  Quantum information brings together theories of physics and computer science.
This synthesis challenges the basic intuitions of both fields. In this thesis,
we show that adopting a unified and general language for process theories
advances foundations and practical applications of quantum information. Our
first set of results analyze quantum algorithms with a process theoretic
structure. We contribute new constructions of the Fourier transform and
Pontryagin duality in dagger symmetric monoidal categories. We then use this
setting to study generalized unitary oracles and give a new quantum blackbox
algorithm for the identification of group homomorphisms, solving the GROUPHOMID
problem. In the remaining section, we construct a novel model of quantum
blackbox algorithms in non-deterministic classical computation. Our second set
of results concerns quantum foundations. We complete work begun by Coecke et
al., definitively connecting the Mermin non-locality of a process theory with a
simple algebraic condition on that theory's phase groups. This result allows us
to offer new experimental tests for Mermin non-locality and new protocols for
quantum secret sharing. In our final chapter, we exploit the shared process
theoretic structure of quantum information and distributional compositional
linguistics. We propose a quantum algorithm adapted from Weibe et al. to
classify sentences by meaning. The clarity of the process theoretic setting
allows us to recover a speedup that is lost in the naive application of the
algorithm. The main mathematical tools used in this thesis are group theory
(esp. Fourier theory on finite groups), monoidal category theory, and
categorical algebra.
