4,676 research outputs found
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
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 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
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 . 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
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