This paper presents first steps of an approach to quantum information processing in the framework of higher category theory from a noncommutative mathematics perspective. The aim is to provide a unifying theory for the structure and dynamics of composite quantum information processing systems, such that states, evolution, entanglement, decoherence are modeled by abstract categorical constructions and vice versa new mathematical structures arising from higher dimensional algebra could be "tested" as computational schemes and possibly realized by physical experiments.
This paper presents first steps of an approach to quantum information processing in the framework of higher category theory from a noncommutative mathematics perspective. The aim is to provide a unifying theory for the structure and dynamics of composite quantum information processing systems, such that states, evolution, entanglement, decoherence are modeled by abstract categorical constructions and vice versa new mathematical structures arising from higher dimensional algebra could be "tested" as computational schemes and possibly realized by physical experiments.
This paper explores the intersection of quantum algebraic geometry and quantum information theory, focusing on how noncommutative algebraic structures provide a powerful framework for understanding and manipulating quantum systems. We investigate the role of noncommutative rings and schemes in encoding quantum states, quantum operations, and entanglement. Furthermore, we examine the application of tools from algebraic geometry, such as sheaves and cohomology, to characterize the complexity and properties of quantum information processing tasks. Specifically, we delve into the use of noncommutative tori and their classifications, alongside explorations of quantum metrics and preordered semialgebras, to build a bridge between abstract mathematical structures and concrete applications in quantum technologies. Our goal is to highlight the potential of quantum algebraic geometry to offer new insights and techniques for advancing the field of quantum information.
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.
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.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Quantum processes with indefinite causal structure emerge when we wonder which are the most general evolutions, allowed by quantum theory, of a set of local systems which are not assumed to be in any particular causal order. These processes can be described within the framework of higher−order quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely, we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems of the higher order quantum maps.
With the development of topological field theory, the mathematical tool of the tensor category was also introduced into physics. Traditional group theory corresponds to a special category,group category. Tensor categories can describe higher-order interactions and symmetric relations, while group theory can only describe first-order interactions. In fact, the quantum circuit itself constitutes a category. However, at present, the field of quantum computing mainly uses group theory as a mathematical tool. If category theory is introduced into the field of quantum simulation, the application scope of quantum computers can be greatly expanded. This paper propose a new dynamic simulation method,categorical quantum simulation. In our paradigm quantum simulation is no longer based on the structure of the group theory, but based on the structure of the tensor category. This could enable many systems that could not be efficiently quantum simulated before.In this article we give an concrete example of the categorical simulation of SU(3) Yang-Mills theory. It shows that categorical quantum simulation provides a new encoding method,emergenism encoding, which saves more qubits resources than reductionism quantum encoding. In addition, many domains can be described in the language of category theory, which allows quantum circuits to directly encode and simulate these domains.