{"id":"29022ba6-53ef-43ea-a323-7f7dddd285a1","arxiv_id":"2505.10161","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Photon-added GHZ-type coherent states are claimed to lower multiparameter phase-estimation errors under simultaneous and nonlinear estimation compared with NOON and entangled-coherent states, in the paper's analytical models.","lead":"This paper proposes a new family of light states, made by adding photons to GHZ-type entangled coherent states, and claims they can estimate several optical phases simultaneously with very high precision. The paper says that measuring all phases at once beats measuring them one by one, especially with nonlinear phase shifts.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equations (40)–(41) rest on a QFIM (38) that assigns mode-0 PACS moments to the ordinary coherent probe modes; direct recomputation of F from Eq. (34) will not reproduce them.","rationale":"The reader correctly identified the QFIM derivation as the weakest load-bearing step, and I agree that Eqs. (40)-(41) are not established. My stress-test refines the diagnosis: the problem is not simply that 'all d+1 modes contribute identical moments,' but that the moments written in Eq. (36) are the mode-0 photon-added coherent state moments even though H_p acts on modes 1..d, which are ordinary coherent states in each branch. Even the n=0 limit of the formulas disagrees in the overlap exponent with a direct evaluation. The paper contains no independent verification (no machine-checked proof, no reproducible numerics, no alternative derivation), and the homodyne approximations (49)-(50) are likewise asserted without derivation. Because the main numerical figures and the abstract's superiority claims are generated from the unverified QFIM expressions, the correct disposition is to reject the submission in its present form, while leaving open that a corrected derivation might salvage the qualitative comparison.","tokens_in":15104,"tokens_out":12913,"duration_ms":126223,"concrete_test":"Take d=2, n=1, |α|=1 (or the simpler n=0 limit) and recompute the 2×2 QFIM for state (34) directly from Eq. (15) with H_p=a†_p a_p, p=1,2, using the explicit expansion (20) and overlaps (23)-(24). Compare the diagonal element F_{11} and the off-diagonal element F_{12} with Eqs. (38)-(39) evaluated at the same parameters. If either element differs — the direct diagonal contains coherent-state Poisson-type moments and e^{-2(d+1)|α|²} interferences while (38) contains mode-0 Laguerre expressions — then Eqs. (39)-(40) are incorrect and the central QCRB claim is unsupported.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's headline quantitative claim (simultaneous QCRBs, Eqs. (40)-(41)) depends entirely on the QFIM structure in Eqs. (36)-(39). For the probe in Eq. (34), only mode 0 is photon-added; the estimated parameters are phases imprinted by H_p = a†_p a_p on modes 1..d, which contain ordinary coherent states in each branch. Eq. (36) nevertheless writes <H_p> = b(α,n,d) h(α,n,d), with h built from mode-0 PACS moments: it contains [(n+1)!L_{n+1}(±|α|²)-n!L_n(±|α|²)]/[n!L_n(-|α|²)] and the overlap factor e^{-2d|α|²}. Direct evaluation with H_p acting on modes 1..d gives different diagonal moments (for example, the n=0 limit contains |α|² and e^{-2(d+1)|α|²} interference, not e^{-2d|α|²}), and <H_p H_q> for p≠q is nonzero because of interference between the two branches. Hence F_{pq}=4[<H_p H_q>-<H_p><H_q>] does not reduce to the one-parameter form 4bg(δ_{pq}-(bh²/g)J) of Eqs. (38)-(39), whose inversion produces the d(√d+1)² prefactor in Eq. (40). Since Eqs. (40)-(41), all figures, and the abstract's 'simultaneous estimation generally outperforms' conclusion rest on this step, the central claim is not established by the submitted derivation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multiparameter phase estimation with GHZ-type photon-added coherent states (PACS). It derives quantum Cramér-Rao bounds for independent estimation (Sec. 3.1), simultaneous linear and non-linear estimation (Sec. 3.2), compares the results with NOON and entangled-coherent states (Sec. 4), and analyzes homodyne detection (Sec. 5). The central quantitative claims are that simultaneous estimation outperforms independent estimation and that PACS-based GHZ states achieve the highest precision, especially as the photon-excitation number n increases. These claims rest on the simultaneous