The space of density states in geometrical quantum mechanics

We present a geometrical description of the space of density states of a quantum system of finite dimension. After presenting a brief summary of the geometrical formulation of Quantum Mechanics, we proceed to describe the space of density states \D(\Hil) from a geometrical perspective identifying the stratification associated to the natural GL(\Hil)--action on \D(\Hil) and some of its properties. We apply this construction to the cases of quantum systems of two and three levels.Comment: Amslatex, 18 pages, 4 figure

Tensorial description of quantum mechanics

Relevant algebraic structures for the description of Quantum Mechanics in the Heisenberg picture are replaced by tensorfields on the space of states. This replacement introduces a differential geometric point of view which allows for a covariant formulation of quantum mechanics under the full diffeomorphism group.Comment: 8 page

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

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.Comment: 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, 200

Tensorial characterization and quantum estimation of weakly entangled qubits

In the case of two qubits, standard entanglement monotones like the linear entropy fail to provide an efficient quantum estimation in the regime of weak entanglement. In this paper, a more efficient entanglement estimation, by means of a novel class of entanglement monotones, is proposed. Following an approach based on the geometric formulation of quantum mechanics, these entanglement monotones are defined by inner products on invariant tensor fields on bipartite qubit orbits of the group SU(2)xSU(2).Comment: 23 pages, 3 figure

Evolution of very low mass pre-main sequence stars and young brown dwarfs under accretion: A phenomenological approach

In the poster presented in Cool Star 15, we analyzed the effect of disk accretion on the evolution of very low mass pre-main sequence stars and young brown dwarfs and the resulting uncertainties on the determination of masses and ages. We use the Lyon evolutionary 1-D code assuming a magnetospheric accretion process, i.e., the material falls covering a small area of the radiative surface, and we take into account the internal energy added from the accreted material as a free parameter $\epsilon$. Even if the approach to this problem is phenomenological, our formalism provides important hints about characteristics of disk accretion, which are useful for improved stellar interior calculations. Using the accretion rates derived from observations our results show that accretion does not affect considerably the position of theoretical isochrones as well as the luminosity compared with standard non-accreting models. See more discussions in a forthcoming paper by Gallardo, Baraffe and Chabrier (2008).Comment: Poster contribution Cool Star 15, St. Andrews, U

Introduction to Quantum Mechanics and the Quantum-Classical transition

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.Comment: Survey paper based on the lectures delivered at the XV International Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11-16, 2006. To appear in Publ. de la RSM

The boundary field theory induced by the Chern-Simons theory

The Chern-Simons theory defined on a 3-dimensional manifold with boundary is written as a two-dimensional field theory defined only on the boundary of the three-manifold. The resulting theory is, essentially, the pullback to the boundary of a symplectic structure defined on the space of auxiliary fields in terms of which the connection one-form of the Chern-Simons theory is expressed when solving the condition of vanishing curvature. The counting of the physical degrees of freedom living in the boundary associated to the model is performed using Dirac's canonical analysis for the particular case of the gauge group SU(2). The result is that the specific model has one physical local degree of freedom. Moreover, the role of the boundary conditions on the original Chern- Simons theory is displayed and clarified in an example, which shows how the gauge content as well as the structure of the constraints of the induced boundary theory is affected.Comment: 10 page

Classical Tensors and Quantum Entanglement I: Pure States

The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.Comment: 16 pages, 1 figure, to appear in Int. J. Geom. Meth. Mod. Phy