Optimized Steering: Quantum State Engineering and Exceptional Points

The state of a quantum system may be steered towards a predesignated target state, employing a sequence of weak $\textit{blind}$ measurements (where the detector's readouts are traced out). Here we analyze the steering of a two-level system using the interplay of a system Hamiltonian and weak measurements, and show that $\textit{any}$ pure or mixed state can be targeted. We show that the optimization of such a steering protocol is underlain by the presence of Liouvillian exceptional points. More specifically, for high purity target states, optimal steering implies purely relaxational dynamics marked by a second-order exceptional point, while for low purity target states, it implies an oscillatory approach to the target state. The dynamical phase transition between these two regimes is characterized by a third-order exceptional point.