Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups thermodynamic processes. We characterise a wide class GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator (a one-parameter semigroup of) Markovian belongs to this class. completely classify visualise them their non-Markovian counterparts case single qubit. Second, use description in framework bilinear control systems reachable sets coherently controllable with switchable coupling bath. The core problem reduces studying hybrid system ("toy model") on standard simplex allowing two types evolution: (i) instantaneous permutations (ii) $d$-stochastic maps. generalise upper bounds set toy model invoking new results thermomajorisation. Using tools theory fully these as well stabilisable states exemplified by exact qutrit systems.