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.
Highlights
The field of classical stochastic processes forms a major branch of mathematics
As we show in Eq (104), such a map can be thought to come from unitary system-environment dynamics, with the caveat that the initial system environment has no correlations [in Eq (104), it was of the form ρS ⊗ |00|]
We moved to quantum stochastic processes, covering the early works from half a century ago to modern methods used to differentiate between
Summary
In both natural and social sciences, are not isolated from their environment. While quantum engineering was advancing, many of the early results in the field of quantum stochastic processes regained importance and new problems have arisen requiring a fresh look at how we characterize and model open quantum systems Central among these problems is the need to understand the nature of memory that quantum environments carry. The progress, both in experimental and theoretical physics has been fast leading to many review papers focusing on different facets of open quantum systems [6,7,8,9,10,11,12] and the complex multilayered structure of memory effects in quantum processes [10] This tutorial adds to this growing literature and has its own distinct focus. While we aim to provide as many references as possible for further reading, we do so without a claim to comprehensiveness, and much of the results that have been found in the field will be left unsaid, and far too much will not even be addressed
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