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arXiv:2505.20382

Time-averaged continuous quantum measurement

Pierre Guilmin, Pierre Rouchon, Antoine Tilloy

Published May 26, 2025• 0 citations
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Abstract

The theory of continuous quantum measurement allows to reconstruct the state $ρ_t$ of a system from a continuous stochastic measurement record $I_t$. However, this truly continuous-time signal $I_t$ is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages $I_k$ over finite intervals of duration $Δt$. In this letter, we take this digitization seriously and define $\barρ_n$ as the best Bayesian estimate of the quantum state given (only) a digitized record $(I_1,\dots,I_n)$. We show that $\barρ_{n+1}$ can be computed recursively from $I_{n+1}$ and $\barρ_n$ using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of $\sqrt{Δt}$. This allows reconstructing quantum trajectories in regimes of coarse $Δt$ where existing methods fail, estimating parameters at fixed $Δt$ without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.

DOI
10.48550/arxiv.2505.20382

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The theory of continuous quantum measurement allows to reconstruct the state ρtρ_tρt​ of a system from a continuous stochastic measurement record ItI_tIt​. However, this truly continuous-time signal ItI_tIt​ is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages IkI_kIk​ over finite intervals of duration ΔtΔtΔt. In this letter, we take this digitization seriously and define ρˉn\barρ_nρˉ​n​ as the best Bayesian estimate of the quantum state given (only) a digitized record (I1,…,In)(I_1,\dots,I_n)(I1​,…,In​). We show that ρˉn+1\barρ_{n+1}ρˉ​n+1​ can be computed recursively from In+1I_{n+1}In+1​ and ρˉn\barρ_nρˉ​n​ using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of Δt\sqrt{Δt}Δt​. This allows reconstructing quantum trajectories in regimes of coarse ΔtΔtΔt where existing methods fail, estimating parameters at fixed ΔtΔtΔt without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.
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.
Random-time sampling of a quantum system is introduced as a new approach to continuous quantum measurements with prospects for ultra-low-perturbation measurements. Random sampling is, e.g., naturally realized in an optical spin noise experiment when weak probe-laser light exhibits random single-photon events in the detector. We show that a direct evaluation of these detector time-traces yields power spectra that are equivalent to those of the usual continuous measurement regime. Surprisingly, this holds true even for average sampling rates much lower than the typical frequency range of the measured quantum dynamics. The third-order quantum polyspectrum (bispectrum) also contains the same information as its continuous counterpart. Many applications of random-time sampling are envisioned for high-resolution spectroscopy, circuit quantum electrodynamics, quantum sensing, and quantum measurements in general.
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.
Continuous measurement of quantum systems gives rise to stochastic dynamics of the conditional quantum state, described by diffusive stochastic master equations. In this paper, we study parameter estimation for such equations when the Hamiltonian and measurement operators depend on unknown parameters. Based on multiple independent observed trajectories with a known initial state, we construct a contrast function using the deterministic averaged state and define a maximum contrast estimator for the unknown parameter. We prove strong consistency and asymptotic normality of the parameter estimator in a fixed-time, many-trajectory asymptotic regime. A key point is that the covariance matrix appearing in the asymptotic normality is given in a form that naturally leads to a consistent covariance estimator. This covariance estimator is computable from the observed data together with the deterministic averaged dynamics, so the asymptotic normality result can be used to construct standard errors and assess uncertainty for the parameter estimator.
Truly continuous quantum trajectories, obtained from homodyne or heterodyne readouts, can only ever be reconstructed approximately. The continuous measurement signal, needed for exact reconstruction, is averaged over bins of finite time ΔtΔtΔt during any analog to digital conversion step. The best reconstruction possible, knowing only this discrete record, was introduced recently and dubbed the Robinet state. In this article, we show how the Robinet state can be computed with a numerical discretization scheme that is completely positive, accurate to arbitrarily high order in ΔtΔtΔt, and that does not rely on any other external solver. Our derivation relies on a dilation of the stochastic master equation into a system + transmission line setup, constructed in such a way that measuring what we call the "zero mode" of the line yields the Robinet state. We test the method on a challenging example with random Hamiltonian and jump operator, and verify its accuracy up to order 101010. Apart from its numerical interest, our approach provides a wealth of physical insights, extending in particular recent results on purity obtained by Wonglakhon, Chantasri, and Wiseman, that would be difficult to obtain in any other way.
We propose a self-contained and accessible derivation of an exact formula for the n -point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on the initial quantum state and on the Stochastic Master Equation (SME) governing the dynamics. This derivation applies to both jump and diffusive evolutions and takes into account common imperfections of realistic measurement devices. We show how these correlations can be efficiently computed numerically for commonly filtered and integrated signals available in practice.