REVIEW 4 major objections 4 minor 29 references
Leggett-Garg inequalities cannot be violated in quantum measurements
T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Leggett-Garg inequalities cannot be violated by any projective measurement, strong or weak; reported violations are artifacts of mixing invasive and non-invasive assumptions.
desk verdict The paper's weak-measurement proof is invalid: the correlator calculation drops pointer-state interference terms, so the main new claim is unsupported; the strong-measurement part is a clean restatement of known critiques. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the joint wave function of a quantum system plus the ancillas that record each measurement. A strong projective measurement is implemented by a controlled-NOT operation that flips the ancilla when the system is in one basis state; a weak projective measurement is the same gate with the flip replaced by a small rotation, leaving the ancilla in $|\epsilon\rangle_2=\sqrt{1-\epsilon^2}|0\rangle_2+\epsilon|1\rangle_2$. Tracing out the quantum system leaves reduced density matrices for the measurement devices, and the correlators $K_{ij}$ are read off as $\operatorname{Tr}(\rho_{ij}\sigma_z\otimes\sigma_z)$. The argument works because the Leggett-Garg combinations are built from the diagonal elements $P(xyz)$ of a joint probability distribution over three binary outcomes, and nonnegative probabilities automatically bound those combinations by 1.
What would settle it
Compute the three correlators for a weak measurement modeled by the full unitary $U=e^{-igP_\theta\otimes\sigma_y}$, keeping the system's coherent superposition instead of replacing the flipped ancilla by $|\epsilon\rangle$; if for any $\epsilon\in(0,1)$ the combination $K_{12}+K_{23}-K_{13}$ exceeds 1, the paper's no-violation claim for weak projective measurements fails. Experimentally, re-analyze the kept and discarded outcome sets of a published 'interaction-free' Leggett-Garg experiment, since the paper predicts both sets give the same correlators and the same apparent violation.
Extended reading notes
Core claim
The paper's central claim is that the Leggett-Garg inequalities, $B_1=K_{12}+K_{23}-K_{13}\le 1$ and its two cyclic variants, are never violated by any projective measurement of a qubit, at any measurement strength and for any initial state. Using a unitary measurement model in which a strong measurement is a controlled-NOT flip of an ancilla and a weak measurement rotates the ancilla to $|\epsilon\rangle_2=\sqrt{1-\epsilon^2}|0\rangle_2+\epsilon|1\rangle_2$, the author derives the correlators $K_{12}=(|\alpha|^2-|\beta|^2)(1-\epsilon^2)+\epsilon^2\cos\theta_1$, $K_{23}=(|\alpha|^2-|\beta|^2)(1-\epsilon^2)\cos\theta_1\cos\theta_2+\epsilon^2\cos\theta_2$, and $K_{13}=\cos\theta_1\cos\theta_2-2\sqrt{1-\epsilon^2}|\alpha||\beta|\sin\theta_1\sin\theta_2$. Substituting these into the Leggett-Garg combinations yields expressions bounded above by 1 for all $\epsilon\in[0,1]$ and all $\alpha,\beta$. The apparent violations reported in the literature follow from using $\epsilon=0$ for the middle measurement when computing $K_{13}$ while using $\epsilon=1$ for the same measurement when computing $K_{12}$ and $K_{23}$.
Load-bearing premise
The argument depends on a model in which every weak measurement completely destroys the measured system's original superposition—even at zero strength, where no measurement actually occurs; if a genuine weak measurement leaves the system mostly intact, the no-violation claim for weak projective measurements may collapse.
Editorial extensions
If this is right
- Every reported experimental violation of a Leggett-Garg inequality would be explained as an artifact of combining a non-invasive assumption for one correlator with an invasive measurement for the others, rather than as evidence against macrorealism.
- The three-point correlator for consecutive strong measurements is $K_{13}=\cos\theta_1\cos\theta_2$, not $\cos(\theta_1+\theta_2)$; experiments that measure all three correlators directly can check this prediction.
- Quantum mechanics would remain an adequate description of macroscopic objects, removing the original motivation for doubting it on Leggett-Garg grounds.
