On symmetries and conserved quantities for quantum dynamics: an algebraic approach

This work focuses on the analysis of the relation between dynamical symmetries and conserved quantities, and the implications of the former on invariance and exact model order reduction, in the context of open quantum systems. The first three chapters serve as a primer on the mathematical framework and preliminary results needed in the rest of the manuscript: finite quantum probability, dynamics of closed and open quantum systems, and a quick recap on the main tools for the study of invariance and structure of fixed-points. Then we present the main known facts on conserved quantities and their link with weak and strong symmetries providing, in some cases, alternative proofs. The main results of this thesis, partially detached from the previous discussion, are twofold: on the one hand, reachability-based model order reduction is connected to symmetries providing an alternative way to compute a suitable algebra for the reduction; second, a new structural result on the spectra of off-diagonal evolution operators in the decomposition of the operator space induced by a strong symmetry is provided, enhancing current symmetry-based methods for the analysis of open quantum systems. The main tools used in this work lie in finite-dimensional operator algebras and non-commutative probability theory, as well as in the theory of open quantum systems.