Bayesian quantum estimation of the separation of two incoherent point sources

Boyu Zhou; Christos N. Gagatsos; Saikat Guha

REVIEW 3 major objections 5 minor 31 references

Bayesian quantum estimation of the separation of two incoherent point sources

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Under a Bayesian prior, direct imaging can outperform SPADE for mid-range source separations, contrary to the Fisherian result.

desk verdict Plausible and novel Bayesian take on superresolution, but the only closed-form check is wrong as printed and the headline numerics are unreproducible; worthy of review but not citable as-is. read the letter →

arxiv 2412.05245 v1 pith:ZWZOVK4F submitted 2024-12-06 quant-ph

Boyu Zhou , Saikat Guha , Christos N. Gagatsos This is my paper · ORCID

classification quant-ph MSC 81P5062F15

keywords BayesianquantumestimationtwoincoherentpointsourcesseparationSPADEdirectimagingminimummeansquareerrorhalf-Gaussianpriorsuperresolution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what happens when a prior probability distribution on the separation of two incoherent point sources is available, and evaluates measurements by the Bayesian minimum mean square error instead of the Fisherian Cramér–Rao bound. It finds that with a displaced half-Gaussian prior, SPADE remains nearly optimal for small likely separations, but in an intermediate range of the prior mean, direct imaging (DI) actually outperforms SPADE. The regime dependence matters because real telescopes and microscopes often have prior knowledge about the objects they image, so the best measurement strategy can depend on that prior.

What carries the argument

The argument rests on two pieces. First, the Bayesian MMSE formalism expresses the optimal single-shot mean squared error as $\delta = \mathrm{tr}(\hat{\Gamma}_2 - \hat{B}\hat{\Gamma}_1)$ with $\hat{\Gamma}_k = \int_0^\infty P(\alpha) \alpha^k \hat{\rho}(\alpha) \, d\alpha$ and $\hat{B}$ the optimal projector. Second, through the Gaussian point-spread function the imaging problem is mapped to a single-mode coherent-state space: the Hermite–Gauss modes in the image plane become Fock states in the hypothetical space, so that photon-number-resolving detection corresponds to SPADE and homodyne detection to direct imaging. This mapping lets the authors compute the MMSE and compare it with the two measurement strategies. For the half-Gaussian prior, $\hat{\Gamma}_0$ becomes a squeezed thermal state, which enables an analytic treatment of the optimal measurement.

What would settle it

A concrete experiment: prepare two incoherent point sources separated by a value drawn from a displaced half-Gaussian prior with mean $\mu_t = 2$ and variance $\sigma_t^2 \approx 1$, then compare the mean squared error of SPADE and direct imaging over many trials; if direct imaging does not beat SPADE in the middle range of the prior, the paper's central regime claim is falsified.

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Extended reading notes

Core claim

The paper's central claim is that SPADE is not the optimal measurement in general for estimating the separation of two incoherent point sources when a prior on the separation is available. Working with the Bayesian minimum mean square error, the authors show that for a displaced half-Gaussian prior, SPADE attains near-optimal performance when the prior mean is small (sources expected close together) and when the prior mean is large both SPADE and direct imaging converge to the fundamental MMSE. In the middle range of the prior mean, however, direct imaging produces a smaller mean squared error than SPADE. This contrasts with the Fisherian approach, in which SPADE achieves the quantum Cramér–Rao bound regardless of separation.

Load-bearing premise

The whole comparison assumes that the Gaussian point-spread function and paraxial approximation make the mapping from imaging space to a single-mode coherent-state space exact, so that PNR detection in the hypothetical space truly equals SPADE and homodyne detection truly equals direct imaging; if the point-spread function is not Gaussian, the quantitative crossover between DI and SPADE may not transfer.

