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Here, we are concerned with comparing estimation schemes for the quantumstate under continuous measurement (quantum trajectories), namely quantum statefiltering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)],quantum state smoothing. Unfortunately, the cumulative errors in the mosttypical simulations of quantum trajectories with a total time of simulation $T$can reach orders of $T \Delta t$. Moreover, these errors may correspond todeviations from valid quantum evolution as described by a completely positivemap. Here we introduce a higher-order method that reduces the cumulative errorsin the complete positivity of the evolution to of order $T\Delta t^2$, whetherfor linear (unnormalised) or nonlinear (normalised) quantum trajectories. Ourmethod also guarantees that the discrepancy in the average evolution betweendifferent detection methods (different `unravellings', such as quantum jumps orquantum diffusion) is similarly small. This equivalence is essential forcomparing quantum state filtering to quantum state smoothing, as the latterassumes that all irreversible evolution is unravelled, although the estimatoronly has direct knowledge of some records. In particular, here we compare, forthe first time, the average difference between filtering and smoothingconditioned on an event of which the estimator lacks direct knowledge: a photondetection within a certain time window. We find that the smoothed state isactually {\em less pure}, both before and after the time of the jump.Similarly, the fidelity of the smoothed state with the `true' (maximalknowledge) 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.

Completely Positive Quantum Trajectories with Applications to Quantum State Smoothing