Quantum Metrology under Coarse-Grained Measurement

Byeong-Yoon Go; Geunhee Gwak; Jiyong Park; Nicolas Treps; Sungho Lee; Young-Do Yoon; Young-Sik Ra

arxiv: 2601.16106 · v1 · submitted 2026-01-22 · 🪐 quant-ph

Quantum Metrology under Coarse-Grained Measurement

Byeong-Yoon Go , Geunhee Gwak , Young-Do Yoon , Sungho Lee , Nicolas Treps , Jiyong Park , Young-Sik Ra This is my paper

Pith reviewed 2026-05-16 11:50 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum metrologycoarse-grained measurementphase estimationsqueezed vacuumHeisenberg scalingFisher informationhomodyne detection

0 comments

The pith

Even two-bin measurements enable Heisenberg-limited phase estimation in interferometry

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates how restricting measurement resolution in quantum metrology affects achievable precision. It analyzes an interferometer fed with squeezed vacuum and coherent light, computing the Fisher information for homodyne detection binned at various coarse levels. The key result is that even binary outcomes support an estimator saturating the Cramér-Rao bound and scaling as the Heisenberg limit rather than the standard quantum limit. Experiments confirm a 1.2 dB quantum enhancement with two bins that grows toward 3.8 dB with more bins. This shows quantum advantage can survive severe detector limitations common in practice.

Core claim

Using a squeezed-vacuum interferometer, the analysis shows that coarse-graining homodyne detection to only two bins still allows an optimal phase estimator, found via the method of moments, that saturates the Cramér-Rao bound and delivers phase variance scaling as the inverse square of the mean photon number.

What carries the argument

Fisher information of the binned homodyne outcomes, optimized by the method of moments to saturate the Cramér-Rao bound for phase estimation

If this is right

Quantum-enhanced phase estimation remains possible even when detectors have extremely limited resolution.
The method of moments supplies a concrete calibration route to implement the optimal strategy in general lab settings.
Enhancement grows monotonically with the number of bins and approaches the ideal homodyne limit.
Coarse graining does not erase the advantage of squeezed light over classical inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same robustness may extend to other sensing tasks such as displacement or frequency estimation.
Treating detector imperfections as effective coarse graining could provide a unified way to budget experimental error sources.
Hardware simplification via lower-resolution detectors becomes a viable design choice for quantum sensors.

Load-bearing premise

An optimal estimator saturating the Cramér-Rao bound can always be found and applied via the method of moments for any level of coarse graining without extra unaccounted noise.

What would settle it

An experiment in which the measured phase variance with two-bin detection scales no better than one over the mean photon number under otherwise ideal conditions.

Figures

Figures reproduced from arXiv: 2601.16106 by Byeong-Yoon Go, Geunhee Gwak, Jiyong Park, Nicolas Treps, Sungho Lee, Young-Do Yoon, Young-Sik Ra.

**Figure 2.** Figure 2: (a) shows the FI ratio fM in Eq. (6) for the two configurations. Under the extremely coarse-grained measurement with M = 2, where the bin boundaries are given by −b1 = b3 = R and b2 = 0, the FI ratio already amounts to about 64%. The FI ratio under optimal binning (f opt M ) increases monotonically with M, as each additional bin boundary introduces an extra parameter to optimize the FI ratio. On the oth… view at source ↗

**Figure 3.** Figure 3: FIG. 3. Examples of optimal weights for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Experimental setup for measuring the interferometric phase [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Experimental determination of the optimal weights and the associated calibration functions. (a) Equal-sized binning and (b) optimal [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Phase estimation error [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Phase estimation error [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

read the original abstract

While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cram\'{e}r-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two-bin coarse graining still yields Heisenberg scaling and 1.2 dB quantum enhancement in this squeezed-vacuum interferometer, with theory and experiment both shown.

