Asymptotic and Non-Asymptotic Analysis for Hidden Markovian Process with Quantum Hidden System

TL;DR

This paper analyzes quantum hidden Markov processes, deriving bounds, limit theorems, and variance expressions using quantum Perron-Frobenius theory, extending classical probabilistic results to quantum systems.

Contribution

It introduces quantum analogs of classical probabilistic bounds and limit theorems for hidden Markov processes, providing new tools for quantum data analysis.

Findings

Derived novel bounds for cumulant generating functions.

Established quantum versions of the central limit theorem and deviation principles.

Provided formulas for asymptotic variance using quantum fundamental matrices.

Abstract

We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data. Using these bounds, we derive the central limit theorem and large and moderate deviations for the tail probability. Then, we give the asymptotic variance is given by using the second derivative of the cumulant generating function. We also derive another expression for the asymptotic variance by considering the quantum version of the fundamental matrix. Further, we explain how to extend our results to a general probabilistic system.