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Phase sensitivity via photon-subtraction operations inside Mach-Zehnder interferometer

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Multi-photon subtraction inside a Mach-Zehnder interferometer can push phase sensitivity below the Heisenberg limit, even under internal photon loss.

desk verdict Competent analytic study of photon subtraction inside an MZI, but the 'Heisenberg-limit breaking' relies on a post-selected photon-number resource count that ignores failed subtraction trials, and the appendix quadrature formulas need correction. read the letter →

arxiv 2505.04499 v1 pith:PJH66YYO submitted 2025-05-07 quant-ph

Qisi Zhou , Qinqian Kang , Tao Jiang , Zekun Zhao , Teng Zhao , Cunjin Liu , Liyun Hu This is my paper · ORCID
classification quant-ph PACS 03.67.-a05.30.-d42.50.Dv03.65.Wj
keywords photonsubtractionMach-ZehnderinterferometerphasesensitivityquantumFisherinformationHeisenberglimithomodynedetectionlossnon-Gaussianoperations
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 proposes putting multi-photon subtraction inside a Mach-Zehnder interferometer, between the first beam splitter and the phase shifter, with a coherent state in one input port and a squeezed vacuum in the other. It claims that this non-Gaussian operation improves phase sensitivity under both intensity and homodyne detection, and that the improvement survives internal photon loss. Under intensity-difference detection, subtracting photons from both arms gives the largest gain; under mode-b homodyne detection, subtracting photons from the phase-carrying arm can push the sensitivity below the Heisenberg limit $1/N$, even at a transmittance of $T=0.7$. The paper also shows that the quantum Fisher information grows with the number of subtracted photons for the best schemes. A sympathetic reader would care because the scheme uses standard ingredients coherent light, squeezed vacuum, and heralded photon subtraction to claim a loss-tolerant route past the standard quantum limit.

What carries the argument

The load-bearing object is the universal moment formula $\langle \hat a^{\dagger p_1}\hat a^{p_2}\hat b^{\dagger q_1}\hat b^{q_2}\rangle = A^2 D_{m,n,p_1,p_2,q_1,q_2}e^M$, which expresses every expectation value needed for phase sensitivity and Fisher information as derivatives of a single generating function. The photon-subtraction operations $\hat a^m\hat b^n$ insert extra annihilation operators into that generating function, and the normalization constant $A$ is fixed by the same formula at zero photon-count moments. Phase sensitivity is then evaluated through the error-propagation formula for the chosen detection observable, and the lossy Fisher information is obtained from the purification bound $F_L = 4\eta\langle \hat n_a\rangle F / [(1-\eta)F+4\eta\langle \hat n_a\rangle]$. This machinery turns the whole scheme into a direct calculation of moments without truncating the state.

What would settle it

Measure or compute the phase sensitivity of scheme A with $m=2,3$ and mode-b homodyne detection using the total input photon number before the heralded subtraction, including discarded events; if $\Delta\phi$ is never below $1/\sqrt{N_{\rm in}}$ at $T=0.7$, the claimed sub-Heisenberg operation is an artifact of the resource accounting. A laboratory version would count the input coherent and squeezed photon flux and the heralding efficiency, then test whether the variance per trial beats the standard quantum limit.

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Extended reading notes

Core claim

The central claim is that multi-photon subtraction performed inside the interferometer, through operations $\hat a^m$, $\hat b^n$, or $\hat a^m\hat b^n$ applied after the first beam splitter, reshapes the conditional output state so that phase estimation improves beyond what the same interferometer achieves without subtraction. For intensity detection, the intensity-difference observable $N_-$ is the best choice and the two-mode scheme $m=n=1$ gives the largest improvement, with schemes A and B identical. For homodyne detection, detecting the quadrature of mode $b$ is best, and only the mode-$a$ subtraction scheme improves sensitivity; at $m=2,3$ this scheme breaks the standard quantum limit and, at $m=3$ in the ideal case and $m=1$ under $T=0.7$ loss, breaks the Heisenberg limit $1/N$. The quantum Fisher information is enhanced by the subtraction, with the symmetric two-mode scheme performing best over wide parameter ranges. All of these conclusions use $N$ defined as the mean photon number of the post-subtraction state inside the interferometer.

Load-bearing premise

The comparison against the standard quantum limit and the Heisenberg limit uses $N$ as the mean photon number of the state after the probabilistic photon subtraction has succeeded, treating the subtracted photons and all failed herald events as free; if one instead counts the full input photon flux before subtraction, the claimed advantage may not survive.

Editorial extensions

If this is right

  • For intensity detection, the intensity-difference observable $N_-$ is the best option, and the symmetric scheme $m=n=1$ gives the largest phase-sensitivity improvement among the three subtraction configurations.
  • For homodyne detection, measuring the $b$-mode quadrature is optimal, and only subtracting photons from mode $a$ improves sensitivity; subtracting from mode $b$ or from both modes degrades it.
  • Increasing the photon-subtraction number $m$ in scheme A improves both phase sensitivity and quantum Fisher information.
  • Under internal photon loss with transmittance $T=0.7$, scheme A with mode-b homodyne detection still breaks the Heisenberg limit, whereas the standard MZI does not even reach the standard quantum limit.
  • The symmetric scheme C gives the highest quantum Fisher information over wide parameter ranges, particularly at small coherent amplitude and larger squeezing.

