Quantum state smoothing when Alice assumes the wrong type of monitoring by Bob

TL;DR

This work probes quantum state smoothing when Alice misidentifies Bob's monitoring setup, revealing that smoothing performance can be unexpectedly robust or even superior to correct smoothing in some regimes. By modeling a coherently driven qubit with three measurement channels and analyzing 27 right/wrong combinations, the authors establish four correlation-based conjectures and validate them numerically, showing that wrong smoothing can produce negative Trace-Square Deviation or higher Fidelity than right smoothing. They derive analytical relations among smoothing powers using state deviations, and identify a correlation coefficient that explains the anomalous cases where wrong smoothing outperforms expectations. The results underscore the nuanced role of measurement backaction and correlation structure in quantum smoothing, with implications for experiments and potential extensions to counterfactual or higher-order analyses.

Abstract

An open quantum system leaks information into its environment. In some circumstances it is possible for an observer, say Alice, to recover that information, as a classical measurement record, in a variety of different ways, using different experimental setups. 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 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.

Figures (4)

• Figure 1: Two examples of individual (true, filtered, and smoothed) trajectories and their corresponding nTrSD and Fidelity to true states. Panels in the first three columns, (a)-(c) and (f)-(h), show the Bloch sphere coordinates of: true state trajectories (light grey), filtered state trajectories (solid blue), valid smoothed state trajectories (dashed red), and wrongly-assumed smoothed state trajectories (dashed black). The last column shows the nTrSD and Fidelity between Alice's estimated states (filtering, valid smoothing, and wrong smoothing) and true states, using the same colour coding.
• Figure 2: Numerical results for the negative Trace Square Deviation smoothing power (${\cal R}_{\cal S}$), defined in Eq. \ref{['eq-ATSDR']}, for all 27 cases, plotted as functions of time (units of $T_\gamma$), where the larger numbers mean better recoveries. The data of 27 cases are separated in 9 panels based on the 9 combinations of dOdV. The colour legends are used in reading the dO, dV, dW setups. The vertical dashed lines show the steady-state region where the transient behaviours from the initial and final conditions are minimal. We also mark "Conj. 1 - 4" to indicate the conjecture each curve correspond to.
• Figure 3: Numerical results for the Fidelity smoothing power (${\cal R}_{\cal F}$), as defined in Eq. \ref{['eq-AFR']}. Other details are as in Figure \ref{['fig-recovD']}. Importantly, the Purity smoothing power (${\cal R}_{\cal P}$), as it does not depend on whether the smoothing was valid or wrong, is exactly the Fidelity power and the TrSD power for the valid case, i.e., ${\cal R}_{\cal P}^{\rm dU} = {\cal R}_{\cal F}^{\rm dU = dV} = {\cal R}_{\cal S}^{\rm dU = dV}$ for a given dO, and can be read from the colored (not gray nor black) curves.
• Figure 4: Numerical results for the correlation coefficient $\alpha$ in Eq. \ref{['eq-powercorr']} or Eq. \ref{['eq-appcorr']}. showing that it is always large for all combinations of observed and unobserved measurement setups. Each panel includes two curves (grey and black) corresponding to two different types of dW for each dOdV. The dOdVdW labels are replicated from Figures \ref{['fig-recovD']} and \ref{['fig-example']} for convenience. The dotdashed grey and dashed black lines are the time-average values of the fluctuating grey and black curves, respectively. The cyan shades show the area of one standard deviation from the averages. The red lines show $\alpha = \sqrt{0.5}$ and $\alpha = 1$ so we can see that the values of $\alpha^2$ is bound in the range $[0.5, 1]$ for the great majority of the time. Note that, by definition, $\alpha$ cannot actually be greater than one so the time spent above 1 represents a statistical fluctuation arising from a finite ensemble size and similar remarks probably apply to the time spent below $\sqrt{0.5}$.