read the letter

The core result is that even two-bin homodyne detection preserves quantum enhancement and Heisenberg scaling in phase estimation for a squeezed-vacuum plus coherent-state interferometer. They back this with Fisher-information analysis that identifies optimal estimators for each coarse-graining level and an experiment that measures 1.2 dB improvement over the classical case with two bins, rising to 3.8 dB as bin count increases. The method-of-moments approach plus calibration steps is presented as a practical way to extract the bound-saturating estimator in real setups. That combination of calculation and lab data is the paper's main strength; it directly tackles a common hardware limitation rather than assuming ideal detectors. The scaling claim is the part that needs the closest look. The stress-test concern about p(φ) saturating near 0 or 1 for large N is reasonable on paper, since fixed bin edges could make dp/dφ grow only linearly while variance stays constant. The abstract states the Fisher information still scales as N², so the authors must be using the squeezing to keep the quadrature distribution from collapsing too quickly or choosing bin placement that avoids the flat region. Without the explicit F versus N curves or the exact bin-boundary strategy in the main text, it is hard to judge how robust that scaling remains once photon number grows. The experimental numbers are given without detailed error budgets or raw histograms, which leaves some room for calibration drift or unmodeled noise. Still, the work is for people building quantum sensors who face imperfect readout hardware. It gives a concrete path to keep quantum advantage when detectors are coarse. The combination of analysis and demonstration is solid enough to send out for refereeing; the central claim is testable and the practical angle is useful even if the scaling needs tighter verification in review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that coarse-grained homodyne detection in a Mach-Zehnder interferometer with squeezed-vacuum plus coherent-state input permits phase estimation beyond the SQL even when the measurement is reduced to only two bins, with the Fisher information still scaling as N² (Heisenberg limit). Theoretical analysis evaluates the Fisher information for different bin numbers, identifies optimal estimation strategies that saturate the Cramér-Rao bound, and employs the method of moments together with calibration procedures. Experimentally, a 1.2 dB quantum enhancement is reported for two bins relative to the classical ideal-measurement case, rising to 3.8 dB as the number of bins increases.

Significance. If the central scaling result holds, the work demonstrates that quantum-metrology advantages remain accessible under severe, realistic measurement imperfections that are ubiquitous in laboratory settings. The provision of explicit calibration procedures for the method of moments and the experimental verification of CR-bound saturation supply a concrete, implementable route to quantum-enhanced sensing without requiring ideal detectors.

major comments (2)

[§3] §3 (Fisher-information analysis): the assertion that two-bin coarse graining preserves F ∝ N² must be accompanied by an explicit asymptotic derivation showing that dp/dφ continues to grow linearly with amplitude while p(1-p) does not collapse to zero for large N; the present text leaves open whether fixed bin boundaries cause the distribution to saturate and revert to SQL scaling.
[Experimental results] Experimental results (Figs. 4–5 and associated text): the reported 1.2 dB enhancement and CR-bound saturation via the method of moments require a quantitative error budget that accounts for possible mismatch between the assumed p(φ) model and the actual coarse-grained histogram; without this, saturation cannot be confirmed.

minor comments (2)

[§4] Clarify the precise location of the two bin boundaries (e.g., relative to the quadrature zero) and whether they are fixed or adaptively chosen with N.
[Discussion] Add a short paragraph comparing the obtained scaling with the ideal (uncoarse-grained) Heisenberg limit to quantify the degradation factor introduced by binning.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive evaluation of the work. We address each major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: §3 (Fisher-information analysis): the assertion that two-bin coarse graining preserves F ∝ N² must be accompanied by an explicit asymptotic derivation showing that dp/dφ continues to grow linearly with amplitude while p(1-p) does not collapse to zero for large N; the present text leaves open whether fixed bin boundaries cause the distribution to saturate and revert to SQL scaling.

Authors: We agree that an explicit asymptotic derivation is required for rigor. In the revised §3 we will insert a dedicated subsection deriving the large-N scaling: with bin boundaries chosen to track the mean shift of the quadrature distribution (which moves linearly with the phase amplitude for the squeezed input), dp/dφ scales as √N (from the squeezed variance) times the amplitude factor, yielding overall linear growth in N after normalization. The binomial factor p(1-p) is shown to approach a nonzero constant (approximately 1/4 at the optimal operating point) rather than vanishing, because the distribution width remains finite relative to the fixed bin separation. Fixed (non-adaptive) boundaries are explicitly contrasted and shown to produce saturation to SQL scaling, confirming that our optimized binning is essential. The full algebraic steps and limiting expressions will be provided. revision: yes
Referee: Experimental results (Figs. 4–5 and associated text): the reported 1.2 dB enhancement and CR-bound saturation via the method of moments require a quantitative error budget that accounts for possible mismatch between the assumed p(φ) model and the actual coarse-grained histogram; without this, saturation cannot be confirmed.

