Phase estimation via delocalized photon subtraction operation inside the SU(1,1) interferometer

Chengzhi Deng; Cunjin Liu; Liyun Hu; Qingqian Kang; Teng Zhao; Zhihao Li

REVIEW 4 major objections 8 minor 51 references

Phase estimation via delocalized photon subtraction operation inside the SU(1,1) interferometer

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a delocalized photon subtraction operation inside an SU(1,1) interferometer, applied between the two amplifiers to a coherent-vacuum input, gives phase sensitivities that beat both localized single-mode versions, get…

desk verdict The ideal-case analysis of D-PSO in an SU(1,1) interferometer looks solid and reasonably novel, but the lossy QFI section mixes two incompatible loss models, so the robustness claim is not yet supported. read the letter →

arxiv 2506.07684 v1 pith:OZNME543 submitted 2025-06-09 quant-ph

Zhihao Li , Qingqian Kang , Teng Zhao , Cunjin Liu , Liyun Hu , Chengzhi Deng This is my paper

classification quant-ph PACS 03.67.-a05.30.-d42.50.Dv03.65.Wj

keywords SU(11)interferometerdelocalizedphotonsubtractionphaseestimationquantumFisherinformationlossCramér-Raoboundintensitydetectionnon-Gaussianoperation

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 tries to show that a delocalized photon subtraction operation (D-PSO) inside an SU(1,1) interferometer improves phase measurement precision with intensity detection, using only a coherent state and vacuum as inputs. The operation is a weighted combination of subtracting photons from mode a and mode b, $(s a + t b)^m$ with $s+t=1$; the two limits are the localized operations on each mode. Across the parameters studied, the delocalized operation covers the optimal phase-sensitivity ranges of both localized operations, improves the quantum Fisher information, and approaches the quantum Cramér-Rao bound more closely as the order $m$ grows. The authors also argue that D-PSO is more robust to internal photon loss than localized photon subtraction. If correct, this gives a practical route to more accurate phase measurements without changing the input states or detection scheme of ordinary SU(1,1) interferometry.

What carries the argument

The central object is the delocalized photon subtraction operation $(s a + t b)^m$ inserted behind the first OPA, with real weights $s$ and $t$ normalized by $s+t=1$; $s=1$ gives subtraction from mode $a$ only, $t=1$ from mode $b$ only, and intermediate $s$ gives delocalized subtraction. The calculation is carried by a generating function $Q_{m,x_1,y_1,x_2,y_2}$ that turns all needed expectation values of normally ordered products into derivatives of one exponential, giving closed-form expressions for phase sensitivity, total photon number, and quantum Fisher information. In the lossy case the authors use the minimized-Kraus bound, which reduces to a formula in the mean and variance of the photon number in mode $a$. The operation's tunable superposition of mode $a$ and mode $b$ subtraction is what lets it inherit the best behavior of both localized operations.

What would settle it

Compute the lossy quantum Fisher information with the same two-mode fictitious beam-splitter loss used in the phase-sensitivity calculation, placing the loss both before and after the phase shift; if the D-PSO advantage over L-PSO disappears in either placement, the paper's robustness claim is not universal.

Watch

Extended reading notes

Core claim

The central claim is that applying $(s a + t b)^m$ between the two optical parametric amplifiers of an SU(1,1) interferometer beats doing the same photon subtraction on only one mode. The paper derives the output state exactly through a generating-function method, computes phase sensitivity from error propagation on the total photon number, and computes the quantum Fisher information directly for the ideal case and via a minimized Kraus bound under photon loss. Its main findings are that D-PSO's optimal working range in phase shift, coherent amplitude, gain, and transmissivity encompasses the optimal ranges of both mode-a-only and mode-b-only subtraction; that its quantum Fisher information is larger than either localized operation in a broad parameter region; and that as the order $m$ increases its sensitivity comes closer to the quantum Cramér-Rao bound while still beating the standard quantum limit and, in part of parameter space, the Heisenberg limit.

Load-bearing premise

The loss-robustness conclusion relies on assuming that the single-mode loss parameter in the Fisher-information calculation describes the same physical loss as the two-mode transmissivity in the phase-sensitivity calculation, with the loss position fixed; the paper does not state this link.

