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 $\Delta t$. Guilmin \emph{et al.} (unpublished) derived such a dynamical map for the (experimentally relevant) case where only the average current $I_t$ over each interval is available.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 $(\Delta t)^{3}$. We show that, nevertheless, the typical distance of $\rho_\text{\faFaucet}$ from $\hat{\psi}_{\text{F}; \vec y_t}$ -- the projector for the pure state conditioned on the full current -- is as large as $(\Delta t)^{3/2}$. We introduce another finite-interval dynamical map (``$\Phi$-map''), which requires only one additional real statistic, $\phi_t$, of the current in the interval, that gives a conditioned state $\hat{\psi}_\Phi$ which is only $(\Delta t)^{2}$-distant from $\hat{\psi}_{\text{F}; \vec y_t}$. We numerically verify these scalings of the error (distance from the true states) for these two maps, as well as for the lowest-order (It\^o) 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 $\Phi$-map gives a smaller error than any other.