- Weak measurements cannot rescue Leggett-Garg violations because the measurement devices are fully projected even when the measured system is not; therefore weak-measurement tests are subject to the same non-invasiveness criticism.
- The inequalities should be read as constraints on classical measurement records, not on the quantum system: classical objects cannot adequately describe quantum objects, not vice versa.
Reading between the lines
- A natural extension of the paper's diagnosis is that any temporal inequality built from correlators obtained under different measurement-strength assumptions—entropic Leggett-Garg bounds, temporal Bell inequalities, or no-signaling-in-time conditions—should show the same pattern of spurious violation when $\epsilon=0$ and $\epsilon=1$ data are mixed.
- The decisive test of the weak-measurement claim would be a model that keeps the system's coherence proportional to the coupling; if Leggett-Garg violations reappear there, the paper's conclusion would be specific to its fully dephasing weak-measurement scheme rather than to all projective measurements.
- The paper's broader moral, that temporal correlation bounds constrain classical records rather than quantum evolution, suggests a route to no-violation theorems for higher-dimensional systems and longer measurement sequences using the same positivity-of-probabilities argument.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that Leggett-Garg inequalities cannot be violated by any projective measurement, strong or weak, and argues that previously reported violations (e.g., Knee et al. 2012, Goggin et al. 2011) are artifacts of mixing invasive and non-invasive assumptions. The author introduces a model with three binary measurement devices, derives correlation functions K12, K23, K13 from a sequence of one weak and one strong measurement on a qubit, and concludes that the LGI combinations B1, B2, B3 are always bounded by 1 for any measurement strength and any initial state.
Significance. If the central claim were correct, the paper would overturn a large body of experimental and theoretical work on Leggett-Garg violations and would support the view that quantum mechanics adequately describes macroscopic objects. However, the new contribution—the no-violation result for weak measurements—rests on an incorrect reduced-density-matrix calculation. The strong-measurement discussion is largely consistent with earlier critiques of non-invasive measurability, but it is not new, and the explanatory framework based on an assumed joint distribution is circular. The paper does not provide a valid derivation of the weak-measurement correlators, so the headline result is not established.
major comments (4)
- [Eqs. (18)–(21)] The derivation of K23 and K13 drops the off-diagonal elements of the pointer state. From the paper's own Eq. (18), the branch containing |θ2>|0>3 has device 2 in the superposition cos(θ1/2)cos(θ2/2)|0>2 − sin(θ1/2)sin(θ2/2)|ε>2. Because <0|ε> = √(1−ε^2) ≠ 0, the reduced density matrix ρ23 obtained by tracing out the quantum system contains cross terms |0>2<ε| and |ε>2<0|. Equation (19) omits these terms, effectively treating |0>2 and |ε>2 as orthogonal. Including them gives K23 = (1−ε^2)cosθ1cosθ2 + ε^2cosθ2 − √(1−ε^2)sinθ1sinθ2 and K13 = cosθ1cosθ2 − √(1−ε^2)sinθ1sinθ2 for the preparation |+>. Consequently Eq. (21) is false, and the inequalities (22)–(24) and (31)–(33) are not derived from the model.
- [Comment after Eq. (21)] The text states that K13 = cos(θ1+θ2) would follow if ε=0, but Eq. (21) itself gives K13 = cosθ1cosθ2 for every ε, including ε=0. This internal contradiction shows that the derivation of K13 is not reliable. The contradiction is not merely semantic: the correct calculation for ε=0 in the model yields cosθ2 (since the first measurement is not performed), not cos(θ1+θ2), so the paper's interpretation of the standard LGI expression is itself muddled.
- [Eq. (13) and the interaction Hamiltonian] The weak-measurement state is not derived from the stated interaction. The paper defines U1 = e^{-iH(g)} with H = gPθ⊗σ_y, for which the ancilla evolves to |ε> on the |θ1> branch and stays at |0> on the |θ̄1> branch. The replacement in Eq. (13) of |1>2 by |ε>2 instead places |ε> on the |θ̄1> branch, since Eq. (13) is the strong-CNOT state with the flip on |θ̄1>. Thus the model is not the stated weak measurement, and the claim that the conclusions apply to 'weak projective measurements' is not supported by the model's dynamics.
