… she sprinkled her with the juice of Hecate's herb, and immediately at the touch of this dark poison, Arachne's hair fell out. With it went her nose and ears, her head shrank to the smallest size, and her whole body became tiny. Her slender fingers stuck to her sides as legs, the rest is belly, from which she still spins a thread, and, as a spider, weaves her ancient web. – Ovid, The Metamorphoses, 8 AD Most quantum protocols rely on the interplay between quantum systems and classical data. For instance, measurements extract classical data from a quantum system, whereas controlled operations use classical data to affect a quantum system. Moreover, given that truth is in the eye of the beholder, we want to understand quantum theory relative to our perception of reality, which is classical, and hence want to understand how the two relate. Somewhat surprisingly, it turns out to be much easier to represent the classical world relative to quantum processes, than the other way around. One way to get a handle on this interaction is to express as much of it as possible in a purely diagrammatic form. Previously, we have drawn diagrams of quantum processes, then used some sort of ‘external’ means of describing the classical data flow, i.e. with indices and brackets, which can't really be plugged together like pieces of a diagram. Even worse, in most standard textbooks, classical data is not even part of the actual formalism, but is described in words. Rather than describing this interplay of quantum systems and classical data using lots of ‘blah blah blah’, or a cross-breed between diagrams and symbols, can we instead just give a diagram of all of the devices involved and how they are wired together? For example, suppose we have a device ‘Bell’ that prepares Bell states, another device ‘Bell-M’ that performs Bell measurements, and a third device ‘Bell-C’ that does Bell corrections, depicted very realistically as follows: Now, suppose we want to describe to a technician how to wire these devices together to do teleportation: We could describe this using a specification language , that is, a diagrammatic language where the boxes correspond to devices and the wires correspond to literally ‘wiring up’ the devices: where we now distinguish quantum wires and classical wires.
… she sprinkled her with the juice of Hecate's herb, and immediately at the touch of this dark poison, Arachne's hair fell out. With it went her nose and ears, her head shrank to the smallest size, and her whole body became tiny. Her slender fingers stuck to her sides as legs, the rest is belly, from which she still spins a thread, and, as a spider, weaves her ancient web. – Ovid, The Metamorphoses, 8 AD Most quantum protocols rely on the interplay between quantum systems and classical data. For instance, measurements extract classical data from a quantum system, whereas controlled operations use classical data to affect a quantum system. Moreover, given that truth is in the eye of the beholder, we want to understand quantum theory relative to our perception of reality, which is classical, and hence want to understand how the two relate. Somewhat surprisingly, it turns out to be much easier to represent the classical world relative to quantum processes, than the other way around. One way to get a handle on this interaction is to express as much of it as possible in a purely diagrammatic form. Previously, we have drawn diagrams of quantum processes, then used some sort of ‘external’ means of describing the classical data flow, i.e. with indices and brackets, which can't really be plugged together like pieces of a diagram. Even worse, in most standard textbooks, classical data is not even part of the actual formalism, but is described in words. Rather than describing this interplay of quantum systems and classical data using lots of ‘blah blah blah’, or a cross-breed between diagrams and symbols, can we instead just give a diagram of all of the devices involved and how they are wired together? For example, suppose we have a device ‘Bell’ that prepares Bell states, another device ‘Bell-M’ that performs Bell measurements, and a third device ‘Bell-C’ that does Bell corrections, depicted very realistically as follows: Now, suppose we want to describe to a technician how to wire these devices together to do teleportation: We could describe this using a specification language , that is, a diagrammatic language where the boxes correspond to devices and the wires correspond to literally ‘wiring up’ the devices: where we now distinguish quantum wires and classical wires.
When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. – Erwin Schrödinger, 1935 By 1935, Schrödinger had already realised that the biggest gulf between quantum theory and our received classical preconceptions is that, when it comes to quantum systems, the whole is more than the sum of its parts. In classical physics, for instance, it is possible to totally describe the state of two systems – say, two objects sitting on a table – by first totally describing the state of the first system then totally describing the state of the second system. This is a fundamental property one expects of a classical, separable universe. However, as Schrödinger points out, there exist states predicted by quantum theory (and observed in the lab!) that do not obey this ‘obvious’ law about the physical world. Schrödinger called this new, totally non-classical phenomenon Verschränkung , which later became translated to the dominant scientific language as entanglement . Quantum picturalism is all about studying the way parts compose to form a whole. In the good company of Schrödinger, we believe that the role of multiple interacting systems – especially non-separable systems – should take centre stage in the study of quantum theory. We shall see in the next section that it is easy to say what it means for a process to be separable in terms of diagrams. Literally, it means that it can be broken up into pieces that are not connected to each other. On the other hand, enforcing non -separability requires us to refine our diagrammatic language. To this end, we introduce special states and effects called cups and caps , respectively. Intuitively, cups and caps act like pieces of wire that have been ‘bent sideways’.
