{"id":"6c0cf732-3757-4c8f-894d-ef0047d85b65","arxiv_id":"2507.03440","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A spatial three-party Leggett-Garg inequality, Eq. (1), is violated by a Heisenberg spin chain; first-violation times scale linearly with distance between measurement sites.","lead":"The paper defines a spatial version of the Leggett-Garg inequality, where three distant observers each measure the same spin observable in a chain at successive times. For a Heisenberg chain the inequality is violated, and the first violation time grows linearly with distance, tracing the light cone of correlation spreading.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The numerical K_n in Fig. 2 is computed from undisturbed Heisenberg two-point functions, while the described protocol uses sequential projective imaging; for J≠0 these differ, so the reported violations and linear τ(n) may not be the measured signal.","rationale":"The most load-bearing step in the paper is not the algebraic derivation of the spatial bound, which under macrorealism plus noninvasive measurability follows in one line from assigning ±1 to the three measured quantities. The real soft spot is the evaluation of the quantum correlators appearing in Eq. (1). The authors define CX,Y as a sequential expectation value but compute it as the real part of a Heisenberg two-point function. These two objects coincide for ideal noninvasive measurements, but not for the projective imaging described as the experimental protocol, once interactions are present. Since the central claim is that the spatial LGI is violated and that the first-violation time grows linearly with distance, a mismatch between the computed quantity and the measured quantity would undermine both the numerical evidence and the experimental promise. The reader's weakest-assumption point about the unproven inequality bounds is real but less decisive; the missing derivation can be supplied trivially under standard assumptions. The more consequential issue is that the plotted K_n is not obviously the quantity an experimentalist would measure. The concrete test above would settle this by computing the projective-protocol K_n directly and checking whether the linear scaling survives. The arbitrary threshold Kn>1.02 and the inaccessible code are secondary concerns; they can be addressed by releasing data and by testing threshold sensitivity, but they do not by themselves threaten the physics as directly as the correlator mismatch. The verdict should remain conditional: the paper's central result is plausible and worth publishing once the authors either adopt an explicit noninvasive-measurement protocol consistent with the Re-formula and show how to implement it, or recompute the violations under the actual projective protocol and confirm that the linear τ(n) scaling persists.","tokens_in":8292,"tokens_out":17224,"duration_ms":230980,"concrete_test":"Use TEBD (or exact diagonalization for n≤7) to simulate the actual sequential projective protocol for J=1, h=1: evolve |+⟩⊗N from 0 to t, apply a Lüders projection of the optimized σ_v at site n, continue evolution to 2t, measure σ_v at site N, and average over outcomes to obtain C_n,N^proj; form K_n^proj = C_1,n + C_n,N^proj − C_1,2n−1 for n=2..7. Repeat with J=0 as a control. If K_n^proj differs from the values in Fig. 2, or if the threshold-crossing times no longer scale linearly with n, the reported violations and the Lieb-Robinson velocity estimate are not established for the described measurement protocol.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Eq. (1) is introduced as the correlator of sequential measurements at sites 1, n, 2n−1 at times 0, t, 2t, and the paper states that these are 'readily computed as CX,Y = Re{⟨Ψ|QX(tX)QY(tY)|Ψ⟩}' (main text; App. B). For a genuine Lüders projective readout, the two-time correlation is E_{X,Y} = Σ qX qY Tr[Π_{qY}^{(Y)}(tY) U_{tX→tY} Π_{qX}^{(X)}(tX) ρ_{tX} Π_{qX}^{(X)}(tX) U†_{tX→tY}]. This equals Re{⟨QX(tX)QY(tY)⟩} only when the state at tX is an eigenstate of QX(tX) or the two Heisenberg observables commute. At t_A=0 the first condition holds because |Ψ⟩=|+⟩⊗N, so C_1,n and C_1,N are fine. But at t_B=t, neither condition holds once J≠0: the interaction makes [σv_n(t), σv_N(2t)] nonzero, and ρ_t is not an eigenstate of σv_n(t). Thus Figs. 2 and 3 report the ideal noninvasive (weak) correlation, not the 'spin state is imaged' projective protocol described as the experiment. Spatial separation does not prevent invasiveness in an interacting chain: a strong measurement at site n at time t can alter the subsequent state at site N. Therefore the central numerical claim and the experimental proposal are mismatched, and the reported linear τ(n) may reflect the