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String diagram rewrite theory II: Rewriting with symmetric monoidal structure
- Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, Fabio Zanasi
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- Mathematical Structures in Computer Science / Volume 32 / Issue 4 / April 2022
- Published online by Cambridge University Press:
- 29 September 2022, pp. 511-541
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Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.
String diagram rewrite theory III: Confluence with and without Frobenius
- Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Paweł Sobociński, Fabio Zanasi
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- Mathematical Structures in Computer Science / Volume 32 / Issue 7 / August 2022
- Published online by Cambridge University Press:
- 13 June 2022, pp. 829-869
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In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.
Causal inference via string diagram surgery: A diagrammatic approach to interventions and counterfactuals
- Bart Jacobs, Aleks Kissinger, Fabio Zanasi
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- Mathematical Structures in Computer Science / Volume 31 / Issue 5 / May 2021
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- 16 November 2021, pp. 553-574
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Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.
12 - Quantum Computation
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 30 March 2017
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- 16 March 2017, pp 679-736
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Summary
In the Name of the Pasta, and of the Sauce, and of the Holy Meatballs …
– Bobby Henderson, The Gospel of the Flying Spaghetti Monster, 2006After the conceptual comes the practical. While the study of quantum foundations is as old as quantum theory itself, the field of quantum computing is relatively new. So new in fact that large-scale, practical quantum computing is still not a reality. A typical ‘quantum computer’ takes many months to set up before performing such astounding tasks as factoring 6 into 3 × 2. Nonetheless, if those machines would exist, we know that we would gain amazing speed-ups in solving some hard (classical) computational problems, such as those involved in breaking a huge portion of cryptographic systems in use today.
Before we get into ‘quantum computing’, we should say a couple of things about ‘computing’. What is computing? Our answer by now probably won't come as such a shock: it's a process theory! Computation is indeed all about wiring the inputs and outputs of small processes together to make bigger processes. More specifically, a computation consists of a (finite) set of basic processes, which are wired together according to some (also finite) instructions, which we refer to as an algorithm or simply a program.
The only essential difference between classical and quantum computation is the contents of the basic processes. For classical computation, these operations consist of things like logical operations (e.g. XOR) or reading/writing locations in memory. For quantum computation, we can extend this with quantum processes and classical-quantum interactions such as measurements. So quantum computing is all about figuring out how to write new kinds of programs that exploit these new building blocks to build faster algorithms or accomplish new kinds of tasks that aren't possible classically.
The first quantum algorithms were ‘proofs of concept’, in the sense that they solved some problem much faster than a classical computer, but the kinds of problems they solved were not particularly interesting in their own right. However, this changed drastically with the advent of Grover's quantum search and Shor's factoring algorithms.
Appendix Some Notations
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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13 - Quantum Resources
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 30 March 2017
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- 16 March 2017, pp 737-789
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In my life, I have prayed but one prayer: oh Lord, make my enemies ridiculous. And God granted it.
– Voltaire, letter to Étienne Noël Damilaville, 1767Although bankers on Wall Street can have pretty much whatever they want in whatever quantity they want, this chapter is for the rest of us cheapskates. Throughout this book, we have assumed that we, like quantum bankers, have had access to whatever processes we need in whatever quantity. For instance, in teleportation, we assumed we could obtain Bell states at will and do joint measurements on pairs of systems. For non-locality, we assumed we had lots of GHZ states around, so we could make enough measurements to see our assumptions about locality cave in. And for MBQC, we assumed that we had huge graph states around to implement universal quantum computation.
As you might expect, these things ain't cheap! Quantum processes involving multiple systems typically take some very special equipment and many hours to implement. So, when it comes to such resources, it behooves one to think about doing as much as possible with as little as possible. This is of course true not just for quantum states, but for any kind of resource: coal, oil, nuclear, wind, and solar energy; certain chemicals; or just a bit of affection.
What is important about all resources is what we can do with them. If we think of the benefits of resources (e.g. warm, comfy houses or secure communication) as being resources themselves, we see that pretty much all questions about resources boil down to the following two:
1. Can a given resource be converted into some other resource?
2. How much of resource X to do we need to obtain resource Y?
These are exactly the questions a resource theory aims to answer.
One place this idea of ‘conversion of resources’ appears very explicitly is in the study of chemical reactions, where one finds expressions like these:
6CO2 + 6H2O + light -→ C6H12O6 + 6O2
C6H12O6 -→ 2C2H5OH + 2CO2
Such an expression tells us that we can convert the stuff on the LHS to the stuff on the RHS. Even though the expression doesn't provide any details about how this conversion is actually done, it does provide us with two very useful pieces of information.
