The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moir'e phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum

arXiv:2608.18886 · quant-ph · Submitted 2026-08-19 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "The exact price of local realism in CHSH experiments".

Mira: The gist: This work determines the minimal measurement dependence required for local hidden-variable accounts to reproduce observed CHSH values as a function of detection efficiency,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So to wrap up this part of our discussion about "The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moir'e phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum," the paper is essentially determined to find the minimal measurement dependence M required for faithful local hidden variable accounts to reproduce observed singles rates, coincidence rates, and unbiased marginals of a polarization singlet with symmetric detector efficiency eta eff.

Mira: They claim they've derived a formula for this minimal measurement dependence as M(S, eta eff) = zero eta eff((S+two) eta eff-four)/six and this surface has three landmark edges: Hall’s tight bound at eta eff = one the Garg–Mermin detection threshold, and the postselection ceiling S = four/eta eff - two <ref:2608.18886#pg1,1$, the Garg–Mermin detection threshold, and the postselection ceiling $S = 4>.

Lev: It matters because they also solved for a unique detection profile D(m) = sqrt m h(m) u under which a deterministic sign model reproduces the singlet exactly, and this proves that exact quantum correlations and angle-independent coincidence rates are jointly impossible in this class.

Kai: They also certify the quantum point exactly at S = two sqrt two and eta eff = nine/ten to be M = (twenty-seven sqrt two - thirty-three)/one hundred which they verified using primal and dual certificates in Q(sqrt two) arithmetic <ref:2608.18886#pg1>.

Mira: The paper then follows the mathematics to an experimental consequence by solving the detection sector exactly, which leads them to a proof of impossibility: no pure-detection model can yield both the exact singlet correlation E = - two and an angle-independent coincidence rate <ref:2608.18886#pg1>.

Lev: What this means for us is that we’re not just dealing with a simple violation; we’re dealing with a complex trade-off where the measurement dependence budget depends on both how much you violate the inequality and how efficient your detectors are.

Conclusion: Kai: So looking at the title of this paper again, "The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moir'e phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum," it captures all these complicated results.

Mira: Exactly. It’s about quantifying how much measurement dependence you need to account for when you look at CHSH experiments as a function of detector efficiency, while also pointing out this specific unmeasured fringe in the fourfold coincidence sum that exists even at the violating angles.

Lev: The paper shows that there is a specific price surface M(S, eta eff) that dictates exactly how much measurement dependence you need to keep your local models faithful to what an experimenter actually observes.

Kai: And the implication for the broader community is that it sets a rigorous benchmark for testing local hidden variable theories against real experimental realities, moving beyond just checking if a simple inequality is violated.

Mira: It’s about showing that pure-detection models are fundamentally constrained; they can't reproduce both exact quantum correlations and angle-independent coincidence rates at the same time.

Lev: So ultimately, this work gives us a clear picture of the constraints imposed by efficiency and measurement dependence on local realism, even if we don't get a perfect quantum result every single time.

quant-ph

Submitted: 2026-08-19

Updated: 2026-10-18

Comments: 14 pages, 5 figures. Code and exact certificates at https://github.com/RandomInternetPreson/moire-phase-space-sampler. v4: Secs. 9-10 rewritten after independent review; the claims that existing data do not constrain the fringe and that flatness forces measurement dependence are withdrawn. Theorem 2 restated (equal values, not zero, at the CHSH angles). Mathematical results unchanged

Code: https://github.com/RandomInternetPreson/moire-phase-space-sampler

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: The gist: This work determines the minimal measurement dependence required for local hidden-variable accounts to reproduce observed CHSH values as a function of detection efficiency, revealing an

Key concepts

Measurement Dependence ($M$): This is a measure of how much the local hidden-variable theory must rely on the settings chosen for the measurements to match experimental results. The paper uses Hall's measure, $M$, derived from linear programming to find the minimum dependence required for a theory to reproduce observed CHSH values ($S$) given detector efficiency ($\eta_{\rm eff}$).
Price Surface $M(S, \eta_{\rm eff})$: This surface plots the minimal measurement dependence ($M$) required as a function of the desired CHSH value ($S$) and detector efficiency ($\eta_{\rm eff}$). It defines a 'price' for achieving specific quantum correlations under local realism, showing how measurement strategy must adapt to detectability.
Moiré Phase Locking: This is a specific local mechanism found that can saturate the theoretical bound on measurement dependence. It involves a single hidden phase evolving deterministically, with randomness only introduced by frozen offsets and flight times in an ensemble, allowing the system to reach minimal dependence ($M=0$) at high efficiency ($\eta_{\rm eff} \approx 0.80$).
Fourfold Coincidence Sum Fringe: This refers to a specific observable in experiments involving four detectors. The paper shows that the detection profile of local realism predicts a fringe in this sum proportional to $\cos 4(\Delta a - \Delta b)$, which vanishes at the CHSH angles, proving that four-point experiments are blind to this feature unless measurement dependence is high.

