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

TL;DR

The paper addresses quantum trajectories under time-binned measurements, showing that conditioning only on $I_t$ yields typical errors of $O((\Delta t)^{3/2})$ relative to fully conditioned states. It introduces the $\Phi$-map, which uses an additional statistic $\phi_t$ alongside $I_t$ to achieve $O((\Delta t)^2)$ accuracy, and provides a rigorous Kraus-operator and superoperator formulation to compare with existing maps. Analytic results and numerical simulations across various measurement schemes demonstrate that the $\Phi$-map generally offers the smallest trajectory error, except in special cases such as QND with normal $\hat{c}$ where robinet can be exact. The findings suggest practical routes to improved finite-interval quantum trajectory simulations by experimentally accessing the dual record $\{I_t,\phi_t\}$, or by dual-record simulation when only $I_t$ is available. The work advances understanding of information loss in coarse-grained quantum measurements and provides a concrete, higher-accuracy framework for quantum state estimation under finite-time resolution.

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 $Δt$. Guilmin \emph{et al.} (unpublished) derived such a dynamical map for the (experimentally relevant) case where only the average current $I_t$ over each interval is available.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 $(Δt)^{3}$. We show that, nevertheless, the typical distance of $ρ_\text{\faFaucet}$ from $\hatψ_{\text{F}; \vec y_t}$ -- the projector for the pure state conditioned on the full current -- is as large as $(Δt)^{3/2}$. We introduce another finite-interval dynamical map (``$Φ$-map''), which requires only one additional real statistic, $φ_t$, of the current in the interval, that gives a conditioned state $\hatψ_Φ$ which is only $(Δt)^{2}$-distant from $\hatψ_{\text{F}; \vec y_t}$. 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 $φ_t$ can be extracted from experiment along with $I_t$, then the $Φ$-map gives a smaller error than any other.

Figures (3)

• Figure 1: Bloch representation of pure-state estimates conditioned on different types of records. The hypothetical true state, $\hat{\psi}_{\text{F}; \vec{y}_t}^\text{T}$, is represented by the fixed black arrows. (a) Quantum states conditioned only on $I_t$. The possible states conditioned on $I_t$, $\hat{\psi}_{\text{F}; \vec{y}_t|I_t}$, are indicated by the light-green shaded area. The robinet state $\rho_{\text{\faFaucet}}$, represented by the green arrow, typically lies at a distance of order $(\Delta t)^{3/2}$ from $\hat{\psi}_{\text{F}; \vec{y}_t}^\text{T}$. The crudest estimate, $\hat{\psi}_\text{I}$ (Itô), is denoted by the blue arrow with a distance of order $\Delta t$. (b) Quantum states conditioned on both $I_t$ and $\phi_t$. The possible states conditioned on $\{I_t, \phi_t\}$, $\hat{\psi}_{\text{F}; \vec{y}_t|I_t,\phi_t}$, are indicated by the light-magenta shaded area. The nearly exact state $\hat{\psi}_\Phi$ (magenta arrow) achieves a typical distance of order $(\Delta t)^2$ from $\hat{\psi}_{\text{F}; \vec{y}_t}^\text{T}$.
• Figure 2: True and finite-time trajectories: This example depicts the qubit $z$-measurement with $\hat{c}=\sqrt{\gamma/2}{\hat{\sigma}}_z$. The true record ($y_\tau$) and its corresponding trajectory ($z_\text{T}(t_j)=\text{Tr}[\hat{\sigma}_z\hat{\psi}_\text{T}(t_j)]$) are shown in the blue curves, while the coarse-grained record ($I_t$) and its corresponding trajectory, $z_\text{R}(t_j)$, are denoted by the orange curves. The trajectories are simulated using $\hat{M}_\text{R}(I_{t_j})$. The green curve indicates the trace distance of the trajectory from the coarse-grained record comparing with the true one, $D_{\text{R};k}(t_j)$. Parameters: $\eta=1,$ and $\gamma =1$.
• Figure 3: Time-average trace-squared errors: the histograms show $\sigma_{\text{A};k}^2$ computed via Eq. \ref{['TSE1']} (log-scaled) of individual trajectories, $|{\psi_\text{A}}\rangle$, generated by each method comparing with the true state, $|{\psi_\text{T}}\rangle$, of five special quantum measurement examples. The MTrSEs ($\bar{\sigma}^2_\text{A}$) and MTrAEs ($\bar{D}_\text{A}$) for method A are reported in the plots. The subscript texts are A $\in\{\text{I, R, W, \faFaucet}, \Phi\}$, used to indicate the numerical results from the maps: $\hat{M}_\text{I}$, $\hat{M}_\text{R}$, $\hat{M}_\text{W}$, $\hat{M}_\text{\faFaucet}$ and $\hat{M}_\Phi$ ($\Phi$), respectively. Parameters: $\hat{H}=0,$ and $\gamma=1$ for all cases.