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Quantum Entanglement and Projective Ring Geometry
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2006, том 2
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- Symmetry, Integrability and Geometry: Methods and Applications, 2006, том 2, випуск за цей рік
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Quantum Entanglement and Projective Ring Geometry
Інший ID:
2000 Mathematics Subject Classification: 81P15; 51C05; 13M05; 13A15; 51N15; 81R05
Посилання:
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.
Підтримка:
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.
Дата:
2006
Переглядів:
578
Завантажень:
231
Анотація:
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.
