pith. sign in

arxiv: 2510.04818 · v2 · pith:ZBH5EB2Ynew · submitted 2025-10-06 · 🪐 quant-ph

Super-resolution of partially coherent bosonic sources

Joaqu\'in L\'opez-Su\'arez , Michalis Skotiniotis This is my paper

Pith reviewed 2026-05-21 21:21 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum super-resolutionpartially coherent sourcesbosonic imagingquantum Fisher informationnuisance parametersBloch vectorsub-Rayleigh regime
0
0 comments X

The pith

Super-resolution for estimating the separation of two partially coherent bosonic sources remains possible even with nuisance parameters present, but only for balanced sources, and persists more broadly when some parameters are known.

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

The paper derives the ultimate quantum limits for jointly estimating the separation, location, relative intensity, and coherence factor of two partially coherent sources. It establishes that super-resolution, meaning quantum precision better than the classical Rayleigh limit for separation, holds in the presence of unknown nuisance parameters only when the sources have equal intensity. When some parameters are assumed known, super-resolution applies across a wider range of intensity ratios and fails only when the sources are perfectly correlated with coherence factor equal to one. The precision is set primarily by the photon statistics that arise from interference between the sources and varies strongly with the coherence degree. In the sub-Rayleigh regime the problem maps to the state of an effective two-level quantum system whose Bloch vector encodes every unknown parameter.

Core claim

Super-resolution in the separation is achievable both in the presence of nuisance parameters as well as when some of the parameters are assumed known. Nuisance parameters restrict super-resolution to balanced sources, whereas for known parameters super-resolution persists over a broader range of relative intensities and is lost only for perfectly correlated sources, i.e., γ=1. The achievable precision is governed primarily by interference-induced photon statistics and depends strongly on the degree of coherence. In the sub-Rayleigh regime, the imaging problem reduces to an effective two-dimensional Hilbert space description, provided a consistent reference position is used, with all params.

What carries the argument

The effective two-dimensional Hilbert space description of the sub-Rayleigh regime, in which a consistent reference position allows all unknown parameters to be encoded in the Bloch vector of the image-plane state.

If this is right

  • Super-resolution in separation estimation holds with unknown nuisance parameters only for sources of equal intensity.
  • When parameters are known, super-resolution holds for a wider range of intensity ratios and vanishes only at perfect correlation γ=1.
  • Estimation precision is governed by interference-induced photon statistics and changes markedly with the degree of coherence.
  • Indirect schemes that use the purity of the image-plane state are suboptimal for any nonzero coherence and work only in the fully incoherent limit.

Where Pith is reading between the lines

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

  • The Bloch-vector encoding may allow simpler numerical optimization of estimation protocols in related optical setups.
  • Adjusting source coherence could serve as a practical control knob for improving sub-diffraction precision in laboratory imaging.

Load-bearing premise

The sub-Rayleigh imaging problem reduces to an effective two-dimensional Hilbert space description provided a consistent reference position is used, with all parameters encoded in a Bloch vector representation.

What would settle it

An experiment that measures the quantum Fisher information for separation when the coherence factor equals one and finds that the precision remains bounded by the classical Rayleigh limit as separation approaches zero.

Figures

Figures reproduced from arXiv: 2510.04818 by Joaqu\'in L\'opez-Su\'arez, Michalis Skotiniotis.

Figure 1
Figure 1. Figure 1: FIG. 1. The diagonal entries of the lower bound of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. QFI matrix diagonal elements (of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: also demonstrates that it is harder to estimate the separation between two perfectly coherent point sources, as expected. The intensity centroid is the natural choice for a frame of reference as it is the only point in the image plane we can know of—at small separations—prior to measuring the separation s. Information about the intensity centroid can be easily gained via intensity measurements. The same ca… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison between the true QFI for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Qubit and extra contributions to the diagonal element [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. FI for the separation from the aligned SPADE [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Purity as a function of the separation for two values of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Relative difference for separation estimation between [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

We consider imaging of two partially coherent sources and derive the ultimate quantum limits for estimating the separation, location, relative intensity, and coherence factor. We show that super-resolution in the separation is achievable both in the presence of nuisance parameters as well as when some of the parameters are assumed known. Nuisance parameters restrict super-resolution to balanced sources, whereas for known parameters super-resolution persists over a broader range of relative intensities and is lost only for perfectly correlated sources, i.e., $\gamma=1$. The achievable precision is governed primarily by interference-induced photon statistics and depends strongly on the degree of coherence. In the sub-Rayleigh regime, the imaging problem reduces to an effective two-dimensional Hilbert space description, provided a consistent reference position is used, with all parameters encoded in a Bloch vector representation. Finally, indirect estimation schemes based on the purity of the image-plane state are generally suboptimal for all non-zero coherence and become valid only in the incoherent limit.

