We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.

These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum information-flow ’ which enables protocols such as quantum teleportation. 1 To this means we defined strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. • ‘Postulates for an abstract quantum formalism ’ in which classical informationflow (e.g. token exchange) is part of the formalism. As an example, we provided a purely formal description of quantum teleportation and proved correctness in abstract generality. 2 In this formalism types reflect kinds, contra the essentially typeless von Neumann formalism [25]. Hence even concretely this formalism manifestly improves on the usual one. • ‘A high-level approach to quantum informatics’. 3 Indeed, the above discussed work can be conceived as aiming to solve: von Neumann quantum formalism � high-level language low-level language. I also provide a brief discussion on how classical and quantum uncertainty can be mixed in the above formalism (cf. density matrices). 4

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Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics.

Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a category of tame formal distributions with coefficients in a commutative associative algebra and show that it is a dagger category. This gives access to a broad new class of models, with the abstract scalars in the sense of Abramsky being the elements of the algebra. We will also consider a subcategory of local formal distributions, based on the ideas of Kac. Locality has been of fundamental significance in various formulations of quantum field theory. Thus our work may provide the possibility of extending the abstract framework to QFT. We also show that these categories of formal distributions are monoidal and contain a nuclear ideal, a weak form of adjunction appropriate for analyzing categories

in a somewhat unconventional manner. Our main focus will be on monoidal categories, mainly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category of sets and relations, posetal categories, diagrammatic calculi, strictification, compact categories, biproduct categories and abstract matrix calculi, internal structures, and topological quantum field theories. In our attempt to complement the existing literature we (on purpose) omitted some very basic topics for which we point to other available sources. 0 Prologue: cooking with vegetables Consider a raw potato. Conveniently, we refer to it as A. Raw potato A admits several states e.g. ‘dirty’, ‘clean’, ‘skinned’,... We usually don’t eat raw potatoes so we need to process A such that it becomes eatable. We refer to this cooked version of A as B. Also B admits several states e.g. ‘boiled’, ‘fried’, ‘baked with skin’, ‘baked without skin’,... Correspondingly, there are several ways to turn raw potato A into cooked potato B e.g. ‘boiling’, ‘frying’, ‘baking’, respectively referred to as f, f ′ and f ′ ′. We make the fact that these cooking processes apply to raw potato A and produce cooked potato B explicit by labelled arrows: A f ✲ B A f ′

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In this statement we provide some examples of transdisciplinary journeys, from one field to another, and back. In particular, the quantum informatic endeavor is not just a matter of feeding physical theory into the general field of natural computation, but also one of using high-level methods developed in Computer Science to improve on the quantum physical formalism itself, and the understanding thereof. We highlight a seemingly contradictory phenomenon: passing to an abstract, categorical quantum informatic formalism leads directly to a simple and elegant graphical formulation of quantum theory itself, which for example makes the design of some important quantum informatic protocols completely transparent. It turns out that essentially all of the quantum informatic machinery can be recovered from this graphical calculus. But in turn, this graphical formalism provides a bridge between methods of logic and computer science, and some of the most exciting developments in the mathematics of the past two decades: namely those arising from the Jones polynomial invariant of knots and links, the Temperley-Lieb Algebra and related structures.

Summary. In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure (FdHilb). We also provide a detailed discussion of the category which has sets as objects, relations as morphisms, and the cartesian product as its monoidal structure (Rel), and thirdly, categories with manifolds as objects and cobordisms between these as morphisms (2Cob). While sets, Hilbert spaces and manifolds do not share any non-trivial common structure, these three categories are in fact structurally very similar. Shared features are diagrammatic calculus, compact closed structure and particular kinds of internal comonoids which play an important role in each of them. The categories FdHilb and Rel moreover admit a categorical matrix calculus. Together these features guide us towards topological quantum field theories. We also discuss posetal categories, how group representations are in fact categorical constructs, and what strictification and coherence of monoidal categories is all about. In our attempt to complement the existing literature we omitted some very basic topics. For these we refer the reader to other available sources. 1.0 Prologue: cooking with vegetables Consider a ‘raw potato’. Conveniently, we refer to it as A. Raw potato A admits several states e.g. ‘dirty’, ‘clean’, ‘skinned’,... Since raw potatoes don’t digest well we need to process A into ‘cooked potato ’ B. We refer to A and B as kinds or types of food. Also B admits several states e.g. ‘boiled’, ‘fried’, ‘baked with skin’, ‘baked without skin’,... Correspondingly, there are several ways to turn raw potato A into cooked potato B e.g. ‘boiling’, ‘frying’, ‘baking’, to which we respectively refer as f, f ′ and f ′ ′. We make the fact that each of these cooking processes applies to raw potato A and produces cooked potato B explicit via labelled arrows: A f ✲ B A f ′

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley–Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley–Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.