Poissonian twin beam states and the effect of symmetrical photon subtraction in loss estimations

J. C. F. Matthews; J. G. Rarity; N. Samantaray

REVIEW 3 major objections 4 minor 31 references

Poissonian twin beam states and the effect of symmetrical photon subtraction in loss estimations

T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read A single modal-averaging parameter tunes twin beams from thermal to Poissonian statistics while preserving their non-classical correlation, and symmetric photon subtraction triples loss-measurement precision at fixed squeezing.

desk verdict Tunable Poissonian twin-beam model is built on a non-canonical transformation and is unphysical for beta<1; the beta=1 estimator comparison is solid but the paper's central new claim fails. read the letter →

arxiv 2009.11586 v1 pith:3GG23HQW submitted 2020-09-24 quant-ph

N. Samantaray , J. C. F. Matthews , J. G. Rarity This is my paper · ORCID

classification quant-ph

keywords twinbeamstatesphotonsubtractionlossestimationsub-shot-noiseFanofactormodalaveragingquantummetrologyPoissonianstatistics

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

The paper tries to establish that one can tune the individual-beam photon statistics of a twin-beam state from thermal to Poissonian, using a single modal-averaging parameter β, without destroying the non-classical photon-number correlation between the beams. It then claims that symmetric photon subtraction improves loss-measurement precision: at fixed squeezing the low-absorption uncertainty drops by a factor $(m+1)$, up to threefold for two subtracted photons, while at fixed per-photon exposure most of the gain disappears except for a residual 20% advantage in some regimes. The states are characterized by two numbers, the Fano factor $F$ and the noise reduction factor $\sigma$, and the paper identifies which of three absorption estimators—number difference, optimized balanced, and ratio—uses those resources best. If correct, the result matters because laboratory twin beams are often averaged into Poissonian statistics, and the model would show that the quantum correlation needed for sub-shot-noise absorption sensing survives that averaging and that photon subtraction can still buy a factor of three in precision.

What carries the argument

The central object is the $\beta$-generalized Bogoliubov transformation of Eqs. (6)–(7), with $\beta=1/M$ for $M$ collected modes. It drives the argument by making the local Fano factor $F=1+\beta\eta\lambda$ for the un-subtracted state while leaving the noise reduction factor $\sigma=1-\eta$ exactly independent of $\beta$, so the correlation resource is unchanged while local statistics interpolate between thermal and Poissonian. The second mechanism is symmetric photon subtraction, realized in the model by seeding an $(m+1)$-component photon-number superposition state into the nonlinear crystal; its normalization constant is $m!(-i\sqrt{\lambda})^m P_m(i\sqrt{\lambda})$, with $P_m$ the $m$th Legendre polynomial. The subtraction lowers $F$ toward sub-Poissonian values without altering $\sigma$, and the paper's asymptotic formulas show that the fixed-squeezing uncertainty in both the optimized and ratio estimators carries the factor $1/(m+1)$ at low absorption, producing the claimed threefold improvement for $m=2$.

What would settle it

Compute the commutation relation $[\hat{c}_1,\hat{c}_1^\dagger]=1+(\beta-1)\lambda$ for $\beta<1$; if it differs from 1, test whether the state defined by Eqs. (6)–(7) can be produced by any physical unitary or completely positive map, which would settle whether the $\beta=0$ results are properties of a real correlated Poissonian twin beam or artifacts of the transformation. An independent experiment with a many-mode SPDC source could look for the predicted constant $\sigma=1-\eta$ as $\beta$ is reduced toward 0.

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Extended reading notes

Core claim

At the center of the paper is a $\beta$-family of correlated twin beams defined by $\hat{c}_1=\hat{a}_1\sqrt{1+\beta\lambda}+\hat{a}_2^\dagger\sqrt{\lambda}$ and $\hat{c}_2=\hat{a}_2\sqrt{1+\beta\lambda}+\hat{a}_1^\dagger\sqrt{\lambda}$, with $\beta$ the inverse number of collected modes. The detected means are $\langle N_P\rangle=\langle N_R\rangle=\eta\lambda$; the individual variances interpolate from $\eta\lambda+\eta^2\lambda^2$ at $\beta=1$ (thermal) to $\eta\lambda$ at $\beta=0$ (Poissonian), while the noise reduction factor stays $\sigma=1-\eta$ for all $\beta$. The paper claims this keeps the non-classical correlation intact and that symmetric photon subtraction—implemented by seeding a superposition state instead of using beam splitters—lowers the Fano factor without changing $\sigma$. At fixed squeezing, the loss uncertainty in all three estimators is sub-shot-noise, and the low-absorption limit improves by factors 1, 2, and 3 for $m=0,1,2$ subtracted photons, giving the headline threefold advantage. At fixed per-photon exposure, the subtraction advantage mostly subsides, but the number-difference estimator still shows up to 20% advantage over the un-subtracted correlated Poissonian beam at low loss and high $\lambda$, and the optimized estimator shows about 10% at 50% detection loss.

