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REVIEW 3 major objections 5 minor 31 references

Spatial Leggett-Garg Inequalities

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A promising spatial LGI with plausible numerics, but the quantity computed differs from the proposed projective protocol; that mismatch needs fixing. read the letter →

arxiv 2507.03440 v1 pith:UCJ6THLV submitted 2025-07-04 quant-ph cond-mat.quant-gas

Guillem Müller-Rigat , Donato Farina , Maciej Lewenstein , Andrea Tononi This is my paper · ORCID
classification quant-phcond-mat.quant-gas
keywords Leggett-GarginequalityspatialHeisenbergspinchainLieb-Robinsonboundmacrorealismquantumcorrelationspreadingsequentialmeasurementsmany-bodydynamics
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $K_n$.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [Eq. (1) and following paragraph] 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.
  2. [Main text, definition of C_{X,Y} and final experimental paragraph] 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.
  3. [Fig. 3, Table I, and text defining tau] 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.
minor comments (5)
  1. [Appendix A] 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}.
  2. [Fig. 5 caption] The caption should specify the Hamiltonian parameters and chain length used; as printed, only the distance n=5 is stated.
  3. [Code and data statement] Reference [27] says 'To be published online'; please provide a working repository link or DOI in the revised version.
  4. [Fig. 3 caption and main text] 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.
  5. [Conclusion] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical violations and τ(n) scaling are computed directly from the Hamiltonian, with no fitted target values and no self-citation chain.

full rationale

The central quantity K_n is defined from Heisenberg-picture correlators and computed directly from the Hamiltonian in Eq. (2) for the initial product state |+⟩^⊗N, using exact diagonalization, TEBD, and TDVP. The first-violation time τ is read off from these computed curves, not obtained by fitting to the Lieb-Robinson velocity or to any target value. The linear τ-versus-n relation is a numerical observation that the authors then attribute to Lieb-Robinson physics; that attribution is interpretive and does not make the result an input. The optimization over measurement directions in Appendix B is a legitimate search over experimental settings, not a fit of the predicted violation to data. The choice of the Heisenberg Hamiltonian is explicitly described as the outcome of a numerical exploration, which is model selection rather than circularity. The paper contains no load-bearing self-citations: the Lieb-Robinson bound and the standard LGI are external results. The main caveats—that the −3 ≤ K_n ≤ 1 bounds in Eq. (1) are asserted rather than derived for spatially separated sequential measurements, and that the two-point Heisenberg correlator may not match a projective imaging protocol—are correctness and validity concerns, not cases where the output reduces to the input by construction. Therefore no significant circularity is found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new entities or fitted constants used to force the central result. The free parameters are the Hamiltonian couplings, the violation threshold, and the fit range for the tau-n linearity claim.

free parameters (3)
  • Coupling constants J, h = 1, 1
    Chosen as the demonstration case; the authors state the Heisenberg chain was selected after 'a numerical exploration over various possible couplings' because it gives the most appreciable violations.
  • Violation threshold for tau = 1.02
    The first-violation time is defined by K_n > 1.02; this threshold is arbitrary and affects the extracted times and linear fits.
  • Linear fit range for tau vs n = n = 2 to 6
    Straight lines fit points 2 ≤ n ≤ 6 to claim proportionality; small sample and no error estimates.
assumptions (4)
  • domain assumption Macrorealism and noninvasive measurability imply a joint probability distribution over the three measurement outcomes, yielding -3 ≤ K ≤ 1.
    Invoked when stating Eq. (1) holds for the spatial protocol; the proof is not given.
  • standard math For noninteracting Hamiltonians, the three observables Q_1(0), Q_n(t), Q_{2n-1}(2t) commute pairwise.
    Proven in Appendix A; relies on local evolution.
  • domain assumption The Lieb-Robinson bound limits the speed of correlation spreading in the chain.
    Borrowed from [14] and used to interpret the numerical scaling.
  • domain assumption TEBD and TDVP simulations in TeNPy are numerically accurate for the system sizes studied.
    No convergence or truncation-error data are reported.

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Cite this review

Pith. "Pith review of Spatial Leggett-Garg Inequalities." pith.science (2026). https://pith.science/paper/UCJ6THLV

@misc{pith2026250703440,
  author       = {Pith},
  title        = {Pith review of: Spatial Leggett-Garg Inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCJ6THLV}},
  note         = {Machine review of arXiv:2507.03440}
}
read the original abstract

We formulate a spatial extension of the Leggett-Garg inequality by considering three distant observers locally measuring a many-body system at three subsequent times. The spatial form, in particular, is specially suited to analyze propagation of quantum perturbations through spin chains, by capturing how a measurement at one site can later affect distant sites due to the interactions. We illustrate our proposal for a Heisenberg chain in a magnetic field, showing indeed that the first inequality-violation time scales proportionally to the distance between measuring parties. We attribute this phenomenon to Lieb-Robinson physics and, confirming this connection, we find that violations are anticipated when increasing the interaction range. The inequality violation, readily observable in current experiments, demonstrates the incompatibility between two-point correlation functions and the macrorealistic hypothesis. In outlook, spatial Leggett-Garg inequalities constitute a new tool for analyzing the non-relativistic dynamics of many-body quantum systems.

Figures

Figures reproduced from arXiv: 2507.03440 by the authors.

Figure 1
Figure 1. FIG. 1. Measurement setup of the spatial Leggett-Garg [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dynamics of the spatial Leggett-Garg inequality [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Dynamics of the Leggett-Garg correlator for mea [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

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