Authors: We accept that a quantitative error budget is needed to substantiate CR-bound saturation. In the revised manuscript we will add a new subsection (or appendix) that (i) overlays the measured two-bin histograms against the theoretical p(φ) model for the full range of phases, (ii) computes the pointwise residual and its contribution to the Fisher-information uncertainty via Monte-Carlo propagation, and (iii) reports the resulting error bars on the extracted 1.2 dB enhancement together with a statement of the maximum model mismatch tolerated before the observed saturation would be compromised. This analysis will be performed for both the two-bin and higher-bin data sets. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of Heisenberg scaling under coarse-graining

full rationale

The paper computes Fisher information directly from the theoretical probability p(φ) for the squeezed-vacuum + coherent-state input under fixed bin boundaries, using standard quantum-optics expressions for the quadrature distribution. The claim that F scales as N² even for two bins follows from this explicit model rather than from any fitted parameter or self-citation. The method-of-moments estimator is shown to saturate the Cramér-Rao bound by construction from the same model probabilities, without circular re-use of data-derived quantities. No load-bearing self-citation, ansatz smuggling, or renaming of known results is present; the derivation remains self-contained against external quantum-metrology benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard quantum optics and information theory with no explicit free parameters, invented entities, or ad-hoc axioms mentioned.

axioms (1)

standard math Standard quantum mechanics and Fisher information theory apply directly to coarse-grained homodyne detection outcomes.
Invoked to evaluate Fisher information and determine optimal estimation strategies that saturate the Cramér-Rao bound.

pith-pipeline@v0.9.0 · 5540 in / 1226 out tokens · 71834 ms · 2026-05-16T11:50:20.518354+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

FI for coarse-grained measurement is given by F_M(φ)=∑_{k=1}^M (1/P_k) (∂P_k/∂φ)^2 ... f_M = F_M(0)/F_id(0) ... even with M=2 ... Heisenberg scaling
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

optimal weight w_φ = Γ^+_φ ∂⟨o⟩_φ/∂φ ... saturates the CRB

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[1] [1]

We consider two representative coarse-grained measurement configurations for a bin number M: equal-sized bins and optimized bins

This ratio quan- tifies the fraction of the ideal FI that remains accessible to coarse-grained measurement. We consider two representative coarse-grained measurement configurations for a bin number M: equal-sized bins and optimized bins. In the former, all Mbins have an equal size (i.e.b k+1 −b k =2R/M∀k). In the latter, the bin sizes are optimized to max...

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[2] [2]

Giovannetti, S

V . Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit, Science 306, 1330 (2004)

work page 2004

[3] [3]

Giovannetti, S

V . Giovannetti, S. Lloyd, and L. Maccone, Advances in quan- tum metrology, Nat. Photon.5, 222 (2011)

work page 2011

[4] [4]

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, 30 years of squeezed light generation, Phys. Scr.91, 053001 (2016)

work page 2016

[5] [5]

C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D23, 1693 (1981)

work page 1981

[6] [6]

J. A. H. Nielsen, J. S. Neergaard-Nielsen, T. Gehring, and U. L. Andersen, Deterministic quantum phase estimation be- yond N00N states, Phys. Rev. Lett.130, 123603 (2023)

work page 2023

[7] [7]

Pezz ´e and A

L. Pezz ´e and A. Smerzi, Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light, Phys. Rev. Lett.100, 073601 (2008). 9 M w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 2 0.707−0.707 3 0.707 0−0.707 4 0.676 0.206−0.206−0.676 5 0.637 0.307 0−0.307−0.637 6 0.601 0.354 0.116−0.116−0.354−0.601 7 0.569 0.376 0.186 0−0.186−0.376−0.569 8 0.5...

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