Feynman diagrams have fascinated physicists and philosophers since they were introduced to the world about 70 years ago. Clearly, they help in calculation; they have allowed nearly impossible problems to be solved with relative ease. This is agreed by all, but that is probably where the consensus ends. Are they pictures of physical processes? Are they just devices for keeping track of mathematical formulae, that do the real work? Are they some sort of mix of both?They are almost as famous as representations of the Bohr atom, even though most people could not give even a sketchy account of what they do. In spite of this they have taken on a role as a cultural icon. Feynman used them to decorate his own van (Fig. 1). The van was even used in an episode of the popular TV series The Big Bang Theory. They have been used as art work in everything from posters, to coffee mugs, to wallpaper. But none of this is particularly helpful in understanding how they work.If we ask working physicists who use Feynman diagrams what it is they are doing, we get a variety of responses. One is puzzlement; a shrug of indifference. Even those very skilled at using Feynman diagrams might be hard pressed to explain (to a philosopher’s satisfaction) what’s happening. After all, most of us can ride a bike but would be similarly hard pressed to explain in any detail what we’re doing. A more common response might go like this: The diagram, or series of diagrams, is a schematic picture of some physical process, say, two electrons interacting, which we then link to mathematical terms. After writing out the mathematical terms in the form of a series, we calculate and get the probability of an interaction happening. This sounds good initially, but runs into trouble when we ask what a “picture of a physical process” could mean? This is, after all, the quantum world where things do not move continuously through space and time. It seems doubtful the diagrams could be diagrams of things in the physical realm. They are connected to physical reality and they are connected to wonderfully accurate predictions. And yet, mystery remains.The following articles on Feynman diagrams address several timely issues, both historical and contemporary. These are metaphysical and epistemological questions related to models and how we represent the world. The papers are united by these themes, but otherwise are independent of one another. They are arranged alphabetically and can be read in any order.James Robert Brown, “How Do Feynman Diagrams Work?” takes the view that they are not representations or models of the world at all. Like others, he stresses that they cannot be pictures of reality, since, thanks to the Heisenberg relations, there can be no such thing as a trajectory. Rather than seek some other more abstract type of representation, he introduces a distinction between descriptive and prescriptive flow charts. The latter could be a representation of physical reality, but not the former. A prescriptive flow chart is more like a recipe to bake a cake. He takes Feynman diagrams to be this sort of thing.Mauro Dorato and Emanuele Rossanese, “The Nature of Representation in Feynman Diagrams,” also reject the idea that Feynman diagrams can be considered to be pictures or depictions of actual physical processes. But unlike Brown, they do not abandon the thought that they can be models in some sense. They appeal to a famous account by RIG Hughes, known as the Denotation, Deduction, and Interpretation theory of models, where “models” are to be interpreted as merely inferential, non-representational devices constructed in given social contexts by the community of physicists.Letitia Meynell, “Picturing Feynman Diagrams and the Epistemology of Understanding,” assumes that Feynman diagrams are pictorial representations but agrees with Brown and Dorato and Rossanese that they do not represent actual physical processes. She then takes up the following puzzle: If Feynman diagrams represent states of affairs, but do not do so truthfully, what can their epistemic value be? She argues that Feynman diagrams have been epistemically powerful (at least in part) because, as pictorial representations, they facilitate an understanding of QED, and particle physics more generally. Appealing to Feynman’s own remarks, she draws out what it might mean to have an understanding of something that is not factive. Although her approach allows for a thin sense of substantively non-factive epistemic success, it is continuous with a factive sense of understanding that is more familiar in the sciences.Michael Stöltzner, “Feynman Diagrams: Modeling between Physics and Mathematics,” analyzes Feynman diagrams within the context of recent debates about models in science and against the backdrop of other diagrammatic methods in mathematical physics when dealing with infinite or asymptotic series. This allows one to avoid the dichotomy of whether Feynman diagrams represent mathematical or physical objects, or a mere tool mediating between them. Along those lines one does not, however, obtain a universal answer to the question as to what Feynman diagrams represent. A single Feynman diagram, in actual scientific practice, can stand for a single mathematical expression or for a physical phenomenon depending on whether the diagram stands for a single term in an infinite series or for a subseries that is given a physical interpretation. Stöltzner’s reading of the representation problem also derives support from the historical fact that Feynman was initially motivated by the Breit-Schrödinger model of a quivering electron and from interpreting this model within