Limit theorems for Quantum Trajectories

TL;DR

This paper establishes advanced limit theorems for quantum trajectories, a class of Markov processes modeling quantum systems under repeated measurements, under certain assumptions.

Contribution

It proves Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, and Moderate Deviation Principle for quantum trajectories, extending previous results.

Findings

Proved LLN for quantum trajectories.

Established CLT and other limit theorems.

Provided a rigorous mathematical framework for quantum measurement processes.

Abstract

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure - see Benoist et al. Probab. Theory Relat. Fields 2019. In this article we prove, finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.