Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics

TL;DR

This paper develops a quantum analogue of Lebesgue decomposition to establish a theory of contiguity, significantly expanding the framework of quantum statistics and its applications in local asymptotic normality.

Contribution

It introduces a novel quantum Lebesgue decomposition and applies it to develop a theory of contiguity in quantum statistics, extending previous work on quantum local asymptotic normality.

Findings

Established a quantum analogue of Lebesgue decomposition.

Formulated a new theory of contiguity in quantum statistics.

Expanded the scope of quantum local asymptotic normality.

Abstract

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.