{"id":"478e75a7-1e3e-4e14-b8e2-4c4f4b8c9dfe","arxiv_id":"2601.04364","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetry generators give optimal readouts for critical-state quantum sensors, and such sensors stay at or above the standard quantum limit under spin-flip, dephasing, and qubit-loss noise, with non-unitary deformation sometimes enhancing the quantum Fisher information.","lead":"Quantum critical states can act as high-precision sensors; this paper shows the symmetry-based measurements that make them work and how noise affects them. Critical probes stay useful where GHZ states fail, and some non-unitary deformations can even improve precision.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-unitary enhancement (Eq. 24) hinges on the imported decoded-correlator scaling ⟨Z Z⟩_d∼|j−k|^{−1/(2K)} from Ref. [46], which this paper neither derives nor verifies; if that scaling fails, the central 'enhancement' claim collapses.","rationale":"The reader's weakest_assumption identifies exactly the same point: the non-unitary enhancement rests on the decoded-correlator scaling from Ref. [46] and on the operational meaning of the averaged QFI. I agree that this is the most load-bearing concern. The symmetry-readout claim for internal symmetries is self-contained (App. A gives a rigorous proof for parity), and the reflection operator in the spatial case is Hermitian, so that part is on solid ground. The bit-flip decoherence result is exact and numerically verified; the dephasing result matches known no-go bounds; the qubit-loss scaling is supported by numerical collapse, and the paper itself notes the Ising case is 'mostly conceptual'. The non-unitary enhancement, by contrast, is the only claim with no verification in this manuscript: it is a direct corollary of a scaling result from a previous paper. If that scaling is incorrect, the enhancement vanishes, and the paper loses its most striking conclusion. However, there is no positive evidence that the scaling is wrong, so the appropriate verdict remains CONDITIONAL, requiring verification of the imported scaling before full acceptance. My stress-test does not change the reader's verdict.","tokens_in":35433,"tokens_out":12891,"duration_ms":123538,"concrete_test":"Perform exact diagonalization (or DMRG) of the ladder Hamiltonian (21) with local symmetry-preserving perturbations tuned to a known Luttinger parameter K>1 (e.g., K=2). For system sizes L=10,12,14,16, simulate projective measurements of X_{j,1} on the first chain, sample outcomes s with Born probabilities p_s, and compute the decoded correlator ⟨Z_{j,2}Z_{k,2}⟩_d = Σ_s p_s ⟨Z_{j,2}Z_{k,2}⟩_s s_{j+1}...s_k for all pairs (j,k). Fit the power-law exponent χ in ⟨Z Z⟩_d ∼ |j−k|^{−χ}. If χ deviates from 1/(2K) by more than 10%, Eq. (24) is unsupported. Alternatively, directly compute the averaged QFI through Eq. (23)-(24) and check the L-scaling L^{2[1−1/(4K)]}; a mismatch would invalidate the enhancement claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's most striking result is Eq. (24): F_Q[ρ] ∼ L^{2[1−1/(4K)]} for K>1, which claims that non-unitary deformation of a critical state can improve sensing precision beyond the pristine state. This conclusion rests entirely on the decoded-correlator scaling ⟨Z_{j,2}Z_{k,2}⟩_d ∼ |j−k|^{−1/(2K)} imported from Ref. [46] (same group). The present manuscript provides no derivation, no independent check, and no numerical evidence for this scaling; it simply states that the Kennedy-Tasaki mapping allows one to 'deduce' it. In Sec. IV, the per-outcome QFI in Eq. (23) and its Born-probability average in Eq. (24) are correct only if that decoded-correlator scaling holds. Additionally, the claim that parity on the second chain saturates the CRB for every outcome s relies on the state preserving a spin-flip symmetry after deformation; while plausible, this is asserted without explicit proof. If the decoded correlator decays faster than |j−k|^{−1/(2K)} for K>1, the enhancement disappears and the protocol's advantage over the pristine critical state is lost. Because this is the headline result of the abstract and Sec. IV, it is the most load-bearing assumption in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops interferometric quantum sensing protocols based on critical many-body states. It proposes a symmetry-based algorithm for choosing optimal measurements: for internal symmetries, a symmetry generator anticommuting with the encoding operator saturates the quantum Cramér–Rao bound; for spatial symmetries, translation/reflection operators measured via Hadamard test are proposed. The paper then analyzes non-unitary deformed critical states, claiming that outcome-dependent decoding can enhance the QFI scaling beyond the pristine critical state (Eq. 24). Finally, it studies decoherence: