Exploring the Limits of Open Quantum Dynamics I: Motivation, New Results from Toy Models to Applications

Which quantum states can be reached by controlling open Markovian $n$-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature $T$. The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case $T=0$ (amplitude damping) we present new results for $0 <T < \infty$ using methods of $d$-majorisation.Comment: Extended abstract, accepted to conference MTNS 2020. The results by d-majorisation are new and superseed arXiv:1905.01224 published as DOI: 10.1109/CDC40024.2019.9029452 . Part-II is arXiv:2003.04164 . arXiv admin note: text overlap with arXiv:1905.0122