Reading between the lines

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

  • If the resource accounting is changed to include the photons consumed by the heralded subtraction and the failed subtraction events, the claimed beating of the SQL and HL may shrink or disappear; that accounting test is the most direct check of the practical advantage.
  • The loss tolerance of scheme A suggests photon subtraction acts as a conditional non-Gaussian filter that reshapes the state toward one with higher Fisher information; comparing it against photon addition or photon catalysis at equal post-selected photon number would show how specific the effect is.
  • The generating-function method used here can be applied directly to other non-Gaussian operations placed inside the interferometer, such as photon addition or number-conserving operations, and to other interferometric layouts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript proposes a phase-estimation scheme in which m and n photons are subtracted from the two modes of a Mach-Zehnder interferometer after the first beam splitter, using a coherent state in mode a and a squeezed vacuum in mode b. It derives a generating-function expression for normally ordered moments of the output state and uses it to compute phase sensitivity for intensity and homodyne detection, together with ideal and lossy quantum Fisher information and quantum Cramér-Rao bounds. The central numerical claims are that photon subtraction improves phase sensitivity, that intensity-difference detection can surpass the standard quantum limit even with internal loss, and that mode-b homodyne detection can break the Heisenberg limit 1/N even under loss.

Significance. The analytic generating-function framework for arbitrary subtraction orders m and n is a useful methodological contribution, and the paper covers a broad parameter space in a systematic way, including both ideal and lossy operation. The manuscript contains no fitted parameters; all plotted curves are direct evaluations of the derived expressions, which is a positive feature. If the central claims survive a fair-resource accounting, the scheme would be interesting because it combines Gaussian inputs with experimentally feasible non-Gaussian operations and compares detection strategies under loss. However, the headline sub-Heisenberg claim currently rests on a conditional resource metric that needs to be tested, and the homodyne covariance formula in the appendix is incorrect.

major comments (2)
  1. [Eq. (12), Figs. 5 and 9, abstract] The SQL/HL comparison defines N by Eq. (12) as the mean photon number of the conditional post-subtraction state inside the interferometer. Because the subtraction a^m b^n is probabilistic, the resources actually consumed include all rejected trials; in the ideal projection model the success probability is p_success = 1/A^2, and the input mean photon number is |α|^2 + sinh^2 r. With the current definition, the claimed breaking of the Heisenberg limit in Fig. 9 and the abstract is a comparison against a limit evaluated for a smaller resource count than the one invested. A fair comparison should use N_fair = (|α|^2 + sinh^2 r)/p_success, or should include p_success in the QCRB in Eq. (21), e.g. Δϕ ≥ 1/sqrt(p_success v F). The manuscript does not provide such a check, so the headline loss-tolerant sub-Heisenberg claim is not supported as stated. In the lossy case this is especially acute, because loss further reduces the post-selected mean photon number used to draw the Heisenberg-limit curve.
  2. [Appendix A, Eqs. (A6)-(A8)] The homodyne formulas are inconsistent with the definition in Eq. (17). For X_a = (a+a†)/√2, the second moment is (1/2)(⟨a^2⟩+⟨a†^2⟩+2⟨a†a⟩+1), whereas Eq. (A6) uses the unnormalized form X = a+a†, and similarly for Eq. (A7). In addition, Eq. (A8) for cov[X_a, X_b] is identical in form to the intensity covariance in Eq. (A5); the true quadrature covariance must involve terms such as ⟨a b⟩, ⟨a b†⟩, ⟨a† b⟩, and ⟨a† b†⟩ minus the corresponding product of means. The single-quadrature cases X_a and X_b used in Figs. 6-9 are invariant under the missing global 1/√2 factor in the error-propagation ratio, so this may not change those curves, but the general formula as presented is wrong and would corrupt any optimized combination c2,d2 of quadratures. Please correct these expressions and confirm the numerical results with the corrected formulas.
minor comments (5)
  1. [Sec. II and Fig. 12] The operator order in Eq. (1) places the loss operator B_Lw after the photon subtraction a^m b^n, while Sec. IV.B and Fig. 12 are ambiguous about whether the loss occurs before or after the subtraction; please clarify the terminology and make the notation consistent.
  2. [Fig. 3 caption] The caption labels the panels as (a), (b), (c) but the text describes panel (a) twice; the panel labels should be corrected.
  3. [Eqs. (A5) and (A8)] There are typographical errors in the last factors, e.g. 'Dm,n1,1,0,0,eM' should be 'D_{m,n,1,1,0,0}e^M'; please fix these expressions.
  4. [References] Reference [24] is dated 1900; it should be 1990. Also, 'phonton losses' in Sec. II and 'experimentlly' in Sec. II should be corrected.
  5. [Sec. III.A.1.c] The text refers to 'Fig. 6(a)' and 'Fig. 6(b)' when discussing the SQL/HL comparison for intensity detection, but the relevant figure appears to be Fig. 5; the cross-references should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all computed phase sensitivities and QFI values are direct analytic evaluations of the stated input states and operations; the SQL/HL comparison is a resource-accounting choice, not a circular reduction.