undisturbed correlation rather than any measured violation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates a spatial extension of the Leggett-Garg inequality for three parties measuring the same local observable at sites 1, n, and 2n-1 of a spin chain at times 0, t, and 2t. The quantity K_n = C_{1,n} + C_{n,2n-1} - C_{1,2n-1} is asserted to satisfy -3 <= K_n <= 1 under macrorealistic assumptions. For the Heisenberg chain with J=h=1, exact diagonalization and TEBD/TDVP are used to compute unperturbed two-time correlation functions, reporting violations, a first-violation time tau that grows linearly with n, and a reduced slope when next-nearest-neighbor couplings are added. The paper interprets this linear scaling as a Lieb-Robinson-type light-cone effect and proposes several experimental platforms for observing the violations.","tokens_in":8650,"tokens_out":10951,"duration_ms":145994,"significance":"If the central claims held, the paper would introduce a useful new many-body witness: spatial Leggett-Garg inequalities would allow probing correlation spreading and estimating a lower bound on the Lieb-Robinson velocity using the same simple correlator structure as temporal LGIs. The work has clear strengths: the numerical K_n is computed directly from the Hamiltonian without fitting target values; the noninteracting no-violation proof is analytic; the measurement optimization is well motivated; and the manuscript includes a code/data statement. However, two load-bearing issues currently block acceptance: the bound in Eq. (1) is not derived for the spatial interacting scenario, and the numerical correlators do not match the projective imaging protocol described as the experiment. These issues are substantial but appear fixable within the manuscript's scope.","major_comments":[{"comment":"The bound -3 <= K_n <= 1 is asserted for spatially separated sequential measurements without a derivation. In the standard LGI the bound follows from the assumption that a single macrorealistic variable Q(t) has definite values at three times and that the intermediate measurement is noninvasive. Here the observable is measured at three different sites, and the paper does not state the corresponding macrorealism and noninvasiveness assumptions or prove that they imply a joint distribution for (Q_1(0), Q_n(t), Q_{2n-1}(2t)). The proof in Appendix A for noninteracting Hamiltonians relies on pairwise commutativity and joint measurability, not on macrorealism, so it does not cover the interacting case that is the central result. The claim that 'measuring in different sites helps preventing the clumsiness loophole' is not substantiated and is questionable for an interacting chain, because a projective measurement at site n at time t can influence the later statistics at site 2n-1 through the couplings. Without this derivation, the reported violations cannot be interpreted as a test of macrorealism.","section":"Eq. (1) and following paragraph"},{"comment":"The correlators are computed as C_{X,Y} = Re{<Psi|Q_X(t_X)Q_Y(t_Y)|Psi>}, which is the undisturbed two-time correlation function, not the correlation obtained by the 'spin state is imaged' projective protocol described in the final section. For sequential Lueders measurements the two-time correlation is E_{X,Y} = sum_{qX,qY} qX qY Tr[Pi_{qY}^{(Y)}(t_Y) U_{tX->tY} Pi_{qX}^{(X)}(t_X) rho_{tX} Pi_{qX}^{(X)}(t_X) U^dagger_{tX->tY}], which equals Re{<Q_X(t_X)Q_Y(t_Y)>} only when rho_{tX} is an eigenstate of Q_X(t_X) or the two Heisenberg observables commute. At t_A=0 the first condition holds, so C_{1,n} is fine, but at t_B=t neither condition holds once J != 0: the interaction makes [sigma_v_n(t), sigma_v_{2n-1}(2t)] nonzero, and rho_t is not an eigenstate of sigma_v_n(t). Thus Figs. 2 and 3 report weak/noninvasive correlations, not the signal of the proposed projective experiment, and spatial separation does not make the measurement noninvasive in an interacting chain. The experiment proposal and the numerical claim are mismatched.","section":"Main text, definition of C_{X,Y} and final experimental paragraph"},{"comment":"The first-violation time tau is defined by the arbitrary criterion K_n > 1.02, and no error bars or sensitivity analysis are provided. Since Table I shows that for n=6 the maximal optimized K_n is only about 1.024 for both NN and NNN cases, the extracted tau for large n is extremely sensitive to the threshold and to numerical truncation errors in the TEBD/TDVP data. In addition, the linear fits include the points n=2 and n=3 for the NNN case, where the text states that the inequality is violated for any finite t>0 because the measured sites are directly coupled; these points are not in the propagating regime and may bias the quoted slopes. Please