Neither weak nor strong entropic Leggett-Garg inequalities can be violated

Christoph Adami

REVIEW 3 major objections 4 minor 1 cited by

Neither weak nor strong entropic Leggett-Garg inequalities can be violated

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Entropic Leggett-Garg inequalities cannot be violated by either strong or weak quantum measurements when the full three-time joint density matrix is used consistently.

desk verdict The strong-measurement half is a legitimate challenge to published entropic LG violations, but the weak half is wrong as written and that is the headline claim. read the letter →

arxiv 1908.03124 v3 pith:T5NJJZIS submitted 2019-08-08 quant-ph

Christoph Adami This is my paper · ORCID

classification quant-ph PACS 03.65.Ta

keywords Leggett-Garginequalitiesentropicweakmeasurementsstrongquantummeasurementdevicesthree-timejointdensitymatrixentropyVenndiagrammacrorealism

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

This paper argues that entropic Leggett-Garg inequalities, the information-theoretic cousins of the standard temporal correlation inequalities, cannot be violated by quantum measurements, whether strong or weak. The apparent violations reported elsewhere are traced to a consistency error: mixing two-time data from runs without the middle measurement with the three-time joint density matrix of consecutive measurements. Once the three detectors are described by one joint density matrix, the inequalities hold, because they constrain the classical measurement devices rather than the quantum system. If correct, this removes a supposed clash between quantum mechanics and macrorealism and shifts the lesson to the unreliability of devices as descriptions of quantum states.

What carries the argument

The load-bearing object is the entropy Venn diagram for the three measurement devices, whose region sizes are set by the pairwise entropies $S_{12}$, $S_{13}$, and $S_{23}$ computed from the joint density matrix $\rho_{123}$ of detectors $A_1$, $A_2$, and $A_3$. The diagram converts consistency of $\rho_{123}$ into non-negativity of combinations such as $S_{12}+S_{23}-S_{13}$. Weak measurements enter through the coupling parameter $\epsilon$, with the middle ancilla displaced to $\sqrt{1-\epsilon^2}|0\rangle+\epsilon|1\rangle$, changing the shared-entropy regions and adding the binary entropy $H[\epsilon]=-x\log_2 x-(1-x)\log_2(1-x)$ to the inequality. The Venn diagram is the bridge from the density matrix to the algebraic form of $B'_1(\epsilon)$.

What would settle it

Evaluate the paper's own expressions (30)-(32) at $\theta_1=\pi/2$, $\theta_2=0$, $\epsilon=1/2$, and compute $B'_1(\epsilon)=S_{12}-S_{13}+S_{23}-H[\epsilon]$; the result is approximately $-0.73$, which would directly contradict the claim that weak entropic Leggett-Garg inequalities cannot be violated.

Watch

Extended reading notes

Core claim

The central claim is that the entropic inequalities $B^\star_1 = S_{12}+S_{23}-S_{13}\ge 0$, $B^\star_2 = S_{12}+S_{13}-S_{23}\ge 0$, and $B^\star_3 = S_{13}+S_{23}-S_{12}\ge 0$ cannot be violated by strong quantum measurements, and that the weak-measurement counterpart $B'_1(\epsilon)=S_{12}-S_{13}+S_{23}-H[\epsilon]\ge 0$ cannot be violated either. The paper computes the three pairwise entropies from the explicit three-device joint density matrix for consecutive measurements on a maximally mixed qubit, with the middle measurement at relative angle $\theta_1$ and the third at $\theta_2$. For strong measurements, $B^\star_1$ is non-negative by convexity, while $B^\star_2$ and $B^\star_3$ reduce to a positive entropy. For a weak middle measurement of strength $\epsilon$, the Venn diagram is modified so that only part of the device entropy is shared with the quantum system, producing the extra $-H[\epsilon]$ term. The interpretive claim is that reported violations disappear when the full three-time joint probability $P(x,y,z)$ is used, because the inequalities constrain the measurement devices, not the measured system.