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.

Referee Report

1 major / 2 minor

Summary. The paper derives ultimate quantum limits on estimating the separation, centroid location, relative intensity, and coherence factor γ of two partially coherent bosonic sources. It claims that super-resolution of the separation remains possible both when nuisance parameters are present (but only for balanced sources) and when some parameters are known a priori (over a wider intensity range, failing only at γ=1). The analysis rests on reducing the sub-Rayleigh imaging problem to an effective two-dimensional Hilbert space whose Bloch-vector components encode all parameters, provided a consistent reference position is chosen. Indirect estimation via image-plane purity is shown to be suboptimal except in the fully incoherent limit.

Significance. If the effective 2D reduction is rigorously justified, the work supplies a useful multi-parameter extension of quantum super-resolution theory to partially coherent sources. It clarifies how interference statistics and the value of γ control the achievable precision and distinguishes the impact of nuisance parameters from the case of known parameters. The explicit comparison with purity-based indirect schemes adds practical insight. The manuscript does not supply machine-checked proofs or reproducible code, but the parameter-free character of the QFI derivations (once the reduction is accepted) is a strength.

major comments (1)
  1. [Sub-Rayleigh regime reduction (abstract and main derivation section)] The central claim that super-resolution persists (with or without known parameters) depends on the exactness of the reduction to a two-dimensional Hilbert space whose Bloch vector encodes separation, location, intensity ratio, and γ. The manuscript states that this reduction holds in the sub-Rayleigh regime with a consistent reference position, but does not provide an explicit derivation showing that the mapping from the full image-plane density operator remains exact once nuisance parameters vary; residual coupling to higher-order PSF modes could alter the off-diagonal elements that determine the QFI for separation. This issue is load-bearing for the reported regimes of super-resolution.
minor comments (2)
  1. Notation for the coherence factor γ and the Bloch-vector components should be introduced with a single consistent definition early in the text rather than appearing piecemeal.
  2. Figure captions would benefit from explicit statements of the parameter values used (e.g., specific γ and intensity ratios) to allow direct comparison with the analytic QFI expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for a more explicit justification of the effective two-dimensional Hilbert space reduction. We address this point below and will strengthen the presentation in the revised version.

read point-by-point responses

  1. Referee: The central claim that super-resolution persists (with or without known parameters) depends on the exactness of the reduction to a two-dimensional Hilbert space whose Bloch vector encodes separation, location, intensity ratio, and γ. The manuscript states that this reduction holds in the sub-Rayleigh regime with a consistent reference position, but does not provide an explicit derivation showing that the mapping from the full image-plane density operator remains exact once nuisance parameters vary; residual coupling to higher-order PSF modes could alter the off-diagonal elements that determine the QFI for separation. This issue is load-bearing for the reported regimes of super-resolution.

    Authors: We agree that an explicit derivation of the reduction, including the effect of varying nuisance parameters, would improve rigor. In the sub-Rayleigh limit the image-plane density operator is expanded in a Taylor series of the PSF about a consistently chosen reference position (the centroid). The linear terms in the small separation parameter span an effective two-dimensional subspace whose Bloch-vector components directly encode the separation, centroid, intensity ratio, and coherence factor γ. Higher-order PSF modes are orthogonal to this subspace and contribute only at O(δ²) or higher, where δ is the normalized separation; these corrections do not mix into the off-diagonal coherences that determine the quantum Fisher information for separation. Because the nuisance parameters appear only as coefficients within the same two-dimensional Bloch vector, they do not induce additional coupling to higher modes. We will add a self-contained appendix that carries out this expansion step by step for both the known-parameter and nuisance-parameter cases, thereby confirming that the reported super-resolution regimes remain valid. revision: yes