refit using n>=4 and report the fit parameters with uncertainties.","section":"Fig. 3, Table I, and text defining tau"}],"minor_comments":[{"comment":"In the expression for K_n in the noninteracting case, the second correlator is written C_{n,2n} but should be C_{n,2n-1}.","section":"Appendix A"},{"comment":"The caption should specify the Hamiltonian parameters and chain length used; as printed, only the distance n=5 is stated.","section":"Fig. 5 caption"},{"comment":"Reference [27] says 'To be published online'; please provide a working repository link or DOI in the revised version.","section":"Code and data statement"},{"comment":"The criterion K_n > 1.02 is introduced only in the Fig. 3 caption; please define tau and the threshold in the main text and state whether the same threshold is used for the NN and NNN curves.","section":"Fig. 3 caption and main text"},{"comment":"The claim that the violation is 'readily observable' would be strengthened by an estimate of the required coherence time and measurement fidelity for a specific experimental platform.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The two major issues are likely fixable: supply a short but explicit proof of Eq. (1) under clearly stated macrorealism/noninvasiveness assumptions, and either reformulate the experiment as weak/ideal-negative measurements or compute genuine projective sequential correlators. If the revised numerics change Fig. 3, the linear scaling claim should be rechecked. The manuscript has a solid numerical core and the spatial extension is potentially interesting, but it is not yet ready for publication in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report on 2507.03440. The core idea is genuinely nice: extend the Leggett-Garg inequality to three spatially separated sites and use the first violation time as a probe of Lieb-Robinson light-cone dynamics. The numerical observation that tau grows linearly with the distance n in a Heisenberg chain is plausible and the connection to Lieb-Robinson physics is well argued. The optimization over measurement directions is clean, and the proof that noninteracting Hamiltonians cannot give violations is complete. If the central claim holds up, this is a useful tool for quantum simulators.\n\nThe paper has one big soft spot. The quantity they compute in Figs. 2 and 3 is Re<QX(tX)QY(tY)>, the undisturbed Heisenberg two-point function. That is the correct correlation for a noninvasive (weak) measurement, but the protocol they describe to experimentalists is projective imaging of the spin state. In an interacting chain these are not the same. At t_A=0 the initial state is an eigenstate of sigma_x, so the first correlator is fine, but for C_n,2n-1 the projective readout at site n at time t updates the state, and by time 2t that disturbance can propagate to site N. So the reported violations and the linear tau(n) scaling may not correspond to what the described experiment would actually measure. Spatial separation does not automatically make the measurement noninvasive, and the paper's claim that measuring different sites prevents the clumsiness loophole needs more than a sentence.\n\nA related, smaller gap is that the bounds -3<=K<=1 are asserted for the spatial case without a derivation. It is likely a straightforward generalization of the usual LGI argument, but the paper should either prove it or state clear assumptions. Minor issues: the 1.02 threshold for defining tau is arbitrary, there are no error bars on the numerical plots, and the code/data link is currently a placeholder.\n\nWho is this for? People working on LGI tests in many-body systems and on light-cone characterization in quantum simulators. The idea deserves a serious referee, but the revision needs to address the measurement mismatch. If the authors can show the projective-measurement correlation also violates the inequality and preserves the scaling, or specify that they intend weak measurements, the paper would be solid. As it stands, the central claim is not yet established for the proposed experiment.","headline":"A promising spatial LGI with plausible numerics, but the quantity computed differs from the proposed projective protocol; that mismatch needs fixing.","tokens_in":9115,"tokens_out":6431,"would_cite":false,"duration_ms":75809,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A spatial version of the Leggett-Garg inequality is violated by an interacting Heisenberg spin chain, with the first-violation time growing linearly with the distance between measuring parties.","keywords":["Leggett-Garg inequality","spatial Leggett-Garg inequality","Heisenberg spin chain","Lieb-Robinson bound","macrorealism","quantum correlation spreading","sequential measurements","many-body quantum