Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information
arXiv:0806.4701 · doi:10.1142/S0219887808003156
Abstract
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger 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.
AmsLaTeX, 37 pages, 8 figures. This paper is an expanded version of some lectures delivered by one of us (G. M.) at the ``Advanced Winter School on the Mathematical Foundation of Quantum Control and Quantum Information'' which took place at Castro Urdiales (Spain), February 11-15, 2008
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- Geometrical Description of Quantum Mechanics - Transformations and Dynamics
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- Tensorial dynamics on the space of quantum states
- Ehrenfest statistical dynamics in chemistry: study of decoherence effects
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- Geometric realizations of Kaehler and of para-Kaehler curvature models
- Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
- Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation
- Klein's programme and Quantum Mechanics