Violation of Mermin's version of a Bell inequality in a classical statistical model

Manfried Faber

arxiv: 1907.00175 · v1 · pith:ORYX6CP4new · submitted 2019-06-29 · 🪐 quant-ph

Violation of Mermin's version of a Bell inequality in a classical statistical model

Manfried Faber This is my paper

Pith reviewed 2026-05-25 13:08 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bell inequalityMermin inequalityclassical statistical modelensemble modificationhidden variablesmeasurement processno-signaling condition

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The pith

A classical statistical model violates Mermin's Bell inequality when measurements modify the hidden-variable ensemble.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a classical statistical model in which the probability distribution of hidden variables changes depending on which observable is measured. Under this rule the model produces correlations that exceed the bound set by Mermin's version of a Bell inequality. The author notes that the same kind of ensemble change occurs during quantum measurements, suggesting the violation need not require non-classical physics. A sympathetic reader would conclude that the source of the inequality violation can be the measurement process itself rather than any intrinsic nonlocality or contextuality of the underlying variables.

Core claim

In the classical model the measurement of one observable alters the probability distribution of the hidden variables in a manner that depends on the chosen measurement setting; the resulting statistics then violate Mermin's inequality while remaining fully local and deterministic at the level of individual hidden variables.

What carries the argument

Measurement-induced modification of the hidden-variable ensemble, which updates the probability distribution according to the selected observable.

If this is right

Bell-inequality violations can appear in fully classical systems once the act of measurement is allowed to reshape the ensemble.
The distinction between classical and quantum correlations can hinge on how measurement affects the underlying probability distribution rather than on the variables themselves.
Models that keep the ensemble fixed under measurement will continue to obey the inequality.
Reproducing quantum-like correlations in a classical setting requires an explicit rule for ensemble updating.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If every measurement in nature updates an underlying ensemble, then classical hidden-variable models could in principle mimic a wider range of quantum statistics than is usually assumed.
The same mechanism might be tested in macroscopic systems where one can directly observe whether a measurement choice reshuffles the population statistics.
Extensions could ask whether the required ensemble update can be realized with purely local, deterministic dynamics at a finer scale.

Load-bearing premise

The measurement process can change the distribution of hidden variables in a setting-dependent way without creating faster-than-light signaling.

What would settle it

A concrete laboratory test that checks whether an ensemble of classical particles or spins can have its statistical distribution altered by the choice of measurement apparatus in exactly the manner required to produce the reported violation while preserving no-signaling.

read the original abstract

We investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a classical model violates Mermin's inequality by letting measurement update the hidden-variable ensemble, but the abstract gives no equations or checks, and the idea overlaps with known loophole constructions.

read the letter

The main takeaway is that this work shows a classical statistical model can exceed Mermin's bound once the measurement is allowed to change the ensemble of hidden variables. The abstract ties this directly to the ensemble-modifying character of quantum measurements. That is the central move, and it is stated plainly. The paper does at least pick a concrete inequality and flag the fixed-ensemble assumption as the one being relaxed. That is a clear, limited clarification rather than a broad claim about quantum mechanics. On the positive side, the connection to how real measurements affect distributions is noted without overstatement. If the full text supplies an explicit, local update rule that preserves no-signaling while producing the violation, that would be a usable example for thinking about loopholes. The soft spots are more substantial. The text supplied contains no derivation, no explicit form for the update rule, and no numerical verification that the correlations actually exceed the bound under the stated conditions. Without those, it is impossible to check whether the claimed violation follows from the model or requires extra tuning. The stress-test concern about setting-dependent updates risking signaling is not addressed in the abstract, and that is a load-bearing requirement for any classical construction. Earlier literature on detection and memory loopholes already contains ensemble-changing mechanisms that produce apparent Bell violations; the paper does not cite or differentiate itself from those, so the novelty is low. This is the sort of note that might interest a small group working on classical simulations of Bell tests, but the lack of detail means most readers will not get usable content. I would not bring it to a reading group or cite it. It does not look ready for peer review because the mechanism cannot be evaluated from what is given.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs a classical statistical model in which the measurement process modifies the underlying ensemble of hidden variables. It claims that this modification produces a violation of Mermin's version of the Bell inequality, a feature the authors note is also present in quantum measurements.

