Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

Details Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

2008-06-28

arxiv.org/abs/0806.4701v1

Int.J.Geom.Meth.Mod.Phys.5:989-1032,2008

Physics

Quantum Physics

AmsLaTeX, 37 pages, 8 figures. This paper is an expanded version of some lectures delivered by one of us (G. M.) at the ``Adva

Scientific paper

10.1142/S0219887808003156

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

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