Quantum stochastic processes and quantum non-Markovian phenomena

TL;DR

This paper reviews the recent progress in understanding quantum stochastic processes, clarifying their structure using quantum combs, and addressing longstanding issues that hinder the development of a consistent quantum theory of stochastic phenomena.

Contribution

It introduces a comprehensive framework for quantum stochastic processes using quantum combs, resolving ambiguities and outlining future research directions.

Findings

Clarified the structure of quantum stochastic processes.

Connected classical and quantum stochastic theories.

Identified open problems in the field.

Abstract

The field of classical stochastic processes forms a major branch of mathematics. They are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences. When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers. Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities. This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum information theory. We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain. Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes. We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.