Editorial extensions

If this is right

For small likely separations, SPADE remains the measurement of choice, approaching the Bayesian MMSE.
For intermediate prior knowledge, direct imaging can give lower mean squared error than SPADE, so measurement choice should be adapted to the prior.
For large likely separations, both measurements approach the MMSE and the Rayleigh limit becomes irrelevant.
The Bayesian MMSE gives a guaranteed single-shot attainable bound, unlike the Fisherian Cramér–Rao bound which is asymptotic.
The result suggests that prior information can be exploited to relax the requirement of exotic measurements in some regimes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The qualitative regime dependence (DI beating SPADE in a middle band) likely extends to non-Gaussian point-spread functions, but the boundaries of the band would shift; a robustness check with a different PSF would be a natural next step.
In adaptive settings, a receiver could first perform a coarse direct-imaging measurement to localize the prior, then switch to SPADE; the paper's MMSE comparison suggests such hybrid strategies could beat either fixed measurement.
For astronomical applications, the prior could come from a source catalog or magnitude distribution; the paper's framework suggests that the optimal instrument design depends on that catalog.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: this paper adds a genuinely new Bayesian layer to the SPADE-versus-direct-imaging question. Personick's MMSE framework applied to two-point separation with half-Gaussian priors hasn't been done before, and the observation that the optimal measurement is prior-dependent is worth taking seriously. The single-source appendix is clean and the imaging-space mapping in Sec. V C is clearly explained.

The problem is Eq. (29). As typeset it gives a negative MSE for σ beyond about 0.76. Substituting σ=1 gives roughly -0.61, which is not an estimator error. The correct expression, from a direct PNR calculation using Eqs. (26)-(28), is σ² - (2σ²)/(π√(σ²+1))[1 + σ arcsin(σ/√(σ²+1))]. The printed version has √(σ²+1) in the numerator instead of the denominator. That matters because Eq. (29) is the only closed-form check in the paper; Fig. 1 is built on it.

The displaced-prior results in Figs. 2 and 3, which carry the central DI-over-SPADE claim, come from a truncated-Fock numerical calculation with no code and no convergence data. The stated cutoff (35) is not sufficient. I don't think the misplaced square root directly invalidates the displaced-prior crossover—those curves are computed independently—but it makes the pipeline look unsafe, and the headline claim is exactly the kind of quantitative statement that needs a reproducible numerics package or an independent check.

The paper is otherwise fairly careful: the Personick/Helstrom formalism is applied correctly, the squeezed-thermal structure of Γ0 is a nice touch, and the authors do not oversell their conclusion—they explicitly note SPADE is not optimal in general. The literature coverage is adequate.

Bottom line: this is a legitimate paper for the quantum-imaging audience, and it deserves a serious referee. But it should not be accepted until Eq. (29) is fixed and the displaced-prior numerics are released or independently reproduced. I would not cite it in its current form.

Referee Report

3 major / 5 minor

Summary. The paper addresses Bayesian estimation of the separation of two incoherent point sources with a Gaussian PSF. For a half-Gaussian prior and a displaced half-Gaussian prior on the separation parameter, it compares the Bayesian MMSE with the MSEs of SPADE (modeled as photon-number-resolving detection in a single-mode coherent-state space) and direct imaging (modeled as homodyne detection). The non-displaced case is treated analytically through a squeezed-thermal-state representation of the operator Γ0 and a closed-form PNR MSE, while the displaced case is treated numerically by truncating a Fock expansion at up to 35 photons. The main conclusion is that SPADE is not universally optimal under a Bayesian prior: for intermediate prior means, direct imaging can outperform SPADE, in contrast to the Fisherian result.

Significance. If the numerical results are correct, the paper provides a useful Bayesian extension of superresolution imaging, showing that the choice between SPADE and direct imaging depends on the prior. The mapping between the hypothetical single-mode coherent-state space and the imaging space (Table I) is clearly laid out, and the squeezed-thermal-state treatment of the non-displaced prior is elegant. The paper also gives a closed-form expression for the photon-number-resolving MSE, which can serve as a benchmark. However, the paper does not ship code or convergence data for its central numerical claims, and there are demonstrable errors in two analytic statements that support those claims. The significance is therefore conditional on correction and independent verification.

major comments (3)