quantum Fisher information matrix in Eqs. (36)-(39) and the total-variance formulas in Eqs. (40)-(41).","tokens_in":15497,"tokens_out":22984,"duration_ms":216921,"significance":"If the bounds in Eqs. (40)-(41) were correct, the paper would provide a useful extension of multiparameter optical metrology to photon-added coherent states and a concrete comparison with existing NOON and ECS results. The state construction is natural, the independent-estimation expressions in Sec. 3.1 are explicit, and the paper addresses three different protocols, which is a useful framing. However, the central simultaneous-estimation result is not established: the QFIM is computed from moments that do not correspond to the actual probe state, and the final variance formulas do not follow from the matrix that is inverted. Because the abstract and the conclusions draw their main message from these formulas, the paper in its current form does not support its advertised conclusions.","major_comments":[{"comment":"The QFIM is not a property of the probe state in Eq. (34). In that state, mode 0 is a photon-added coherent state and modes 1,...,d are ordinary coherent states, but the moments in Eqs. (36)-(37) are n-dependent PACS moments and are used as if they applied to every mode. For H_p = a†_p a_p with p = 1,...,d, a direct calculation gives, writing x=|α|² and S = cos(lπ)e^{-2(d+1)x} L_n(x)/L_n(-x) with 2N_l² = 1/(1+S): ⟨H_p⟩ = x(1-S)/(1+S), ⟨H_p²⟩ = x² + x(1-S)/(1+S), and ⟨H_p H_q⟩ = x² for p≠q. These expressions do not have the structure ⟨H_p H_q⟩ = δ_pq b g and ⟨H_p⟩ = b h with b,g,h given by Eq. (37); in particular, the branch-overlap factor in Eq. (37) is e^{-2d|α|²}, whereas the normalization in Eq. (35) and the true branch overlap require e^{-2(d+1)|α|²}. Thus Eq. (38) assigns PACS-mode moments to the coherent modes and omits the nonzero off-diagonal second moments, so the inversion leading to Eq. (40) is not a consequence of the defined state.","section":"3.2, Eqs. (36)-(39)"},{"comment":"Even accepting Eq. (38), the total-variance formula (40) does not follow. With F = 4bg(I - aJ), a = bh²/g, the inverse has eigenvalues 1/[4bg(1-ad)] and 1/(4bg) (multiplicity d-1), so Tr(F^{-1}) = d[1 - a(d-1)]/[4bg(1-ad)]. This is not equal to d(√d+1)²h²/(4g²) for the quantities defined in Eq. (37). The prefactor d(√d+1)² is the known NOON-state result from Ref. [48] and appears here without derivation; Eq. (41) inherits the same problem. A concrete consistency check is d=1: for n=0 and large |α|², Eq. (40) gives |δφ|² ≈ 1/|α|^4, whereas the exact single-parameter QFI for a phase shift on a coherent mode is 4|α|², giving a bound at most 1/(4|α|²). Equation (40) is therefore not only underived but numerically incompatible with the single-parameter limit.","section":"3.2, Eqs. (40)-(41)"},{"comment":"The output state written for the linear protocol is not the result of the d-parameter unitary U = exp(i Σ_{p=1}^d H_p φ_p). Equation (43) contains a single phase φ applied only to the PACS mode, |α e^{iφ}, n⟩_0, while all d coherent modes remain unshifted. This is a one-parameter phase shift, not a simultaneous d-parameter encoding. Consequently, the homodyne probability distribution in Eq. (47) and the variance approximations in Eqs. (49)-(50) do not describe the multiparameter simultaneous estimation problem analyzed in Sec. 3.2, and the comparison made in Sec. 5 is not with the protocol claimed in the abstract.","section":"5, Eq. (43)"}],"minor_comments":[{"comment":"The Laguerre polynomial is written as L_n = Σ ... x^k ... with no argument on the left-hand side; it should be L_n(x). In addition, the text says 'order m' but the subscript is n.","section":"2.1, Eq. (19)"},{"comment":"The line 'a α = |α|e^{iϕ}' should read α = |α|e^{iϕ}, and the denominator '(i)2' in the overlap sum should be '(i!)