Linear Gaussian quantum state smoothing: Understanding the optimal unravelings for Alice to estimate Bob's state

Kiarn T. Laverick, Areeya Chantasri, and Howard M. Wiseman

Phys. Rev. A 103, 012213 – Published 19 January 2021

Abstract

Quantum state smoothing is a technique to construct an estimate of the quantum state at a particular time, conditioned on a measurement record from both before and after that time. The technique assumes that an observer, Alice, monitors part of the environment of a quantum system and that the remaining part of the environment, unobserved by Alice, is measured by a secondary observer, Bob, who may have a choice in how he monitors it. The effect of Bob's measurement choice on the effectiveness of Alice's smoothing has been studied in a number of recent papers. Here we expand upon the Letter which introduced linear Gaussian quantum (LGQ) state smoothing [Phys. Rev. Lett. 122, 190402 (2019)]. In the current paper we provide a more detailed derivation of the LGQ smoothing equations and address an open question about Bob's optimal measurement strategy. Specifically, we develop a simple hypothesis that allows one to approximate the optimal measurement choice for Bob given Alice's measurement choice. By “optimal choice” we mean the choice for Bob that will maximize the purity improvement of Alice's smoothed state compared to her filtered state (an estimated state based only on Alice's past measurement record). The hypothesis, that Bob should choose his measurement so that he observes the back-action on the system from Alice's measurement, seems contrary to one's intuition about quantum state smoothing. Nevertheless, we show that it works even beyond a linear Gaussian setting.

Received 31 August 2020
Accepted 23 November 2020

DOI:https://doi.org/10.1103/PhysRevA.103.012213

Physics Subject Headings (PhySH)

Open quantum systems Quantum measurements Quantum states of light Quantum stochastic processes

Quantum Information, Science & Technology

Authors & Affiliations

Kiarn T. Laverick ¹, Areeya Chantasri ^1,2, and Howard M. Wiseman¹

¹Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4111, Australia
²Optical and Quantum Physics Laboratory, Department of Physics, Faculty of Science, Mahidol University, Bangkok 10140, Thailand

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Vol. 103, Iss. 1 — January 2021

