Scale-Stable Two-Component Entropic Complexity on Classical and Quantum Law Spaces

This preprint introduces a two-component notion of entropic complexity for discrete-time implementations on abstract law spaces that simultaneously cover classical probability measures and quantum states on von Neumann algebras. The first component, the internal law–time complexity JintJᵢₙₜJint, is defined as a pathwise sum of divergences between consecutive internal laws. The second component, the interface complexity CifaceCifaceCiface, is the Shannon entropy rate of a stationary, ergodic boundary process on a finite alphabet. The paper works entirely at the level of law-space Markov operators and divergence families satisfying data-processing inequalities, so that classical Wasserstein settings and quantum Bures–HK / Fibered Bures–HK (FBHK) geometries fit into a common framework. For directed refinement nets of implementations linked by observation-compatible coarse-grainings, the author defines scale-enveloped invariants (Jint∞,Ciface∞)(Jᵢₙₜ^∞, Ciface^∞)(Jint∞,Ciface∞) and proves invariance under a strong notion of data-processing equivalence, in the spirit of Blackwell equivalence for statistical experiments but lifted to law-space trajectories. On the analytic side, the work connects JintJᵢₙₜJint to metric gradient flows via discrete evolution variational inequalities (EDIs) and a “compatibility inequality” between divergence and metric. Under these assumptions, the internal complexity admits non-circular finite-action bounds in terms of free-energy-like functionals. The framework is illustrated by classical Fokker–Planck–type gradient flows (via the JKO scheme) and quantum examples based on FBHK flows and AF/UHF CAR nets with local measurements. A comparison with boundary-only measures such as statistical complexity, excess entropy, and Kolmogorov–Sinai entropy highlights how internal law–time complexity quantifies representation-dependent dynamical structure that is invisible at the boundary level.