Markov invariant geometry on manifolds of states

Abstract

This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers.Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states.A description is given of all invariant Riemannian metrics on manifolds of sectorial states.The equations of the geodesics for the entire family of invariant linear connections A = ~A, yEIR, are integrated on sets of classical probability distributions.A description is given of the projective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor T(7 -I), with the Riemann-Christoffel curvature tensor.I. The collection of all probability measures P(.) on a measurable space (~, ~) of elementary outcomes ~ is a convex subset of the semi-ordered linear space of all countably additive charges R, i.e., measures of bounded variation on (~,~). Compatible with the semiorder relation of this space is the natural norm -the total variation of the charge

Quantum Covariance, Quantum Fisher Information, and the Uncertainty Relations

“…2 From coarse-graining to Fisher information and covariance Heuristically, coarse-graining implies loss of information, therefore Fisher information should be monotone under coarse-graining. This was proved in [3] in probability theory and a similar approach was proposed in [18] for the quantum case. The approach was completed in [21], where a class of quantum Fisher information quantities was introduced, see also [22].…”

A Novel Approach to Canonical Divergences within Information Geometry

“…However, it turns out that, in order to solve the geodesic Equation (32), both normalized curves have to be reparametrized. More precisely, it has been shown in [8] (Theorems 14.1. and 15.1.) that, with appropriate reparametrizations τ p,X and τ p,q , we have the following form of the α-geodesic in the simplex S n−1 :…”

“…We shall now generalize the relation Equation (9) between the squared distance D p and the difference of two points p and q to the more general setting of a differentiable manifold M. Given a fixed point p ∈ M, we want to define a vector field q → X(q, p), at least in a neighbourhood of p, that corresponds to the difference vector field Equation (8). Obviously, the problem is that the difference p − q is not naturally defined for a general manifold M. We need an affine connection ∇ in order to have a notion of a difference.…”

“…If we attach to each point q ∈ M the difference vector X(q, p), we obtain a vector field that corresponds to the vector field Equation (8) in R n . In order to interpret this vector field as a negative gradient field of a (squared) distance function, and thereby generalize Equation (9), we need a Riemannian metric g on M. Given such a metric, we assume integrability of X and ∇, respectively, in the sense that for all p there exists a function D p satisfying…”

On Grover’s search algorithm from a quantum information geometry viewpoint

“…The requirement that the distance between density matrices expresses quantum statistical distinguishability implies that this distance must decrease under coarsegraining (stochastic maps). Unlike the classical case, it turns out that there are infinitely many monotone Riemannian metrics satisfying this requirement [24][25][26].…”