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We associate here the relationship between de-coherence to the statistical notion of disequilibrium with regards to the dynamics of a system that reflects the interaction between matter and a given field. The process is described via information geometry. Some of its tools are shown here to appropriately explain the process' mechanism. In particular we gain some insight into what is the role of the uncertainty principle (UP) in the pertinent proceedings.
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In this work, we study quantum decoherence as reflected by the dynamics of a system that accounts for the interaction between matter and a given field. The process is described by an important information geometry tool: Fisher's information measure (FIM). We find that it appropriately describes this concept, detecting salient details of the quantum-classical changeover (qcc). A good description of the qcc report can thus be obtained; in particular, a clear insight into the role that the uncertainty principle (UP) plays in the pertinent proceedings is presented. Plotting FIM versus a system's motion invariant related to the UP, one can also visualize how anti-decoherence takes place, as opposed to the decoherence process studied in dozens of papers. In Fisher terms, the qcc can be seen as an order (quantum)-disorder (classical, including chaos) transition.
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The Fisher-Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expression in general, by gathering known results (closed forms for submanifolds and bounds) and derive expressions for the distance between distributions with the same covariance matrix and between distributions with mirrored covariance matrices. An application of the Fisher-Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.
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Consider µ a probability measure and P µ the set of µ -equivalent strictly positive probability densities. To endow P µ with a structure of a C ∞ -Banach manifold we use the φ -connection by an open arc, where φ is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and κ -exponential functions. Moreover, we find the tangent space of P µ at a point p, and as a consequence the tangent bundle of P µ . We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and κ -exponential functions can be used to generalize of Rényi divergence.
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In this paper, we investigate the mixture arc on generalized statistical manifolds. We ensure that the generalization of the mixture arc is well defined and we are able to provide a generalization of the open exponential arc and its properties. We consider the model of a φ -family of distributions to describe our general statistical model.
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This paper proposes a method for the beta pricing model under the consideration of non-Gaussian returns by means of a generalization of the mean-variance model and the use of principal curves to define a divergence model for the optimization of the pricing model. We rely on the q-exponential model so consider the properties of the divergences which are used to describe the statistical model and fully characterize the behavior of the assets. We derive the minimum divergence portfolio, which generalizes the Markowitz's (mean-divergence) approach and relying on the information geometrical aspects of the distributions the Capital Asset Pricing Model (CAPM) is then derived under the geometrical characterization of the distributions which model the data, all by the consideration of principal curves approach. We discuss the possibility of integration of our model into an adaptive procedure that can be used for the search of optimum points on finance applications.