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

This work introduces the concept of the quantum structural triad—a foundational relationship between energy fluctuations, von Neumann entropy, and quantum mutual information. It establishes that in any physically realizable, locally coupled quantum system, these three quantities form an inseparable structure whose joint vanishing is forbidden on average. From first principles of locality and non-stationarity, two quantitative inequalities are derived: a Hamiltonian-capped threshold, governed by minimal local energy scales and subsystem dimensions, and a speed-based threshold, expressed through the Bures geometric rate and entropic–correlational floors. Together, they define explicit, falsifiable bounds for subsystem viability and provide a measure of quantum structural integrity—the minimal resource combination required for a subsystem to remain dynamically and informationally active. The framework applies to both closed and open quantum systems, extending naturally to Lindbladian dynamics and even to topological quantum orders. Practical implications are demonstrated for qubit certification, gate verification, and error-correction prioritization. All analytical derivations are accompanied by formal Coq/Rocq proof verification and reproducible numerical simulations. The results suggest a unified structural principle underlying coherence, correlation, and entropy in quantum systems, offering a first-principles foundation for assessing the physical viability of quantum technologies. Reproducibility materials, including code and proofs, are hosted on GitHub. Access can be granted upon request via [email protected].