# Invariant Measure for Quantum Trajectories

## TL;DR

This paper investigates the long-term behavior of quantum systems under repeated measurements, establishing conditions for unique invariant measures and demonstrating geometric convergence using novel techniques.

## Contribution

It introduces new conditions for the existence and uniqueness of invariant measures for quantum trajectory Markov chains and develops innovative methods for analyzing their convergence.

## Key findings

- Unique invariant measure exists under certain conditions.
- Proven geometric convergence in Wasserstein metric.
- Developed new techniques beyond standard random matrix theory.

## Abstract

We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.