  Physical theories can be characterized in terms of their state spaces and
their evolutive equations. The kinematical structure and the dynamical
structure of finite dimensional quantum theory are, in light of the
Choi-Jamio{\l}kowski isomorphism, one and the same --- namely the homogeneous
self-dual cones of positive semi-definite linear endomorphisms on finite
dimensional complex Hilbert spaces. From the perspective of category theory,
these cones are the sets of morphisms in finite dimensional quantum theory as a
dagger compact closed category. Understanding the intricate geometry of these
cones and charting the wider landscape for their host category is imperative
for foundational physics. In Part I of this thesis, we study the shape of
finite dimensional quantum theory in terms of quantum information. In Part II
of this thesis, we move beyond quantum theory within the vein of Euclidean
Jordan algebras. In posting this thesis on the arXiv, we hope that it might
serve as a useful resource for those interested in its subjects.
