Quantum Stochastic Processes: A Case Study

TL;DR

This paper investigates quantum phase space Brownian motion, a quantum stochastic process derived from open quantum systems, highlighting its properties and potential applications in quantum dissipative phenomena.

Contribution

It introduces a detailed model of quantum phase space Brownian motion and constructs its dilation, linking quantum stochastic processes to classical analogs.

Findings

Quantum phase space Brownian motion has independent, identically distributed increments.

The process can be derived as a Markovian limit of a simple open quantum system model.

Potential applications in understanding dissipative effects in quantum Hall systems.

Abstract

We present a detailed study of a simple quantum stochastic process, the quantum phase space Brownian motion, which we obtain as the Markovian limit of a simple model of open quantum system. We show that this physical description of the process allows us to specify and to construct the dilation of the quantum dynamical maps, including conditional quantum expectations. The quantum phase space Brownian motion possesses many properties similar to that of the classical Brownian motion, notably its increments are independent and identically distributed. Possible applications to dissipative phenomena in the quantum Hall effect are suggested.