Two-Stage Estimation Method

• Two-stage estimation is a framework that splits parameter estimation into a preliminary stage for initial estimations and a refined stage for optimal adjustment.
• It leverages the initial estimate to tailor a second-stage measurement, achieving improved efficiency and near-optimal performance in scenarios like quantum-enhanced transmittance sensing.
• Recent advances relax strict regularity conditions, expanding the method's applicability across quantum, econometric, and engineering applications.

A two-stage estimation method is a statistical framework that partitions the estimation of parameters into two sequential steps, often to overcome structural, computational, or information bottlenecks in a single-stage approach. These methods are especially powerful when direct estimation is encumbered by dependence on nuisance parameters, complex models, or nontrivial optimality conditions such as those encountered in quantum, econometric, or engineering settings. The essential principle is to use the outcome of a first-stage estimator—typically based on a smaller or simpler subproblem—to adapt or calibrate the (potentially optimal) estimator used in the second stage, with the goal of improving statistical efficiency, robustness, computational tractability, or controlling error probabilities.

1. Formal Framework and Methodology

The formal structure of a two-stage estimation method divides the parameter estimation process into:

1. Preliminary (first-stage) estimation: A subset of the data, or an alternative measurement or model, is used to compute an initial estimate (pre-estimator) of the parameter or nuisance quantities. This estimator is often deliberately chosen for tractability or independence from the parameter of interest, albeit at the cost of sub-optimality.
2. Refined (second-stage) estimation: The preliminary estimate is used to construct or adapt the measurement, likelihood, or model employed for the main estimation step. In classical settings, this might mean plugging in nuisance parameter estimates to form adaptive estimators or likelihoods. In quantum settings, it enables construction of an optimal measurement that depends on the unknown parameter. The estimator applied to this refined data is then shown, under suitable regularity conditions, to achieve optimal or near-optimal efficiency in large samples.

Mathematically, the procedure is often represented as follows for quantum parameter estimation:

• Let $\{\sigma(\theta)\}$ be a parametric family of quantum states, with unknown parameter $\theta$.
• Stage 1: Apply a measurement $M_1$ (independent of $\theta$) to a subset $f(n)$ of $n$ copies to obtain $\hat{\theta}_p$.
• Stage 2: Use $\hat{\theta}_p$ to configure the optimal measurement (e.g., the projector onto the spectral decomposition of the symmetric logarithmic derivative, SLD) $M_2({\hat{\theta}_p})$ for the remaining $n-f(n)$ copies, producing the refined estimate $\hat{\theta}_r$.

This structure mirrors classical two-step estimators in situations with nuisance parameters, such as profile likelihood or adaptive GMM estimators.

2. Quantum Cramér–Rao Bound and Motivation

In quantum parameter estimation, the quantum Cramér–Rao bound (QCRB) provides a lower limit for the variance of any unbiased estimator of a parameter encoded in a quantum state. The QCRB for an unbiased estimator $\hat{\theta}$ of a parameter $\theta_0$ is

$$V_{\theta_0}(\hat{\theta}) \ge \mathcal{J}_{\theta_0}^{-1}$$

where $\mathcal{J}_{\theta_0}$ is the quantum Fisher information (QFI), defined via the symmetric logarithmic derivative (SLD) $\Lambda(\theta)$ by

$$\partial_{\theta} \sigma(\theta) = \frac{1}{2} [\Lambda(\theta)\sigma(\theta) + \sigma(\theta)\Lambda(\theta)],$$

$$\mathcal{J}_{\theta_0} = \text{Tr}\left(\Lambda(\theta_0)^2 \sigma(\theta_0)\right).$$

The ultimate theoretical limit, $\mathcal{J}_{\theta_0}^{-1}$, can only be attained if the measurement is matched precisely to the true (but unknown) value of $\theta_0$. This circularity motivates two-stage methods: the first stage delivers a preliminary estimate to "tune" the measurement in the second stage so as to achieve the QCRB in the asymptotic regime.

