Completely positive quantum trajectories with applications to quantum state smoothing

To compare quantum estimation theory schemes we must acknowledge that, in some cases, the quantitative difference between them might be small and hence sensitive to numerical errors. 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 time step $\Delta t$ and total simulation time $T$ can scale as $T\Delta t$. 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 the order of $T{\left(\Delta t\right)}^2$, whether for linear (unnormalized) or nonlinear (normalized) quantum trajectories. Our method also guarantees that the discrepancy in the average evolution between different detection methods (different “unravelings,” 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 unraveled, although the estimator only has direct knowledge of some records. In particular, here we compare 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 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.