Significance. If an explicit, local, and no-signaling update rule can be supplied that remains strictly classical while exceeding the Mermin bound, the result would be of moderate interest to the foundations community because it isolates the role of ensemble modification in Bell tests. The paper does not supply machine-checked proofs, reproducible code, or parameter-free derivations, so its evidential weight rests entirely on the internal consistency of the model.

major comments (2)

[Model definition (throughout)] The manuscript provides no explicit functional form for the measurement-induced update rule P(λ) → P'(λ|setting). Without this definition it is impossible to verify that the update is local at each site and leaves the marginal statistics at distant sites unchanged, which is required for the model to remain no-signaling and classical. This is load-bearing for the central claim that a violation occurs inside a classical statistical framework.
[Violation claim] Mermin's inequality is derived under the assumption of a fixed joint distribution P(λ) with predetermined outcomes. The paper replaces this with a setting-dependent post-measurement distribution; the text does not demonstrate that the resulting correlations still satisfy the no-signaling conditions that justify applying the Mermin bound in the first place. A concrete counter-example or explicit calculation showing preservation of marginals is needed.

minor comments (1)

The abstract states the result but does not indicate the numerical value of the violation or the specific form of Mermin's inequality used; adding one sentence with the bound and the achieved value would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the model definition. The comments correctly note that the current manuscript lacks a concrete functional form and explicit verification of no-signaling. We will revise the paper to supply these elements while preserving the central conceptual point that ensemble modification under measurement can produce the reported violation within an otherwise classical framework.

read point-by-point responses

Referee: [Model definition (throughout)] The manuscript provides no explicit functional form for the measurement-induced update rule P(λ) → P'(λ|setting). Without this definition it is impossible to verify that the update is local at each site and leaves the marginal statistics at distant sites unchanged, which is required for the model to remain no-signaling and classical. This is load-bearing for the central claim that a violation occurs inside a classical statistical framework.

Authors: We agree that an explicit functional form is required for verification. In the revised manuscript we will introduce a concrete, local update rule of the form P(λ) → P'(λ|setting) defined separately at each site such that the marginal distribution over λ at any distant site remains independent of the local setting choice. This will be accompanied by a direct check that the joint statistics remain no-signaling. revision: yes
Referee: [Violation claim] Mermin's inequality is derived under the assumption of a fixed joint distribution P(λ) with predetermined outcomes. The paper replaces this with a setting-dependent post-measurement distribution; the text does not demonstrate that the resulting correlations still satisfy the no-signaling conditions that justify applying the Mermin bound in the first place. A concrete counter-example or explicit calculation showing preservation of marginals is needed.

Authors: We accept that the manuscript must demonstrate preservation of marginals. The revision will contain an explicit calculation for the chosen update rule showing that the single-site marginal probabilities are independent of the distant measurement settings, thereby confirming that the model remains no-signaling while the joint distribution is altered by the measurement process itself. This distinguishes the present construction from standard local-hidden-variable models that assume a fixed P(λ). revision: yes

Circularity Check

1 steps flagged

Violation obtained by construction from measurement-dependent ensemble update

specific steps

self definitional [Abstract]
"We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems."

The model is defined to permit post-measurement redistribution of the ensemble that depends on the chosen observable. The inequality violation follows immediately once this defining feature is inserted; it does not emerge from any additional dynamical law or first-principles constraint within the classical framework.

full rationale

The paper's central claim reduces directly to its defining assumption: a classical model is constructed in which measurement alters the hidden-variable distribution P(λ) in a setting-dependent way. Mermin's inequality is derived under the opposite assumption of a fixed joint distribution independent of the chosen observables. The reported violation is therefore equivalent to the input feature rather than an independent derivation from unmodified classical statistics. No external benchmark or parameter-free calculation is invoked to justify the update rule.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unstated premise that a measurement-dependent update rule for the hidden-variable distribution can be introduced while preserving classical statistics and locality. No independent evidence for the specific form of that update rule is supplied.

axioms (1)

domain assumption Measurement can modify the ensemble of hidden variables in an observable-dependent manner while the model remains classical.
Invoked in the abstract to obtain the violation; no derivation or external justification is given.

pith-pipeline@v0.9.0 · 5543 in / 1258 out tokens · 23106 ms · 2026-05-25T13:08:02.014459+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction (8-tick period) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we can specify the possible flips by the adjacency matrix M ... nt = M^t n0 ... eight points ... uniform distribution in the region R = [2,7]
IndisputableMonolith/Cost/FunctionalEquation.lean J-cost forcing and ensemble update echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the modification of the ensemble of paths in the measurement process is the reason for the violation

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