Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 066, 14 pages      quant-ph/0605239      https://doi.org/10.3842/SIGMA.2006.066

Quantum Entanglement and Projective Ring Geometry

Michel Planat a, Metod Saniga b and Maurice R. Kibler c
a) Institut FEMTO-ST, CNRS/Université de Franche-Comté, Département LPMO, 32 Avenue de l'Observatoire, F-25044 Besançon Cedex, France
b) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
c) Institut de Physique Nucléaire de Lyon, IN2P3-CNRS/Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France

Received June 13, 2006, in final form August 16, 2006; Published online August 17, 2006

Abstract
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.

Key words: quantum entanglement; two spin-½ particles; finite rings; projective ring lines.

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