[Sec. V A, Eq. (29)] The formula as typeset is incorrect: the second term has √(σ²+1) in the numerator rather than the denominator. For σ=1, the printed expression gives 1 − (2/π)√2(π/4+1) ≈ −0.607, and it is negative for all σ ≳ 0.76, which is impossible for an MSE. Repeating the Gaussian integral from Eqs. (26)–(28) gives σ² − (2σ²/[π√(σ²+1)])[σ arcsin(σ/√(σ²+1)) + 1]. Because this equation is the analytic anchor of the SPADE curve in Fig. 1, the equation must be corrected and Fig. 1 regenerated or explicitly confirmed to have used the corrected formula.
[Sec. VI, after Eq. (37)] The statement "We can easily simplify the equation and find that trΓ₂=σ²" is incorrect for the displaced half-Gaussian prior. The trace of Γ₂ equals the second moment of the prior, ∫ q² P(q)dq = μ_t² + σ_t² (with μ_t and σ_t² as defined in Eqs. (35)–(36)), not the scale parameter σ². For example, with μ=1 and σ=1, the second moment is approximately 2.86, not 1. This error is load-bearing: δ = trΓ₂ − tr(BΓ₁), so if the numerical MMSE calculations use trΓ₂=σ², the MMSE curves in Figs. 2 and 3 are shifted. Please correct this claim and specify how trΓ₂ was actually evaluated numerically.
[Sec. VI, Figs. 2 and 3] The central claim that DI outperforms SPADE in the middle range of μ_t rests entirely on numerical evaluations of Eqs. (26), (30), and (37), but the paper provides no code, no pseudocode, no quadrature details for the homodyne integrals, and no convergence study beyond the statement that the maximum truncated number is 35 for Fig. 2. The demonstrable typo in Eq. (29) weakens confidence that the numerics use the correct analytic anchors. The authors should provide the code or a step-by-step numerical recipe with tolerances and convergence data for both Figs. 2 and 3.

minor comments (5)

[Abstract and Sec. II] There are typos: "non-dispalced" in the abstract and "assumtions" in Sec. II; also "conjuction" appears in Sec. II.
[Eq. (17)] The second integral in Eq. (17) is over qα from −∞ to 0, while the text states "we always set α = qα/√2 ≥ 0". Please clarify the notation by defining α=|qα|/√2 after the change of variables.
[Sec. V A] The derivation of Eq. (29) from Eqs. (26)–(28) is not shown; please include the intermediate integral steps in an appendix, especially since this closed form is used to validate Fig. 1.
[Sec. VI] The paper states the maximum truncated number is 35 for Fig. 2 but gives no cutoff or convergence information for Fig. 3 or for the homodyne quadrature integration in Eq. (30). Please state the numerical parameters used.
[Sec. VII] The sentence "SPADE has advantage when the variance is larger than 1.1" is difficult to reconcile with Figs. 2(c)–(d) and with the following sentence about increasing the variance; please rephrase the non-monotonic behavior more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MMSE and measurement MSEs are computed from external quantum estimation formulas with prior parameters as inputs.

full rationale

The derivation chain is self-contained rather than circular. The Bayesian MMSE is computed from external results of Personick (Ref. 20) and Helstrom (Ref. 27), and the per-measurement MSEs follow from Born's rule with the prior PDF treated as an input. No parameter is fitted to the target quantity and then renamed as a prediction. The mapping between the hypothetical coherent-state space and the imaging space (SPADE/PNR and DI/homodyne) is an explicitly stated unitary/basis correspondence under the Gaussian-PSF assumption, not a definition that forces the comparative conclusion. Self-citations appear only as background or future-directions references and do not carry the load-bearing argument. The possible typo or numerical issue in Eq. (29) noted in reviewer skepticism is a correctness and reproducibility concern, not a circularity of the kind where an output equals an input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard quantum estimation theory plus a specific source and PSF model. The only hand-chosen inputs are the prior parameters, which are used to map regimes rather than to fit a target result. No new entities are introduced.

free parameters (2)

prior width sigma = varied (Figs. 1-3)
Width of the half-Gaussian prior on q_alpha. It is an input to the problem, but the qualitative results depend on it and it is chosen by hand for the plots.
prior mean mu (displaced half-Gaussian) = varied (Figs. 2-3)
Location parameter of the displaced half-Gaussian prior. Together with sigma it fixes the prior mean mu_t and variance sigma_t^2, and the superiority regimes are mapped against these values.

