Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics.

We herein develop a theory of contiguity in the quantum domain based upon a novel quantum analogue of the Lebesgue decomposition. The theory thus formulated is pertinent to the weak quantum local asymptotic normality introduced in the previous paper [Yamagata, Fujiwara, and Gill, \textit{Ann. Statist.}, \textbf{41} (2013) 2197-2217.], yielding substantial enlargement of the scope of quantum statistics.