Lyapunov-Bounded Quantum Information Processing: A Control-Theoretic Framework for Stable Recursive Dynamics via Information Curvature

We construct a Lyapunov stability framework for quantum information processing systems thatundergo recursive self-modification of their control parameters. The central object is a Lyapunovfunction V = 12 η2 + 12 ∥∇ϕ∥2 + 12 LQEC, defined over an entropy–field–loss state space rooted inquantum Fisher information geometry. We prove that its time derivative is negative semi-definiteunder three jointly necessary mechanisms: (i) an information-curvature metric Λq derived from thequantum Fisher information that modulates the effective growth exponent via attention-temperaturegating, (ii) an entropic drag Γent that compounds with accumulated processing capacity, and (iii) ahysteresis-controlled state reset triggered when a collapse metric χ exceeds a safety threshold. Underthe full control nexus, we prove that the system’s capability trajectories remain in a compact setalmost surely (Theorem 9), and that ablation of any single mechanism permits unbounded growth(Corollary 10). A three-action hysteresis controller (DEEPEN/ECHO/RESET) with analyticallyderived dead-band prevents Zeno-type switching instabilities. We discuss connections to quantumerror correction, the GKSL master equation, non-Markovian noise kernels, and propose falsifiableexperimental protocols on superconducting qubit platforms and variational quantum eigensolvers.Supplementary data and code are archived at Zenodo (DOI: 10.5281/zenodo.18969642)