Quantum state smoothing: when the types of observed/unobserved measurement matter*

We consider a problem of quantum state estimation of a dynamical system interacting with a lossy environment that is being partially monitored. Using the technique called Quantum state smoothing as introduced by Guevara and Wiseman [Phys. Rev. Lett. 115, 180407], a proper quantum state at any time t, of a two-level system undergoing a Rabi oscillation, can be optimally estimated using observed records both before and after the estimation time. We here extend the original study, which only considered measured homodyne records and smoothing with unobserved photon counting records, to other combinations of bath observations. We find that the quality of the smoothing, as determined by the purity of the estimated state, clearly depends on what types of records are available to the observer and what type of records were lost. We have also investigated the record correlation functions between different types of observation and used them as a guide to predict the quality of the smoothing in different regimes.