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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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