local bit-flip channels (exact QFI formula, Eq. 26), local dephasing (SQL scaling), and qubit loss (sub-SQL precision from subsystem parity). The conclusions compare critical states favorably to GHZ states under realistic noise.","tokens_in":35768,"tokens_out":10121,"duration_ms":92291,"significance":"If correct, the results provide practical, symmetry-informed measurement strategies for critical-state sensors and identify a surprising non-unitary enhancement of QFI scaling. The exact bit-flip QFI formula (Eq. 26) and its numerical verification are clear strengths, as is the systematic comparison with GHZ and spin-squeezed states. However, the spatial-symmetry measurement claim lacks a supporting derivation, and the non-unitary enhancement relies on an imported correlator scaling and on an average of per-outcome QFIs whose operational meaning needs clarification.","major_comments":[{"comment":"The claim that the Hadamard-test POVM for the translation operator T yields classical Fisher information scaling L^{2(1−Δ)} is not supported by Appendix A. Appendix A treats only Hermitian parity A with A^2=I and projective outcomes P±=(I±A)/2. For T (unitary, T^2≠I), the binary POVM has probabilities [1±Re⟨T⟩]/2; its Fisher information is (∂θ Re⟨T⟩)^2/(1−(Re⟨T⟩)^2), which is not covered by the appendix. Also, for odd-L reflection the anticommutation condition fails. A separate derivation or numerical verification is needed before claiming that spatial-symmetry measurements saturate the CR bound.","section":"§III.B / Appendix A"},{"comment":"Equation (24) writes F_Q[ρ]=Σ_s p_s F_s^Q, identifying the QFI of the averaged state with the Born-probability average of per-outcome QFIs. This equality is not generally true: by convexity, F_Q(Σ_s p_s |ψ_s⟩⟨ψ_s|) ≤ Σ_s p_s F_Q(|ψ_s⟩). If the protocol conditions on the known classical outcomes s, then the average is the relevant conditional Fisher information, but this operational interpretation must be stated explicitly. As written, Eq. (24) conflates the QFI of a mixture with the average of conditional QFIs and overstates what is proven.","section":"§IV, Eq. (24)"},{"comment":"The enhancement in Eq. (24) rests on the imported scaling ⟨Z_{j,2}Z_{k,2}⟩_d ∼ |j−k|^{-1/(2K)} from Ref. [46]. The manuscript gives only a one-sentence Kennedy–Tasaki argument and no independent numerical check. Since this is the headline result of Sec. IV, please provide a self-contained derivation in an appendix or a numerical verification (e.g., DMRG for the relevant K values). Without this, the reader cannot assess whether the enhancement survives beyond the assumed correlator decay.","section":"§IV, decoded correlator"}],"minor_comments":[{"comment":"s_j ∈ ±1 should be s_j ∈ {±1}.","section":"§IV, Eq. (20)"},{"comment":"Specify the parity operator as A = ∏_j X_{j,2}.","section":"§IV after Eq. (24)"},{"comment":"The text has an incomplete sentence: 'since we that the coefficients...' — presumably 'assume' or 'show' is missing.","section":"§Appendix E, Eq. (E3)"},{"comment":"For odd-L chains, 'we observe a similar scaling' is stated without data or derivation; add numerical details or a reference.","section":"§III.B"},{"comment":"Eq. (B2) defines C(t) with an overline for ensemble average, but the notation is not explained; clarify.","section":"§Appendix B"},{"comment":"References [65] and [76] appear to refer to the same work by Chai and Yang; verify.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The internal-symmetry algorithm and the noiseless, bit-flip, and dephasing analyses are solid and likely correct. The non-unitary enhancement and spatial-symmetry measurement claims need additional support (derivations or numerics) and an explicit statement of the conditional-measurement setting for Eq. (24). The paper should be revisable within its scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The strongest, cleanest part is the symmetry-informed optimal measurement recipe. For internal symmetries it's rigorous (Appendix A), and the spatial-symmetry Hadamard-test extension is a genuinely useful trick for Rydberg systems. I buy the argument that parity/reflection/translation expectation values give essentially identical θ-dependence, and the bit-flip QFI (Eq. 26) is exact and numerically verified. The decoherence map — critical states fall back to SQL rather than dying like GHZ — is coherent and consistent with known no-go results.\n\nThe soft spot is Section IV. The non-unitary enhancement claim (Eq. 24) is the flashiest result but carries two unproven loads. First, the decoded-correlator scaling ⟨Z Z⟩_d ∼ |j−k|^{−1/(2K)} is imported from the same group's earlier paper; this manuscript gives no derivation or numerical check. That is a real gap. Second, the average QFI over outcomes is treated as the sensing resource without explicitly discussing the conditional estimation protocol. I think that is fixable, since outcome-dependent imprinting gives you a classical label, so the average of conditional QFIs is the right figure of merit. But the paper should say so.