A Fluctuation-Sensitive Entropy Functional and Channel Framework withApplication to Genetic Systems

Entropy and entanglement are central to quantum information, yet useful, variance-sensitive generalizations remain limited. We introduce a fluctuation-sensitive entropy-like functional G = √pq + 2pq for bi-allelic systems and show it closely approximates Shannon entropy with negligible Kullback–Leibler divergence across allele-frequency regimes. Unlike standard heterozygosity, G explicitly includes a variance term, placing it among generalized entropies in the spirit of Rényi and Tsallis while preserving interpretability for stochastic populations. To study noisy dynamics we construct a family of circulant transition matrices that act as CPTP maps and recover identity, permutation, and depolarizing channels as limiting cases, fitting naturally into the operator-sum formalism. Mapping classical fluctuation to a quantum-noise parameter, we quantify entanglement decay using Werner states and concurrence, and demonstrate that increasing fluctuation drives the entanglement–separability crossover. Independent analyses yield closely matched thresholds, gene regulatory networks (GRNs) predictability collapses near p ≈ 0.854 while entanglement vanishes near p ≈ 0.873, suggesting a narrow operational Goldilocks zone for information-preserving regulation. This formalism rigorously demonstrates that the complex channel is a realization of the Ry rotation, explicitly embedding allele frequencies as the parameters that define a fundamental quantum gate. As a case study, we apply the framework to genetic systems to illustrate its utility for analyzing diversity, decoherence, and information transfer in complex stochastic dynamics.