Minimal-Action Information Geometry: Unified Information Distance Dinfo

抄録

The unification of internal (gauge) and external (spacetime) coordinates remains a central challenge in unifying fundamental interactions. Several geometric approaches, including quantum Riemannian metrics, information-geometric distances, and Wasserstein transport, have been explored. However, no single metric spans classical probability distributions, quantum states, and heterogeneous information carriers. Therefore, we sought a minimal-action geometric framework capable of unifying these regimes within a single state space. Starting from an abstract, information-theoretic action constrained by locality, additivity, reparametrization invariance, joint convexity, and contractivity under admissible channels, we demonstrate that the action necessarily assumes a quadratic-norm form, inducing a parameterization-independent geodesic distance. Requiring a faithful embedding into a real Hilbert space identifies Jensen-Shannon-type divergences as the canonical metrics compatible with these axioms. The resulting unified information distance, Dinfo, recovers the square root of the Jensen-Shannon divergence for classical distributions and its quantum analogue for density operators. It also extends naturally to heterogeneous and hybrid systems. This minimal-action construction establishes Dinfo as a derived metric rather than a postulated one. Consequently, Dinfo provides a rigorous, coordinate-independent foundation for analyzing entropy-driven interactions, information contraction, and multiscale informational structures spanning physical, biological, and computational contexts.