Canonical divergence for measuring classical and quantum complexity

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.Comment: 17 page

A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians

We give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution. When the parameter of the distribution varies, it defines a parametric curve that we call an escort-path. The R\'enyi divergence appears as a natural by-product of the setting. We study the dynamics of the Fisher information on this path, and show in particular that the thermodynamic divergence is proportional to Jeffreys' divergence. Next, we consider the problem of inferring a distribution on the escort-path, subject to generalized moments constraints. We show that our setting naturally induces a rationale for the minimization of the R\'enyi information divergence. Then, we derive the optimum distribution as a generalized q-Gaussian distribution

Extension of information geometry for modelling non-statistical systems

In this dissertation, an abstract formalism extending information geometry is introduced. This framework encompasses a broad range of modelling problems, including possible applications in machine learning and in the information theoretical foundations of quantum theory. Its purely geometrical foundations make no use of probability theory and very little assumptions about the data or the models are made. Starting only from a divergence function, a Riemannian geometrical structure consisting of a metric tensor and an affine connection is constructed and its properties are investigated. Also the relation to information geometry and in particular the geometry of exponential families of probability distributions is elucidated. It turns out this geometrical framework offers a straightforward way to determine whether or not a parametrised family of distributions can be written in exponential form. Apart from the main theoretical chapter, the dissertation also contains a chapter of examples illustrating the application of the formalism and its geometric properties, a brief introduction to differential geometry and a historical overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and Prof. dr. Jacques Tempere, December 2014, 108 page

Information geometry in quantum field theory: lessons from simple examples

Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems. By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography. We begin by demonstrating that the symmetries of the physical theory under study play a strong role in the resulting geometry, and that the appearance of an AdS metric is a relatively general feature. We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion theory, and find that the curvature diverges precisely at the phase transition on both sides. We discuss the differences that result from placing a metric on the space of theories vs. states, using the example of coherent free fermion states. We compare the latter to the metric on the space of coherent free boson states and show that in both cases the metric is determined by the symmetries of the corresponding density matrix. We also clarify some misconceptions in the literature pertaining to different notions of flatness associated to metric and non-metric connections, with implications for how one interprets the curvature of the geometry. Our results indicate that in general, caution is needed when connecting the AdS geometry arising from certain models with the AdS/CFT correspondence, and seek to provide a useful collection of guidelines for future progress in this exciting area.Comment: 36 pages, 2 figures; added new section and appendix, miscellaneous improvement