Preface
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp xiii-xviii
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Glad you made it here! This book is about telling the story of quantum theory entirely in terms of pictures. Before we get into telling the story itself, it's worth saying a few words about how it came about. On the one hand, this is a very new story, in that it is closely tied to the past 10 years of research by us and our colleagues. On the other hand, one could say that it traces back some 80 years when the amazing John von Neumann denounced his own quantum formalism and embarked on a quest for something better. One could also say it began when Erwin Schrödinger addressed Albert Einstein's concerns about ‘spooky action at a distance’ by identifying the structure of composed systems (and in particular, their non-separability) as the beating heart of quantum theory.
From a complementary perspective, it traces back some 40 years when an undergraduate student named Roger Penrose noticed that pictures out-classed symbolic reasoning when working with the tensor calculus.
But 80 years ago the authors weren't around yet, at least not in human form, and 40 years ago there wasn't really that much of us either, so this preface will provide an egocentric take on the birth of this book. This also allows us to wholeheartedly acknowledge all of those without whom this book would never have existed (as well as some who nearly succeeded in killing it).
Things started out pretty badly for Bob, with a PhD in the 1990s on a then completely irrelevant topic of contextual ‘hidden variable representations’ of quantum theory – which recently have been diplomatically renamed to ontological models (Harrigan and Spekkens, 2010; Pusey et al., 2012). After a period of unemployment and a failed attempt to become a rock star, Bob ventured into the then even more irrelevant topic of von Neumann's quantum logic (Birkhoff and von Neumann, 1936) in the vicinity of the eccentric iconoclast Constantin Piron (1976).
It was there that category theory entered the picture, as well as serious considerations on the fundamental status of composition in quantum systems – something that went hand-in-hand with bringing quantum processes (rather than quantum states) to the forefront …
10 - Quantum Theory: The Full Picture
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 624-654
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8 - Picturing Classical-Quantum Processes
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 405-509
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… 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 ADMost 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.
References
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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7 - Quantum Measurement
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 345-404
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A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.
– Max Planck, 1936The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation. A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’
‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense.
Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history!
Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.
5 - Hilbert Space from Diagrams
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 154-250
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I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.
– John von Neumann, letter to Garrett Birkhoff, 1935We have now seen how processes described by string diagrams already exhibit some quantum-like features. It is natural to ask how much extra work is needed to go from string diagrams to Hilbert spaces and linear maps, the mathematical tools von Neumann used to formulate quantum theory in the late 1920s. The answer is: not that much.
We start by considering what it takes for two processes to be equal. In many process theories, it suffices for them to agree on a relatively small number of states. This feature leads very naturally to the notion of basis, and we can use adjoints to identify a particularly handy type of basis, called an orthonormal basis (ONB). When all types admit a basis, any process can be completely described by a collection of numbers called its matrix.
Now, such a matrix identifies a particular process uniquely, but for any matrix to represent a process we need to add a bit more structure. Therefore, we allow processes with the same input/output wires to be combined into one, or summed together. If a process theory admits string diagrams, has ONBs for every type, and has sums of processes, we can describe sums, sequential composition, parallel composition, transpose, conjugate, and adjoint all in terms of operations on matrices. We call this the matrix calculus of a process theory.
Thus, by adding ONBs and sums, we have very nearly recovered the full power of linear algebra, but with the added generality that the numbers λ are still very unrestricted (in particular, they need not be the elements of some field like the real or complex numbers). In fact, a matrix calculus for relations makes perfect sense, where the numbers are booleans.
The final step towards Hilbert spaces and linear maps consists of requiring the numbers of the process theory to be the complex numbers.
11 - Quantum Foundations
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 655-678
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Summary
Mermin once summarized a popular attitude towards quantum theory as ‘Shut up and calculate.’ We suggest a different slogan: ‘Shut up and contemplate!’
– Lucien Hardy and Rob Spekkens, 2010This chapter is dedicated to the foundations of quantum theory, or as it's more fashionably called these days, quantum foundations. Here, we will use all of the things we've learned so far to probe some very deep questions:
1. What features of nature are imposed on us by quantum theory?
2. Conversely, what features of a physical theory are imposed on us by (our current understanding of) nature?
3. Which of these features are ‘properly quantum’, in the sense that they have no counterpart in any classical physical theory?