Editorial extensions

If this is right

A standard SU(1,1) interferometer with coherent and vacuum inputs can reach sub-SQL, and in some regions sub-Heisenberg-limit, phase sensitivity by adding an $m$-th-order delocalized photon subtraction before the phase shift.
Choosing the delocalization weight $s$ optimally makes the interferometer avoid the need to choose which single mode to subtract from, because the D-PSO optimal phase-shift, amplitude, gain, and transmissivity ranges cover both localized operations.
As the order $m$ grows, the intensity-detection phase sensitivity moves closer to the quantum Cramér-Rao bound, so higher-order D-PSO is a practical way to close the gap to the ultimate precision limit.
Under photon loss, D-PSO keeps a phase-sensitivity advantage over both localized operations across the transmissivity range, meaning the precision gain is not bought at the cost of fragility.
The quantum Fisher information advantage of D-PSO over L-PSO widens with order, loss parameter, and coherent amplitude, which identifies favorable operating conditions for the scheme.

Reading between the lines

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

Beyond the paper, the same tunable-weight construction could be applied to other non-Gaussian operations, such as photon addition or photon catalysis inside SU(1,1), where a delocalized superposition might again combine the strengths of the two modes; the paper does not explore these variants.
The underlying mechanism suggested by the results is that D-PSO acts as a coherent superposition of two channel operations, so it should generate controllable two-mode correlations after the first amplifier; a direct measurement of entanglement or discord versus the weight $s$ would test this picture.
A testable experimental prediction is that the optimal phase sensitivity occurs at an interior value of $s$, not at $s=0$ or $s=1$, for fixed coherent amplitude, gain, order, and loss; scanning $s$ would confirm that delocalization itself, rather than simply more photon subtraction, is responsible for the improvement.

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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

The new thing here is the delocalized photon subtraction operation (D-PSO) placed inside the SU(1,1) interferometer, with coherent plus vacuum inputs, and a systematic comparison against localized subtraction (L-PSO) on each mode. That specific combination is new, and the ideal-case treatment is genuinely useful: the phase sensitivity and QFI calculations are explicit, the parameter optimization over t and m is clearly described, and the figures show D-PSO consistently outperforming L-PSO in the no-loss regime. Appendix A gives enough detail that a competent reader could rederive the results. I believe the ideal-case part is correct and worth publishing.

The soft spots are all in the lossy QFI section, and they are real. Section II models internal loss with two fictitious beam splitters of transmissivity T on both modes, and the phase-sensitivity curves (Figs. 5, 12, 13) use that model. Section IV B, however, switches to Escher's single-mode formula with a parameter η, never states the relation between η and T, never fixes the loss position relative to the phase shift (the λ in Eq. (12)), and considers loss only on mode a. The text also misstates η: it calls η=1 the "complete lossy" case, but Eq. (12) gives the lossless limit there, and the figures confirm that QFI increases with η. On top of that, the moments ⟨n⟩ and ⟨Δn²⟩ fed into Eq. (13) are computed with Q that already includes U_B, so if T<1 is used, the state being characterized already has loss, and the Escher formula would double-count it. This is not a minor issue: the abstract's claim of "the ability to resist internal loss" leans directly on these QFI results. The phase-sensitivity part under the T-model is self-consistent, but the QFI-based support for the robustness claim is not established until the model is clarified.

Who gets value: people working on SU(1,1) interferometry and non-Gaussian operations will want the ideal-case comparison and the analytic machinery. It deserves a serious referee, not a desk reject, because the novel operation and the explicit ideal-case analysis are worth examining even though the lossy section needs a major overhaul: state the mapping between η and T (or use the same two-mode loss model for QFI), correct the η=1 misstatement, and either compute a proper two-mode-loss QFI or clearly restrict the claim to single-mode loss.

My recommendation: send it to peer review with the expectation of heavy revision on the loss model. If the authors fix that, the paper is a solid contribution to the non-Gaussian metrology literature.

Referee Report

4 major / 8 minor

Summary. The manuscript proposes inserting a delocalized photon-subtraction operation (D-PSO), written as (s a + t b)^m with s+t=1, between the two OPAs of an SU(1,1) interferometer whose inputs are a coherent state and vacuum. Using error propagation on the photon-number-sum measurement, it derives the phase sensitivity and separately computes the quantum Fisher information with a generating-function technique in Appendix A. The paper compares D-PSO with the two localized subtraction operations (on mode a only or mode b only) in ideal and lossy conditions, and benchmarks the results against the SQL, HL, and QCRB. The central claims are that D-PSO improves phase sensitivity and QFI relative to L-PSO and is more robust against internal photon loss.

