Quantum Entanglement and Projective Ring Geometry

Michel R. P. Planat (1) , Metod Saniga (1, 2) , Maurice R. Kibler (3)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 Astronomical Institute of the Slovak Academy of Sciences
3 IPNL - Institut de Physique Nucléaire de Lyon

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. 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.

Domaines: Physique Quantique [quant-ph]; Physique mathématique [math-ph]

Michel R. P. Planat, Metod Saniga, Maurice R. Kibler. Quantum Entanglement and Projective Ring Geometry. SIGMA, 2006, 2, 066, 14 p. ⟨10.3842/SIGMA.2006.066⟩. ⟨hal-00077093v4⟩