assumptions (5)

domain assumption The two-source single-photon state is rho = (1/2)|alpha><alpha| + (1/2)|-alpha><-alpha| (Eq. 9), assuming weak sources, incoherence, known centroid, and equal brightness.
This is the standard model from Tsang et al. [3] and Bisketzi et al. [6]; it limits the results to Gaussian PSF, paraxial, single-photon regime.
standard math Personick's MMSE formula (Eqs. 12-14) gives the minimum achievable mean square error over all projective measurements.
Used without proof, cited to [20]; it is the foundation of all MMSE computations.
standard math Squeezed displaced Fock-basis projection formula (Eq. 24, from [28]) is correct.
The key matrix element that produces the non-displaced MMSE depends on this external formula; a sign or normalization error there would change the curves.
domain assumption The prior PDF on q_alpha is exactly half-Gaussian (Eq. 15) or displaced half-Gaussian (Eq. 33), with the physical separation linearly related to q_alpha.
All results are conditional on this prior; if the true prior is different, the numerical comparisons do not apply.
ad hoc to paper Truncating the Fock expansion at 35 modes gives converged MMSE in the displaced case (Sec. VI).
No convergence test is shown; the cutoff is stated for the example but not justified.

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Cite this review

Pith. "Pith review of Bayesian quantum estimation of the separation of two incoherent point sources." pith.science (2026). https://pith.science/paper/ZWZOVK4F

@misc{pith2026241205245,
  author       = {Pith},
  title        = {Pith review of: Bayesian quantum estimation of the separation of two incoherent point sources},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWZOVK4F}},
  note         = {Machine review of arXiv:2412.05245}
}

read the original abstract

We address the estimation problem of the separation of two arbitrarily close incoherent point sources from the quantum Bayesian point of view, i.e., when a prior probability distribution function (PDF) on the separation is available. For the non-dispalced and displaced half-Gaussian prior PDF, we compare the performance of SPADE and direct imaging (DI) with the Bayesian minimum mean square error and by varying the prior PDF's parameters we discuss the regimes of superiority of either SPADE or DI.

Figures

Figures reproduced from arXiv: 2412.05245 by the authors.

**Figure 2.** FIG. 2. The dependence of the MMSE, MSE for SPADE, and MSE for direct imaging (DI) on the variance [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. The dependence of the MMSE, MSE for SPADE, and MSE for direct imaging (DI) on the mean [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

For convenience, we change again the estimation parameter to qα = √ 2α

In our case α ≥ 0, therefore we set pα = 0. For convenience, we change again the estimation parameter to qα = √ 2α. Now, the task is to estimate the real and non- negative parameter qα, encoded in ˆρ, when the prior probability distribution P (qα) is the half-Gaussian distribution given by, P (qα) = 2 σ √ 2π e− q2 α 2σ2 , qα ≥ 0, (15) where σ is known. We...

work page

[2] [2]

Max Born and Emil Wolf, Principles of optics: elec- tromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013)

work page 2013

[3] [3]

Xxxi. investigations in optics, with spe- cial reference to the spectroscope,

Rayleigh, “Xxxi. investigations in optics, with spe- cial reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879)

work page

[4] [4]

Quantum theory of superresolution for two incoher- ent optical point sources,

Mankei Tsang, Ranjith Nair, and Xiao-Ming Lu, “Quantum theory of superresolution for two incoher- ent optical point sources,” Phys. Rev. X 6, 031033 (2016)

work page 2016

[5] [5]

Fundamental limit of resolving two point sources limited by an arbitrary point spread function,

Ronan Kerviche, Saikat Guha, and Amit Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in 2017 IEEE International Symposium on Information Theory (ISIT) (2017) pp. 441–445. 11

work page 2017

[6] [6]

Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: Semiclas- sical treatment,

Mankei Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: Semiclas- sical treatment,” Phys. Rev. A 97, 023830 (2018)

work page 2018

[7] [7]

Quantum limits of localisation microscopy,

Evangelia Bisketzi, Dominic Branford, and Animesh Datta, “Quantum limits of localisation microscopy,” New Journal of Physics 21, 123032 (2019)

work page 2019

[8] [8]

Attaining the quantum limit of su- perresolution in imaging an object’s length via pre- detection spatial-mode sorting,

Zachary Dutton, Ronan Kerviche, Amit Ashok, and Saikat Guha, “Attaining the quantum limit of su- perresolution in imaging an object’s length via pre- detection spatial-mode sorting,” Phys. Rev. A 99, 033847 (2019)

work page 2019