The first half of this thesis is dedicated to the study of generalizations of probability
theory which include quantum theory and many of its alternatives. This research
program seeks to understand quantum theory by exploring it ‘from the outside’,
where it sits within a landscape of all operationally defined theories. We approach
this problem with a new perspective: generalized decoherence. Quantum theory
recovers classical theory via decoherence, where a ‘classical limit’ is achieved by
a quantum system interacting with its environment, losing its quantum coherence.
Typically it is the emergence of the classical world from the quantum that is studied.
We reverse this paradigm and ask - how does the existence of a classical limit define
the structure of quantum theory, or any other post-classical theory? We derive the
existence of entangled states as a necessary feature of any theory that allows for the
emergence of the classical world in this way. We then use our framework for generalized decoherence to explore the properties of theories that can ‘hyperdecohere’
to quantum theory. We find that any such theory must be very exotic compared to
quantum theory, violating either the postulate of tomographic locality, reversibility,
causality or purity. In the second half of this thesis we turn our attention to the
second law of thermodynamics. We present a framework for deriving the second
law under general constraints and explore two pertinent examples, constraining the
fluctuations in work and constraining the size of the thermal bath and deriving the
second law in these cases. Bounding fluctuations results in a unified free energy that
contains the single-shot and Helmholtz free energies as limiting cases. Bounding the
size of the thermal bath results in a finite-bath second law that is more general that
previous attempts and draws connections between non-asymptotic thermodynamics
and second-order information theory.