A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

Michel R. P. Planat (1) , Haret Rosu (2) , Serge Perrine (3) , Metod Saniga (4)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 Dept of Applied Mathematics
3 FT R&D - France Télécom Recherche & Développement
4 Astronomical Institute of the Slovak Academy of Sciences

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

Mots clés: Finite Geometries; Galois Fields; Fourier Transforms; Quantum Information Theory

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

Michel R. P. Planat, Haret Rosu, Serge Perrine, Metod Saniga. A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements. Foundations of Physics, 2006, 36, pp.1662-1680. ⟨hal-00002833v3⟩