Foundations of quantum local asymptotic normality via information geometry

Local asymptotic normality is a cornerstone of classical asymptotic statistics. Extending it to quantum statistical models entails formidable conceptual and technical challenges arising from their noncommutative nature, above all, the definition of a quantum likelihood ratio, whose very choice dictates the subsequent theoretical development. Employing quantum information geometry based on the symmetric logarithmic derivative, we formulate a quantum Lebesgue decomposition and the square-root likelihood ratio. These constructions provide a natural framework for a quantum analogue of Le Cam’s third lemma, the formulation of quantum local asymptotic normality, and the asymptotic representation results developed in our earlier work linking regular quantum statistical models to quantum Gaussian shift models. In this way, quantum information geometry emerges as a natural and unifying foundation for asymptotic quantum statistics.