Generated Polar Spaces and (Multi-)Qudits
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...
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| Видавець: | Інститут математики НАН України |
|---|---|
| Дата: | 2009 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2009
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149117 |
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| Цитувати: | 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. |
Репозиторії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149117 |
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irk-123456789-1491172019-02-20T01:26:15Z Factor-Group-Generated Polar Spaces and (Multi-)Qudits Havlicek, H. Odehnal, B. Saniga, M. 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. 2009 Article 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. 1815-0659 2000 Mathematics Subject Classification: 20C35; 51A50; 81R05 http://dspace.nbuv.gov.ua/handle/123456789/149117 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
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. |
| format |
Article |
| author |
Havlicek, H. Odehnal, B. Saniga, M. |
| spellingShingle |
Havlicek, H. Odehnal, B. Saniga, M. Factor-Group-Generated Polar Spaces and (Multi-)Qudits Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Havlicek, H. Odehnal, B. Saniga, M. |
| author_sort |
Havlicek, H. |
| title |
Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_short |
Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_full |
Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_fullStr |
Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_full_unstemmed |
Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_sort |
factor-group-generated polar spaces and (multi-)qudits |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149117 |
| citation_txt |
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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT havlicekh factorgroupgeneratedpolarspacesandmultiqudits AT odehnalb factorgroupgeneratedpolarspacesandmultiqudits AT sanigam factorgroupgeneratedpolarspacesandmultiqudits |
| first_indexed |
2023-05-20T17:32:10Z |
| last_indexed |
2023-05-20T17:32:10Z |
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1796153521908744192 |