  We derive generic upper bounds on the rate of purity change and entropy
increase for open quantum systems. These bounds depend solely on the generators
of the nonunitary dynamics and are independent of the particular states of the
systems. They are thus perfectly suited to investigate dephasing and
thermalization processes of arbitrary systems. We apply these results to single
and multiple dephasing channels, to a problem of quantum control in the
presence of noise, and to cooling.