²'.","section":"2.1, Eq. (21)"},{"comment":"The same symbol I is used for both the identity matrix and the all-ones matrix; the latter should be denoted by J or a calligraphic symbol to avoid confusion.","section":"3.2, Eq. (39)"},{"comment":"The captions for Figures 2 and 3 contain duplicated or mismatched panel labels; for example, Figure 2 lists '(b) d=5 and l=1' and then '(b)|α|²=4 and n=1', and Figure 3 similarly mislabels panel (d).","section":"Figure captions, Figs. 2 and 3"},{"comment":"The comparison with NOON and ECS states should state the resource constraint explicitly. If the total photon number N̄ is not fixed across the states being compared, the claim that PACS-based GHZ states offer 'maximum precision' for larger n may be a trivial consequence of using more photons.","section":"4, resource comparison"}],"recommendation":"reject","confidential_remarks":"The flaws in Sec. 3.2 are load-bearing and not typographical: the QFIM does not correspond to the probe state, and the final formulas appear to be transplanted from the NOON-state literature rather than derived from the stated model. The homodyne section addresses a single-parameter protocol, not the d-parameter simultaneous protocol. A corrected manuscript would require recomputing the central simultaneous-estimation results from scratch, and the conclusions would likely change. I do not see a local fix that preserves the paper's central claims, so I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a modest extension of known multiparameter optical metrology to photon-added GHZ-type coherent states, and the extension is not carried through correctly. The independent-estimation section is plausible and internally consistent; the simultaneous-estimation section, which contains the paper's main claims, is not.\n\nWhat is new: replacing the coherent modes in a GHZ-type ECS with a photon-added coherent state in one mode, and writing the resulting QFI expressions with Laguerre polynomials. At n=0 the state reduces to the generalized ECS of Liu et al. [55]; for n>0 the expressions are new. The comparison to NOON and ECS is a reasonable exercise, and the homodyne analysis, while brief, is a sensible addition.\n\nWhere it breaks: the QFIM in Eqs. (36)–(38) is computed for the wrong operator moments. In the probe (34), only mode 0 is photon-added; modes 1,...,d are ordinary coherent states. Yet g and h in (37) contain only mode-0 PACS moments and an e^{-2d|α|²} factor. Direct evaluation with H_p = a_p†a_p on mode p gives ⟨H_p⟩ proportional to |α|² (1 − cos(lπ) e^{-2(d+1)|α|²} L_n/L_n(-)), not the e^{-2d} form, and it gives nonzero off-diagonal second moments ⟨H_p H_q⟩ from cross-branch interference. So F does not have the one-parameter-plus-J form of Eq. (39), and the inversion leading to Eq. (40) is unsupported. The d(√d+1)² prefactor looks imported from the NOON/ECS results rather than derived here. Eq. (43) also phase-encodes only mode 0, which is inconsistent with a d-parameter protocol if the generator is supposed to act on every mode. The homodyne approximations (49)–(50) are asserted without derivation; these are secondary, but the QFIM issue is load-bearing because the abstract's main conclusion rests on Eqs. (40)–(41).\n\nProportion: the rest of the paper is organized honestly and cites the relevant literature. This is not a case of invented data or circular fitting; it is a calculation error that invalidates the central quantitative claims.\n\nWho this is for: a reader interested in PACS-based probes might find the independent estimates and the state definitions useful, but the simultaneous results should not be used. I would not send this to referees as submitted. The authors need to redo the QFIM computation, show the off-diagonal terms, and either re-derive or withdraw the simultaneous bounds. After that revision it might be worth another look.","headline":"The independent-estimation part is credible, but the simultaneous QFIM is miscomputed: Eq. (36) assigns PACS moments to ordinary coherent modes and drops off-diagonal covariances, so the paper's headline QCRBs do not follow.","tokens_in":15983,"tokens_out":6715,"would_cite":false,"duration_ms":61800,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Photon-added GHZ-type coherent states can estimate multiple phases at once with precision that beats estimating each phase separately.","keywords":["multiparameter quantum metrology","photon-added coherent states","GHZ-type coherent states","quantum Cramér-Rao bound","simultaneous vs independent estimation","nonlinear phase estimation","homodyne detection","NOON states"],"falsifier":"Evaluate the normalized state of Eq. (34) for $n=1$, $d=2$ and compute the $2\\times 2$ quantum Fisher information matrix by direct differentiation without imposing equal moments; if $\\mathrm{Tr}(F^{-1})$ differs from Eq. (40) at any value of $|\\alpha|^2$, the central bound is not exact.","tokens_in":14826,"feed_emoji":"⚛️","tokens_out":6076,"duration_ms":62495,"temperature":0.7,"pith_summary":"This