Load-bearing premise

The model's entire $\beta<1$ family—including the Poissonian endpoint and the claimed 20% advantage—rests on treating Eqs. (6)–(7) as a valid physical mode transformation for every $\beta$ between 0 and 1; for $\beta<1$ the output operators satisfy $[\hat{c}_1,\hat{c}_1^\dagger]=1+(\beta-1)\lambda\neq 1$, so those states are not automatically physical twin beams.

Editorial extensions

If this is right

At fixed squeezing, symmetric subtraction of $m$ photons improves the low-absorption loss-measurement uncertainty by a factor $(m+1)$, giving a threefold advantage for $m=2$ over the un-subtracted twin beam; this holds across the full $\beta$ range.
The noise reduction factor $\sigma=1-\eta$ is independent of both $\beta$ and $m$, so the photon-number correlation that enables sub-shot-noise sensing is preserved when local statistics are made Poissonian and when photons are subtracted.
For the ratio estimator, uncertainty depends only on $\sigma$ and not on the Fano factor, so it reaches sub-shot-noise performance for any mean photon number and any $\beta$, but it gains nothing from photon subtraction at fixed per-photon exposure.
Under fixed per-photon exposure, most subtraction advantage disappears, but the number-difference estimator retains up to 20% advantage over the un-subtracted Poissonian twin beam at low detection loss and high $\lambda$, and the optimized estimator retains about 10% advantage at 50% detection loss.
Among the three estimators, the optimized balanced estimator is the best overall: it matches the ratio estimator at low loss and outperforms it at high detection loss because its uncertainty carries $\sqrt{1-\eta^2}$ rather than $\sqrt{1-\eta}$.

Reading between the lines

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

Inference: the residual 20% advantage at fixed per-photon exposure is the regime where the model's physicality matters most, because that endpoint depends on $\beta=0$; a mode-resolved experimental test would be needed before counting that advantage as real.
Inference: if the $(m+1)$-fold scaling at fixed squeezing holds beyond $m=2$, symmetric subtraction of three or more photons from each arm should give a fourfold uncertainty reduction in the same low-absorption, low-$\lambda$ limit, which the paper's method can test directly.
Inference: the ratio estimator's independence from Fano factor suggests it may be the estimator of choice when source statistics are drifting, since its uncertainty is fixed by $\sigma$ alone and therefore by detection efficiency rather than by modal averaging.
Inference: the paper assumes balanced detection losses; extending the comparison to unbalanced losses, which the authors leave for future work, would determine whether the ranking of estimators changes when the reference arm is noisier than the probe arm.

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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

Short version: the paper's headline claim—a twin-beam model that tunes individual beam statistics from thermal to Poissonian while preserving correlation—is not physically defined for beta<1. The transformation in Eqs. (6)-(7) gives non-bosonic operators unless beta=1: the commutator is 1+(beta-1)lambda. The authors call beta=1/M a modal averaging parameter, but they never show the transformation follows from an actual M-mode TMSV. It reproduces the first two moments of the multimode state, but that doesn't make it a valid mode map, and for photon subtraction they apply it to non-vacuum inputs. So the Poissonian-regime results, including the 20% advantage, are computed from an unphysical operator model.

What's actually good: the paper gives a systematic comparison of three loss estimators and derives clean asymptotic scalings at fixed squeezing. For beta=1 (standard TMSV), the m+1 improvement for m-photon subtraction is plausible and consistent with known results. The estimator comparison, especially the ratio estimator, is a legitimate extension. The authors also note the balanced-detection limitation.

Soft spots, in order: the non-canonical transformation is load-bearing; without it, the tunable model collapses. The photon-subtraction equivalence is cited to the authors' own Ref. [21] and no independent argument is given for beta<1. And the Fano factor plots for subtracted states are graphical only, with expressions omitted—annoying but minor compared to the first issue.

