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Categorical Quantum Mechanics an Introduction Categorical Quantum Mechanics An Introduction Lectured by David Reutter and Jamie Vicary Notes by Chris Heunen and Jamie Vicary Department of Computer Science, University of Oxford 16 Lectures, Hilary Term 2019 1 The idea for this book came from a mini-course at a spring school in 2010, aimed at beginning graduate students from various fields. We first developed the notes in 2012, when they formed the basis for a graduate course at the Department of Computer Science in Oxford, which has run every year since. As of 2016 a version of this course has also run in the School of Informatics in Edinburgh. The text formed the basis for postgraduate summer and winter schools in Dalhousie, Pisa, and Palmse as well, and has improved with feedback from all these students. In selecting the material, we have strived to strike a balance between theory and application. Applications are interspersed throughout the main text and exercises, but at the same time, the theory aims at maximum reasonable generality. The emphasis is slightly on the theory, because there are so many applications that we can only hint at them. Proofs are written out in full, excepting a handful that are beyond the scope of this book, and a couple of chores that are left as exercises. We have tried to assign appropriate credit, including references, in historical notes at the end of each chapter. Of course the responsibility for mistakes remains entirely ours, and we hope that readers will point them out to us. To make this book as self-contained as possible, there is a zeroth chapter that briefly defines all prerequisites and fixes notation. This text would not exist were it not for the motivation of Bob Coecke. Between inception and publication of this book, he and Aleks Kissinger wrote another one, that targets a different audience but covers similar material [44]. Thanks are also due to the students who let us use them as guinea pigs for testing out this material. Finally, we are enormously grateful to John-Mark Allen, Pablo Andres Martinez, Miriam Backens, Bruce Bartlett, Oscar Cunningham, Brendan Fong, Pau Enrique Moliner, Tobias Fritz, Peter Hines, Martti Karvonen, Alex Kavvos, Aleks Kissinger, Bert Lindenhovius, Daniel Marsden, Alex Merry, Vaia Patta, David Reutter, Francisco Rios, Peter Selinger, Sean Tull, Dominic Verdon, Linde Wester and Vladimir Zamdzhiev for careful reading and useful feedback on early versions. Chris Heunen and Jamie Vicary Edinburgh and Oxford, December 2018 Introduction Physical systems cannot be studied in isolation. After all, we can only observe their behaviour with respect to other systems, such as a measurement device. The central premise of this book is that the ability to group individual systems into compound systems should be taken seriously. We adopt the action of grouping systems together as a primitive notion, and investigate models of quantum theory from there. The judicious language for this story is that of categories. Category theory teaches that a lot can be learned about a given type of mathematical species by studying how specimens of the species interact with each other, and it provides a potent instrument to discover patterns in these interactions. No knowledge of specimens’ insides is needed; in fact, this often only leads to tunnel vision. The methods of categories might look nothing like what you would expect from a treatise on quantum theory. But a crucial theme of quantum theory naturally fits with our guiding principle of compositionality: entanglement says that complete knowledge of the parts is not enough to determine the whole. In providing an understanding of the way physical systems interact, category theory draws closely on mathematics and computer science as well as physics. The unifying language of categories accentuates connections between its subjects. In particular, all physical systems are really quantum systems, including those in computer science. This book applies its foundations to describe protocols and algorithms that leverage quantum theory. Operational foundations An operational scientist tries to describe the world in terms of operations she can perform. The only things one is allowed to talk about are such operations – preparing a physical system in some state, manipulating a physical system via some experimental setup, or measuring a physical quantity – and their outcomes, or what can be derived from those outcomes by mathematical reasoning. This is a very constructive way of doing science. After all, these operations and outcomes are what really matter when you build a machine, design a protocol, or otherwise put your knowledge of nature to practical use. But traditional quantum theory contains many ingredients for which we have no operational explanation, even though many people have tried to find one. For example, if states are unit vectors in a Hilbert space, what are the other vectors? If a measurement is a hermitian operator on a Hilbert space, what do other operators signify? Why do we work with complex numbers, when at the end of 2 3 the day all