dynamics"],"falsifier":"A direct computation of $K_n$ under a local hidden-variable model that reproduces the claimed sequential correlations would falsify the macrorealistic interpretation; more concretely, an experiment on a Rydberg or trapped-ion chain measuring $K_n$ for $n=2,\\dots,7$ that finds the first-violation time $\\tau$ does not grow linearly with $n$ would falsify the claimed connection to the Lieb-Robinson velocity.","tokens_in":8137,"feed_emoji":"⚛️","tokens_out":7054,"duration_ms":71978,"temperature":0.7,"pith_summary":"The paper proposes a spatial extension of the Leggett-Garg inequality in which three distant observers measure the same spin-1/2 chain at three successive times, obtaining the correlator $K_n = C_{1,n} + C_{n,2n-1} - C_{1,2n-1}$ that macrorealistic theories must keep between $-3$ and $1$. It shows that for a Heisenberg chain in a magnetic field, interactions are necessary for violation: for noninteracting local evolution the relevant operators commute and $K_n$ never crosses the bound. With interactions, $K_n$ does violate the inequality, and the first-violation time $\\tau$ grows linearly with the distance $n$ between the measuring sites, which the authors read as a finite-speed, Lieb-Robinson-like propagation of quantum correlations. The result turns a three-time correlation test into a tool for reconstructing the light cone of quantum information spreading in many-body systems, with the violation experimentally accessible in current cold-atom, Rydberg, and trapped-ion platforms.","feed_headline":"Spatial Leggett-Garg test clocks quantum signal speed","feed_subtitle":"First-violation time grows linearly with observer distance, tracking the Lieb-Robinson light cone.","key_machinery":"The central object is the spatial Leggett-Garg correlator $K_n = C_{1,n} + C_{n,2n-1} - C_{1,2n-1}$, built from two-time correlators $C_{X,Y} = \\langle Q_X(t_X) Q_Y(t_Y)\\rangle$ for three sites spaced by $n-1$ lattice steps and times spaced by $t$; macrorealism is claimed to bound it by $-3 \\le K_n \\le 1$. The argument is carried by three pieces: the pairwise commutation of $Q_1(0)$, $Q_n(t)$, and $Q_{2n-1}(2t)$ for noninteracting evolution, which forces $K_n$ into the classical range; the bilinear structure $K_n(\\vec v)=\\vec v^T \\mathcal K_n \\vec v$, whose largest eigenvalue gives the optimal measurement direction; and the Lieb-Robinson bound, invoked to explain why the first-violation time grows with $n$ and why longer-range interactions accelerate violation.","core_discovery":"The central claim is that the spatial Leggett-Garg inequality of Eq. (1), applied to three parties measuring the same observable $\\sigma^x$ on sites $1$, $n$, and $2n-1$ of a Heisenberg chain at times $0$, $t$, and $2t$, is violated for interacting dynamics, and that the time $\\tau$ of first violation increases linearly with the chain distance $n$ (Fig. 3). For the isotropic Heisenberg Hamiltonian $H=J\\sum_i \\vec\\sigma_i\\cdot\\vec\\sigma_{i+1} - (h/2)\\sum_i \\sigma^z_i$ with $J=1$, $h=1$ and initial product state $|\\Psi\\rangle=|+\\rangle^{\\otimes N}$, the correlator $K_n$ rises above $1$ near the maxima of the noninteracting curve and is largest for small $n$; optimizing the measured spin direction $\\vec v\\cdot\\vec\\sigma$ further strengthens the violation. The paper proves that for noninteracting Hamiltonians the three relevant observables commute pairwise, so $K_n$ is confined to $[-3,1]$ for any initial state and local measurement, establishing that interactions are required for a spatial LGI violation. The linear $\\tau$ versus $n$ relation, together with the observation that next-nearest-neighbor couplings reduce the slope of that relation, is presented as evidence that the spatial LGI tracks Lieb-Robinson light-cone physics and provides a lower-bound estimate of the Lieb-Robinson velocity.","pith_inferences":["A natural next step is to prove rigorously the bounds $-3\\le K_n\\le1$ for spatially separated sequential measurements; if those bounds fail under a joint-probability assumption at different sites, the reported violations would test the measurement protocol rather than macrorealism.","The linear $\\tau$-$n$ relation could be compared quantitatively with independent Lieb-Robinson velocity bounds computed from the same Hamiltonian, turning a qualitative light-cone reconstruction into a calibrated velocity measurement.","The same inequality structure could be applied to disordered or long-range-interacting systems, where light-cone velocities are modified, to test whether first-violation times track the predicted velocity changes.","Since the optimized measurement requires inferring the full matrix $\\mathcal K_n$ from sequential expectation values over all spin