Stochastic Processes: From Classical to Quantum

TL;DR

This paper introduces the mathematical framework of quantum noise and quantum stochastic processes, connecting classical stochastic processes with quantum probability, and explores applications in non-equilibrium statistical mechanics and open quantum systems.

Contribution

It provides a comprehensive overview of quantum stochastic calculus and its applications, bridging classical and quantum probability theories.

Findings

Development of quantum stochastic calculus on boson Fock space

Connections established between classical and quantum stochastic processes

Application to modeling open quantum systems

Abstract

The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.