Significance. If the claims hold, the paper offers a compact non-Gaussian operation that improves SU(1,1)-interferometer phase estimation and systematically clarifies the difference between delocalized and localized subtraction. The main strengths are the explicit analytic expressions in Appendix A, the direct error-propagation and QFI calculations, and the comparison against external bounds rather than fitted parameters; the optimized coefficient t is a control parameter, not a fit to a target sensitivity. However, the lossy-QFI section is internally inconsistent and disconnected from the loss model used for phase sensitivity, so the robustness claim in the abstract and conclusion is not currently established. Because the issue is localizable and fixable, the paper is a viable candidate for publication after a major revision.

major comments (4)

[§IV B, Eq. (13) and text after Eq. (12)] The sentence stating that η=1 and η=0 represent the situations of complete lossy and absorption is contradicted by Eq. (13), which gives F_L=4⟨Δn²⟩ at η=1 (the ideal pure-state QFI value) and F_L=0 at η=0, and by Figs. 9–11, where F_L increases with η. This suggests η is meant to be the transmission efficiency, not the loss probability; the text and notation must be corrected, and the relation of η to the transmissivity T used elsewhere must be stated.
[§II and §IV B] The phase-sensitivity calculation in Sec. II models loss by two fictitious beam splitters with transmissivity T placed after the D-PSO and before the phase shift, and Figs. 5, 12 and 13 use this T. The lossy QFI in Sec. IV B, by contrast, uses a single-mode parameter η and explicitly considers loss only in mode a. No relation such as η=T is stated, and no justification is given for ignoring loss in mode b for the QFI. Therefore the claimed ability to resist internal photon loss is not supported by the manuscript as written; the authors should either state and justify the mapping or recompute F_L under the same two-mode loss model.
[§IV B, Eq. (12)] The parameter λ in the Kraus operator is left unspecified in the reported F_L; the text says λ=0 and λ=−1 correspond to loss before and after the phase shift, but Eq. (13) and Figs. 9–11 do not state which value is used. If F_L depends on λ, the QFI values and hence the QCRB used in Fig. 13 are ambiguous; the authors must fix λ or minimize over it.
[§IV C, Fig. 13] Fig. 13 compares the phase sensitivity computed at T=0.7 with a QCRB derived from F_L, but the value of η used for the QCRB is not stated. Figs. 10 and 11 also use 'T=0.7' in the captions although Eq. (13) is expressed in terms of η. If η is intended to equal T, that identification must be explicit; otherwise the comparison mixes two unrelated loss models and the claim that D-PSO approaches the QCRB under loss is not meaningful.

minor comments (8)

[§III, after Fig. 5] The sentence 'It is found from Fig. 4 as follows' appears in the paragraph discussing the T-dependence and should refer to Fig. 5.
[throughout] The notation for the phase is inconsistent: equations and the text use ϕ, while most figure captions and some textual passages use φ; please unify the symbol.
[Figs. 10 and 11] The captions state T=0.7, but the plotted quantity F_L is defined by Eq. (13) in terms of η; the value of η used (presumably η=0.7) should be stated explicitly.
[Figs. 2–13] The value of the delocalization coefficient t is not reported for the plotted curves; for reproducibility, state whether t is optimized pointwise or fixed for each figure.
[Eq. (2) and Appendix A] The quantity Q_{m,x1,y1,x2,y2} is used in Eq. (2) but defined only in Appendix A; a brief forward reference or a one-line definition in Sec. II would improve readability.
[throughout] There are several typographical errors, including 'gardually' for 'gradually', 'equivals' for 'equivalent', and 'differerence' for 'difference'.
[References] The reference formatting is inconsistent, e.g., 'Phys. Rev, A', 'Phys. Rev. L.', and some entries with incomplete author lists; please standardize the bibliography style.
[Abstract and Conclusion] The phrase that D-PSO can 'cover and even exceed the advantages of the L-PSO on two modes' is stronger than the parameter-dependent numerical evidence; it should be qualified with the ranges of φ, α, g, m, and T for which the comparison holds.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central phase-sensitivity and QFI results are direct evaluations of standard error-propagation and quantum-Fisher-information definitions, with external benchmarks; the flagged loss-model mismatch is a consistency concern, not a circular derivation.

full rationale

The paper's central derivation is self-contained. The phase sensitivity in Eq. (3) is the standard error-propagation formula, and the QFI in Eqs. (4)-(7) uses the standard pure-state definition; all expectation values are computed by direct operator calculus from the model state in Eq. (1) via the generating-function method in Appendix A. The comparison benchmarks (SQL, HL, QCRB) are external standards, and the parameter t is a control parameter of the proposed D-PSO operation, not a parameter fitted to a target sensitivity. The lossy-QFI formula Eq. (13) is taken from Escher et al. (Ref. [50]), an external established result, so it provides independent support rather than relying on a self-citation. One minor caveat is that D-PSO contains the two L-PSO operations as endpoints of Eq. (1) (s=1, t=0 and s=0, t=1), and the D-PSO curves are obtained by optimizing t; therefore the statement that D-PSO can 'cover' the L-PSO advantages is partly guaranteed by construction. However, the nontrivial claims of exceeding L-PSO and approaching the QCRB still require the explicit calculations, and the central derivation does not reduce to its inputs by definition. The inconsistency between the single-mode loss parameter eta in Sec. IV B and the two-beam-splitter transmissivity T in Sec. II is a correctness and modeling concern, not a circularity, and does not affect the ideal-case results.