\n\nThe spatial-symmetry Fisher-information scaling is also asserted with a pointer to Appendix A, which only treats Hermitian parity. For translation (non-Hermitian), the Hadamard-test POVM needs its own derivation. Minor, but worth fixing.\n\nThe qubit-loss section is honest — they note the Ising window is mostly conceptual — and the power-counting is supported by iDMRG collapse.\n\nBottom line: this is a well-written, honest paper with several contributions that will survive scrutiny. The non-unitary enhancement is the one thing I'd want checked before promoting it. I'd send it to a good referee, and ask that referee to focus on Section IV and the spatial-symmetry derivation. Not a desk reject. I'd cite the symmetry-readout and decoherence parts.","headline":"A practical symmetry-based readout recipe for critical-state sensors that mostly holds up, but the headline non-unitary enhancement rests on an imported correlator scaling that should be verified before full endorsement.","tokens_in":36235,"tokens_out":3158,"would_cite":true,"duration_ms":34681,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Symmetry measurements saturate the quantum Cramér-Rao bound for critical-state sensors, and non-unitary deformation can push precision beyond the pristine critical state, approaching the Heisenberg limit.","keywords":["quantum sensing","quantum Fisher information","quantum criticality","symmetry-informed measurements","decoherence","non-unitary deformation","GHZ states","Cramér-Rao bound"],"falsifier":"Compute exactly, for small system sizes, the decoded correlator ⟨Z_{j,2}Z_{k,2}⟩_d on the non-unitarily deformed ladder state at K>1 and compare the extracted power-law exponent with −1/(2K); any deviation invalidates the predicted F_Q ∼ L^{2[1−1/(4K)]} enhancement. Alternatively, experimentally implement the projective-measurement decoding protocol U_j(θ)=e^{iθ s_1...s_j Z_{j,2}} and check whether the outcome-averaged QFI follows that scaling.","tokens_in":35302,"feed_emoji":"⚛️","tokens_out":11804,"duration_ms":104302,"temperature":0.7,"pith_summary":"The paper asks whether many-body quantum critical states are practical resources for interferometric sensing—states that can be prepared, read out optimally, and survive noise. Its central claim is that the optimal readout is dictated by symmetry: when the phase-imprinting operator O has a definite charge under a symmetry generator A of the critical state (i.e., {A,O}=0), measuring A saturates the quantum Cramér-Rao bound, as demonstrated for the Ising chain and Rydberg-atom arrays. The paper then quantifies how imperfections affect the quantum Fisher information. Symmetry-preserving bit-flip noise only renormalizes the prefactor, keeping the QFI scaling; local dephasing pushes the critical state back to the standard quantum limit, but never worse, while GHZ states (superpositions of two macroscopically distinct configurations) decay exponentially; qubit loss still allows sub-SQL precision via subsystem parity measurements. Most notably, non-unitary deformations of a critical wavefunction—arising from weak measurements or imperfect teleportation—can make correlations decay more slowly, and with a decoding protocol the average QFI scales as L^{2[1−1/(4K)]}, exceeding the pristine state for K>1 and approaching Heisenberg scaling. If these claims hold, critical states offer a robust and increasingly feasible route to Heisenberg-limited sensing.","feed_headline":"Non-unitary noise can boost critical-state sensing precision","feed_subtitle":"Symmetry readouts saturate the bound; noisy critical states degrade to at worst the SQL, while GHZ states fail.","key_machinery":"The central object is the quantum Fisher information F_Q[ρ]=2∑_{λ_i+λ_j>0}(λ_i−λ_j)²/(λ_i+λ_j)|⟨i|O|j⟩|², which for pure states reduces to 4Var(O). Two identities carry the argument. The anticommutation relation {A,O}=0 makes the symmetry generator A an optimal observable: it forces ⟨O⟩=0 and yields δθ=1/(2√Var(O)) + O(θ²), converting optimal measurement design into a symmetry-group classification problem. The decoded-correlator identity ⟨Z_{j,2}Z_{k,2}⟩_d = Σ_s p_s ⟨Z_{j,2}Z_{k,2}⟩_s s_{j+1}⋯s_k ∼ |j−k|^{−1/(2K)}, established through a non-local duality transformation, converts non-unitary deformation into slower correlation decay; from it the QFI scaling F_Q∼L^{2[1−1/(4K)]} follows directl","core_discovery":"On the paper's own terms, the central discovery is twofold. First, optimal readout of a critical-state sensor is a symmetry problem: if the imprinting operator O has a well-defined charge