Introduction Our starting point is a particular “canvas” aimed to “draw” theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this chapter, we consider the conceptual foundations for this canvas and how these can then be converted into mathematical structure. With very little structural effort (i.e., in very abstract terms) and in a very short time span, the categorical quantum mechanics (CQM) research program, initiated by Abramsky and this author, has reproduced a surprisingly large fragment of quantum theory [3, 45, 48, 56, 60–62, 170, 179]. It also provides new insights both in quantum foundations and in quantum information —for example, in [49–51, 58, 59, 64, 79, 80]—and has even resulted in automated reasoning software called quantomatic [71–73], which exploits the deductive power of CQM, which is indeed a categorical quantum logic [77]. In this chapter, we complement the available material by not requiring prior knowledge of category theory and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere—for example, in representation theory [74], quantum algebra [176], knot theory [187], topological quantum field theory (TQFT) [132], and several other areas.
Introduction Our starting point is a particular “canvas” aimed to “draw” theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this chapter, we consider the conceptual foundations for this canvas and how these can then be converted into mathematical structure. With very little structural effort (i.e., in very abstract terms) and in a very short time span, the categorical quantum mechanics (CQM) research program, initiated by Abramsky and this author, has reproduced a surprisingly large fragment of quantum theory [3, 45, 48, 56, 60–62, 170, 179]. It also provides new insights both in quantum foundations and in quantum information —for example, in [49–51, 58, 59, 64, 79, 80]—and has even resulted in automated reasoning software called quantomatic [71–73], which exploits the deductive power of CQM, which is indeed a categorical quantum logic [77]. In this chapter, we complement the available material by not requiring prior knowledge of category theory and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere—for example, in representation theory [74], quantum algebra [176], knot theory [187], topological quantum field theory (TQFT) [132], and several other areas.
Abstract Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.
. A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories , and categorical quantum mechanics . The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how non-locality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and no-signalling models are characterized in these terms. Introduction. A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories [48, 41, 52, 47], and categorical quantum mechanics [6, 7]. • Operational theories allow general formulations of results in quantum foundations and quantum information [11, 12, 10]. They also play a prominent rôle in current work on axiomatizations of quantum mechanics [36, 19, 49, 25]. • Categorical quantum mechanics enables a high-level approach to quantum information and quantum foundations, which can be presented in terms of string-diagram representations of structures in monoidal categories [7]. This has proved very effective in providing a conceptually illuminating and technically powerful perspective on a range of topics, including quantum protocols [6], entanglement [24], measurement-based quantum computing [29], no-cloning [1], and non-locality [22]. The aim of the present paper is to establish strong connections between these two formalisms. We shall begin by reviewing operational theories. We then show how a proper formulation of compound systems within the operational framework leads to a view of operational theories as representations of monoidal categories of a particular form. We call these operational representations .
Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.
Applied category theory provides powerful mathematical tools for modelling processes and their composition. Symmetric monoidal categories, which involve series and parallel composition, are particularly well-suited for describing the composition of processes in space and time. Also called process theories, they admit string diagrams, which constitute a visually intuitive, mathematically rigorous, expressive and flexible syntax that is applicable to wide-ranging scientific domains. In this thesis, we employ string diagrams to investigate a selection of topics in the areas of quantum foundations, computing, and natural language processing: (1) We formalise constructor theory as a process theory. In the context of quantum physics, we also demonstrate the conflict between constructor-theoretic principles of locality and composition. Moreover, we argue that if the principle of locality is rejected, categorical quantum mechanics (CQM) can be conceived as a constructor theory of quantum physics. (2) We develop a formalism for wave-based logic circuits with phase encoding. We motivate the formalism using the example of spin-wave circuits, and then demonstrate its utility in design, analysis and optimisation of Boolean logic circuits. (3) We investigate the elimination of inter-language grammatical bureaucracy in the distributional compositional circuits (DisCoCirc) framework. In particular, we develop a hybrid grammar for a restricted fragment of the Urdu language, and show that Urdu text endowed with this hybrid grammar maps surjectively to DisCoCirc text circuits. Furthermore, we show that for the same language fragment, Urdu and English text circuits become the same up to gate-level translation. The aforementioned work supports the view that a process-relational outlook in science is well-supported by applied category-theoretic tools, particularly string diagrams.
The report deals with the axiomatic approach to the phenomena of quantum computation and algorithms. It gives an overview of the classical approach (due to von Neumann) using the structures of Hilbert spaces, as well as mentions some major results in the field and discusses the flaws of the model and its current dominance. This leads to the introduction of a new axiomatization, first proposed by S. Abramsky and B. Coecke in 2007, which relies on the modern concepts of category theory. It is a field of mathematics originally created in the 1940s by specialists in algebraic topology but currently gaining popularity in other branches because of its abstract nature and emphasis on operations rather than objects. Most importantly, category theory is a powerful tool in functional analysis allowing mathematicians to reformulate the statements of Hilbert space theory in a more intuitive way, without matrix calculus. Finally, the logic behind quantum operations is discussed: it is revealed that the ideas of quantum algorithms have a strong connection to Jean-Yves Girard’s linear logic.
This paper explores the application of category theory to provide a semantic framework for quantum vector spaces, which are fundamental to quantum mechanics and quantum information. We move beyond the traditional algebraic and analytical descriptions of Hilbert spaces to adopt a more abstract, compositional, and graphical approach rooted in monoidal categories. This categorical perspective allows for a unified understanding of quantum phenomena, including superposition, entanglement, and measurement, by representing quantum systems as objects and processes as morphisms. We delve into the concept of compact closed categories and dagger symmetric monoidal categories as the appropriate mathematical structures to capture the essential features of quantum theory, such as duality, adjoints, and the tensor product for composite systems. Furthermore, the paper discusses how graphical calculi, arising from these categorical structures, offer an intuitive and rigorous method for reasoning about quantum protocols and computations, facilitating the design and verification of quantum algorithms. By providing a high-level, axiomatic framework, categorical semantics clarifies the foundational principles of quantum mechanics and opens new avenues for developing quantum programming languages and understanding the interplay between quantum information and computation.