Minimal-Action Information Geometry: Unified Information Distance Dinfo

抄録

Distances are typically introduced as static measures of distinguishability between states. These quantities are often assumed to reflect an underlying metric or probabilistic structure, yet the fundamental relationship between distance and metric is rarely examined. This raises the question of whether distance should be viewed as a secondary quantity derived from a metric, or as a primary geometric concept from which metrics may emerge. Here, we revisit information geometry from a distance-first perspective, treating distance as the primary geometric object. As a concrete realization, we introduce a unified information distance, D_info, defined consistently with a least-length (variational) principle, and develop a geometric framework that takes this distance as fundamental. Within this framework, the metric tensor arises as the local infinitesimal expansion of the distance, enabling a coherent treatment of classical, quantum, and mixed systems within a single formalism. Importantly, distance is not assumed to encode physical interactions directly; rather, it acts as a structural constraint on admissible geometries and variational configurations. By elevating distance to a primary role, this approach reorganizes the foundations of information geometry and clarifies its interface with physical theory.