Picturing quantum processes: A first course on quantum theory and diagrammatic reasoning

Abstract

We provide a self-contained introduction to quantum theory using a unique diagrammatic language. Far from simple visual aids, the diagrams we use are mathematical objects in their own right, which allow us to develop from first principles a completely rigorous treatment of ‘textbook’ quantum theory. Additionally, the diagrammatic treatment eliminates the need for the typical prerequisites of a standard course on the subject, making it suitable for a multi-disciplinary audience with no prior knowledge in physics or advanced mathematics. By subscribing to a diagrammatic treatment of quantum theory we place emphasis on quantum processes, rather than individual systems, and study how uniquely quantum features arise as processes compose and interact across time and space. We introduce the notion of a process theory, and from this develop the notions of pure and mixed quantum maps, measurements and classical data, quantum teleportation and cryptography, models of quantum computation, quantum algorithms, and quantum non-locality. The primary mode of calculation in this tutorial is diagram transformations, where simple local identities on diagrams are used to explain and derive the behaviour of many kinds of quantum processes. This tutorial roughly follows a new textbook published by Cambridge University Press in 2017 with the same title.