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Figure 1
A diagrammatic representation of the quantum state smoothing formalism. Bob, who has access to both the observed record $O$ and the unobserved record $U$ , is able to obtain the best estimate of the quantum state, the true state $ρ_{T} : = ρ_{\overset{\leftarrow}{O} \overset{\leftarrow}{U}}$ of the quantum system $Q$ . Alice, on the other hand, has access to only the observed record $O$ . If Alice does not know of the existence of the $U$ , then her best estimate would be the filtered estimate $ρ_{F} : = ρ_{\overset{\leftarrow}{O}}$ . However, if Alice knows the measurement setting Bob used to obtain $U$ , she can utilize the full past-future observed record to obtain the smoothed state $ρ_{S} : = ρ_{\overset{\leftrightarrow}{O}}$ , which is a more accurate estimate of Bob's true state than the filtered state.
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Figure 2
A sample realization of the OPO system's state trajectory, where $η_{o} = η_{u} = 0.5, θ_{o} = π / 4$ , and $θ_{u} = - π / 8$ , where the time $t$ is in units of $χ^{- 1}$ and the total run time $T = 4$ . We have set $ℏ = 2$ for this simulation. The evolutions in the $q$ and $p$ quadratures, in panels (a) and (b), respectively, clearly show that the smoothed mean (red) outperforms the filtered mean (blue) in terms of estimating the true mean (black). The SWV mean (green), on the other hand, does a terrible job of estimating the true mean, as expected. The disparity between the SWV state and the remaining states can clearly be seen in the phase-space diagrams, plotted at four snapshots in time in the panels (c)–(f). In (c), the filtered, smoothed, and true states all begin at the same point, with the same covariance (where the ellipse indicates the 1-SD region of the Wigner function). However, the mean of the SWV state (green dot) is largely displaced from the rest and its covariance is significantly smaller. As time progresses, the filtered, smoothed, and true states begin to separate and the covariances decrease, where the smoothed covariance sits somewhere between the filtered and true covariance. At the final time $T$ , only the true state is displaced from the remaining states, which are all the same, as there is no future record left.
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Figure 3
Contour plots of (a) the measurement overlap Eq. (65), (b) the unobserved overlap Eq. (69), (c) the observed overlap Eq. (71), and (d) the RPR for the noisy linear attenuator system in the steady state for different values of the observed (Alice) and unobserved (Bob) homodyne phases. In this example, the range of the unobserved and observed homodyne phases are $Θ_{u} = [- π / 2, π / 2)$ and $Θ_{o} = [- π / 2, π / 2)$ , respectively. Note that while (a), (b), and (c) look identical, the scales of the contours are very different due to Alice and Bob measuring different channels. In (d) we see that the RPR closely resembles the objective functions in (a)–(c), and the optimal RPR (solid white line), obtained numerically, perfectly matches the maximum of the objective functions. In all plots we consider the case where $γ_{↑} = 0.999 γ_{↓}$ . Alice perfectly measures the attenuation channel ( $η_{↓, o} = 1, θ_{↓, o} = θ_{o}$ ), and Bob perfectly measures the amplification channel ( $η_{↑, u} = 1, θ_{↓, u} = θ_{u}$ ). We have set $ℏ = 2$ .
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Figure 4
Contour plots of (a) the measurement overlap Eq. (65), (b) the unobserved overlap Eq. (69), (c) the observed overlap Eq. (71), and (d) the RPR for the on-threshold OPO in steady state for different values of the observed (Alice) and unobserved (Bob) homodyne phases. In this example, the range of the unobserved and observed homodyne phases are $Θ_{u} = [- π / 2, π / 2)$ and $Θ_{o} = [0, π)$ , respectively. In (a), we immediately see that the optimal measurement strategy according to hypothesis A (dashed black line) is very different from the optimal measurement strategy, obtained numerically, for RPR [solid black line in (d)], indicating that it is incorrect. In (b), both the solution to Eq. (70) (dashed black line) and the unobserved overlap behave very differently compared to the optimal measurement strategy and the RPR, respectively, in (d). On the contrary, in (c) the solution to Eq. (66) (dashed black line) gives a close approximation to the optimal measurement strategy. Furthermore, the square overlap has developed some of the characteristics of the RPR. In all plots, both Alice and Bob measure the same damping channel (with homodyne phases $θ_{o}$ and $θ_{u}$ , respectively) but with $η_{o} = η_{u} = 0.5$ . We have set $ℏ = 2$ .
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Figure 5
The hypothesized and optimal unobserved measurement phases (left-hand-side axis) and the RPR (right-hand-side axis) for the OPO system in the steady state with varying observed measurement efficiency $η_{o}$ ( $η_{u} = 1 - η_{o}$ ), for two fixed observed measurement phases (top: $θ_{o} = π / 8$ , bottom: $θ_{o} = 3 π / 8$ ). We consider two hypotheses of the optimal measurement strategy for Bob, hypothesis A, Eq. (66) (blue dotted line), and hypothesis C, Eq. (72) (red dashed line), comparing to the numerically obtained optimal strategy (black solid line). The results show that the strategy in Eq. (72) gives a very close approximation to the optimal RPR.
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Figure 6
Analysis of hypothesis A, B, and C and the relative average purity recovery ( $R$ ) for the example of a driven qubit coupled dissipatively to a bosonic bath. We restrict Alice and Bob to only two measurement choices, either $x$ or $y$ homodyne. The numerical values in tables (a), (b), and (c) are the objective functions for the respective hypotheses. For B and C this required stochastic simulated, and we used 3000 records each. The qubit's relative average purity recovery [Table (d)] is obtained using the numerical techniques presented in Ref. [22], simulating 3000 observed and 10 000 unobserved records for both measurement settings. Here the colored cells indicate good (green), moderate (yellow), and bad (red) improvement. Only hypothesis C [Table (c)] correctly predicts the pattern of the relative average purity recovery.
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