The Open Quantum Brownian Motion and continual measurements

TL;DR

This paper provides a rigorous mathematical analysis of the Open Quantum Brownian Motion, demonstrating convergence across multiple descriptions and introducing a general framework for continual measurement of non-demolition observables.

Contribution

It establishes convergence results for the Open Quantum Brownian Motion in various frameworks and develops a broad framework for continual measurement of non-demolition observables.

Findings

Proves convergence of quantum trajectory, Langevin, and Lindbladian descriptions.

Introduces a general framework for continual measurement of non-demolition observables.

Applies the framework to position measurement of the Open Quantum Brownian Motion.

Abstract

This article is a mathematical analysis of the Open Quantum Brownian Motion. This object was introduced by Bernard, Bauer, Benoist and Tilloy as the limit of a family of Open Quantum Random Walks on the discrete line. We prove the convergence for the three possible descriptions of this object: the quantum trajectory satisfying a Belavkin Equation, the unitary evolution on the Fock space satisfying a quantum Langevin Equation, and the Lindbladian evolution. We introduce a very general framework for the continual measurement of non-demolition observables, which is applied to the measurement of the position of the Open Quantum Brownian Motion, and we probe some questions related to the convergence of processes in this context.