Circularity Check

0 steps flagged

Minor self-citation on 2D reduction; central QFI derivation remains independent

full rationale

The paper derives quantum Fisher information bounds for separation, location, intensity ratio and coherence factor γ from an effective two-mode model in the sub-Rayleigh limit. The reduction to a 2D Hilbert space with Bloch-vector encoding is stated to follow from a consistent reference position and is used to obtain the reported regimes of super-resolution. No equation in the provided text shows a fitted parameter being relabeled as a prediction, nor does any load-bearing step collapse to a self-citation whose validity is presupposed by the present work. The central claims therefore retain independent content from the QFI calculation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum optics assumptions for bosonic fields and a partial-coherence model parameterized by γ; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Bosonic sources obey standard second-quantized field operators with a coherence factor γ between 0 and 1.
    Invoked throughout the abstract as the physical model for the two sources.
  • domain assumption Sub-Rayleigh regime permits reduction to an effective two-dimensional Hilbert space when a consistent reference position is chosen.
    Stated explicitly as the condition under which the Bloch-vector description holds.

pith-pipeline@v0.9.0 · 5691 in / 1322 out tokens · 58788 ms · 2026-05-21T21:21:00.532103+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Any good estimator must satisfy the following four conditions [33]: i

    Classical parameter estimation Consider n independent and identically distributed (i.i.d.) samples x = (x1,...,xn) drawn from a random variable X∼p(x|θ), where the unknown parameters θ∈Θ are to be estimated using anestimator f :Xn→Θ, which maps the data to an estimate ˆθ =f (x). Any good estimator must satisfy the following four conditions [33]: i. Unbias...

    work page

  2. [2]

    qubit” term and an “extra

    Quantum parameter estimation The transition from classical to quantum parameter estimation follows from the realization that the random 6 variable X corresponds to the outcome of a quantum measurement—described by the resolution of the identity into a set of positive operators {Ex > 0|∑ x∈XEx = 1l}— whose probability distribution, p(x|θ), is given by Born...

    work page

  3. [3]

    Rayleigh, Lond.Edinb.Dubl.Phil.Mag.10, 116 (1880)

    L. Rayleigh, Lond.Edinb.Dubl.Phil.Mag.10, 116 (1880)

    work page

  4. [4]

    Tsang, R

    M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X6, 031033 (2016)

    work page 2016

  5. [5]

    Nair and M

    R. Nair and M. Tsang, Opt. Express24, 3684 (2016)

    work page 2016

  6. [6]

    Lupo and S

    C. Lupo and S. Pirandola, Phys. Rev. Lett.117, 190802 (2016)

    work page 2016

  7. [7]

    S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A95, 063847 (2017)

    work page 2017

  8. [8]

    ˇRehaˇ cek, Z

    J. ˇRehaˇ cek, Z. Hradil, B. Stoklasa, M. Pa´ ur, J. Grover, A. Krzic, and L. L. S´ anchez-Soto, Phys. Rev. A96, 062107 (2017)

    work page 2017

  9. [9]

    ˇReh´ aˇ cek, Z

    J. ˇReh´ aˇ cek, Z. Hradil, D. Koutn` y, J. Grover, A. Krzic, and L. L. S´ anchez-Soto, Phys. Rev. A98, 012103 (2018)

    work page 2018

  10. [10]

    X.-M. Lu, H. Krovi, R. Nair, S. Guha, and J. H. Shapiro, npj Quantum Information4, 64 (2018)

    work page 2018

  11. [11]

    Napoli, S

    C. Napoli, S. Piano, R. Leach, G. Adesso, and T. Tufarelli, Phys. Rev. Lett.122, 140505 (2019)

    work page 2019

  12. [12]

    Prasad and Z

    S. Prasad and Z. Yu, Phys. Rev. A99, 022116 (2019)

    work page 2019

  13. [13]

    Tsang, Phys

    M. Tsang, Phys. Rev. A99, 012305 (2019)

    work page 2019

  14. [14]

    C. Lupo, Z. Huang, and P. Kok, Phys. Rev. Lett.124, 080503 (2020)

    work page 2020

  15. [15]

    J. O. De Almeida, J. Ko/suppress lody´ nski, C. Hirche, M. Lewen- stein, and M. Skotiniotis, Phys. Rev. A103, 022406 (2021)

    work page 2021

  16. [16]

    F. Yang, A. Tashchilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica3, 1148 (2016)

    work page 2016

  17. [17]

    W.-K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett.118, 070801 (2017)

    work page 2017

  18. [18]

    F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A96, 063829 (2017)

    work page 2017

  19. [19]

    J. M. Donohue, V. Ansari, J. ˇReh´ aˇ cek, Z. Hradil, B. Stok- lasa, M. Pa´ ur, L. L. S´ anchez-Soto, and C. Silberhorn, Phys. Rev. Lett.121, 090501 (2018)

    work page 2018

  20. [20]

