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 $T$ 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.
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 \ensuremathΔt and total simulation time T can scale as T\ensuremathΔt. 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(\ensuremathΔt)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.
Abstract We investigate the estimation technique called quantum state smoothing introduced in (Guevara and Wiseman 2015 Phys. Rev. Lett. 115 180407), which offers a valid quantum state estimate for a partially monitored system, conditioned on the observed record both prior and posterior to an estimation time. The technique was shown to give a better estimate of the underlying true quantum states than the usual quantum filtering approach. However, the improvement in estimation fidelity, originally examined for a resonantly driven qubit coupled to two vacuum baths, was also shown to vary depending on the types of detection used for the qubit’s fluorescence. In this work, we analyse this variation in a systematic way for the first time. We first define smoothing power using an average purity recovery and a relative average purity recovery, of smoothing over filtering. Then, we explore the power for various combinations of fluorescence detection for both observed and unobserved channels. We next propose a method to explain the variation of the smoothing power, based on multi-time correlation strength between fluorescence detection records. The method gives a prediction of smoothing power for different combinations, which is remarkably successful in comparison with numerically simulated qubit trajectories.
Theoretical tools used in processing continuous measurement records from real experiments to obtain quantum trajectories can easily lead to numerical errors due to a noninfinitesimal time resolution. In this work we propose a systematic assessment of the accuracy of a map. We perform error analyses for diffusive quantum trajectories, based on single-time-step Kraus operators proposed in the literature, and find the orders in time increments $\mathrm{\ensuremath{\Delta}}t$, to which such operators satisfy the conditions for valid average quantum evolution (completely positive, convex linear, and trace preserving), and the orders to which they match the Lindblad solutions. Given these error analyses, we propose a Kraus operator that satisfies the valid average quantum evolution conditions and agrees with the Lindblad master equation, to second order in $\mathrm{\ensuremath{\Delta}}t$, thus surpassing all other existing approaches. In order to test how well our proposed operator reproduces exact quantum trajectories, we analyze two examples of qubit measurement, where exact maps can be derived: a qubit subjected to a dispersive (z-basis) measurement and a fluorescence (dissipative) measurement. We show analytically that our proposed operator gives the smallest average trace distance to the exact quantum trajectories compared to existing approaches.
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 are dynamical equations for quantum states conditioned on the results of a time-continuous measurement, such as a continuous-in-time current yt. 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 It, is available. Remarkably, this coarse-grained record It 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, φt, 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 φt can be extracted experimentally alongside It, the nearly exact map provides the most accurate state estimate among the finite-interval maps considered.
Abstract An open quantum system leaks information into its environment. We are interested in physical situations where an observer, say Alice, can recover some of that information, as a classical measurement record. The optimal way for Alice to estimate the quantum state at time t from the record before t is known as quantum filtering. Recently, a version of quantum smoothing, in which Alice estimates the state at time t using her record on both sides of t , has been developed. It requires Alice to make optimal inferences about the pre- t record of a second observer, say Bob, who recovers whatever information Alice does not. But for Alice to make this inference, she needs to know Bob’s measurement setup. In this paper we consider what happens if Alice is mistaken in her assumption about Bob’s setup. We show that the accuracy—as measured by the trace-squared-deviation, of Alice’s estimate of the true state (i.e. the state conditioned on her and Bob’s pre- t records)—depends strongly on her setup, Bob’s actual setup, and the wrongly assumed setup. Using resonance fluorescence as a model system, we show numerically that in some cases the wrong smoothing is almost as accurate as the right smoothing, but in other cases much less accurate, even being less accurate than Alice’s filtered estimate. Curiously, in some of the latter cases the fidelity of Alice’s wrong estimate with the true state is actually higher than that of her right estimate. We explain this, and other features we observe numerically, by some simple analytical arguments.
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 Δt 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 Δt, 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 10. 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.