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

The central claim rests on standard interferometer and QFI formalism plus one ad hoc assumption: that the loss parameter eta in the Fisher information formula maps directly onto the transmissivity T used in the phase sensitivity model. This mapping is never stated, so the loss robustness claim depends on an unverified identification.

free parameters (2)

t = optimized numerically, not reported
The D-PSO operation (s a + t b)^m includes coefficients s and t with s+t=1; the paper optimizes t to minimize phase sensitivity but never gives the optimal values or a closed form.
m = 1, 2, 3
The order of the photon subtraction operation is scanned as a control parameter; higher m improves sensitivity but the paper does not derive an optimal m.

assumptions (5)

domain assumption The SU(1,1) interferometer is modeled as two balanced OPAs with a phase shift between them, described by two-mode squeezing transformations.
Sec. II, Eqs. (1) and the setup; this is the standard model from Yurke et al.
domain assumption The D-PSO operation (s a + t b)^m is physically realizable as a delocalized photon subtraction across both modes, following delocalized photon-addition experiments.
Sec. II; the paper cites Biagi et al. for experimental feasibility, but no experimental implementation of subtraction is given.
standard math The QFI for a pure state with a phase shift generated by n_a is F = 4 Var(n_a).
Eq. (5) in Sec. IV A.
standard math The lossy QFI obeys the Escher et al. formula F_L = 4*eta*<n>*<Delta n^2> / ((1-eta)*<Delta n^2> + eta*<n>).
Eq. (13) in Sec. IV B, citing [50]. The derivation assumes a specific loss channel and phase-shift ordering (lambda).
ad hoc to paper Internal photon loss is modeled by fictitious beam splitters with transmissivity T, and the loss parameters T and eta are treated equivalently.
Sec. II uses T for phase sensitivity; Sec. IV B uses eta for QFI. The paper never states eta = T or fixes the loss location lambda.

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Cite this review

Pith. "Pith review of Phase estimation via delocalized photon subtraction operation inside the SU(1,1) interferometer." pith.science (2026). https://pith.science/paper/OZNME543

@misc{pith2026250607684,
  author       = {Pith},
  title        = {Pith review of: Phase estimation via delocalized photon subtraction operation inside the SU(1,1) interferometer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZNME543}},
  note         = {Machine review of arXiv:2506.07684}
}

read the original abstract

We propose a theoretical scheme to improve the precision of phase measurement using intensity detection by implementing delocalized photon subtraction operation (D-PSO) inside the SU(1,1) interferometer, with the coherent state and the vacuum state as the input states. We compare the phase sensitivity and the quantum Fisher information between D-PSO and localized photon subtraction operation (L-PSO) under both ideal and photon-loss cases. It has been found that the D-PSO can improve the measurement accuracy of the SU(1,1) interferometer and enhance its robustness against internal photon loss. And it can cover and even exceed the advantages of the L-PSO on two modes, respectively. In addition, by comparing the standard quantum limit, the Heisenberg limit and quantum Cram\'er-Rao bound, we find that the phase sensitivity of the D-PSO can get closer to the quantum Cram\'er-Rao bound and has the ability to resist internal loss.

Figures

Figures reproduced from arXiv: 2506.07684 by the authors.

**Figure 2.** FIG. 2: The comparison between D-PSO and L-PSO, about [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: The comparison between D-PSO and L-PSO, about [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: The comparison between D-PSO and L-PSO, about [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 7.** Figure 7: FIG. 7: The comparison between D-PSO and L-PSO, on QFI as [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: FIG. 8: The calculation model of the Fisher information under [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: FIG. 9: The comparison between D-PSO and L-PSO, on [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: FIG. 10: The comparison between D-PSO and L-PSO, on [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]

**Figure 11.** Figure 11: FIG. 11: The comparison between D-PSO and L-PSO, on [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]

**Figure 13.** Figure 13: FIG. 13: The comparison of the phase sensitivity [PITH_FULL_IMAGE:figures/full_fig_p007_13.png]

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