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

TL;DR

This paper compares different finite-interval quantum trajectory maps, showing that including an additional current statistic improves the closeness of the conditioned state to the fully conditioned quantum trajectory, with potential experimental advantages.

Contribution

It introduces the $\

Findings

The impurity of the binned state scales as $(\Delta t)^3$.

The distance of the binned state from the full conditioned state scales as $(\Delta t)^{3/2}$.

The $\

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 Guilmin \emph{et al.} (unpublished) derived such a dynamical map for the (experimentally relevant) case where only the average current $I_t$ over each interval is available.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.