Serge Perrine | Searcharxiv

arXiv2006

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

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.

quant-ph↗

arXiv2003

Recherches autour de la theorie de Markoff

The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic geometry. This algebraic approach is completed with an analytical development concerning fuchsian groups. The link with the Teichmuller theory for punctured toruses is completely described, giving their classification with a reduction theory. More general considerations about Riemann surfaces, geodesics and their hamiltonian study are quoted, together with applications in physics, 1/f noise and zeta function. Ideas about important conjectures are presented. Reasons why the Markoff theory appears in different geometrical contexts are given, thanks to decomposition results in the group GL(2,Z).

math-ph↗