Quantum Entanglement and Projective Ring Geometry
The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 1...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2006 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2006
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146101 |
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| Cite this: | Quantum Entanglement and Projective Ring Geometry / M. Planat, M. Saniga, M.R. Kibler // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 33 назв. — англ. |
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Planat, M. Saniga, M. Kibler, M.R. 2019-02-07T13:18:01Z 2019-02-07T13:18:01Z 2006 Quantum Entanglement and Projective Ring Geometry / M. Planat, M. Saniga, M.R. Kibler // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 33 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81P15; 51C05; 13M05; 13A15; 51N15; 81R05 https://nasplib.isofts.kiev.ua/handle/123456789/146101 The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF(2), with n = 2, 3 and 4. This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT–51–012704, the VEGA project # 2/6070/26 (both from Slovak Republic) and by the trans-national ECO-NET project # 12651NJ “Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases” (France). We are grateful to Dr. Petr Pracna for a number of fruitful comments/remarks and for creating the last two figures. One of the authors (M.S.) would like to thank the warm hospitality extended to him by the Institut FEMTO-ST in Besan¸con and the Institut de Physique Nucl´eaire in Lyon. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Entanglement and Projective Ring Geometry Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Quantum Entanglement and Projective Ring Geometry |
| spellingShingle |
Quantum Entanglement and Projective Ring Geometry Planat, M. Saniga, M. Kibler, M.R. |
| title_short |
Quantum Entanglement and Projective Ring Geometry |
| title_full |
Quantum Entanglement and Projective Ring Geometry |
| title_fullStr |
Quantum Entanglement and Projective Ring Geometry |
| title_full_unstemmed |
Quantum Entanglement and Projective Ring Geometry |
| title_sort |
quantum entanglement and projective ring geometry |
| author |
Planat, M. Saniga, M. Kibler, M.R. |
| author_facet |
Planat, M. Saniga, M. Kibler, M.R. |
| publishDate |
2006 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF(2), with n = 2, 3 and 4.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146101 |
| citation_txt |
Quantum Entanglement and Projective Ring Geometry / M. Planat, M. Saniga, M.R. Kibler // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 33 назв. — англ. |
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| first_indexed |
2025-12-07T15:43:02Z |
| last_indexed |
2025-12-07T15:43:02Z |
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1850864757793357824 |