    Parniak, S

    M. Parniak, S. Bor´ owka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrza´ nski, Phys. Rev. Lett.121, 250503 (2018)

    work page 2018

  21. [21]

    Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, Optica6, 534 (2019)

    work page 2019

  22. [22]

    Boucher, C

    P. Boucher, C. Fabre, G. Labroille, and N. Treps, Optica 7, 1621 (2020)

    work page 2020

  23. [23]

    Rouvi` ere, D

    C. Rouvi` ere, D. Barral, A. Grateau, I. Karuseichyk, G. Sorelli, M. Walschaers, and N. Treps, Optica11, 166 (2024)

    work page 2024

  24. [24]

    Larson and B

    W. Larson and B. E. A. Saleh, Optica5, 1382 (2018)

    work page 2018

  25. [25]

    Tsang and R

    M. Tsang and R. Nair, Optica6, 400 (2019)

    work page 2019

  26. [26]

    Larson and B

    W. Larson and B. E. A. Saleh, Optica6, 402 (2019)

    work page 2019

  27. [27]

    Hradil, J

    Z. Hradil, J. ˇReh´ aˇ cek, L. S´ anchez-Soto, and B.-G. Englert, Optica6, 1437 (2019)

    work page 2019

  28. [28]

    Hradil, D

    Z. Hradil, D. Koutn´ y, and J.ˇReh´ aˇ cek, Opt. Lett.46, 1728 (2021)

    work page 2021

  29. [29]

    Liang, S

    K. Liang, S. A. Wadood, and A. N. Vamivakas, Phys. Rev. A104, 022220 (2021)

    work page 2021

  30. [30]

    Karuseichyk, G

    I. Karuseichyk, G. Sorelli, M. Walschaers, N. Treps, and M. Gessner, Phys. Rev. Res.4, 043010 (2022)

    work page 2022

  31. [31]

    S. A. Wadood, K. Liang, Y. Zhou, J. Yang, A. I. Alonso, X.-F. Qian, T. Malhotra, M. S. Hashemi Rafsanjani, A. N. Jordan, R. W. Boyd, and A. N. Vamivakas, Opt. Express 29, 22034 (2021)

    work page 2021

  32. [32]

    Chrostowski, R

    A. Chrostowski, R. Demkowicz-Dobrzanski, M. Jarzyna, and K. Banaszek, Int. J. Quantum Inform.15, 1740005 (2017)

    work page 2017

  33. [33]

    de Almeida, M

    J. de Almeida, M. Lewenstein, and M. Skotiniotis, arXiv preprint arXiv:2110.00986 (2021)

    work page arXiv 2021

  34. [34]

    Born and E

    M. Born and E. Wolf,Principles of Optics: Electromag- netic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, 1999)

    work page 1999

  35. [35]

    S. S. Wilks,Mathematical Statistics(John Wiley & Sons, New York, 1962)

    work page 1962

  36. [36]

    Cram´ er,Mathematical Methods of Statistics, Prince- ton paperbacks Princeton landmarks in mathematics and physics (Princeton University Press, New Jersey, 1999)

    H. Cram´ er,Mathematical Methods of Statistics, Prince- ton paperbacks Princeton landmarks in mathematics and physics (Princeton University Press, New Jersey, 1999)

    work page 1999

  37. [37]

    R. A. Fisher and E. J. Russell, Philos. Trans. R. Soc. A 222, 309 (1922)

    work page 1922

  38. [38]

    H. L. Van Trees,Detection, Estimation, and Modulation Theory: Part I: Detection, Estimation, and Linear Modu- lation Theory(Wiley, New York Chichester, 2001)

    work page 2001

  39. [39]

    S. L. Braunstein and C. M. Caves, Phys. Rev. Lett.72, 3439 (1994)

    work page 1994

  40. [40]

    S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza´ nski, Phys. Rev. A94, 052108 (2016)

    work page 2016

  41. [41]

    Schwartz, J

    O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, Nano Lett.13, 5832 (2013)

    work page 2013

  42. [42]

    Li and Z

    W. Li and Z. Wang, Opt. Express30, 12684 (2022)

    work page 2022

  43. [43]

    Lodahl, S

    P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys.87, 347 (2015)

    work page 2015

  44. [44]

    Utzat, W

    H. Utzat, W. Sun, A. E. K. Kaplan, F. Krieg, M. Gin- terseder, B. Spokoyny, N. D. Klein, K. E. Shulenberger, C. F. Perkinson, M. V. Kovalenko, and M. G. Bawendi, Science363, 1068 (2019)

    work page 2019