The motivation for this thesis was to recast quantum self-testing [MY98,MY04]
in operational terms. The result is a category-theoretic framework for
discussing the following general question: How do different implementations of
the same input-output process compare to each other? In the proposed framework,
an input-output process is modelled by a causally structured channel in some
fixed theory, and its implementations are modelled by causally structured
dilations formalising hidden side-computations. These dilations compare through
a pre-order formalising relative strength of side-computations. Chapter 1
reviews a mathematical model for physical theories as semicartesian symmetric
monoidal categories. Many concrete examples are discussed, in particular
quantum and classical information theory. The key feature is that the model
facilitates the notion of dilations. Chapter 2 is devoted to the study of
dilations. It introduces a handful of simple yet potent axioms about dilations,
one of which (resembling the Purification Postulate [CDP10]) entails a duality
theorem encompassing a large number of classic no-go results for quantum
theory. Chapter 3 considers metric structure on physical theories, introducing
in particular a new metric for quantum channels, the purified diamond distance,
which generalises the purified distance [TCR10,Tom12] and relates to the Bures
distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for
causality in terms of '(constructible) causal channels' and 'contractions'. It
simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in
monoidal categories [JSV96]. The formalism allows for the definition of 'causal
dilations' and the establishment of a non-trivial theory of such dilations.
Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus
pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN
978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of
December 2020, defended 11th of February 2021. Keywords: dilations, applied
category theory, quantum foundations, causal structure, quantum self-testing.
242 pages, 1 figure. Comments are welcom