the context of stochastic processes.Adrian Wüthrich, “The Exigencies of War and the Stink of a Theoretical Problem,” approaches these topics from an historian’s point of view, shedding light on the issues by focussing on debates in this historical context. As he put it in his subtitle, Wüthrich’s aim is to understand the genesis of Feynman’s electrodynamics as mechanistic modelling at different levels. Feynman’s idiosyncratic method of dealing with quantum electrodynamic phenomena has been cast as an instance of a non-rigorous, modular and rule-based style of theorizing. Given such a characterization, it seems natural that significant changes in Feynman’s style were triggered by work he was assigned during World War II, where quick fixes may have counted more than derivations from first principles. Indeed, Feynman’s theoretical proposals of an improved electrodynamics do bear traces of Feynman’s war-related work, which immediately preceded it. However, Wüthrich explains, the attempt to apply the vocabulary of the “new mechanistic philosophy” to lesser known pieces of archival evidence brings to the fore a new kind of shift in Feynman’s style. Feynman’s disposition for such a shift may also have been fostered during the war, but it was forced upon him by a specific theoretical difficulty, which was largely unrelated to his war-time research.These articles contain some technicalities, but they should all be sufficiently accessible to general readers who would like to know more about Feynman diagrams and how they relate to theories, models, mathematics, and the world. The best philosophy of science is intimately linked to and draws upon current science and its history. It pays close attention to details: the details of theorizing, the details of models and how they function, the details of experimentation and measurement, and so on. We hope this collection of five articles on Feynman diagrams, by focussing closely on detail, will contribute to a better understanding of these strange but highly productive ciphers and symbols.
This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part, we focus on classical–quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioral properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by ‘spiders’ which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of ‘genuine quantumness’, from which special processes called ‘phase spiders’ arise, and get a first glimpse of quantum nonlocality.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
We haven't really paid much attention to thought as a process. We have engaged in thoughts, but we have only paid attention to the content, not to the process. – David Bohm and David Peat, 1987 In this chapter we provide a practical introduction to basic diagrammatic reasoning, namely how to perform computations and solve problems using diagrams. We also demonstrate why diagrams are far better in many ways than traditional mathematical notation. The development and study of diagrammatic languages is a very active area of research, and intuitively obvious aspects of diagrammatic reasoning have actually taken many years to get right. Luckily, the hard work needed to formalise the diagrams in this book has already been done! So, all that remains to do is reap the benefits of a nice, graphical language. Along the way, we will encounter Dirac notaton . Readers who have previously studied quantum mechanics or quantum information theory may have already seen Dirac notation used in the context of linear maps. Here, we'll explain how it arises as a one-dimensional fragment of the two-dimensional graphical language. Thus, readers not familiar with Dirac notation will learn it as a special case of the graphical notation we use throughout the book. We also introduce the notion of a process theory , which provides a means of interpreting diagrams by fixing a particular collection of (physical, computational, mathematical, edible, etc.) systems and the processes that these systems might undergo (being heated up, sorted, multiplied by two, cooked, etc.). As we pointed out in Chapter 1, taking process theories as our starting point represents a substantial departure from standard practice in many disciplines. Rather than forcing ourselves to totally understand single systems before even thinking about how those systems compose and interact, we will seek to understand systems primarily in terms of their interactions with others. Rather than trying to understand Dave the dodo by dissecting him (at which point, he'll look pretty much like any other fat bird), we will turn him loose in the world and see what he does. This turns out to be very close in spirit to the aims of category theory , which we will meet briefly in the advanced material at the end of this chapter.
The unique features of the quantum world are explained in this book through the language of diagrams, setting out an innovative visual method for presenting complex theories. Requiring only basic mathematical literacy, this book employs a unique formalism that builds an intuitive understanding of quantum features while eliminating the need for complex calculations. This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations. Written in an entertaining and user-friendly style and including more than one hundred exercises, this book is an ideal first course in quantum theory, foundations, and computation for students from undergraduate to PhD level, as well as an opportunity for researchers from a broad range of fields, from physics to biology, linguistics, and cognitive science, to discover a new set of tools for studying processes and interaction.