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arXiv:2601.10937 · Physical Review A

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

Nattaphong Wonglakhon, Areeya Chantasri, Howard M. Wiseman

quant-phSubmitted Jan 16, 2026• Announced Jan 19, 2026• 0 citations• Revised Oct 5, 2026 (v3)
v1 · Jan 16, 2026v2 · Feb 11, 2026v3 · Oct 5, 2026
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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 finite-interval dynamical maps for quantum trajectories, where the measurement record is available only over intervals of duration $Δt$. Guilmin \emph{et al.}\ (unpublished) derived such a map for the experimentally relevant case in which only a coarse-grained current (time-binned), denoted by $I_t$, is available. Remarkably, this coarse-grained record $I_t$ still yields a conditioned state, termed the \emph{robinet} state, whose impurity scales as small as $(Δt)^{3}$. We show, however, that the \emph{typical} trace distance between the robinet state and the fully conditioned state (obtained from the complete measurement record) scales as $(Δt)^{3/2}$. We then introduce a higher-order finite-interval dynamical map, the ``nearly-exact'' map, which requires only one additional real statistic, $φ_t$, from each interval. We analytically show that the resulting conditioned state is much closer to the fully conditioned state, with a distance scaling as $(Δt)^{2}$. We further numerically verify the distance scalings, measured relative to highly accurate approximations of the fully conditioned state, for the robinet map, our nearly-exact map, the lowest-order approximate Itô map, and two existing higher-order maps. Our results show that, for generic systems, whenever $φ_t$ can be extracted experimentally alongside $I_t$, the nearly exact map provides the most accurate state estimate among the finite-interval maps considered.

Journal reference
Phys. Rev. A 114, 042415 (2026)
DOI
10.1103/9gbb-7lk6
Comments
16 pages, 3 figures, 3 tables

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Abstract For quantum diffusive measurements, the system’s dynamics can be described by the Itô stochastic master equation, which works well for an infinitesimal time resolution. However, in practical quantum experiments, one cannot make a time step to be infinitesimal, as it can introduce correlation of noise in time, making the Markovian assumption invalid. On the other hand, increasing a time step can cause errors from non-commuting operations describing the system’s dynamics. We therefore consider implementing the Euler-Milstein and relevant approaches, namely the Itô map, the high-order completely positive map, and the quantum Bayesian, to simulate quajntum trajectory for a quantum system under diffusive continuous measurements. In particular, we numerically simulate trajectories for a qubit measurement in z basis. We show the comparison of individual trajectories and their averaged trajectories among these approaches. We find that the high-order completely positive map approach yields the most accurate averaged quantum trajectory. Furthermore, we also investigate the trace distance from true stochastic quantum trajectories, comparing the four approaches using the numerical simulation. We show that, for a realistic time resolution (as in a superconducting qubit, experiment), the high-order map does give the most accurate estimate of the qubit trajectories.
Quantum trajectories of a Markovian open quantum system arise from the back-action of measurements performed in the environment with which the system interacts. In this work, we consider counting measurements of quantum jumps, corresponding to different representations of the same quantum master equation. We derive necessary and sufficient conditions under which these different measurements give rise to the same unravelled quantum master equation, which governs the dynamics of the probability distribution over pure conditional states of the system. Since that equation uniquely determines the stochastic dynamics of a conditional state, we also obtain necessary and sufficient conditions under which different measurements result in identical quantum trajectories. We then consider the joint stochastic dynamics for the conditional state and the measurement record. We formulate this in terms of labelled quantum trajectories, and derive necessary and sufficient conditions under which different representations lead to equivalent labelled quantum trajectories, up to permutations of labels. As those conditions are generally stricter, we finish by constructing coarse-grained measurement records, such that equivalence of the corresponding partially-labelled trajectories is guaranteed by equivalence of the trajectories alone. These general results are illustrated by two examples that demonstrate permutation of labels, and equivalence of different quantum trajectories.
To compare quantum estimation theory schemes we must acknowledge that, in some cases, the quantitative difference between them might be small and hence sensitive to numerical errors. Here, we are concerned with comparing estimation schemes for the quantum state under continuous measurement (quantum trajectories), namely quantum state filtering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)], quantum state smoothing. Unfortunately, the cumulative errors in the most typical simulations of quantum trajectories with time step $\mathrm{\ensuremath{\Delta}}t$ and total simulation time TTT can scale as $T\mathrm{\ensuremath{\Delta}}t$. Moreover, these errors may correspond to deviations from valid quantum evolution as described by a completely positive map. Here we introduce a higher-order method that reduces the cumulative errors in the complete positivity of the evolution to the order of $T{(\mathrm{\ensuremath{\Delta}}t)}^{2}$, whether for linear (unnormalized) or nonlinear (normalized) quantum trajectories. Our method also guarantees that the discrepancy in the average evolution between different detection methods (different ``unravelings,'' such as quantum jumps or quantum diffusion) is similarly small. This equivalence is essential for comparing quantum state filtering to quantum state smoothing, as the latter assumes that all irreversible evolution is unraveled, although the estimator only has direct knowledge of some records. In particular, here we compare the average difference between filtering and smoothing conditioned on an event of which the estimator lacks direct knowledge: a photon detection within a certain time window. We find that the smoothed state is actually less pure, both before and after the time of the jump. Similarly, the fidelity of the smoothed state with the ``true'' (maximal knowledge) state is also lower than that of the filtered state before the jump. However, after the jump, the fidelity of the smoothed state is higher.
Real-world experiments on quantum systems are always performed with measurement apparatus whose interaction times with the systems are finite. This restricts the observable quantum states to the space of time-coarse-grained density matrices ${\overline{\ensuremath{\rho}}(t)}$, providing motivation for the time-coarse-graining (TCG) approach that solves for $\overline{\ensuremath{\rho}}(t)$ without ever referring to the unobservable $\ensuremath{\rho}(t)$ of infinite time resolution. Phenomenologically, this implies that coherent transitions far outside the bandwidth would be filtered out, leaving only their effective impacts on the ``slow'' dynamics resolvable by the finite time resolution of the measurements. Therefore, the TCG framework provides rigorous justification for many existing effective Hamiltonian methods in the literature that aim at capturing the unitary part of the long-time dynamics. However, since time-coarse-graining is fundamentally irreversible, the resulting effective model allows for secular loss of information and dissipation of energy in general, which cannot be captured by any unitary effective models and has to be treated with explicit time-coarse-graining. Such incoherent effects are particularly prominent in driven nonlinear quantum systems where exchange of information and energy with the drive gives rise to incoherent effective dynamics at all finite time resolutions in general. While rigorous in principle, existing TCG methods in the literature can be applied only to simple systems by one iteratively solving superoperator equations at low orders in the coupling strengths. The complexity of such methods prevents systematic study of the time-coarse-grained dynamics and limits analytical results to the IR (low-resolution) limit in most cases. We address these limitations in this paper, presenting a systematic time-coarse-graining method that overcomes the complexities and restrictions of current techniques, offering a comprehensive and accurate modeling framework for driven nonlinear quantum systems. We derive closed-form formulas as well as diagrammatic representations for both unitary and nonunitary contributions, in the form of an effective Hamiltonian and nonunitary dissipators at arbitrary order in the coupling strengths, and complement this with a computer-algebra software package. We demonstrate the effectiveness of the method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.