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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149117 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862739075965911040 |
|---|---|
| author | Havlicek, H. Odehnal, B. Saniga, M. |
| author_facet | Havlicek, H. Odehnal, B. Saniga, M. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T20:07:08Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149117 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:07:08Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Havlicek, H. Odehnal, B. Saniga, M. 2019-02-19T17:30:06Z 2019-02-19T17:30:06Z 2009 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 https://nasplib.isofts.kiev.ua/handle/123456789/149117 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. This work was carried out in part within the “Slovak-Austrian Science and Technology Cooperation Agreement” under grants SK 07-2009 (Austrian side) and SK-AT-0001-08 (Slovak side), being also partially supported by the VEGA grant agency projects Nos. 2/0092/09 and 2/7012/27. The final version was completed within the framework of the Cooperation Group “Finite Projective Ring Geometries: An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics, and Chemistry of Coupling” at the Center for Interdisciplinary Research (ZiF), University of Bielefeld, Germany. The authors are grateful to Wolfgang Herfort (Vienna) for his suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Factor-Group-Generated Polar Spaces and (Multi-)Qudits Article published earlier |
| spellingShingle | Factor-Group-Generated Polar Spaces and (Multi-)Qudits Havlicek, H. Odehnal, B. Saniga, M. |
| title | 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_short | Factor-Group-Generated Polar Spaces and (Multi-)Qudits |
| title_sort | factor-group-generated polar spaces and (multi-)qudits |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149117 |
| work_keys_str_mv | AT havlicekh factorgroupgeneratedpolarspacesandmultiqudits AT odehnalb factorgroupgeneratedpolarspacesandmultiqudits AT sanigam factorgroupgeneratedpolarspacesandmultiqudits |