Bottom line: the paper is a decent write-up of a plausible but flawed model. The beta=1 parts are fine, but the stated goal is the tunable regime, and that's exactly where the physics goes off the rails. I would send it to a serious referee because the issue is subtle—maybe a proper multimode derivation can save the photon-subtraction claims—but as it stands, the Poissonian results are not supported. For a reading group, it's a useful case study in why non-canonical transformations are invalid.

Recommendation: engage with it only for the beta=1 estimator comparison; cite nothing from the beta<1 regime.

Referee Report

3 major / 4 minor

Summary. The paper proposes a phenomenological model, Eqs. (6)-(7), in which a parameter beta interpolates the individual-beam photon statistics of a twin-beam state from thermal (beta=1) to Poissonian (beta=0) while keeping the photon-number correlation, quantified by the noise reduction factor sigma, intact. The authors use this model to study three estimators of optical loss (number difference, optimized balanced, and ratio estimators), and then incorporate symmetric photon subtraction, implemented by seeding an (m+1)-component superposition state into the nonlinear crystal. They report asymptotic scaling of the loss uncertainty that improves by a factor (m+1) with m-photon subtraction in the low-absorption limit, claim up to a factor-three advantage for two-photon subtraction at fixed squeezing, and find about 20% advantage in some per-photon-exposure regimes. The abstract and conclusion present these as experimentally realizable quantum enhancements.

Significance. If the central model were valid, the paper would offer a useful interpolation between thermal and Poissonian twin-beam statistics and a systematic comparison of three absorption estimators, with explicit asymptotic scaling factors and a clear separation of the roles of the Fano factor and the noise reduction factor. The asymptotic expressions in Eqs. (11)-(14) are a strength, and the comparison among estimators is clearly laid out. However, the entire beta<1 regime, including the claimed Poissonian-case advantages, rests on a mode transformation that is not a canonical bosonic transformation, so the states and observables used in that regime are not defined as quantum-mechanical objects. The central claim of an experimentally realizable model with tunable statistics is therefore not established by the manuscript as written.

major comments (3)

[Section III, Eqs. (6)-(7)] The transformation c1 = a1 sqrt(1+beta lambda) + a2^dagger sqrt(lambda) and c2 = a2 sqrt(1+beta lambda) + a1^dagger sqrt(lambda) does not preserve the bosonic commutation relations for beta<1. Direct calculation gives [c1,c1^dagger] = 1 + (beta-1) lambda, which equals 1 only for beta=1; for beta=0 the commutator is 1-lambda and becomes negative for lambda>1. Consequently c1 and c2 cannot be interpreted as annihilation operators of physical modes, and c1^dagger c1 cannot be interpreted as a photon number operator. No unitary or completely positive transformation produces this map for beta<1. Since all results for beta<1, including the Poissonian (beta=0) regime and the claimed 20% advantage in Section III.B.2, are computed by applying these non-canonical operators to vacuum, the associated states and uncertainties are not defined as physical quantum predictions. A legitimate multimode derivation must be supplied before these results can be assessed.
[Section III.A] The equivalence between seeding an (m+1)-component superposition state and performing m-photon subtraction is assumed from Ref. [21], an arXiv preprint by the same first author, without a self-contained derivation or a peer-reviewed citation. This equivalence underpins all m=1 and m=2 results, so the manuscript should either provide a direct proof of the equivalence within the present model or cite a published derivation that covers the beta<1 case. As written, the photon-subtraction results for the Poissonian regime inherit both the unphysical transformation problem and an unverified input-state equivalence.
[Section III.B] The paper states that the analytic expressions for the Fano factors are 'too cumbersome to present' and gives only graphical results, with no code, data, or supplementary derivation. Since the central quantitative claims, such as the factor-three and 20% advantages, are read off from these plots, the authors should make the underlying expressions or reproducible numerical code available, or include the required formulas in an appendix.

minor comments (4)

[Section III.B.1] The text says the fixed-squeezing comparison is carried out 'at the same mean energy (photon number exposure) as the un-subtracted state' and then says r is fixed; these two statements are in tension because photon subtraction increases the mean photon number for fixed r. The authors should clarify the exact balancing condition used for each figure.
[Figures 6-8] The dotted lines are described as 'uncertainties evaluated using classical resources with average energies of m photon subtracted state,' but the classical states used for comparison are not defined. Please specify whether these are coherent states with the same mean photon number, or some other reference.
[Throughout] There are several typographical errors that should be corrected: 'Bougolibov' should be 'Bogoliubov', 'wavemixing' should be 'wave mixing', 'Poisonian' should be 'Poissonian', and 'uncerianty' should be 'uncertainty'.
[Section III, Eq. (8)] The notation ⟨Delta(N_P,N_R)⟩ for the covariance is nonstandard and could be confused with an uncertainty product; a standard notation such as Cov(N_P,N_R) or Delta^2(N_P,N_R) would improve clarity.

