Closed quantum systems are described by unitary evolution. Open quantum systems in Markovian approximation are described by completely positive maps. We introduce this framework and argue that it is natural for quantizing classically dissipative systems. Then we show three variants of the baker map: the usual one which preserves area both locally and globally, ``sloppy'' baker map, which preserves area locally but not globally, and ``Gaussian'' baker map, which preserves area globally, but not locally. We will discuss the quantizations of the first two models as well as the failed attempts at the quantization of the third one. We will explain why we are so obstinate to get it and present our current attempts and ideas. This is a work in progress, all comments are welcome.