22078 articles – 15904 references  [version française]
 HAL: hal-00077093, version 2
 arXiv: quant-ph/0605239
 Available versions: v1 (2006-05-29) v2 (2006-06-12) v3 (2006-08-15) v4 (2006-08-18)
 Quantum Entanglement and Projective Ring Geometry
 Michel R. P. Planat 1, Metod Saniga 1, 2
 (2006-05-29)
 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$\times$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. All lines in each pencil carry mutually commuting operators; in one of the pencils, which we call the kernel, the observables on two lines share a base of maximally entangled states. The three operators on any line in each pencil represent a row or column of some Mermin's \lq\lq magic" square, thus revealing an inherent geometrical nature of the latter. 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.
 1: Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (FEMTO-ST) CNRS : UMR6174 – Université de Franche-Comté – Université de Technologie de Belfort-Montbeliard – Ecole Nationale Supérieure de Mécanique et des Microtechniques 2: Astronomical Institute, Slovak Academy of Sciences (ASTRINSTSAV) Astronomical Institute, Slovak Academy of Sciences 3: Institut de Physique Nucléaire de Lyon (IPNL) CNRS : UMR5822 – IN2P3 – Université Claude Bernard - Lyon I
 Subject : Physics/Quantum PhysicsPhysics/Mathematical Physics
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 hal-00077093, version 2 http://hal.archives-ouvertes.fr/hal-00077093 oai:hal.archives-ouvertes.fr:hal-00077093 From: Michel Planat <> Submitted on: Monday, 12 June 2006 14:39:10 Updated on: Monday, 12 June 2006 15:04:53