Circularity Check

1 steps flagged · score 4.0 of 10

Photon-subtraction results depend on a load-bearing equivalence imported from a same-author citation; the underlying statistics are computed from an explicit model, not fitted.

self citation load bearing [Section III.A 'Symmetrical photon subtraction', after Eq. (10); see also Fig. 1 caption]
"Alternatively by injecting m+1 component superposition state [21] to the NL in place of vacuum, equivalently executes deterministically the m photon subtraction operation as shown in the right image of fig 1."

Reference [21] is N. Samantaray et al., arXiv:1809.10706, whose first author is the first author of the present paper. The equivalence between seeding a superposition state and deterministic m-photon subtraction is the only bridge from the standard non-unitary subtraction expression (Eq. 9) to the seeded model used in all subsequent m>0 calculations (Figs. 4-12 and Eqs. (11)-(14)). That equivalence is not derived or independently verified within this paper; it is imported from the authors' own prior work. The claimed photon-subtraction advantages (factor three at fixed squeezing; 20% in the per-photon-exposure regime) therefore rest on a load-bearing self-citation rather than on a proof contained in this paper.

full rationale

The paper's statistical core is analytic rather than fitted: once the transformation in Eqs. (6)-(7) and the input superposition state are assumed, the variances, Fano factors, and estimator uncertainties follow by direct computation. There is no fitting of parameters to experimental data, so the central uncertainty formulas are not circular in the fitted-input sense. The thermal-to-Poissonian interpolation is presented explicitly as a phenomenological model, with beta as a defined control parameter; that is a modeling assumption rather than a predicted match to external data. The non-canonical commutator for beta<1 is a physical-consistency concern, not a circularity, and is therefore not scored here. The one genuinely load-bearing circular element is the reliance on Ref. [21], a same-first-author preprint, for the equivalence that turns photon subtraction into a seeded superposition process; all m>0 advantage claims inherit that premise. Because the statistical computation after that premise has independent content, the appropriate score is moderate: some self-citation with a load-bearing role, but not a wholesale reduction of the derivation to its inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper's central claim depends on three main inputs: the beam-splitter model for loss, the beta-interpolated mode transformation, and the equivalence between superposition-state seeding and photon subtraction. The first is standard; the second is an ad hoc non-unitary model that is load-bearing for the Poissonian regime; the third is taken from the authors' own prior work without independent validation. No new physical entities are introduced.

free parameters (1)

beta = 0 to 1 (swept)
Modal averaging parameter beta = 1/M in Eqs. 6-7, introduced ad hoc to interpolate photon statistics from thermal (beta=1) to Poissonian (beta=0). It is not derived from a unitary model and is not fixed by experiment. All results are parameterized by beta.

assumptions (4)

domain assumption The beam-splitter model tau = eta (1-gamma) correctly represents sample absorption and detection loss for both beams.
Used in Section II for all estimators and uncertainty formulas.
ad hoc to paper The operators defined in Eqs. 6-7 describe physical twin-beam modes for all beta in [0,1], including preserving the non-classical correlation.
This is the central assumption that carries the Poissonian regime; it is not justified and fails canonical commutation for beta<1.
ad hoc to paper Seeding an (m+1)-component superposition state into the nonlinear crystal exactly reproduces m-photon subtraction (from Ref [21]).
Invoked in Section III A to compute Fano factors and uncertainties for subtracted states; it is cited to the authors' own prior work rather than independently derived here.
standard math The error-propagation formulas for the three estimators (Eqs. 1, 3, 5) are correct for states with Fano factor F and noise reduction factor sigma.
Standard propagation of uncertainty, but the specific expressions are not derived in the paper.

