The Alicki–Lindblad–Fannes dynamical (ALF) entropy measures the rate at which new information is gathered about a quantum system by inspecting its long-time evolution. After extending the ALF entropy to open quantum dynamics, we propose it as a measure of back-flow of information from the environment. We show that the behavior of the open-system ALF entropy is a stronger witness than existing approaches based only on the open system reduced dynamics. In the case of a qubit collisionally coupled to a classical spin chain, we obtain an exact expression for the open-system ALF entropy explicitly depending on the environment correlations. An extreme case shows how the information flow from environment to system corresponds to vanishing entropy production as for reversible finite quantum systems.