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Quantum Structures and Category Theory

Roman Frič

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The methods and constructions of category theory are suitable tools for studying quantum structures and generalized probability. The aim of the present paper is to support our claim via several examples and to motivate further research in this area.

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Kvantové štruktúry a teória kategórií

Roman Frič

The methods and constructions of category theory are suitable tools for studying quantum structures and generalized probability. The aim of the present paper is to support our claim via several examples and to motivate further research in this area.

•Jan 1, 2004•0 citations

Categories of quantum cpos

Andre Kornell, Bert Lindenhovius, Michael W. Mislove

Abstract This paper unites two research lines. The first involves finding categorical models of quantum programming languages with recursion and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to finding noncommutative generalizations (also called quantum generalizations) of these structures. Using a quantization method called discrete quantization , which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of $\omega$ -complete partial orders (cpos), called quantum cpos . Cpos are central in domain theory and are widely used to construct categorical models of programming languages with recursion. We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages, which is illustrated with some examples. Because of their noncommutative character, quantum cpos may form the backbone of a future quantum domain theory that provides structural methods for the denotational semantics of recursive quantum programming languages.

arXiv:2406.01816•Mathematical Structures in Computer Science•Jan 1, 2026•0 citations

Quantum Probabilities

There are well known mathematical models of quantum structures ([3]) including probability aspects of the theories. One of the successful ways for the theories can be in using MV-algebras ([2]). Recall that MV-algebras play in multi-valued logics the same role as Boolean algebras in two-valued logics. The probability theory on MV-algebras was reviewed in [8] and [9]. The aim of this contribution is in using the methods of MV-algebra probability theory in some structures dierent

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Generalized probability theories: what determines the structure of quantum theory?

Peter Janotta, Haye Hinrichsen B

The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical structure of quantum theory. This review paper tries to present the framework and recent results to a broader readership in an accessible manner. To achieve this, we follow a constructive approach. Starting from few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. In this framework classical and quantum theory appear as special cases, and the aim is to understand what distinguishes quantum mechanics as the fundamental theory realized in nature. It turns out non-classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features.

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Algebraic Structures of Quantum Physics

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We examine a series of developments in mathematics used in quantum physics. These include Hilbert space. We then examine how several developments in mathematics can be used in application to Quantum physics.

Open Journal of Philosophy•Jan 1, 2021•0 citations

Categories for Quantum Theory

Chris Heunen, Jamie Vicary

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.

•Nov 19, 2019•103 citations
Abstract This paper unites two research lines. The first involves finding categorical models of quantum programming languages with recursion and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to finding noncommutative generalizations (also called quantum generalizations) of these structures. Using a quantization method called discrete quantization , which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of ω\omegaω -complete partial orders (cpos), called quantum cpos . Cpos are central in domain theory and are widely used to construct categorical models of programming languages with recursion. We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages, which is illustrated with some examples. Because of their noncommutative character, quantum cpos may form the backbone of a future quantum domain theory that provides structural methods for the denotational semantics of recursive quantum programming languages.
The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical structure of quantum theory. This review paper tries to present the framework and recent results to a broader readership in an accessible manner. To achieve this, we follow a constructive approach. Starting from few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. In this framework classical and quantum theory appear as special cases, and the aim is to understand what distinguishes quantum mechanics as the fundamental theory realized in nature. It turns out non-classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features.
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.