full rationale

The paper's derivation chain is self-contained. The output state is defined in Eq. (1) as a sequence of beam splitters, phase shift, losses, and photon-subtraction operations; the universal expectation formula Eq. (2) follows by differentiation of a generating function constructed from those operators. Phase sensitivities in Eqs. (A1)-(A8) are direct applications of the error-propagation formula Eq. (10) to the expectation values of intensity or quadrature operators, and the QFI in Eqs. (20) and (29) comes from the standard pure-state QFI formula Eq. (18) and the Escher et al. loss formula Eq. (28), both cited to independent external literature. No parameter is fitted to a subset of data and then renamed a prediction; the plotted curves are evaluations of the derived formulas with chosen input parameters (α, r, m, n, T). The SQL/HL comparison uses N defined in Eq. (12) as the mean photon number of the normalized post-subtraction state inside the MZI before the second beam splitter. That is a stated resource metric; whether it should instead count the photon flux consumed in failed heralding trials is a substantive physical criticism about post-selection accounting, but it is not a circularity, because N is not defined in terms of the quantity being predicted and the comparison does not presuppose the claimed advantage. Refs. [45] and [46] are by overlapping authors, but they are used only as motivation for studying internal photon operations, not as the mathematical basis for the new formulas; the universal formula, phase sensitivities, and QFI expressions are derived within this paper. Therefore there is no load-bearing circular step and the central result retains independent content.

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

The scheme uses standard ingredients (coherent and squeezed states, beam splitters, phase shift, vacuum loss channels) plus idealized photon subtraction. The main unproven inputs are the applicability of the Escher loss formula, the treatment of subtraction as resource-free, and the correctness of the universal moment formula. No new physical entities are introduced.

assumptions (5)
  • domain assumption The fictitious-beam-splitter loss model with vacuum environment correctly describes internal photon losses in the interferometer.
    Introduced in Sec. II with operators B_La, B_Lb and transmissivity T_k; all lossy phase sensitivity results rely on this noise model.
  • domain assumption The Escher purification-limit formula FL = 4η⟨n_a⟩F / ((1-η)F + 4η⟨n_a⟩) applies to the loss configuration considered here.
    Used in Sec. IV.B (Eq. (28), citing Ref. [66]) without deriving the loss-position dependence or specifying the loss factor γ; this formula drives the lossy QFI claims.
  • ad hoc to paper The photon-subtraction operation â^m b^n is noiseless and its probabilistic success does not add to the resource count.
    Used in Eq. (1) and Eq. (12); the SQL/HL comparison defines N after subtraction, implicitly ignoring the success probability of the heralded subtraction.
  • standard math The error propagation formula (Eq. (10)) is a valid estimator of phase sensitivity for the states and measurements used.
    Standard in quantum metrology; used for all phase sensitivity plots.
  • standard math The universal generating-function expression (Eqs. (2)-(8)) correctly computes the moments of the output state.
    The paper does not prove this formula in the main text; all intensity and homodyne results depend on it.

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Cite this review

Pith. "Pith review of Phase sensitivity via photon-subtraction operations inside Mach-Zehnder interferometer." pith.science (2026). https://pith.science/paper/PJH66YYO

@misc{pith2026250504499,
  author       = {Pith},
  title        = {Pith review of: Phase sensitivity via photon-subtraction operations inside Mach-Zehnder interferometer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJH66YYO}},
  note         = {Machine review of arXiv:2505.04499}
}
read the original abstract

Based on the conventional Mach-Zehnder interferometer, we propose a metrological scheme to improve phase sensitivity. In this scheme, we use a coherent state and a squeezed vacuum state as input states, employ multi-photon-subtraction operations and make intensity-detection or homodyne-detection. We study phase sensitivity, quantum Fisher information and quantum Cram\'er-Rao bound under both ideal and lossy conditions. The results indicate that choosing an appropriate detection method and photon subtraction scheme can significantly enhance the phase sensitivity and robustness against photon losses. Even under lossy conditions, the multi-photon subtraction schemes can surpass the standard quantum limit. Notably, the homodyne detection method can even break through the Heisenberg limit. Moreover, increasing the number of photon-subtracted can enhance both phase sensitivity and quantum Fisher information. This research highlights the significant value of this scheme in quantum precision measurement.

Figures

Figures reproduced from arXiv: 2505.04499 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of a MZI with multi-PSSs. The two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The phase sensitivity of multi-PSSs based on intensity [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The phase sensitivity of scheme A based on [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The phase sensitivity of scheme A as a function of trans [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The phase sensitivity of the multi-PSSs based on the ho [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The phase sensitivity of scheme A as a function of trans [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of the phase sensitivity of scheme A based [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The phase sensitivity of scheme A based the homodyne [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The QFI as a function of (a) the coherent amplitude [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Schematic diagram of the photon losses on mode [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

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