This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics. First up are dilations, covered in chapter 2. In chapter 3"diamond, andthen, dagger"we turn to the second topic: effectus theory. Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own.

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categori...

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categori...

Many examples of penetrating wounds of the heart and pericardium have been recorded (Barrett, 1950). The phrenic nerve has occasionally been divided in the neck as a result of knifing (Reichl, 1932) or of gun-shot wounds (Neuhofer, 1922), but the literature contains no reports of injuries to the phrenic nerve in the chest in which the patient has survived. The following case is therefore of int...

We show that Rob Spekken’s toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact v...