We'll address these questions by looking at one of those most celebrated (and historically controversial) properties of quantum theory: quantum non-locality. First, we will give a precise definition of non-locality and prove that it exists within the theory of quantum processes and in fact already within the comparatively tiny subtheory of causal Clifford maps. Then, we will present a new process theory called spek, which has locality built right in. A remarkable thing is that the two theories of Clifford maps and spek are identical in every respect except one: the phase group of a single system. And (another spoiler alert!) it is indeed this one difference that kills the proof of non-locality that works for quantum theory.
Quantum Non-locality
Quantum non-locality is probably still the least understood of all the new quantum features, in both philosophical and structural terms. Our upbringing in a seemingly ‘classical’ world, and especially our undeniably corrupting ‘classical’ scientific education, tends to make us expect two things from a physical theory:
1. Realism: physical systems have real pre-existing properties, and hence the outcome of ‘measuring’ such a property is fixed in some way prior to the measurement.
2. Locality: it is impossible for one system to affect another distant system instantaneously.
3 - Processes as Diagrams
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 28-82
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Summary
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, 1987In 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.
Index
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- 16 March 2017, pp 822-827
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9 - Picturing Phases and Complementarity
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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- 16 March 2017, pp 510-623
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2 - Guide to Reading This Textbook
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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- Picturing Quantum Processes
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Summary
Nothing endures but change.
– Heraclitus of Ephesus, 535–475 BCWho Are You and What Do You Want?
While there is already a plethora of textbooks on quantum theory and its features, this one is unique because it is based on quantum picturalism.
Prerequisites. There are hardly any prerequisites to this textbook. We do not expect our readers to have a background in physics or computer science or to have any profound background in mathematics. In principle, some basic secondary-school mathematics should be sufficient.
For example, linear algebra (and of course quantum theory) is presented from scratch in a diagrammatic manner. However, this does not mean that the first half of this book will be a boring read for the specialist, since these presentations from scratch are radically different from the usual ones.
Target audience. Given its low entrance fee, as well as its unique form and content, this textbook should appeal to a broad audience, ranging from students to experts from a wide range of disciplines including physicists, computer scientists, mathematicians, logicians, philosophers of science, and researchers from other areas with a multidisciplinary interest, such as biologists, engineers, cognitive scientists, and educational scientists.
One particular target audience for this book consists of students and researchers in quantum computation and quantum information, as we will apply the tools from quantum picturalism directly to these areas. A practising quantum computing researcher may discover a new set of tools to attack open problems where traditional methods have failed, and a student may find some subjects explained in a manner that is much easier to grasp.
Another target audience consists of students and experts with an interest in foundations and/or philosophy of physics, who can read in this book about a process-oriented approach to physics that takes composition of systems as a first-class citizen, rather than a derived notion. In particular, this is the first book that uses diagrammatic language to capture the idea of a process theory and puts such process theories forward as a new foundation for quantum theory, in which all standard quantum theoretical notions can be expressed.
4 - String Diagrams
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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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, 1935By 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’.
Contents
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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1 - Introduction
- Bob Coecke, University of Oxford, Aleks Kissinger, Radboud Universiteit Nijmegen
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Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition.
– Thomas Kuhn, The Essential Tension, 1977.Quantum theory has been puzzling physicists and philosophers since its birth in the early 20th century. However, starting in the 1980s, rather than asking why quantum theory is so weird, many people started to ask the question:
What can we do with quantum weirdness?
In this book we not only embrace this perspective shift, but challenge the quantum icons even more. We contend that one should not only change the kinds of questions we ask about quantum theory, but also:
change the very language we use to discuss it!
Before meeting this challenge head-on, we will tell a short tale to demonstrate how the quantum world defies conventional intuitions …
The Penguins and the Polar Bear
Quantum theory is about very special kinds of physical systems – often very small systems – and the ways in which their behaviour differs from what we observe in everyday life. Typical examples of physical systems obeying quantum theory are microscopic particles such as photons and electrons. We will ignore these for the moment, and begin by considering a more ‘feathered’ quantum system. This is Dave:
He's a dodo. Not your typical run-of-the-mill dodo, but a quantum dodo.We will assume that Dave behaves in the same manner as the smallest non-trivial quantum system, a two-level system, which these days gets referred to as a quantum bit, or qubit. Let's compare Dave's state to the state of his classical counterpart, the bit. Bits form the building blocks of classical computers, whereas (we will see that) qubits form the building blocks of quantum computers. A bit:
1. admits two states, which we tend to label 0 and 1,
2. can be subjected to any function, and
3. can be freely read.
Here, ‘can be subjected to any function’ means that we can apply any function on a bit to change its state. For example, we can apply the ‘NOT’ function to a bit, which interchanges the states 0 and 1, or the ‘constant 0’ function which sends any state to 0.