Load-bearing premise

The load-bearing premise is that the correct weak-measurement entropic inequality is $B'_1(\epsilon)=S_{12}-S_{13}+S_{23}-H[\epsilon]\ge 0$, with the $-H[\epsilon]$ term read off the Venn diagram rather than derived; if that premise is false, the weak no-violation result collapses.

Editorial extensions

If this is right

If the paper is right, entropic Leggett-Garg inequalities cannot serve as tests of macrorealism; they test whether three measurement devices have a consistent joint description.
Reported violations, including the nuclear-magnetic-resonance experiment cited in the paper, would be artifacts of combining three-time joint data with two-time data from runs without the middle measurement.
A weak or zero-strength middle measurement makes $S_{12}$ and $S_{23}$ tend to one, so the inequalities become trivial rather than violated when the middle measurement is effectively omitted.
The no-violation result extends the author's earlier conclusion for standard Leggett-Garg inequalities to entropy-form inequalities, supporting the view that quantum mechanics consistently describes classical devices.

Reading between the lines

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

The weak-measurement case is directly checkable: the modified inequality $B'_1(\epsilon)$ is read off the Venn diagram rather than derived independently, so evaluating the paper's own expressions at $\theta_1=\pi/2$, $\theta_2=0$, $\epsilon=1/2$ would decide whether the no-violation claim survives in that region.
The same consistency argument, if extended to temporal or entropic Bell inequalities, suggests that sequential full joint distributions should always satisfy the corresponding entropic bounds, unifying the time and space cases under one principle.
A tomographic experiment that reconstructs $\rho_{123}$ for three consecutive weak measurements on a single photon or spin could directly test the paper's central distinction by comparing $S_{13}$ from the full state with $S_{13}$ from a run without the middle measurement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The one thing to know: this paper has two halves. The strong-measurement half is a real contribution. It computes the three-time joint density matrix for consecutive projective measurements and shows that the entropic quantity B1* = S12 + S23 − S13 ≥ 0 holds. The point that previous derivations of S13 marginalize over the middle measurement, effectively omitting it, is a fair and important criticism of Refs. [20] and [29]. I believe that half deserves serious attention.

The weak-measurement half does not survive contact with the paper's own equations. The paper finds that the unmodified B1 can be violated for ε < 1, then reads off a modified inequality B'1(ε) = S12 − S13 + S23 − H[ε] ≥ 0 from a Venn diagram. The extra term is asserted, not derived, and it is not the entropy of the weak detector: the paper itself gives S(A2) = H[(1 + √(1−ε²))/2], which for ε = 1/2 is about 0.355, not H[1/2] = 1. Worse, the proposed inequality is numerically false. Using the paper's Eqs. (30)–(33) at θ1 = π/2, θ2 = 0, ε = 1/2 gives S12 ≈ 1.355, S23 = 1, S13 = 2, so B'1 ≈ 1.355 − 2 + 1 − 1 = −0.645. The stress-test note says −0.726; the arithmetic is slightly off, but the sign is the point. The central no-violation claim for weak measurements is unsupported as stated.

The rest of the discussion is largely interpretive. The claim that the inequalities describe the measurement devices rather than the quantum system might be right, but the weak proof is the only thing tying it to the entropic case. The citation pattern is reasonable; the self-citations to [1] and [33] are appropriate because [1] is the companion argument and [33] is the source of the Venn-diagram structure. I did not check every line of the strong proof, but nothing there jumps out as wrong.

Who gets value from this? Someone working on entropic temporal inequalities or on the Katiyar experiment would want to read the strong section. The weak section would mislead them if they do not check it.

Recommendation: this is not a desk-reject. It deserves a serious referee because the strong argument challenges published experimental claims and the mistake is instructive. My own verdict is reject in current form: Section III should be withdrawn or replaced by a properly derived inequality. If the author can prove a corrected weak bound, the paper has a chance.

