  The idea that symmetries simplify or reduce the complexity of a system has
been remarkably fruitful in physics, and especially in quantum mechanics. On a
mathematical level, symmetry groups single out a certain structure in the
Hilbert space that leads to a reduction. This structure is given by the
irreducible representations of the group, and in general it can be identified
with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The
primary focus of this thesis is the extension of the framework of reductions
from symmetries to operator algebras, and its applications in
finite-dimensional quantum mechanics. Finding the irreducible representations
structure is the principal problem when working with operator algebras. We will
therefore review the representation theory of finite-dimensional operator
algebras and elucidate this problem with the help of two novel concepts:
minimal isometries and bipartition tables. One of the main technical results
that we present is the Scattering Algorithm for analytical derivations of the
irreducible representations structure of operator algebras. For applications,
we will introduce a symmetry-agnostic approach to the reduction of dynamics
where we circumvent the non-trivial task of identifying symmetries, and
directly reduce the dynamics generated by a Hamiltonian. We will also consider
quantum state reductions that arise from operational constraints, such as the
partial trace or the twirl map, and study how operational constraints lead to
decoherence. Apart from that, we will extend the idea of reduction beyond
operator algebras to operator systems, and formulate a quantum notion of
coarse-graining that so far only existed in classical probability theory. We
will also characterize how the uncertainty principle transitions to the
classical regime under coarse-grained measurements and discuss the implications
in a finite-dimensional setting.
