Purity and entropy evolution speed limits for open quantum systems

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.