3. Theoretical Advances and Broadening of Applicability

Earlier theoretical treatments (e.g., Hayashi, Matsumoto) imposed stringent regularity conditions—such as fast decay of large deviations probability and uniform integrability—on both the preliminary and refined estimators. These limited practical applicability, particularly for nonlinear estimators like MLEs often used on quantum measurement outcomes.

Recent advances (Gong et al., 27 Feb 2024) relax these conditions, showing that it is sufficient for the refined estimator, applied to the measurement outcomes from the SLD-eigenbasis measurement, to be consistent and asymptotically normal. This allows broad classes of estimators (including MLEs) to be encompassed in the two-stage framework with only a slightly weaker asymptotic property (e.g., the relaxation of uniform integrability requirements). As a result, the two-stage strategy is now robust to both classical and quantum estimation irregularities as long as consistency and asymptotic normality hold.

4. Practical Application: Quantum-enhanced Transmittance Sensing

A concrete application addressed in (Gong et al., 27 Feb 2024) is quantum-enhanced estimation of the transmittance $\theta$ in a bosonic channel (optical or microwave). The measurement strategy is:

• Stage 1: Probe the channel with a coherent (laser) state and perform homodyne measurement; compute the MLE $\hat{\theta}_p$ from this data.
• Stage 2: Use a quantum probe state (such as a two-mode squeezed vacuum) and construct the refined SLD-based measurement using the preliminary estimate. Measurement outcomes (from, e.g., a photon-number resolving detector) are processed with an estimator (MLE or similar) to yield $\hat{\theta}_r$.

This method attains estimation precision approaching the quantum limit dictated by the QCRB, and the analysis supports optimality under the less restrictive new regularity conditions.

Furthermore, the methods generalize to other quantum-enhanced metrological protocols (such as super-resolution imaging) and demonstrate robustness to the presence of nuisance parameters and measurement imperfections.

5. Asymptotic Properties and Statistical Inference

The main asymptotic results can be formulated as:

• As $n \to \infty$ (number of quantum state copies), and with $f(n)\in \omega(1)\cap o(n)$ (i.e., a vanishing fraction of samples used in the preliminary stage), the refined estimator $\hat{\theta}_r$ satisfies
  • Weak consistency: $\hat{\theta}_r \xrightarrow{P} \theta_0$
  • Asymptotic normality:

  $$\sqrt{n - f(n)} (\hat{\theta}_r - \theta_0) \xrightarrow{D} \mathcal{N}(0, \mathcal{J}_{\theta_0}^{-1})$$

Thus, the two-stage estimator achieves the quantum optimal mean square error in the limit, under general conditions.

The relaxation of previous regularity conditions (e.g., uniform boundedness of conditional MSE) replaces them with achievable requirements (consistency and asymptotic normality) for the estimator processing quantum measurement outcomes. This sharply increases the relevance and flexibility of the two-stage method in real laboratory or field settings, and supports the derivation of confidence intervals and error bounds via standard asymptotic statistics.

6. Analogy with Classical Two-Step Estimation and Broader Implications

The quantum two-stage method closely mirrors classical two-step estimators, particularly in the presence of nuisance parameters. In the classical setting, preliminary estimation of nuisance parameters facilitates formation of an adaptive or profile likelihood, removing the dependence of the optimal estimator on unknown or unobservable variables. The same structure applies in quantum settings, where the optimal measurement itself requires knowledge of the parameter, and is realized by a preliminary estimation step followed by an adaptive allocation of the optimal (QCRB-achieving) measurement.

This analogy highlights the foundational unity of two-stage estimation as a technique for breaking the circularity inherent in estimators whose optimal performance depends (directly or indirectly) on the very parameter being estimated.

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In sum, two-stage estimation is a general paradigm for constructing asymptotically optimal estimators when the optimal procedure cannot be implemented directly due to dependence on unknown parameters, high computational cost of joint optimization, or the need to adapt measurement strategies. Its successful application to quantum-enhanced transmittance sensing and other quantum metrology tasks, under mild asymptotic regularity conditions, showcases its central role in the modern statistical methodology for both quantum and classical inference (Gong et al., 27 Feb 2024).