Quantum speed limit of a non-decoherent open system

We study the minimum time related to the quantum speed limit that characterizes the evolution of an open quantum system with the help of a simple model in the short and long time limits. We compare in particular the situation corresponding to a unique state and several states in the environment space. For short time intervals the results show a sensible difference in the behaviour of the system for different strengths of the interaction between the system and its environment.