Licentiate thesis defense: Holonomy in Quantum Information Geometry

In this thesis we provide a uniform treatment of two non-adiabatic geometric
phases for dynamical systems of mixed quantum states, namely those of
Uhlmann and of Sjöqvist et al. We develop a holonomy theory for the latter
which we also relate to the already existing theory for the former. This makes
it clear what the similarities and differences between the two geometric phases
are. We discuss and motivate constraints on the two phases. Furthermore,
we discuss some topological properties of the holonomy of ‘real’ quantum
systems, and we introduce higher-order geometric phases for not necessarily
cyclic dynamical systems of mixed states. In a final chapter we apply
the theory developed for the geometric phase of Sjöqvist et al. to geometric
uncertainty relations, including some new “quantum speed limits”.