Referee Report

3 major / 4 minor

Summary. The manuscript claims to prove that entropic Leggett-Garg inequalities cannot be violated by either strong or weak sequential quantum measurements. For strong measurements, the author constructs the joint density matrix of three detectors and derives pairwise entropies in Eqs. (20), (25), and (26), concluding that the entropic inequalities B⋆1, B⋆2, and B⋆3 cannot be violated. For weak intermediate measurements, an ancilla rotation parameterized by ε is introduced in Eq. (29), pairwise entropies are stated in Eqs. (30)-(32), and a modified inequality B′1(ε)=S12−S13+S23−H[ε]≥0 is asserted in Eq. (33). The paper concludes that reported violations of entropic Leggett-Garg inequalities are artefacts of using two-point functions that ignore the back-action of the intermediate measurement.

Significance. The question whether entropic Leggett-Garg inequalities can be violated is topical, and the manuscript deserves credit for making its central formulas explicit and checkable: Eqs. (20), (25), (26), and (30)-(32) allow a direct numerical test of the claims. That test, however, is fatal to the main new conclusion. The weak-measurement inequality (33) is contradicted by the paper's own expressions for a simple parameter choice, and the term subtracted in that inequality is not the entropy of the weak detector as computed in the text. The strong-measurement conclusion appears to be repairable, but the proof as printed contains algebraic errors in Eqs. (27)-(28). Because the principal new claim — that weak entropic Leggett-Garg inequalities cannot be violated — is unsupported by the manuscript's own equations, the paper does not establish its central result in its present form.

major comments (3)

[§III, Eq. (33) with Eqs. (30)-(32)] The claimed weak-measurement inequality is violated by the paper's own formulas. Take θ1=π/2, θ2=0, and ε=1/2. Eq. (30) gives S12 = 1 + H[(1+√3/2)/2] ≈ 1.274. Eq. (31) gives S23 = 1 because its binary-entropy argument is evaluated at p=1. Eq. (32) gives S13 = 1 + H[1/2] = 2. Substituting into the paper's Eq. (33) yields B′1 ≈ 1.274 − 2 + 1 − H[1/2] = −0.726 < 0, directly contradicting the asserted inequality B′1(ε)≥0. This is precisely the weak-measurement regime the theorem is meant to cover. The same parameter choice shows that the subtracted term H[ε]=1 is not the entropy of the weak detector, which the text computes as S(A2)=H[(1+√(1−ε²))/2]≈0.274. Thus the 'read off' of Eq. (33) from Fig. 6b is not a legitimate derivation, and the weak no-violation claim collapses as stated.
[§II, Eqs. (27)-(28)] The strong-measurement formulas printed in Eqs. (27)-(28) contain arithmetic errors. Using Eqs. (20), (25), and (26) with θ1=θ2=θ, the exact expression is B⋆1 = S12 + S23 − S13 = 1 + 2H[cos²θ] − H13, not 2H[cos²θ] − H13 as written in Eq. (27). At θ=π/2 the printed expression evaluates to −1, while the exact expression evaluates to 0. Similarly, Eqs. (28) state B⋆2 = B⋆3 = H13, whereas the correct values are B⋆2 = B⋆3 = 1 + H13. These corrections do not by themselves overturn the strong no-violation conclusion, since the corrected expressions are nonnegative and satisfy the required bounds, but the proof as printed is not correct and must be rewritten.
[§III, after Eq. (33)] The assertion that B′1(ε) is 'guaranteed to be positive owing to strong subadditivity of quantum entropies' is not justified. Strong subadditivity states S(A1A2)+S(A2A3) ≥ S(A1A2A3)+S(A3), which does not directly involve the term −S13 appearing in B′1. Since the Venn diagram in Fig. 6b is introduced without defining its regions in terms of the measured density matrix, the modified inequality is an additional assumption rather than a consequence of the weak-measurement model. This is load-bearing because, as the paper itself notes, the unmodified inequality (9) can be violated for weak measurements; the entire weak-measurement argument rests on the validity of the unproven Eq. (33).

minor comments (4)

