Pour décrire les vibrations à l'intérieur des molécules diatomiques, le potentiel de Morse est une meilleure approximation que le système de l'oscillateur harmonique. Ainsi, en se basant sur la définition des états cohérents et comprimés donnée dans le cadre du problème de l'oscillateur harmonique, la première partie de ce travail suggère une construction des états cohérents et comprimés pour le potentiel de Morse. Deux types d’états seront construits et leurs différentes propriétés seront étudiées en portant une attention particulière aux trajectoires et aux dispersions afin de confirmer la quasi-classicité de ces états. La deuxième partie de ce travail propose d'insérer ces deux types d’états cohérents et comprimés de Morse dans un miroir semi-transparent afin d'introduire un nouveau moyen de créer de l'intrication. Cette intrication sera mesurée à l’aide de l’entropie linéaire et nous étudierons la dépendance par rapport aux paramètres de cohérence et de compression. / In order to describe the vibrations inside a diatomic molecule, the Morse potential is a better approximation than the harmonic oscillator system. Thus, based on the definition of the coherent states given in the context of the harmonic oscillator, the first part of this work suggests a construction for the squeezed coherent states of the Morse potential. Two types of states will be constructed and their diverse properties will be studied with special attention to the trajectories and dispersions in order to confirm their quasi-classicity. The second part of this work proposes to insert those two types of Morse squeezed coherent states in a beam splitter in order to introduce a new way of creating entanglement. This entanglement will be measured by the linear entropy and we will study the dependence with the coherence and squeezing parameters.

Pour décrire les vibrations à l'intérieur des molécules diatomiques, le potentiel de Morse est une meilleure approximation que le système de l'oscillateur harmonique. Ainsi, en se basant sur la définition des états cohérents et comprimés donnée dans le cadre du problème de l'oscillateur harmonique, la première partie de ce travail suggère une construction des états cohérents et comprimés pour le potentiel de Morse. Deux types d’états seront construits et leurs différentes propriétés seront étudiées en portant une attention particulière aux trajectoires et aux dispersions afin de confirmer la quasi-classicité de ces états. La deuxième partie de ce travail propose d'insérer ces deux types d’états cohérents et comprimés de Morse dans un miroir semi-transparent afin d'introduire un nouveau moyen de créer de l'intrication. Cette intrication sera mesurée à l’aide de l’entropie linéaire et nous étudierons la dépendance par rapport aux paramètres de cohérence et de compression. / In order to describe the vibrations inside a diatomic molecule, the Morse potential is a better approximation than the harmonic oscillator system. Thus, based on the definition of the coherent states given in the context of the harmonic oscillator, the first part of this work suggests a construction for the squeezed coherent states of the Morse potential. Two types of states will be constructed and their diverse properties will be studied with special attention to the trajectories and dispersions in order to confirm their quasi-classicity. The second part of this work proposes to insert those two types of Morse squeezed coherent states in a beam splitter in order to introduce a new way of creating entanglement. This entanglement will be measured by the linear entropy and we will study the dependence with the coherence and squeezing parameters.

We study minimization problems with respect to a one-parameter family of generalized relative entropies. These relative entropies, which we call relative -entropies (denoted I (P; Q)), arise as redundancies under mismatched compression when cumulants of compression lengths are considered instead of expected compression lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative -entropies behave like squared Euclidean distance and satisfy the Pythagorean property. We explore the geometry underlying various statistical models and its relevance to information theory and to robust statistics. The thesis consists of three parts. In the first part, we study minimization of I (P; Q) as the first argument varies over a convex set E of probability distributions. We show the existence of a unique minimizer when the set E is closed in an appropriate topology. We then study minimization of I on a particular convex set, a linear family, which is one that arises from linear statistical constraints. This minimization problem generalizes the maximum Renyi or Tsallis entropy principle of statistical physics. The structure of the minimizing probability distribution naturally suggests a statistical model of power-law probability distributions, which we call an -power-law family. Such a family is analogous to the exponential family that arises when relative entropy is minimized subject to the same linear statistical constraints. In the second part, we study minimization of I (P; Q) over the second argument. This minimization is generally on parametric families such as the exponential family or the - power-law family, and is of interest in robust statistics ( > 1) and in constrained compression settings ( < 1). In the third part, we show an orthogonality relationship between the -power-law family and an associated linear family. As a consequence of this, the minimization of I (P; ), when the second argument comes from an -power-law family, can be shown to be equivalent to a minimization of I ( ; R), for a suitable R, where the first argument comes from a linear family. The latter turns out to be a simpler problem of minimization of a quasi convex objective function subject to linear constraints. Standard techniques are available to solve such problems, for example, via a sequence of convex feasibility problems, or via a sequence of such problems but on simpler single-constraint linear families.