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Pith. "Pith review of Poissonian twin beam states and the effect of symmetrical photon subtraction in loss estimations." pith.science (2026). https://pith.science/paper/3GG23HQW

@misc{pith2026200911586,
  author       = {Pith},
  title        = {Pith review of: Poissonian twin beam states and the effect of symmetrical photon subtraction in loss estimations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3GG23HQW}},
  note         = {Machine review of arXiv:2009.11586}
}

read the original abstract

We have devised an experimentally realizable model generating twin beam states whose individual beam photon statistics are varied from thermal to Poissonian keeping the non-classical mode correlation intact. We have studied the usefulness of these states for loss measurement by considering three different estimators, comparing with the correlated thermal twin beam states generated from spontaneous parametric down conversion or four-wave mixing. We then incorporated the photon subtraction operation into the model and demonstrate their advantage in loss estimations with respect to un-subtracted states at both fixed squeezing and per photon exposure of the absorbing sample. For instance, at fixed squeezing, for two photon subtraction, up to three times advantage is found. In the latter case, albeit the advantage due to photon subtraction mostly subsides in standard regime, an unexpected result is that in some operating regimes the photon subtraction scheme can also give up to 20% advantage over the correlated Poisson beam result. We have also made a comparative study of these estimators for finding the best measurement for loss estimations. We present results for all the values of the model parameters changing the statistics of twin beam states from thermal to Poissonian.

Figures

Figures reproduced from arXiv: 2009.11586 by the authors.

**Figure 2.** Absorption measurement: (a) direct one path imag [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Scheme for generating correlated TBS states: Co [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 5.** Figure 5: (Color Online) 3D plots of detected mean number of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 4.** Figure 4: (Color Online) Plots of measured Fano Factor as a [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 6.** Figure 6: (Color online) Plots of Uncertainty in the number [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: (Color online) Plots of uncertainty in the optimized [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: (Color online) Plots of uncertainty in the ratio mea [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: (Color online) Comparision of uncertainties among [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: (Color online) Plots of Fano factor as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

**Figure 12.** Figure 12: (Color online) Plots of uncertainty in the optimized [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [21]

Strategies for enhancing quantum entanglement by local photon subtraction

T. J. Bartley et.al., "Strategies for enhancing quantum entanglement by local photon subtraction", Phys. Rev. A 87,022313 (2013)

work page 2013

[2] [1]

In this way, r is ﬁxed as mean energy λ (for m = 0)=sinh2r

Fixed squeezing parameter The analysis of comparing uncertainties for diﬀerent m at ﬁxed squeezing has been carried at the same mean energy (photon number exposure) as the un-subtracted (m = 0) state. In this way, r is ﬁxed as mean energy λ (for m = 0)=sinh2r. Uncertainties for number diﬀer- ence measurement is shown in ﬁg 6. It shows SSN for diﬀerent val...

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[3] [2]

This set of equation conﬁrms the standard SNL for the limit γ→ 1 as per the discussion in IIA

and forλ→ 0, the uncertainty scales as ∆γm=0 Diff ≈ 1√ηλ, ∆γm=1 Diff ≈ 1 2√ηλ, (12) ∆γm=2 Diff ≈ 1 3√ηλ. This set of equation conﬁrms the standard SNL for the limit γ→ 1 as per the discussion in IIA. The improve- ment in the standard SNL for the limitγ → 0 comes from a loss dependent factor1−η in the numerator. The factor of improvements in the same limit...

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[4] [3]

and forλ→ 0, the uncertainty scales as ∆γm=0 Opt ≈ √1−γ√ηλ , ∆γm=1 Opt ≈ √1−γ 2√ηλ , (14) ∆γm=2 Opt ≈ √1−γ 3√ηλ . This set of equation says, unlike the number diﬀerence measurement, for γ → 1, the uncertainty scales much better than the standard SNL and it improves further with the number of subtracted photonm. On the other side of the limit γ → 0, apart ...

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[5] [4]

Since there is aβ dependence on mean energy, we further consider the energy balancing at a ﬁxed model parameter

Fixed per photon exposure Fixedperphotonexposureanalysishasbeencarriedby balancing numerically the mean energies of subtracted states so that mean energies for diﬀerentm are equal to energy of (m = 0) un-subtracted state. Since there is aβ dependence on mean energy, we further consider the energy balancing at a ﬁxed model parameter. Before presenting the ...

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[6] [5]

Teleportation improvement by incon- clusive photon subtraction

Olivares et. al, "Teleportation improvement by incon- clusive photon subtraction", Phys. Rev. A 67, 032314 (2003)

work page 2003

[7] [6]

Optimal estimation of losses at the ul- timate quantum limit with non-Gaussian states

Adesso et. al, "Optimal estimation of losses at the ul- timate quantum limit with non-Gaussian states", Phys. Rev. A.79,040305 (2009)

work page 2009

[8] [7]

Recent developments in photon-level opera- tions on travelling light ﬁelds

MS Kim,"Recent developments in photon-level opera- tions on travelling light ﬁelds", J. Phys. B: At. Mol. Opt. Phys.41, 133001 (2008)

work page 2008