  Here, we are concerned with comparing estimation schemes for the quantum
state under continuous measurement (quantum trajectories), namely quantum state
filtering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)],
quantum state smoothing. Unfortunately, the cumulative errors in the most
typical simulations of quantum trajectories with a total time of simulation $T$
can reach orders of $T \Delta t$. Moreover, these errors may correspond to
deviations from valid quantum evolution as described by a completely positive
map. Here we introduce a higher-order method that reduces the cumulative errors
in the complete positivity of the evolution to of order $T\Delta t^2$, whether
for linear (unnormalised) or nonlinear (normalised) quantum trajectories. Our
method also guarantees that the discrepancy in the average evolution between
different detection methods (different `unravellings', such as quantum jumps or
quantum diffusion) is similarly small. This equivalence is essential for
comparing quantum state filtering to quantum state smoothing, as the latter
assumes that all irreversible evolution is unravelled, although the estimator
only has direct knowledge of some records. In particular, here we compare, for
the first time, the average difference between filtering and smoothing
conditioned on an event of which the estimator lacks direct knowledge: a photon
detection within a certain time window. We find that the smoothed state is
actually {\em less pure}, both before and after the time of the jump.
Similarly, the fidelity of the smoothed state with the `true' (maximal
knowledge) state is also lower than that of the filtered state before the jump.
However, after the jump, the fidelity of the smoothed state is higher.
