  We propose an analysis of the time-optimal control of a dissipative two-level
quantum system whose dynamics is governed by the Lindblad equation. This simple
system allows one to use tools of geometric control theory and to construct its
optimal synthesis, i.e. to determine the set of all the optimal trajectories
starting from a given initial point. We study different processes such as
conversion of a pure state into a mixed state and purification of a mixed
state. In particular cases, we show that dissipation is not undesirable and can
help accelerating the control.
