This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in low-level formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly important in this treatment, as is appropriate, since entanglement is a fundamental driver of quantum computation. The benefits two-fold: as well as producing foundational tools for the study of quantum computation this work also connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting. We describe, following Abramsky and Coecke, how quantum mechanics may be carried out without reference to Hilbert space, in a strongly compact closed category. In particular we show how to freely construct a categorical model of abstract quantum mechanics from an arbitrary category. We introduceMultiplicative Categorical Quantum Logic (mCQL), a sequent calculus whose proof rules capture the structure of compact closed categories. This sequent calculus is interpreted in a freely generated compact closed category, and its semantics is sound with respect to cut elimination. We define an equivalent graphical syntax, similar to linear logic’s proof-nets, and prove that these proof-nets provide a full and faithful representation of any freely generated compact closed category. Further analysis of the structure of quantum states which correspond to mCQL proofs using multiplicative linear logic shows that the linear type system describes the quantum entanglement found in such states. We show that the entanglement present in these states is always of a particularly simple form: collections of entangled pairs. In order to tackle arbitrary entanglement, we generalise the work of Kelly and Laplaza to give a representation theorem for the free compact closed category by a polycategory. Such categories are shown to be equivalent to a generalised system of proof-nets whose axioms may have more than one premise
This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in low-level formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly important in this treatment, as is appropriate, since entanglement is a fundamental driver of quantum computation. The benefits two-fold: as well as producing foundational tools for the study of quantum computation this work also connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting. We describe, following Abramsky and Coecke, how quantum mechanics may be carried out without reference to Hilbert space, in a strongly compact closed category. In particular we show how to freely construct a categorical model of abstract quantum mechanics from an arbitrary category. We introduceMultiplicative Categorical Quantum Logic (mCQL), a sequent calculus whose proof rules capture the structure of compact closed categories. This sequent calculus is interpreted in a freely generated compact closed category, and its semantics is sound with respect to cut elimination. We define an equivalent graphical syntax, similar to linear logic’s proof-nets, and prove that these proof-nets provide a full and faithful representation of any freely generated compact closed category. Further analysis of the structure of quantum states which correspond to mCQL proofs using multiplicative linear logic shows that the linear type system describes the quantum entanglement found in such states. We show that the entanglement present in these states is always of a particularly simple form: collections of entangled pairs. In order to tackle arbitrary entanglement, we generalise the work of Kelly and Laplaza to give a representation theorem for the free compact closed category by a polycategory. Such categories are shown to be equivalent to a generalised system of proof-nets whose axioms may have more than one premise
Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.
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.
This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and ∗-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger-linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger-LDC. From every (isomix) dagger-LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger-LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).2 It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.
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.