Information-Theoretic Foundations of Quantum Uncertainty: From Fisher Information and Cramér-Rao Bounds to Modern Error-Disturbance Corrections

This manuscript investigates the relationship between classical statistical estimation theory and quantum uncertainty relations. Starting from Fisher information and the Cramér-Rao inequality, the paper demonstrates how the Heisenberg uncertainty relation can be recovered within an estimation-theoretic framework. The discussion is further extended to quantum Fisher information and modern error-disturbance relations, including the formulations of Ozawa, Branciard, and Busch-Lahti-Werner. Beyond the derivation itself, the manuscript proposes a broader research program connecting quantum measurement theory with information geometry, Bayesian inference, nonparametric estimation, machine learning, and information-theoretic interpretations of measurement disturbance. Particular attention is given to the possibility that modern uncertainty relations may be understood through concepts such as distinguishability, information transfer, information loss, and recoverability. The work is intended as a conceptual bridge between quantum foundations, statistical inference, and information science, and outlines several directions for future theoretical development.