Dagger linear logic for categorical quantum mechanics

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.

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

We show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact PROP. Several consequences arise from this observation:– Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two– A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement.We also enrich the rewrite system so as to get the completeness of the Clifford fragments of both the initial formalism and its enriched version.

Morphisms between categorified spin networks

We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.

The teaching of graphical calculus in engineering schools (1860-1970)

Graphical calculus is the part of numerical calculation that is based on geometrical constructions. It was established as a subject of knowledge and teaching from the 1860s onwards. With its three specialized components, graphical statics, graphical integration and nomography, it was a response to the growing calculation needs of engineers at the time of the industrial revolutions and the development of communication channels. It then became progressively obsolete in the 1960s due to the increasing role of computers and electronic calculators. The aim of the present contribution is to examine the content of graphical calculus courses created in engineering schools and universities between 1860 and 1960. Through a quantitative study of the treatises intended for the teaching of graphical calculus, we will also attempt to analyse, both chronologically and geographically, the diffusion processes of this teaching in Europe and the rest of the world during the period under consideration. Keywords: history of mathematics, mathematics education, mathematics of engineers, graphical calculus, graphical statics, graphical integration, nomography

Categories for Quantum Theory

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.

Monoidal Categories

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory of monoidal categories, including braidings, symmetries and coherence. They form the core of this book, as they provide the basic language with which the rest of the material will be developed. We introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work with them. We also introduce the monoidal categories Hilb of Hilbert spaces and linear maps, Set of sets and functions and Rel of sets and relations, which will be used as running examples throughout the book.