[Abstract and Introduction] There are typographical errors: 'cannot be be violated' appears in the abstract and in the introduction, and §III contains 'It is nor permissible' (should be 'not permissible') and 'assmed' for 'assumed'.
[Fig. 6b and surrounding text] The notation in Fig. 6b and Eq. (33) labels the subtracted term as H[ε], which conflicts with the text's calculation of the weak-device entropy S(A2)=H[(1+√(1−ε²))/2]. The labels and the algebra need to be made consistent, or a separate symbol must be introduced for the detector entropy.
[Fig. 3 and Sec. II] The tripartite Venn diagram in Fig. 3 is taken from Ref. [33], but the individual regions are not defined in the manuscript. Since the derivation of the entropic inequalities (9)-(11) is presented as an inspection of this diagram, providing explicit definitions of the region entropies would substantially improve verifiability.
[§III, unmodified inequality (9)] The sentence 'Using these entropies in (9) would suggest that weak measurements can certainly violate the entropic Leggett-Garg inequality B⋆1' is imprecise: the numerical violation is parameter-dependent, not a violation for all θ and ε. Clarifying the parameter region would help the reader evaluate the subsequent modification.

Circularity Check

1 steps flagged · score 6.0 of 10

Weak-measurement no-violation is built into the modified inequality (33) rather than derived from the measurement model.

self definitional [Section III, after Eq. (32), Eq. (33)]
"The entropic Leggett-Garg inequality for weak intermediate measurement can be read off the Venn diagram in Fig. 6b as B′_1(ϵ)=S12−S13+S23−H[ϵ]≥0, and naturally it cannot be violated because B′_1(ϵ) is guaranteed to be positive owing to strong subadditivity of quantum entropies."

The weak no-violation claim is not derived from the weak-measurement model; it is inserted into the definition of B′_1 by subtracting a term H[ε] that is read off a Venn diagram rather than computed. The paper itself computes the weak detector entropy as S(A2)=H[1/2(1+√(1−ε²))], which is not H[ε], and none of the pairwise entropies (30)–(32) contains H[ε] as a marginal entropy. The modified inequality is introduced only after the paper notes that the unmodified expression 'can certainly violate' the entropic LGI, so the conclusion that the weak inequality 'cannot be violated' is equivalent to the chosen inequality form, not a consequence of the measurement model.

full rationale

The strong-measurement part of the paper is largely self-contained: the pairwise entropies are computed explicitly from the three-qubit density matrix, and the no-violation bound B⋆_1 = 2H[cos²θ]−H13 ≥ 0 follows from an explicit expression. Citations to the author's prior work, including [1], [31], and [33], are context or are backed by the repeated calculation, so they are not load-bearing circularity by themselves. The weak-measurement result, however, is different in kind. The paper finds that the unmodified entropic inequality can be violated in the weak regime, then modifies the inequality by subtracting H[ε], which is read off a Venn diagram rather than derived from the detector entropy. Since the subtracted term is not identified with any computed entropy of the weak detector, and since the resulting inequality is announced as nonnegative 'naturally' by construction, the no-violation theorem for weak entropic Leggett-Garg inequalities is circular as stated: its content is largely the choice of inequality (33). Because the strong claim is independent but the weak claim reduces to a definitional adjustment, the overall circularity score is 6.

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

The derivation leans on the author's own entropy-Venn-diagram formalism (Refs. [31,33]) and on strong subadditivity. No experimental data or external benchmarks are used. The weak-measurement inequality is introduced ad hoc after the unmodified inequality is found violable, which is the main source of circularity burden.

free parameters (3)

theta1
Relative angle between the first and second measurement bases. The claimed inequality is supposed to hold for all theta1; the counterexample uses theta1=pi/2.
theta2
Relative angle between the second and third measurement bases. The counterexample uses theta2=0.
epsilon = 1/2 in counterexample
Weak-measurement strength parameter in the ancilla state |epsilon> = sqrt(1-epsilon^2)|0> + epsilon|1>. The paper claims the weak inequality holds for all epsilon in [0,1].

