Related papers: Information geometry of Euclidean quantum fields
The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via…
Many network information theory problems face the similar difficulty of single letterization. We argue that this is due to the lack of a geometric structure on the space of probability distribution. In this paper, we develop such a…
Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified…
We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial…
We propose a unified information-geometric framework that formalizes understanding in learning as a trade-off between informativeness and geometric simplicity. An encoder phi is evaluated by U(phi) = I(phi(X); Y) - beta * C(phi), where…
Using the generalized entropies which depend on two parameters we propose a set of quantitative characteristics derived from the Information Geometry based on these entropies. Our aim, at this stage, is modest, as we are first constructing…
The search for a potential function $S$ allowing to reconstruct a given metric tensor $g$ and a given symmetric covariant tensor $T$ on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically…
We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product…
We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Mobius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description,…
We present geometric methods for uniformly discretizing the continuous N-qubit Hilbert space. When considered as the vertices of a geometrical figure, the resulting states form the equivalent of a Platonic solid. The discretization…
In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…
Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful…
This article consists of an introduction to the theory of nonassociative geometric classical and quantum information flows defined by star products with R-flux deformations in string gravity. Corresponding nonassociative generalizations of…
Shannon entropy and Fisher information functionals are known to quantify certain information-theoretic properties of continuous probability distributions of various origins. We carry out a systematic study of these functionals, while…
In this chapter we take up the quantum Riemannian geometry of a spatial slice of spacetime. While researchers are still facing the challenge of observing quantum gravity, there is a geometrical core to loop quantum gravity that does much to…
Information geometry is a study of applying differential geometry methods to challenging statistical problems, such as uncertainty quantification. In this work, we use information geometry to study how measurement uncertainties in…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or PPT operators. It provides bounds on measures for statistical…
In this paper we consider the space of those probability distributions which maximize the $q$-R\'enyi entropy. These distributions have the same parameter space for every $q$, and in the $q=1$ case these are the normal distributions. Some…
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…