paper argues that a class of multi-mode optical probes—Greenberger-Horne-Zeilinger-type coherent states in which one mode carries $n$ added photons while $d$ other modes are ordinary coherent states—can estimate several phase shifts simultaneously with precision that surpasses estimating each phase on its own. It derives closed-form quantum Cramér\\-Rao bounds for three strategies: independent estimation, simultaneous linear phase shifts, and simultaneous nonlinear phase shifts. The claimed result is that simultaneous estimation wins, the nonlinear protocol wins among simultaneous ones, and precision improves as the coherent amplitude $|\\alpha|^2$ and the photon-excitation number $n$ grow while the number of estimated parameters $d$ shrinks. Along the way the paper compares these photon-added GHZ states with NOON states and entangled coherent states, finding the photon-added GHZ states give the tightest bounds.","feed_headline":"Photon-added GHZ states beat separate phase estimation","feed_subtitle":"A closed-form bound shows one simultaneous pass wins over d independent runs, and nonlinear shifts sharpen further.","key_machinery":"The central object is the photon-added multi-mode coherent state of GHZ type: one reference mode prepared in a photon-added coherent state $|\\alpha,n\\rangle$, obtained by applying the creation operator $n$ times to a coherent state, entangled with $d$ ordinary coherent modes $|\\alpha\\rangle$ and their sign-flipped counterparts $|-\\alpha\\rangle$. The argument runs through the quantum Fisher information matrix of a pure state under commuting local phase generators $H_p$, which becomes $F_{pq} = 4(\\langle H_p H_q\\rangle - \\langle H_p\\rangle\\langle H_q\\rangle)$. Because all generators commute and act locally, the paper writes $F$ as a diagonal term plus a rank-one term proportional to the all-ones matrix, and inverts that structure to obtain the $d(\\sqrt{d}+1)^2$ factor. The quantities $g$, $h$, $r$, and $s$ encode the photon statistics through Laguerre polynomials $L_n$, and their ratios control the resulting precision bounds.","core_discovery":"The paper claims that for the probe state $|\\Psi_s\\rangle = \\mathcal{N}_l(\\alpha,n,d)[|\\alpha,n\\rangle_0 \\otimes_{i=1}^d |\\alpha\\rangle_i + e^{il\\pi}|-\\alpha,n\\rangle_0 \\otimes_{i=1}^d |-\\alpha\\rangle_i]$, the quantum Fisher information matrix for simultaneous estimation has the structure $F = 4 b g (I - (b h^2/g) \\mathbb{1})$, where $b$, $g$, and $h$ are functions built from Laguerre polynomials. Inverting this matrix gives the total variance bounds $|\\delta\\varphi|^2_L = d(\\sqrt{d}+1)^2 h^2/(4g^2)$ for the linear generators $H_p = a_p^\\dagger a_p$ and $|\\delta\\varphi|^2_{NL} = d(\\sqrt{d}+1)^2 s^2/(4r^2)$ for the nonlinear generators $(a_p^\\dagger a_p)^2$. The paper's main conclusion is that these simultaneous bounds are lower than the independent-estimation bound $|\\delta\\varphi|^2_{\\mathrm{Ind}} = d/F$, especially in the nonlinear protocol; that precision improves as $|\\alpha|^2$ and $n$ increase and as $d$ decreases; and that among the states compared, the photon-added GHZ-type states yield the highest precision.","pith_inferences":["I would expect the $d(\\sqrt{d}+1)^2$ factor to be inherited from the NOON-style multi-mode superposition rather than from the photon-added modification, with added photons mainly rescaling the ratios $h/g$ and $s/r$; a direct numerical inversion of the exact Fisher matrix for $d=2, n=1$ could separate these two contributions.","A resource-fair comparison would fix the total mean photon number across PACS-GHZ, NOON, and entangled coherent states, since adding $n$ photons raises the resource count and may account for part of the reported advantage.","The homodyne approximations suggest a concrete experimental route: prepare single-photon-added coherent states, imprint $d$ small phase shifts, and test whether the variance scales as predicted with $n$, $\\alpha$, and $d$ in the small- and large-amplitude regimes."],"forward_implications":["Simultaneous estimation with these states beats independent estimation in the linear protocol, and the gap becomes much larger in the nonlinear protocol, where the bounds drop to scales around $10^{-3}$.","Adding photons to the reference mode acts as a metrological resource: at fixed coherent amplitude, larger $n$ lowers the Cramér\\-Rao bound in all three protocols.","For large