Quantum trajectories for time-binned data and their closeness to fully conditioned quantum trajectories

Quantum trajectories are dynamical equations for quantum states conditioned on the results of a time-continuous measurement, such as a continuous-in-time current ${\vec{y}}_t$. Recently there has been renewed interest in dynamical maps for quantum trajectories with time-intervals of finite size $\mathrm{\Delta }t$. Guilmin \emph{et al.} (unpublished) derived such a dynamical map for the (experimentally relevant) case where only the average current $I_t$ over each interval is available. Surprisingly, this binned data still generates a conditioned state ${\rho }_{\text{faFaucet}}$ that is almost pure (for efficient measurements), with an impurity scaling as $(\mathrm{\Delta }t{)}^3$. We show that, nevertheless, the typical distance of ${\rho }_{\text{faFaucet}}$ from ${\widehat{\psi }}_{\text{F};{\vec{y}}_t}$ – the projector for the pure state conditioned on the full current – is as large as $(\mathrm{\Delta }t{)}^{3\mathrm{/}2}$. We introduce another finite-interval dynamical map (``$\mathrm{\Phi }$-map’'), which requires only one additional real statistic, ${\phi }_t$, of the current in the interval, that gives a conditioned state ${\widehat{\psi }}_{\mathrm{\Phi }}$ which is only $(\mathrm{\Delta }t{)}^2$-distant from ${\widehat{\psi }}_{\text{F};{\vec{y}}_t}$. We numerically verify these scalings of the error (distance from the true states) for these two maps, as well as for the lowest-order (Itô) map and two other higher-order maps. Our results show that, for a generic system, if the statistic ${\phi }_t$ can be extracted from experiment along with $I_t$, then the $\mathrm{\Phi }$-map gives a smaller error than any other.