- [Inequalities (31)–(33)] The paper asserts that 'it is straightforward to show' that the inequalities (31)–(33) cannot be violated for any state preparation or any measurement strength, but no proof is provided. Given that the correlators feeding into these inequalities are incorrect, the assertion is unsupported. Even within the paper's own framework, the derivation of these bounds would require an explicit demonstration that the right-hand sides never exceed 1 for all θ1, θ2, α, β, and ε.
minor comments (4)
- [Eq. (2)] The expression for K12 contains repeated and incorrectly signed terms; it should be K12 = P(+++) + P(++−) − P(+−+) − P(+−−) − P(−++) − P(−+−) + P(−−+) + P(−−−).
- [Eqs. (18) and (19)] Equation (18) contains stray parentheses and missing kets (for example '⟩|0⟩3' and '⟩|0⟩2'), and Eq. (19) has an apparent index typo in the second term where |0⟩2⟨0| likely should be |0⟩3⟨0|. These typographical errors make the calculation difficult to follow.
- [Reference [4]] The author of reference [4] is spelled 'Ballentine' (L. E. Ballentine), not 'Ballantine' as in the text and reference list.
- [General correlators, Eqs. (25)–(27)] The statement that K13 'will depend on ε for arbitrary preparations' is not demonstrated and is contradicted by the special case β=0 in Eq. (27), where the preparation-dependent term vanishes but the calculation from Eq. (18) still contains a √(1−ε^2)sinθ1sinθ2 term.
Circularity Check
No significant circularity: the weak-measurement no-violation proof is derived from an explicit measurement model, not from its own conclusion.
full rationale
The paper's argument is a coherent consistency proof. It defines the three correlators K12, K23, and K13 as marginals of a joint three-outcome distribution P(x,y,z), derives the Leggett-Garg inequalities from the assumption that P is a probability distribution, and then computes P from a concrete quantum circuit: CNOT unitaries, the explicit weak-measurement replacement |1> -> |epsilon>, and the strong-measurement limit recovering the known K12 = cos(theta1) and K23 = cos(theta2). The conclusion that the inequalities are not violated follows from the computed correlators together with the derived inequalities; it is not an input to the calculation. No parameter is fitted to data and then renamed a prediction. The self-citations [23-26,29] are peripheral remarks about ancilla diagnostics and entropic inequalities, not load-bearing premises. A possible concern that the no-violation result is tautological because any three binary variables with a joint distribution satisfy the inequalities is not a circularity: the paper supplies the joint distribution from an explicit physical model, and the inequalities are derived from that distribution rather than assumed. Mathematical objections to Eqs. (19)-(21), if valid, would be correctness defects, not circular reductions.
Assumptions & free parameters
free parameters (3)
- measurement strength ε
- angles θ1, θ2
- initial state amplitudes α, β
assumptions (3)
- domain assumption Standard quantum mechanics with projective measurements on the system
- domain assumption The measurement device outcomes have a joint probability distribution P(x,y,z) in each run
- ad hoc to paper Weak measurements dephase the system completely even for ε=0
Cite this review
Pith. "Pith review of Leggett-Garg inequalities cannot be violated in quantum measurements." pith.science (2026). https://pith.science/paper/MTH7OSY5
@misc{pith2026190802886,
author = {Pith},
title = {Pith review of: Leggett-Garg inequalities cannot be violated in quantum measurements},
year = {2026},
howpublished = {\url{https://pith.science/paper/MTH7OSY5}},
note = {Machine review of arXiv:1908.02886}
}
read the original abstract
Leggett and Garg derived inequalities that probe the boundaries of classical and quantum physics by putting limits on the properties that classical objects can have. Historically, it has been suggested that Leggett-Garg inequalities are easily violated by quantum systems undergoing sequences of strong measurements, casting doubt on whether quantum mechanics correctly describes macroscopic objects. Here I show that Leggett-Garg inequalities cannot be violated by any projective measurement. The perceived violation of the inequalities found previously can be traced back to an inappropriate assumption of non-invasive measurability. Surprisingly, weak projective measurements cannot violate the Leggett-Garg inequalities either because even though the quantum system itself is not fully projected via weak measurements, the measurement devices are.
Figures
Reference graph
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