The Quantum Structural Triad: Fluctuations, Entropy, and Correlations as Interdependent Primitives

This work develops the concept of a quantum structural triad—a conditional relationship among energy fluctuations, von Neumann entropy, and quantum mutual information. Within the class of physically realizable, locally coupled quantum systems studied here, these quantities appear empirically interdependent, and their simultaneous vanishing has not been observed under time-averaged conditions. Starting from assumptions of locality and non-stationarity, the analysis derives two quantitative inequalities: (1) a Hamiltonian-capped bound determined by local energy scales and subsystem dimensionality, and(2) a speed-based bound expressed through the Bures geometric rate and entropic–correlational baselines. Together, these relations supply operational limits for subsystem persistence and a quantitative indicator of structural integrity—the minimum combination of energetic and informational resources required for sustained dynamical activity. The formalism is applicable to both closed and open quantum settings, including Lindbladian evolution and topological models. Illustrative applications concern qubit validation, gate verification, and error-correction prioritization. All analytical results are accompanied by verified Coq/Rocq proofs and reproducible numerical simulations. The findings support the view that coherence, correlation, and entropy share a common structural constraint in quantum systems, offering a principled basis for assessing the stability of quantum technologies. Reproducibility materials and proof scripts are available on request ([email protected]).