Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channels

In view of controlling finite-dimensional open quantum systems, we provide a unified Lie-semigroup framework describing the structure of completely positive trace-preserving maps. It allows (i) to identify the Kossakowski-Lindblad generators as the Lie wedge of a subsemigroup, (ii) to link properties of Lie semigroups such as divisibility with Markov properties of quantum channels, and (iii) to characterise reachable sets and controllability in open systems. We elucidate when time-optimal controls derived for the analogous closed system already give good fidelities in open systems and when a more detailed knowledge of the open system (e.g. in terms of the parameters of its Kossakowski-Lindblad master equation) is actually required for state-of-the-art optimal-control algorithms. As an outlook, we sketch the structure of a new, potentially more efficient numerical approach explicitly making use of the corresponding Lie wedge.