assumptions (4)

domain assumption Quantum mechanics and the density-matrix formalism for sequential measurements apply to the three-detector setup.
Section II assumes a maximally mixed input qubit purified by a reference R and measurements implemented by CNOT-like unitaries.
standard math Strong subadditivity of quantum entropy guarantees nonnegativity of the relevant Venn-diagram combinations.
Invoked around Eqs. (6)-(8) and again for B'_1 after Eq. (33); however B'_1 is not actually a direct strong-subadditivity expression.
ad hoc to paper The tripartite entropy Venn diagram in Fig. 3, taken from Ref. [33], correctly represents the three detectors.
Fig. 3 and its peculiar form are imported from the author's own previous work without proof in this paper; the weak modification is also read off a Venn diagram.
ad hoc to paper The weak-measurement model with the ancilla state |epsilon> and the resulting entropies in Eqs. (30)-(32) are correct.
These formulas are stated without derivation and are the basis of the weak-inequality claim; the counterexample indicates that at least one of them or the inequality is wrong.

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Pith. "Pith review of Neither weak nor strong entropic Leggett-Garg inequalities can be violated." pith.science (2026). https://pith.science/paper/T5NJJZIS

@misc{pith2026190803124,
  author       = {Pith},
  title        = {Pith review of: Neither weak nor strong entropic Leggett-Garg inequalities can be violated},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5NJJZIS}},
  note         = {Machine review of arXiv:1908.03124}
}

read the original abstract

The Leggett-Garg inequalities probe the classical-quantum boundary by putting limits on the sum of pairwise correlation functions between classical measurement devices that consecutively measured the same quantum system. The apparent violation of these inequalities by standard quantum measurements has cast doubt on quantum mechanics' ability to consistently describe classical objects. Recent work has concluded that these inequalities cannot be violated by either strong or weak projective measurements [1]. Here I consider an entropic version of the Leggett-Garg inequalities that are different from the standard inequalities yet similar in form, and can be defined without reference to any particular observable. I find that the entropic inequalities also cannot be be violated by strong quantum measurements. The entropic inequalities can be extended to describe weak quantum measurements, and I show that these weak entropic Leggett-Garg inequalities cannot be violated either even though the quantum system remains unprojected, because the inequalities describe the classical measurement devices, not the quantum system. I conclude that quantum mechanics adequately describes classical devices, and that we should be careful not to assume that the classical devices accurately describe the quantum system.

Figures

Figures reproduced from arXiv: 1908.03124 by the authors.

**Figure 4.** FIG. 4. General tri-partite entropy Venn diagram defining [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 3.** FIG. 3. Entropy Venn diagram for three quantum ancillae [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. Bipartite Venn diagram for the first two detectors, where [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) A weak projective measurement by [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Leggett-Garg inequalities cannot be violated in quantum measurements
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The paper argues that Leggett-Garg inequalities cannot be violated by any projective measurement when correlators are defined on measurement device outcomes, and that experimental violations arise from inconsistent as...

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+ sin4(θ 2) ] the entropic Leggett-Garg inequalities become B⋆ 1 = 2H[cos2(θ)]−H13≥ 0, (27) B⋆ 2 =B⋆ 3 =H13≥ 0. (28) It is immediately clear (from using Jensen’s inequality) that the entropic Leggett-Garg inequality B⋆ 1 cannot be violated for strong measurements, just as its non-entropic counterpart [1]. Of course, B⋆ 2 and B⋆ 3 also cannot be violated b...

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Indeed, using again θ1 = θ2 = θ, the equation analogous to (27) can be vi- olated when using H13 = H[cos4(θ/2) + sin 4(θ/2)− 2 √ 1−ϵ2 sin2(θ/2) cos2(θ/2)]. However, this apparent vi- olation is due to the fact that in weak measurements, the entropic Leggett-Garg inequality must be modiﬁed. When measurements are weak ( ϵ < 1), the entropy of the weak devic...

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