Loss-independent time-delay estimation via Hong-Ou-Mandel photon bunching detection

The gist: Using a four-detector scheme, enabling the measurement of both coincidence and photon-bunching terms, achieves the maximum precision given by the quantum Fisher information, without the need of characterizing system losses

Theoretical Framework for Parameter Estimation

Parameter estimation in quantum metrology aims at inferring an unknown physical parameter s from measurement outcomes obtained from a parameter-dependent quantum state ρs<ref:2610.22023#pg2>. A standard approach for constructing an estimator is maximum-likelihood estimation (MLE), which maximizes the likelihood function L(s) with respect to s, yielding the estimator sˆML<ref:2610.22023#pg3>. For statistically independent measurements, the precision of the estimation is fundamentally constrained by the classical Cramér–Rao bound (CRB), Var(ˆs) ≥ 1/mF(s)<ref:2610.22023#pg3>. The Fisher information (FI) quantifies the sensitivity of the probability distribution to variations of the parameter and is defined as F(s) = X x i P(xs, ξ)/∂P(xs, ξ)/∂s<ref:2610.22023#pg3>. In quantum metrology, the quantum Fisher information (QFI) is defined as the maximum classical FI attainable over all POVMs, FQ(s) = max M F(s)<ref:2610.22023#pg3>. The QCRB follows directly from the classical CRB, Var(ˆs) ≥ 1/mFQ(s)<ref:2610.22023#pg3>.

Polarization-based HOM Interference and Detection Schemes

The paper considers a polarization-based HOM interferometer where modes H (horizontal) and V (vertical) play the same role as path modes<ref:2610.22023#pg4>. Interference in the polarization degree of freedom is produced by an arrangement analogous to the BS in the traditional implementation, consisting of a polarizing beamsplitter (PBS) followed by a half-wave plate (HWP) and another PBS<ref:2610.22023#pg4>. If the photon pair is indistinguishable and the orientation of the HWP is θ = π/8, the resulting state is a N00N state in the polarization degree of freedom, with N = 2<ref:2610.22023#pg4>. The probe state of the system just after the first PBS can be written as ψ(δτ)⟩ = Z dω1dω2 f(ω1, ω2) e −i δτ 2 (ω1−ω2) × a† H(ω1) a† V(ω2)0⟩<ref:2610.22023#pg4>. The QFI for the time delay can be computed using directly the probe state described in Eq. (9), resulting in FQ = (∆ω1) 2 + (∆ω2) 2 − 2 Cov(ω1, ω2)<ref:2610.22023#pg4>. For a separable Gaussian JSA, the QFI for the time-delay estimation with a separable Gaussian JSA depends only on the coherence time of the pair source: FQ = 8 t squared c<ref:2610.22023#pg4>.

Comparison of Detection Schemes

The measurable events depend on the chosen detection scheme, and two schemes are considered:

1. Two-detector scheme (2Det): This configuration allows the measurement of coincidences (N2) and single-click events (N1), but requires accounting for nodetection events (N0), associated with lost photon pairs<ref:2610.22023#pg4>. The Fisher information for the time delay is given by F2Det(δτ) = 64 t squared c α squared (1 − γ) squared / (1 + γ) δτ tc squared e −8(δτ tc) squared h / [1 − α e −4(δτ tc) squared i h1 + 3γ + α(1 − γ)e −4(δτ tc) squared i]<ref:2610.22023#pg4>. For perfect visibility (α = 1), the maximum Fisher information is max F2Det(δτ) = lim δτ→0 F2Det(δτ) = 8 t squared c (1 − γ) squared = FQ(1 − γ) squared<ref:2610.22023#pg5>.

2. Four-detector scheme (4Det): This configuration incorporates four detectors, allowing the measurement of coincidence events (N11) and photon-bunching events (N20 and N02), restricting the complete measurement set entirely to coincidence detections, which significantly mitigates ambiguities introduced by system losses<ref:2610.22023#pg5>. The Fisher information for the 4Det system is given by F4Det(δτ, δω = 0, θ = π/8) = 64 δτ squared t 4 c e −8 δτ 2/t squared c (1 − e −8 δτ 2/t squared c)<ref:2610.22023#pg5>. Its maximum value is max F4Det(δτ) = 8 t squared c = FQ<ref:2610.22023#pg5>.

Impact of System Losses and Systematic Errors

The four-detector scheme achieves the ultimate precision under the perfect visibility condition without the need of characterizing losses<ref:2610.22023#pg5>. The two-detector scheme requires precise characterization of system efficiency, as losses introduce ambiguities between lost photon pairs and valid coincidences<ref:2610.22023#pg3>. Furthermore, systematic errors like uncertainties in interferometric visibility (α) act as nuisance parameters that introduce non-trivial correlations between successive experimental repetitions<ref:2610.22023#pg3>. When these correlations are accounted for, the precision is limited by the calibration uncertainty of the systematic parameters<ref:2610.22023#pg7>. For instance, in the 4Det scheme, after 3.7 × 103 repetitions both detection schemes converge to estimates that do not differ significantly<ref:2610.22023#pg9>. The four-detector dispersion is still statistically limited and well above its floor of 8 nm or 26 as, which is set only by the experimental determination of α<ref:2610.22023#pg9>.

Conclusion on Precision Enhancement

The four-detector scheme outperforms the two-detector scheme in the precision of time-delay estimation<ref:2610.22023#pg2>. With experimentally determined calibration uncertainties, the four-detector scheme improves the dispersion by a factor of 2 for the repetitions performed, reaching up to a factor of 4 as both schemes approach their systematic floors<ref:2610.22023#pg2>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. By considering these correlations, we show that the four-detector scheme achieves higher precision than the two-detector scheme, which is strongly limited by losses<ref:2610.22023#pg10>. The four-detector scheme requires no characterization of the system loss to perform an estimation, while the two-detector scheme needs an external calibration of the loss probability γ<ref:2610.22023#pg10>. When considering a loss-varying system, the 2Det scheme delivers estimations that are significantly different for the same displacement when fixing the value of γ; only after a new measurement of the systematic parameter γ do the estimates agree, within uncertainties, with those of the 4Det scheme<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>. This provides a practical route to loss-independent time-delay estimation and a framework for evaluating the limitations imposed by systematic parameter uncertainties in repeated quantum measurements<ref:2610.22023#pg10>. The four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, while retaining the information available per detected photon pair<ref:2610.22023#pg10>.

1. textbfProcessing for Loss-Independent Estimation Systems: Develop an estimator insensitive to losses. This allows AI systems to perform high-precision time-delay estimation without needing to characterizing system losses or its associated systematic floors, as the four-detector scheme achieves the maximum precision given by the quantum Fisher information, without the need of characterizing system losses.

2. textbfModel Correlation Limitations: Implement a framework to account for systematic correlations between repetitions. This addresses the limitation that correlations between repetitions arise due to systematic errors and ensures that as data accumulates, precision gains are correctly bounded by how these correlations affect estimation dispersion, rather than assuming statistically independent measurements.

3. textbfAdaptive Precision Scaling: Design an estimation protocol where precision scaling is governed by systematic uncertainty instead of solely by the number of repetitions. This leverages the finding that the achievable precision is determined not only by the number of repetitions, but also by the uncertainty with which the systematic parameters were measured, allowing for a practical route to loss-independent time-delay estimation.

4. textbfLoss Calibration Automation: Create an automated calibration loop for system loss probability based on measurement outcomes. This replaces manual external calibration, as the four-detector scheme does not simply improve photon detection: it removes the need to independently calibrate the system loss probability, ensuring that the two-detector estimator requires the loss probability as an external calibration parameter.