The Gravitational Aspect of Information: The Physical Reality of Asymmetric "Distance"

TL;DR

The paper argues that divergences in information geometry, though asymmetric, encode physical structure and can be understood as generalized distances. It proves that a canonical Brownian bridge on the Gaussian manifold evolves along an m-geodesic, establishing a direct physical realization of a geodesic in information space. By showing equivalences between canonical, Bregman, and KL divergences and identifying the Euclidean limit in the self-dual case, it provides a deep link between stochastic dynamics and information geometry. The work suggests an information-equivalence principle for randomness and outlines pathways to quantum generalizations using alternative geometric frameworks such as Bures-Wasserstein geometry.

Abstract

We show that when a Brownian bridge is physically constrained to satisfy a canonical condition, its time evolution exactly coincides with an m-geodesic on the statistical manifold of Gaussian distributions. This identification provides a direct physical realization of a geometric concept in information geometry. It implies that purely random processes evolve along informationally straight trajectories, analogous to geodesics in general relativity. Our findings suggest that the asymmetry of informational ``distance" (divergence) plays a fundamental physical role, offering a concrete step toward an equivalence principle for information.