Asymptotic statistical theory of irreducible quantum Markov chains

TL;DR

This paper develops an asymptotic statistical framework for irreducible quantum Markov chains, characterizing parameter identifiability, geometric structures, and convergence properties to facilitate optimal quantum parameter estimation.

Contribution

It introduces the orbifold structure of identifiable parameters, analyzes geometric properties, and establishes quantum local asymptotic normality for stationary output models.

Findings

Identifiable parameters form an orbifold structure.

Quantum Fisher information rate is expressed via a canonical inner product.

Stationary output models converge to quantum Gaussian shift models.

Abstract

In this paper we investigate the asymptotic statistical theory of irreducible quantum Markov chains, focusing on identifiability properties and asymptotic convergence of associated quantum statistical models. We show that the space of identifiable parameters for the stationary output is a stratified space called an orbifold, which is obtained as the quotient of the manifold of irreducible dynamics by a compact group of state preserving symmetries. We analyse the orbifold's geometric properties, the connection between periodicity and strata, and provide orbifold charts as the starting point for the local asymptotic theory. The quantum Fisher information rate of the system and output state is expressed in terms of a canonical inner product on the identifiable tangent space. We then show that the joint system and output model satisfies quantum local asymptotic normality while the stationary output model converges to a product between a quantum Gaussian shift model and a mixture of quantum Gaussian shift models, reflecting the underlying periodicity. These strong convergence results provide the basis for constructing asymptotically optimal estimators of dynamical parameters. We provide an in-depth analysis of the model with smallest dimensions, consisting of two-dimensional system and environment units.