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

Abstract

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 $\hat{I}”t$. Guilmin \emph{et al.} (unpublished) derived such a dynamical map for the (experimentally relevant) case where only the average current $I_t$ over each interval is available.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 $ρ_\text{\faFaucet}$ that is almost pure (for efficient measurements), with an impurity scaling as $(\hat{I}”t{)}^3$. We show that, nevertheless, the typical distance of $ρ_\text{\faFaucet}$ from $\hat{\ddot{I}}{ˆ}_{\text{F};{\vec{y}}_t}$ -- the projector for the pure state conditioned on the full current -- is as large as $(\hat{I}”t{)}^{3/2}$. We introduce another finite-interval dynamical map (``$\hat{I}¦$-map''), which requires only one additional real statistic, $\ddot{I}{•}_t$, of the current in the interval, that gives a conditioned state $\hat{\ddot{I}}{ˆ}_{\hat{I}}¦$ which is only $(\hat{I}”t{)}^2$-distant from $\hat{\ddot{I}}{ˆ}_{\text{F};{\vec{y}}_t}$. 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 $\ddot{I}{•}_t$ can be extracted from experiment along with $I_t$, then the $\hat{I}¦$-map gives a smaller error than any other.

14 pages, 3 figures, 2 tables