RIGIDITY OF FURSTENBERG ENTROPY UNDER UCP MAPS

Shuoxing Zhou

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves monotonicity of Furstenberg entropy under state-preserving M-bimodular ucp maps and characterizes equality via a ∗-isomorphism on Radon–Nikodym factors (Theorem 3.2) using modular-operator inequalities and multiplicative domains; the argument is internally consistent and complete in the provided text. The candidate solution attempts an alternative proof via Araki relative entropy and Petz sufficiency, but it hinges on an unproved identification h_ϕ(A,ϕ_A) = S(ρ_A||ω_A) for a specially constructed pair of states on the RN factor, and on the claim that the factor map pushes forward both states. These steps are not justified and, as stated, conflict with standard modular theory, so the proposed reduction is flawed.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The main theorem is proved cleanly via modular-operator techniques and a functional-calculus lemma, yielding both monotonicity and a sharp equality characterization in terms of RN factors. The argument is rigorous and self-contained relative to the preliminaries. The work also draws connections to results in ergodic theory. Minor presentation tweaks could further streamline the exposition.