{"id":"6f8bf3b4-498d-425e-8ca7-7e902f255f41","arxiv_id":"2505.04499","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Multi-photon subtraction inside a Mach-Zehnder interferometer can improve phase sensitivity and, with homodyne detection, push it below the Heisenberg limit even under loss.","lead":"This paper proposes putting photon-subtraction operations inside a Mach-Zehnder interferometer that uses a coherent state and a squeezed vacuum as inputs. The authors calculate that this can improve phase measurement precision and, with the right detection method, beat the standard quantum limit and even the Heisenberg limit, including under photon loss.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline result that homodyne detection breaks the Heisenberg limit (Fig. 9, abstract) uses the post-selected photon number N of Eq. (12) as its resource metric; a fair accounting that includes photons consumed in failed subtraction events is missing and may erase the advantage.","rationale":"The central claim is not merely that the proposed phase sensitivity is small, but that it beats the SQL and the HL under loss. That comparison has metrological meaning only if N counts the resources actually expended. Equation (12) counts the photons in the accepted branch, which is legitimate for describing the conditional state but not for benchmarking against SQL/HL. Because all rejected heralding events are discarded, the relevant per-estimate resource cost is larger by at least 1/p_success; for m>=1 at T=0.7 this factor is appreciable. This is the same weakest assumption flagged by the reader, and it is load-bearing because Figs. 5 and 9 and the abstract all rely on it. The Appendix A covariance expression is indeed questionable, but it does not affect the single-mode X_b curves that carry the headline claim, and the quadrature-variance normalization issue cancels in the error-propagation ratio if the denominator is computed with the same unscaled quadrature convention. The loss-model mismatch between T and eta should also be reconciled, but the resource-counting issue is the one that can invalidate the principal claim. I agree with the reader's conditional verdict: the paper should be revised to add a fair-resource comparison before its quantitative claims are adopted.","tokens_in":18503,"tokens_out":9582,"duration_ms":101439,"concrete_test":"Recompute Figs. 5 and 9 with the SQL and HL defined from N_fair=(|alpha|^2+sinh^2 r)/p_success, where p_success is the success probability of the photon-subtraction herald (the unnormalized norm <in|B1^dag a^dag^m b^dag^n a^m b^n B1|in> for the chosen subtraction), and repeat for the ideal (T=1) and lossy (T=0.7) curves for m=1,2,3. If delta_phi stops falling below 1/sqrt(N_fair) and 1/N_fair, the headline 'break through the Heisenberg limit' claim does not survive fair resource accounting. As a second, less stringent check, replace N by the total input photon number |alpha|^2+sinh^2 r alone, ignoring the success probability, and see whether any SQL/HL crossing survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (12) defines N as the mean photon number of the normalized post-subtraction state inside the interferometer, i.e. A^2 <in|B1^dag a^dag^m b^dag^n (a^dag a + b^dag b) a^m b^n B1|in>. Each m,n-photon subtraction is a heralded, probabilistic operation: the state is accepted only when the tap detector registers the desired subtraction, and all rejected trials still consume the coherent and squeezed inputs. The SQL and HL curves in Figs. 5 and 9 are therefore drawn for a resource count that omits the dominant resource expenditure. In the lossy case (T=0.7) the same post-selected N makes the Heisenberg-limit comparison even more favorable, because loss also reduces the accepted-state mean photon number. The QFI section does not repair this, since Eq. (28) is expressed through the same ideal-state averages. What has to be true for the central claim is that the advantage survives when N_fair=(|alpha|^2+sinh^2 r)/p_success replaces N in the SQL/HL definitions; the paper provides no such computation. This is a resource-metric issue, not a flaw in the formal interferometric calculation itself, but it is the load-bearing premise of the abstract's strongest statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18759,"tokens_out":13514,"duration_ms":141399,"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":[{"comment":"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.","section":"Eq. (12), Figs. 5 and 9, abstract"},{"comment":"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.","section":"Appendix A, Eqs. (A6)-(A8)"}],"minor_comments":[{"comment":"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.","section":"Sec. II and Fig. 12"},{"comment":"The caption labels the panels as (a), (b), (c) but the text describes panel (a) twice; the panel labels should be corrected.","section":"Fig. 3 caption"},{"comment":"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.","section":"Eqs. (A5) and (A8)"},{"comment":"Reference [24] is dated 1900; it should be 1990. Also, 'phonton losses' in Sec. II and 'experimentlly' in Sec. II should be corrected.","section":"References"},{"comment":"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.","section":"Sec. III.A.1.c"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially publishable after major revision if the authors can provide a fair-resource comparison for the SQL/HL claims and correct the homodyne formulas. If the advantage does not survive the fair accounting, the manuscript should be reframed as a parameter study without the sub-Heisenberg claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a competent analytic treatment of photon subtraction inside a Mach-Zehnder, but the headline claim—homodyne detection breaks the Heisenberg limit even with loss—rests on a resource definition that counts only the heralded post-subtraction photons. If you include the photons consumed in failed subtraction events, the advantage may shrink or disappear. The paper doesn't provide that calculation, and the abstract's strongest statement is therefore over-optimistic.\n\nWhat's genuinely new: moving the photon subtraction from the input ports to inside the MZI, and systematically comparing intensity-difference and mode-b homodyne detection. The generating-function machinery is standard for this group, and the three subtraction schemes (a-only, b-only, both) are worked out consistently. The identification of which detection method suits which scheme is a useful practical result. I found the core analytics coherent, and the plots are direct evaluations of the derived formulas—no curve fitting.