Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions

Kiarn T. Laverick, Ivonne Guevara, and Howard M. Wiseman

Phys. Rev. A 104, 032213 – Published 20 September 2021; Erratum Phys. Rev. A 106, 069903 (2022)

Abstract

Quantum state smoothing is a technique to estimate an unknown true state of an open quantum system based on partial measurement information both prior and posterior to the time of interest. In this paper, we show that the smoothed quantum state is an optimal Bayesian state estimator, that is, it minimizes a Bayesian expected cost function. Specifically, we show that the smoothed quantum state is optimal with respect to two cost functions: the trace-square deviation from and the relative entropy to the unknown true state. However, when we consider a related cost function, the linear infidelity, we find, contrary to what one might expect, that the smoothed state is not optimal. For this case, we derive the optimal state estimator, which we call the lustrated smoothed state. It is a pure state, the eigenstate of the smoothed quantum state with the largest eigenvalue. We illustrate these estimates with a simple system, the driven, damped two-level atom.

Received 4 June 2021
Accepted 2 September 2021

DOI:https://doi.org/10.1103/PhysRevA.104.032213

Physics Subject Headings (PhySH)

Open quantum systems Open quantum systems & decoherence Quantum metrology Quantum stochastic processes

Quantum Information, Science & Technology

Erratum

Erratum: Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions [Phys. Rev. A 104, 032213 (2021)]

Kiarn T. Laverick, Ivonne Guevara, and Howard M. Wiseman

Phys. Rev. A 106, 069903 (2022)

Authors & Affiliations

Kiarn T. Laverick , Ivonne Guevara, and Howard M. Wiseman

Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4111, Australia

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Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
Ensemble average of the linear fidelity between the conditioned state, (a) filtered and (c) smoothed, and the true state for the quantum system of Eq. (10), where Alice and Bob are monitoring the same channel with equal measurement efficiency ( $η = 0.5$ ) using $Y$ homodyne detection and photodetection, respectively. The number of trajectories $M$ used to compute the ensemble average for the smoothed quantum state for this simulation is $2 \times 10^{4}$ . The number of trajectories $N$ used to compute the ensemble average for the linear fidelity between the true state and the smoothed quantum state is $3 \times 10^{4}$ . Note that the observed record is fixed in all cases and the ensembles of unobserved records used to generate the smoothed state and to average the linear fidelity are generated independently. (b) and (d) In order to show the equality in Eq. (7), we have plotted the difference between the purity and the expected linear fidelity for $M = 2 \times 10^{3}$ and $N = 3 \times 10^{3}$ unobserved jump trajectories (dashed line) and for $M = 2 \times 10^{4}$ and $N = 3 \times 10^{4}$ (solid line). We can see that the difference between the purity and the average fidelity decreases as the number of trajectories used to compute the averages increases, supporting the equality in Eq. (7). Here $ℏ = 1, Ω = 3 γ$ , and $η = 0.5$ .
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Figure 2
Relative entropy expected cost function for the (a) filtered state and (b) smoothed state for the quantum system described in Eq. (10). We have also computed the difference (thin black line with the scale on the right $y$ axis) between the expected cost function and the von Neumann entropy to illustrate that the equality in Eq. (12) holds, where ${log}_{2}$ has been used for the entropy in both cases. Other details, including the randomly generated observed record $\overset{\leftrightarrow}{O}$ , are as in Figs. 1 and 1.
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Figure 3
Ensemble average of the linear fidelity between the true state and the lustrated conditioned state, (a) filtered and (b) smoothed, and its difference from the maximum eigenvalue of the conditioned state (thin black line with the scale on the right $y$ axis) for the quantum system in Eq. (10) with Alice and Bob using $Y$ homodyne detection and photodetection, respectively. Other details, including the randomly generated observed record $\overset{\leftrightarrow}{O}$ , are as in Figs. 1 and 1.
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Figure 4
The (a) and (c) TrSD and (b) and (d) LI expected cost functions for the conditioned and lustrated conditioned state, respectively, for the system in Eq. (10). We evaluate the TrSD expected cost function for the (a) filtered and (b) smoothed quantum states and evaluate the LI expected cost function for the (c) filtered lustrated and (c) smoothed lustrated states. The bounds on these expected cost functions are calculated for (a) and (c) from Eq. (23) and for (b) and (d) from Eq. (24). Other details, including the randomly generated observed record $\overset{\leftrightarrow}{O}$ , are as in Figs. 1 and 1.
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