Smoothing is an estimation method whereby a classical state (probability distribution for classical variables) at a given time is conditioned on all-time (both earlier and later) observations. Here we define a smoothed quantum state for a partially monitored open quantum system, conditioned on an all-time monitoring-derived record. We calculate the smoothed distribution for a hypothetical unobserved record which, when added to the real record, would complete the monitoring, yielding a pure-state "quantum trajectory." Averaging the pure state over this smoothed distribution yields the (mixed) smoothed quantum state. We study how the choice of actual unraveling affects the purity increase over that of the conventional (filtered) state conditioned only on the past record.

We use quantum diffusive trajectories to prove that the time evolution of two-qubit entanglement under spontaneous emission can be fully characterized by optimal continuous monitoring. We analytically determine this optimal unraveling and derive a deterministic evolution equation for the system's concurrence. Furthermore, we propose an experiment to monitor the entanglement dynamics in bipartite two-level systems and to determine the disentanglement time from a single trajectory.

We investigate the estimation technique called quantum state smoothing introduced in (Guevara and Wiseman 2015 Phys. Rev. Lett. 115 180407), which offers a valid quantum state estimate for a partially monitored system, conditioned on the observed record both prior and posterior to an estimation time. The technique was shown to give a better estimate of the underlying true quantum states than the usual quantum filtering approach. However, the improvement in estimation fidelity, originally examined for a resonantly driven qubit coupled to two vacuum baths, was also shown to vary depending on the types of detection used for the qubit's fluorescence. In this work, we analyse this variation in a systematic way for the first time. We first define smoothing power using an average purity recovery and a relative average purity recovery, of smoothing over filtering. Then, we explore the power for various combinations of fluorescence detection for both observed and unobserved channels. We next propose a method to explain the variation of the smoothing power, based on multitime correlation strength between fluorescence detection records. The method gives a prediction of smoothing power for different combinations, which is remarkably successful in comparison with numerically simulated qubit trajectories. Gesellschaft which is a function of time difference τ and is symmetric under the interchange of the records, i.e.3 ss ss ss 2 using the steady-state correlators defined in equation (21)- (22). We note that the first argument,definition is the record that appear twice in the correlator. Thus we now have correlators in equations (23) and (26) defined with unit-less measurement results and can capture solely the correlation between any two records. We show in figures 4(a) and (b) the two-time and threetime correlators for all combinations of records JK and J d M , as functions of τ. There are only two out of six of the two-time correlators that are zero:  [ ] J J d , d X N 2 , and  [ ] J J d , d X Y 2 . For the three-time correlators, there are three out of nine, i.e.    [ ] [ ] [ ] in equation (23) are shown as functions of τ. The non-vanishing correlators are for the following:in equation (26) are shown as functions of τ, where  is chosen to be one Rabi period. The colour legend is read in the same way as in figure 3, but with dK and dM, representing any two types of records. The values of correlators here are used for the analysis in table 1. Time τ is presented in units of the Rabi period T Ω =2π/Ω. Therefore, we instead focus on a parameter-independent feature, which is the vanishing or non-vanishing property of the correlators. Some correlators, such as, are zero regardless of the values of  and τ. Those that do not vanish identically are non-zero for almost all values of  and τ.In order to predict the power of quantum state smoothing offered by different measurement unravelling combinations, we propose the following principles. Firstly, the stronger the correlation, the better the smoothing power. We quantify ...