components, an experiment that measures only $\\sigma^x$ would likely find weaker violations; reporting the optimized value is therefore the stronger empirical test."],"forward_implications":["Violation of Eq. (1) in an interacting spin chain demonstrates that spatially separated two-time correlation functions are incompatible with macrorealism, so a measurement at one site can affect distant sites through the many-body dynamics.","The linear growth of first-violation time $\\tau$ with distance $n$ provides a direct, experiment-ready lower-bound estimate of the Lieb-Robinson velocity in the chain.","Adding next-nearest-neighbor couplings reduces the slope of $\\tau$ versus $n$, showing that longer-range interactions speed up the onset of spatial LGI violation, as expected for faster correlation spreading.","Under noninteracting local Hamiltonians no violation is possible regardless of the initial state or local measurement, because the three relevant observables commute pairwise and $K_n$ reduces to a convex combination of classical outcomes.","The same protocol can be implemented in current Rydberg-atom, trapped-ion, and ultracold-atom platforms, where imaging the spins at prescribed times and sites yields the statistics needed to infer $K_n$."],"supporting_citations":[{"why":"Defines the original Leggett-Garg inequality and the macrorealism assumptions the spatial version extends.","marker":"[1]"},{"why":"Supplies the standard derivation of the $[-3,1]$ bound and the review of LGI violations that motivates the spatial generalization.","marker":"[2]"},{"why":"Provides the noninvasive-measurability framework used to justify the macrorealistic bound.","marker":"[3]"},{"why":"Gives the optimization approach over measurement settings used to maximize the correlator at each time.","marker":"[6]"},{"why":"States the Lieb-Robinson bound invoked to explain the linear growth of first-violation time with distance.","marker":"[14]"},{"why":"Supports the expression of the quantum correlators as real parts of sequential expectation values.","marker":"[26]"},{"why":"Supplies spin-chain dynamics background used to motivate the choice of the Heisenberg Hamiltonian.","marker":"[11]"}],"fun_headline_variants":["Spatial Leggett-Garg inequality times Lieb-Robinson cone","Measuring quantum signal speed with spatial Leggett-Garg","First LGI violation time grows linearly with spin-chain distance","Spatial LGI illuminates Lieb-Robinson physics in spin chains","Quantum perturbation speed read from spatial Leggett-Garg test"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that three measurements made at different sites at successive times admit a joint probability distribution and that each measurement leaves the later dynamics unchanged, a bound the paper asserts for the spatial case without a rigorous derivation.","fun_headline_variants_meta":{"raw":{"variants":["Spatial Leggett-Garg inequality times Lieb-Robinson cone","Measuring quantum signal speed with spatial Leggett-Garg","First LGI violation time grows linearly with spin-chain distance","Spatial LGI illuminates Lieb-Robinson physics in spin chains","Quantum perturbation speed read from spatial Leggett-Garg test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000181,"raw_usage":{"total_tokens":1338,"prompt_tokens":1007,"completion_tokens":331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":242}},"tokens_in":623,"tokens_out":331,"duration_ms":4502,"temperature":1.0,"reasoning_tokens":242,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:09:34.932634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computation of $K_n$ under a local hidden-variable model that reproduces the claimed sequential correlations would falsify the macrorealistic interpretation; more concretely, an experiment on a Rydberg or trapped-ion chain measuring $K_n$ for $n=2,\\dots,7$ that finds the first-violation time $\\tau$ does not grow linearly with $n$ would falsify the claimed connection to the Lieb-Robinson velocity.","supporting_citations":[{"cited_title":"Spatial Leggett-Garg Inequalities","cited_arxiv_id":"2507.03440","evidence_quote":"Supplies the standard derivation of the $[-3,1]$ bound and the review of LGI violations that motivates the spatial generalization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the optimization approach over measurement settings used to maximize the correlator at each time."},{"cited_title":"Signoles, T","cited_arxiv_id":null,"evidence_quote":"Supports the expression of the quantum correlators as real parts of sequential expectation values."},{"cited_title":"Budroni, G","cited_arxiv_id":null,"evidence_quote":"Supplies spin-chain dynamics background used to motivate the choice of the Heisenberg Hamiltonian."}],"review_version":1}