probabilities are real numbers? Categories offer a way out of these details, while maintaining powerful conclusions from basic assumptions. They let us focus on what is going on conceptually, showcasing the forest rather than the trees. Graphical calculus When thinking operationally, one cannot help but draw schematic pictures similar to flowcharts to represent what is going on. For example, the quantum teleportation protocol, which we will meet many times in this book, has the following schematic representation: Alice Bob correction measurement preparation We read time upwards, and space extends left-to-right. We say “time” and “space”, but we mean this in a very loose sense. Space is only important in so far as Alice and Bob’s operations influence their part of the system. Time is only important in that there is a notion of start and finish. All that matters in such a diagram is its connectivity. Such operational diagrams seem like informal, non-mathematical devices, useful for illustration and intuitive understanding, but not for precise deduction. In fact, we will see that they can literally be taken as pieces of formal category theory. The “time-like” lines become identity morphisms, and “spatial” separation is modeled by tensor products of objects, leading to monoidal categories. Monoidal categories form a branch that does not play a starring role, or even any role at all, in most standard courses on category theory. Nevertheless, they put working with operational diagrams on a completely rigorous footing. Conversely, the graphical calculus is an effective tool for calculation in monoidal categories. This graphical language is perhaps one of the most compelling features that monoidal categories have to offer. By the end of the book we hope to have convinced you that this is the appropriate language for describing and understanding many phenomena in quantum theory. Nonstandard models Once we have made the jump from operational diagrams to monoidal categories, many options open up. In particular, rather than interpreting diagrams in the category of Hilbert spaces, where quantum theory traditionally takes place, we can instead interpret diagrams in a different 4 category, and thereby explore alternatives to quantum theory. Any calculation that was performed purely using the graphical calculus will also hold without additional work in these novel settings. For example, when we present quantum teleportation in the graphical calculus, and represent it in the category of sets and relations, we obtain a description of classical one-time-pad encryption. Thus we can investigate exactly what it is that makes quantum theory ‘tick’, and what features set it apart from other compositional theories. Thus monoidal categories provide a unifying language for a wide variety of phenomena, drawn from areas including quantum theory, quantum information, logic, topology, representation theory, quantum algebra, quantum field theory, and even linguistics. Within quantum theory, categories highlight different aspects than other approaches. Instruments like tensor products and dual spaces are of course available in the traditional Hilbert space setting, but their relevance is heightened here, as they become the central focus. How we represent mathematical ideas affects how we think about them. Outline After this somewhat roundabout discussion of the subject, it is time to stop beating about the bush, and describe the contents of each chapter. Chapter 0 covers the background material. It fixes notations and conventions while very briefly recalling the basic notions from category theory, linear algebra and quantum theory that we will be using. Our running example categories are introduced: functions between sets, relations between sets, and bounded linear maps between Hilbert spaces. Chapter 1 introduces our main object of study: monoidal categories. These are categories that have a good notion of tensor product, which groups multiple objects together into a single compound object. We also introduce the graphical calculus, the visual notation for monoidal categories. This gives a notion of compositionality for an abstract physical theory. The next few chapters will add more structure, so that the resulting categories exhibit more features of quantum theory. Section 1.3 investigates coherence, a technical topic which is essential to the correctness of the graphical calculus, but which is not needed to understand later chapters. To someone who equates quantum theory with Hilbert space geometry – and this will probably include most readers – the obvious next structure to consider is linear algebra. Chapter 2 shows that important notions such as scalars, superposition, adjoints, and the Born rule can all be represented in the categorical setting. Chapter 3 investigates entanglement in terms of monoidal categories, using the notion of dual object, building up to the important notion of compact category. This structure is quite simple and powerful: it gives rise to abstract notions of trace and dimension, and is already enough to talk about the quantum teleportation protocol.
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