  The Bremermann-Bekenstein bound sets a fundamental upper limit on the rate
with which information can be processed. However, the original treatment
heavily relies on cosmological properties and plausibility arguments. In the
present analysis, we derive equivalent statements by relying on only two
fundamental results in quantum information theory and quantum dynamics --
Fannes inequality and the quantum speed limit. As main results, we obtain
Bremermann-Bekenstein-type bounds for the rate of change of the von Neumann
entropy in quantum systems undergoing open system dynamics, and for the rate of
change of the Shannon information over some logical basis in unitary quantum
evolution.
