Back and forth nudging for quantum state reconstruction

We propose an estimation method allowing to identify the initial state of a quantum system based on the continuous weak measurement of a certain physical observable over a fixed interval of time. The algorithm is based on the back-and-forth nudging method consisting in iterative application of Luenberger observers for the time-forward and time-backward dynamics. A clever change of variables unveils the needed symmetry in the observer design leading to the decrease of a certain distance (in an appropriate metric) between the estimator and the main system, both in forward and backward directions.