\n\nSoft spots, in order of severity. (1) The SQL/HL comparison uses N as the mean photon number of the post-subtraction state (Eq. 12). That treats a heralded, probabilistic operation as if it cost nothing. A fair resource count would divide the input photon flux by the success probability. The paper doesn't address this, and it's load-bearing for the 'breaking the Heisenberg limit' claim. (2) Appendix A has real inconsistencies: the quadrature variances in (A6)-(A7) are missing the 1/2 from X=(a+a†)/√2, and (A8) is identical to the intensity covariance (A5) rather than the quadrature covariance. For the single-mode homodyne ratios used in the paper the missing factors cancel, so the plotted curves may survive, but the formulas as written are wrong and would mislead anyone using the general c2,d2 case. (3) The loss model in the phase-sensitivity section (two-mode transmissivity after subtraction, before the phase shifter) is not reconciled with the QFI section's single-mode loss via Escher's bound. Eq. (29) is borrowed from a different setup and its applicability to this geometry isn't argued.\n\nAlso minor: the input coherent+squeezed-vacuum state already beats the HL in the standard MZI for this homodyne detection (their own Fig. 9 shows m=0 crossing the HL). So the incremental advantage from photon subtraction is smaller than the abstract implies.\n\nWho this is for: quantum metrology specialists interested in non-Gaussian operations and detection schemes. It deserves a serious referee, but the revision needs to fix the appendix formulas and at minimum discuss the resource-counting issue. I wouldn't cite the quantitative claims until that's done.","headline":"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.","tokens_in":19331,"tokens_out":6592,"would_cite":false,"duration_ms":57370,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.-a","05.30.-d","42.50.Dv","03.65.Wj"],"model":"deepseek-v4-flash","headline":"Multi-photon subtraction inside a Mach-Zehnder interferometer can push phase sensitivity below the Heisenberg limit, even under internal photon loss.","keywords":["photon subtraction","Mach-Zehnder interferometer","phase sensitivity","quantum Fisher information","Heisenberg limit","homodyne detection","photon loss","non-Gaussian operations"],"falsifier":"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.","tokens_in":18258,"feed_emoji":"🔬","tokens_out":4833,"duration_ms":43662,"temperature":0.7,"pith_summary":"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.","feed_headline":"Photon subtraction beats the Heisenberg limit even with loss","feed_subtitle":"Inside a standard interferometer, subtracting photons from one arm lets homodyne detection outperform the quantum limit.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes that squeezed light injected into an interferometer can beat the SQL, the baseline this paper extends.","marker":"[21]"},{"why":"Introduces the error-propagation formula used to define phase sensitivity and the SU(1,1) interferometer context that motivates internal operations.","marker":"[26]"},{"why":"Shows photon addition inside an SU(1,1) interferometer improves phase sensitivity, the precedent for placing non-Gaussian operations inside the interferometer.","marker":"[45]"},{"why":"Studies multi-photon subtraction inside an SU(1,1) interferometer with homodyne detection, the closest prior scheme this paper adapts to the MZI.","marker":"[46]"},{"why":"Supplies the purification-limit framework used to compute quantum Fisher information under photon loss.","marker":"[65]"},{"why":"Provides the non-Gaussian squeezed vacuum states and their phase-sensitivity improvement that this paper compares with.","marker":"[43]"},{"why":"Demonstrates the experimental feasibility of photon subtraction from squeezed light, grounding the proposed operations.","marker":"[48]"}],"fun_headline_variants":["Photon subtraction boosts phase sensitivity even with loss","Homodyne detection with photon subtraction breaks Heisenberg limit","Subtracting photons inside MZI boosts precision beyond quantum limits","Multi-photon subtraction in interferometry enhances phase sensitivity","Photon subtraction inside interferometer sharpens phase measurement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Photon subtraction boosts phase sensitivity even with loss","Homodyne detection with photon subtraction breaks Heisenberg limit","Subtracting photons inside MZI boosts precision beyond quantum limits","Multi-photon subtraction in interferometry enhances phase sensitivity","Photon subtraction inside interferometer sharpens phase measurement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001067,"raw_usage":{"total_tokens":4465,"prompt_tokens":929,"completion_tokens":3536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":3459}},"tokens_in":545,"tokens_out":3536,"duration_ms":26482,"temperature":1.0,"reasoning_tokens":3459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:28:05.314792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Quantum metrology from a quantum information science perspective[J]","cited_arxiv_id":null,"evidence_quote":"Establishes that squeezed light injected into an interferometer can beat the SQL, the baseline this paper extends."},{"cited_title":"Two-photon interfer- ence in a Mach-Zehnder interferometer[J]","cited_arxiv_id":null,"evidence_quote":"Introduces the error-propagation formula used to define phase sensitivity and the SU(1,1) interferometer context that motivates internal operations."},{"cited_title":"K., & Panigrahi, P","cited_arxiv_id":null,"evidence_quote":"Shows photon addition inside an SU(1,1) interferometer improves phase sensitivity, the precedent for placing non-Gaussian operations inside the interferometer."},{"cited_title":"Enhanced phase estima- tion in parity-detection-based Mach–Zehnder interferom- eter using non-Gaussian two-mode squeezed thermal in- put state","cited_arxiv_id":null,"evidence_quote":"Studies multi-photon subtraction inside an SU(1,1) interferometer with homodyne detection, the closest prior scheme this paper adapts to the MZI."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the non-Gaussian squeezed vacuum states and their phase-sensitivity improvement that this paper compares with."},{"cited_title":"Phase estimation via multi-photon subtraction inside the SU (1, 1) interferometer","cited_arxiv_id":null,"evidence_quote":"Demonstrates the experimental feasibility of photon subtraction from squeezed light, grounding the proposed operations."}],"review_version":1}