Abstract. This special volume celebrates the 40th anniversary of the discovery of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, which is widely used in quantum physics and quantum chemistry. The present contribution aims to celebrate a related discovery -also by Sudarshan -that of Quantum Maps (which had their 55th anniversary in the same year). Nowadays, much like the master equation, quantum maps are ubiquitous in physics and chemistry. Their importance in quantum information and related fields cannot be overstated. Here, we motivate quantum maps from a tomographic perspective, and derive their well-known representations. We then dive into the murky world beyond these maps, where recent research has yielded their generalisation to non-Markovian quantum processes.

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.

In the classical domain, it is well-known that divisibility does not imply that a stochastic process is Markovian. However, for quantum processes, divisibility is often considered to be synonymous with Markovianity. We show that completely positive (CP) divisible quantum processes can still involve non-Markovian temporal correlations, that we then fully classify using the recently developed process tensor formalism, which generalizes the theory of stochastic processes to the quantum domain.

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.