under a symmetry generator A of the critical state ({A,O}=0), then measuring A saturates the Cramér-Rao bound. For the Ising critical chain this is the parity operator A=∏_j X_j with O=Σ_j Z_j, giving δθ=1/(2√Var(O)) up to O(θ²); for Rydberg chains the same role is played by translation or bond-reflection, measurable through a Hadamard test. Second, non-unitary deformation can enhance rather than destroy sensing. In a ladder model that maps to two decoupled Luttinger liquids with parameter K, deforming with","pith_inferences":["A natural extension is to treat weak measurement as an engineered resource: if non-unitary deformation can slow correlation decay, then choosing the deformation strength and measurement basis could let one dial the QFI exponent continuously between the pristine and Heisenberg values, not just in the K>1 ladder model.","The symmetry-readout recipe is likely portable to non-critical symmetry-sector states as well, since it relies only on {A,O}=0 and A acting as an eigenoperator on the probe; topologically ordered states with nonlocal symmetry generators are an immediate candidate.","The paper's own caveat that its dephased-state results are lower bounds (except for bit flips) points to a concrete next step: compute the exact QFI under local dephasing for the Ising chain to confirm that the SQL scaling is not an artifact of the error-propagation bound.","If the decoded enhancement survives finite-size checks, the practical comparison with GHZ states should be redone including preparation cost: log-depth preparation of critical states plus robustness to loss may make them the better choice even when the ideal QFI of a GHZ state is larger."],"forward_implications":["A universal readout recipe emerges: for any critical state with a discrete internal or spatial symmetry, the optimal measurement saturating the Cramér-Rao bound is the symmetry generator anticommuting with the imprinting operator, so optimal readout does not require constructing the symmetric logarithmic derivative.","Symmetry-preserving bit-flip channels do not destroy the sensing enhancement: at Ising criticality the QFI keeps its L^{7/4} scaling with only a prefactor reduction for p<1/2, and for ZZ channels the QFI is exactly unchanged.","Under local dephasing, critical states degrade to at most the SQL, whereas GHZ states acquire exponentially bad phase uncertainty; the same holds for global dephasing.","Partial qubit loss need not end the protocol: parity measurements on a subregion yield δθ_min ∼ L_sub^{−5/8} for Ising criticality and up to Heisenberg-like L_sub^{−1} for the XXZ chain near Δ→1, improving over the SQL within a finite phase window.","Non-unitary deformation with decoding can turn a corrupted critical wavefunction into a better sensor than the original, with outcome-averaged QFI F_Q ∼ L^{2[1−1/(4K)]} exceeding L^{3/2} for K>1 and approaching L²; this can be implemented with local imprinting operations after the deformation."],"fun_headline_variants":["Noise boosts critical-state quantum sensing","Critical states beat GHZ with symmetry readouts","Symmetry readout unlocks critical sensor precision","Non-unitary tweak sharpens quantum critical sensor","Critical wavefunctions outdo GHZ in noisy sensing"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The enhancement result assumes the imported decoded-correlator scaling ⟨Z_{j,2}Z_{k,2}⟩_d ∼ |j−k|^{−1/(2K)} for K>1, together with the assumption that averaging the QFI over measurement outcomes is the right operational figure of merit; if those fail, non-unitary deformation may not help, even though the symmetry-readout recipe stands on its own.","fun_headline_variants_meta":{"raw":{"variants":["Noise boosts critical-state quantum sensing","Critical states beat GHZ with symmetry readouts","Symmetry readout unlocks critical sensor precision","Non-unitary tweak sharpens quantum critical sensor","Critical wavefunctions outdo GHZ in noisy sensing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1131,"prompt_tokens":712,"completion_tokens":419,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":349}},"tokens_in":456,"tokens_out":419,"duration_ms":4062,"temperature":1.0,"reasoning_tokens":349,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:03:22.786504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute exactly, for small system sizes, the decoded correlator ⟨Z_{j,2}Z_{k,2}⟩_d on the non-unitarily deformed ladder state at K>1 and compare the extracted power-law exponent with −1/(2K); any deviation invalidates the predicted F_Q ∼ L^{2[1−1/(4K)]} enhancement. Alternatively, experimentally implement the projective-measurement decoding protocol U_j(θ)=e^{iθ s_1...s_j Z_{j,2}} and check whether the outcome-averaged QFI follows that scaling.","supporting_citations":[],"review_version":1}