coherent amplitudes $|\\alpha|^2$, the linear and nonlinear bounds become comparable and the total average photon number approaches $|\\alpha|^2 - 1$, placing the states near the Heisenberg limit.","Homodyne detection is close to the quantum limit for intense coherent states, but is less effective than optimal linear estimation for small amplitudes and for many estimated parameters.","Among the states compared in the paper, photon-added GHZ-type states give the lowest Quantum Cramér\\-Rao bound, with the antisymmetric version slightly better at large photon-excitation numbers."],"supporting_citations":[{"why":"Supplies the unitary-parametrization formalism and the quantum Fisher information expression used throughout the derivations.","marker":"[41]"},{"why":"Provides the generalized NOON-state simultaneous estimation result and the comparison baseline the paper builds on.","marker":"[48]"},{"why":"Gives the generalized entangled coherent state multiparameter protocol whose Fisher matrix methods are adapted to photon-added states.","marker":"[55]"},{"why":"Establishes the nonlinear phase-shift metrology framework used for the nonlinear protocol and its comparisons.","marker":"[56]"},{"why":"Supplies the entangled coherent state metrology benchmark and Heisenberg-limit context for the state comparisons.","marker":"[58]"},{"why":"Defines photon-added coherent states, the core resource studied here.","marker":"[65]"},{"why":"Demonstrates experimental generation of single-photon-added coherent states, supporting the practical relevance of the probe states.","marker":"[66]"}],"fun_headline_variants":["Simultaneous phase estimation wins with photon-added GHZ states","Photon-added GHZ states sharpen multiparameter bounds","Nonlinear probes boost simultaneous multiphase estimation","Closed-form QCRB for photon-added GHZ metrology","Photon-added GHZ states beat NOON in precision"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation of the simultaneous bounds assumes that all $d+1$ modes contribute identical first and second moments of the phase generators, even though only mode 0 carries the $n$ added photons; if those moments are unequal, the closed-form Fisher-matrix structure and the $d(\\sqrt{d}+1)^2$ variance formula do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Simultaneous phase estimation wins with photon-added GHZ states","Photon-added GHZ states sharpen multiparameter bounds","Nonlinear probes boost simultaneous multiphase estimation","Closed-form QCRB for photon-added GHZ metrology","Photon-added GHZ states beat NOON in precision"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00096,"raw_usage":{"total_tokens":4176,"prompt_tokens":1117,"completion_tokens":3059,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":2978}},"tokens_in":733,"tokens_out":3059,"duration_ms":21848,"temperature":1.0,"reasoning_tokens":2978,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:16:36.751622+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the normalized state of Eq. (34) for $n=1$, $d=2$ and compute the $2\\times 2$ quantum Fisher information matrix by direct differentiation without imposing equal moments; if $\\mathrm{Tr}(F^{-1})$ differs from Eq. (40) at any value of $|\\alpha|^2$, the central bound is not exact.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the unitary-parametrization formalism and the quantum Fisher information expression used throughout the derivations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generalized NOON-state simultaneous estimation result and the comparison baseline the paper builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the generalized entangled coherent state multiparameter protocol whose Fisher matrix methods are adapted to photon-added states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the nonlinear phase-shift metrology framework used for the nonlinear protocol and its comparisons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the entangled coherent state metrology benchmark and Heisenberg-limit context for the state comparisons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines photon-added coherent states, the core resource studied here."},{"cited_title":"Zavatta, S","cited_arxiv_id":null,"evidence_quote":"Demonstrates experimental generation of single-photon-added coherent states, supporting the practical relevance of the probe states."}],"review_version":1}