Two-stage Quantum Estimation and the Asymptotics of Quantum-enhanced Transmittance Sensing

TL;DR

This work tackles the problem of estimating a scalar parameter embedded in a quantum state when the optimal measurement depends on the unknown parameter. It develops a two-stage estimation framework: a parameter-independent preliminary stage followed by a refinement stage that uses the preliminary estimate to implement a QCRB-achieving measurement, with relaxed regularity conditions to widen the class of usable estimators and explicit incorporation of nuisance parameters. The main result shows that, under these conditions, the two-stage estimator is (weakly and strongly) consistent and asymptotically normal, with the limiting Gaussian variance matching the quantum Cramér–Rao bound, even in the presence of nuisance parameters. The framework is then applied to quantum-enhanced transmittance sensing in a lossy bosonic channel with unknown phase, demonstrating strong consistency and asymptotic normality for the transmittance estimator using an optimal receiver and an MLE-based refinement, thereby achieving near-optimal precision in a practically relevant setting.

Abstract

We consider estimation of a single unknown parameter embedded in a quantum state. Quantum Cramér-Rao bound (QCRB) is the ultimate limit of the mean squared error for any unbiased estimator. While it can be achieved asymptotically for a large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. Prior work addresses this paradox using a two-stage approach: in the first stage, a preliminary estimate is obtained by applying, on a vanishing fraction of quantum state copies, a sub-optimal measurement that does not depend on the parameter of interest. In the second stage, the preliminary estimate is used to construct the QCRB-achieving measurement that is applied to the remaining quantum state copies. This is akin to two-step estimators for classical problems with nuisance parameters. Unfortunately, the original analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators for single-parameter problems at the cost of slightly weakening the asymptotic properties of the two-stage method. We also account for nuisance parameters. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.

Figures (3)

• Figure 1: Sensing of unknown transmittance $\theta$. Sensor transmits $n$-mode probes (systems $S$ of bipartite state $\hat{\rho}_{I^nS^n}$) into a lossy thermal-noise bosonic channel $\mathcal{E}^{(\bar{n}_{\rm T},\theta)}$ modeled by a beamsplitter with unknown transmittance $\theta$ mixing signal and a thermal state with mean thermal photon number $\bar{n}_{\rm T}\equiv\frac{\bar{n}_{\rm B}}{1-\theta}$. Additionally, the returned probe undergoes an unknown phase shift $\gamma$. Reference idler systems $I$ are used in the measurement of output state $\hat{\sigma}_{I^nR^n}(\theta)$, with outcomes passed to estimator $\check{\theta}$. Solid and dashed lines denote optical (quantum) and digital (classical) connections, respectively.
• Figure 2: Sensing the unknown transmittance $\theta$ using $n$ two-mode squeezed vacuum (TMSV) states $\left| \psi \right>_{IS}$. When $\left| \psi \right>_{IS}$ is transmitted, a bipartite output state $\hat{\sigma}_{IR}$ occupies a retained idler system $I$ and a corresponding returned probe system $R$. The receiver applies correction in $R$ for the channel phase shift $\gamma$ using its estimate $\check{\gamma}$ from preliminary stage (see Fig. \ref{['fig:coherent_receiver']}). It then applies a two-mode squeezer (TMS) separately to each of the $n$ output states $\hat{\sigma}_{IR}$ with parameter $\omega$ that is calculated using the preliminary estimate $\check{\theta}_{\rm p}$, followed by independent photon-number-resolving (PNR) measurement of each output mode. MLE is used on the resulting classical output to obtain the estimate $\check{\theta}_{\rm r}$. As in Fig. \ref{['fig:setup']}, $\bar{n}_{\rm T}\equiv\frac{\bar{n}_{\rm B}}{1-\theta}$, and solid and dashed lines denote optical (quantum) and digital (classical) connections, respectively.
• Figure 3: Sensing the unknown transmittance $\theta$ and phase shift $\gamma$ in the preliminary stage using $f(n)\in\omega(1)\cap o(n)$ coherent states $\left| \alpha \right>_{S}$ and a heterodyne measurement. The output state $\hat{\sigma}_{R}$ is a displaced thermal state. A pair of Gaussian random variables describe the output of heterodyne measurement weedbrook12gaussianQIrmp,guha04mastersthesis. An MLE $\check{\theta}_{\rm p}$ that uses the heterodyne receiver's output and the value of $\alpha$ as a classical reference is analyzed in Appendix \ref{['app:pre consistency']}. As in Figs. \ref{['fig:setup']} and \ref{['fig:optimal_receiver']}, $\bar{n}_{\rm T}\equiv\frac{\bar{n}_{\rm B}}{1-\theta}$, and solid and dashed lines denote optical (quantum) and digital (classical) connections, respectively.

Theorems & Definitions (10)

• Lemma 1: hayashi2005statistical
• proof
• Theorem 1
• proof
• Definition 1: Separability huber1967behavior
• Remark 1: billingsley95measure
• Lemma 2: Strong uniform law of large numbers tauchen1985diagnostic
• Lemma 3: Strong consistency of the MLE tauchen1985diagnostic
• Lemma 4
• proof : Proof (Lemma \ref{['lemma: interchange']})