Summary

The gist: This work determines the minimal measurement dependence required for local hidden-variable accounts to reproduce observed CHSH values as a function of detection efficiency, revealing an exact price surface and proving that pure-detection models cannot simultaneously reproduce quantum correlations and angle-independent coincidence rates.

How it works

The research establishes a linear programming problem over the complete space of local strategies to find the minimal measurement dependence M required for faithful local hidden-variable accounts to achieve a specific CHSH value S, as described in Section 3. The conjecture for this price surface is given by M(S, ηeff) = max n 0, ηeff(S + 2)ηeff − 4. This surface has three known edges: Hall’s tight bound at ηeff = 1, the Garg–Mermin detection threshold, and the postselection ceiling S = 4/ηeff − 2. The edge law for the detection profile D(m) is derived analytically to show that exact quantum correlations and angle-independent coincidence rates are jointly impossible for any pure-detection model of this class.

The Price Surface and Bounds

The price surface M(S, ηeff) exhibits an interaction where the measurement-dependence budget is amplified by 1/η2eff, acting on the coincidence subensemble. The naive additive combination M = (1+√2)/3−2/(3ηeff), which shares both endpoints, is falsified because the true surface lies strictly below it in the interior. A corollary derived from this analysis states that M = 0 whenever S ≤ 4/ηeff − 2, implying that measurement dependence is only implied by correlations in the violating region, and ordinary (classical) statistics imply none.

Exact Certification at the Quantum Point

Theorem 1 certifies the minimal measurement dependence at S = 2√2 and ηeff = 9/10 to be exactly M = (27√2 − 33)/100. This value is verified by solving the linear program using integer-relation detection and symbolic verification, confirming that the primal candidate satisfies all faithfulness equalities exactly. Furthermore, Corollary 1 provides exact rational values for several points on this surface, such as M(27/10, 9/10) = 69/2000.

Impossibility of Pure Detection Models

The analysis of the detection sector demonstrates that no detection profile D yields both the exact singlet correlation E(∆) = − cos 2∆ and an angle-independent coincidence rate. This impossibility is proven by showing that a constant D leads to a sawtooth correlation, not the cosine required for the singlet. The resulting residue is a concrete observable, identified as a cos 4(a−b) fringe in the total (fourfold-sum) coincidence rate which vanishes identically at the CHSH angles.

Mechanism for Saturation

A specific local mechanism, moir´e phase locking, is exhibited that saturates the bound. This mechanism involves a single hidden phase λ evolving deterministically under a differential equation where all randomness resides in a quenched ensemble of frozen offsets and flight times. With an ensemble of 1024 frozen offsets, the mechanism attains the certified floor exactly at ηeff = 1 at every tested fidelity, reaching zero measurement dependence at ηeff ≈ 0.80.

Observational Consequences and Falsifiable Fingerprints

The exact profile of the detection sector shows that the total coincidence rate obeys Rd(∆) ∝ Cg(2∆), resulting in a cos 4∆-dominated fringe with a relative peak-to-peak amplitude of 12.4%. This fringe vanishes identically at the CHSH angles, meaning four-point CHSH experiments are mathematically blind to this observable. The protocol designed to supply the factorization veto reveals that a flat-rate prediction of quantum mechanics fails if a fringe is observed, forcing M ≳ 95% of Hall’s floor.

Conclusion and Limitations

The paper concludes that local accounts occupy a one-parameter dial between an informed source and an inefficient detector, with every intermediate position priced by the surface M(S, ηeff). The results are limited because the uniform offset density yields zero measurement dependence by translation symmetry, and the derivation of this density from a formation process remains an open problem. All exact values and identities were verified independently of the tool by machine computation as described above.

Code and Data Availability

All linear programs, exact certificates, spectral solvers, and protocol simulations are provided in the repository directory. The pipeline is deterministic end to end (seedless T = 0 dynamics — offset draws in dither sweep.py) and runs on commodity hardware. The ε = 0.01 price is cross-validated by two independently coded solvers on the same register, both returning 1.375 to three decimals. The total wall time on a 56-core workstation is under two hours for the 256-atom pipeline.

AI-Assisted Preparation

Large-language-model assistance (Claude, Anthropic) was used during this research as a tool for checking derivations, drafting text, and writing verification code under the sole direction of the author. All exact values and identities were verified independently of the tool by machine computation as described above. The research is supported by references [1] through [32].

References

[1] J. S. Bell, Physics Physique Fizika 1, 195 (1964)

[2] J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett. 23, 880 (1969)

[3] B. S.-Cirel’son, Lett Math Phys 4, 93 (1980)

[4] M. J. W. Hall, Phys. Rev. Lett. 105, 250404 (2010)

[5] M. J. W. Hall, Phys Rev A 84, 022102 (2011)

[6] A. S.-Friedman, A. H. Guth, M. J.-W Hall, D.-I Kaiser, and J Gallicchio Phys Rev A 99, 012121 (2019)

[7] A Garg and N. D. Mermin Phys Rev D 35, 3831 (1987)

[8] P. H. Eberhard Phys Rev A 47, R747 (1993)

[9] P. M. Pearle Phys Rev D 2, 1418 (1970)

[10] N. Gisin and B. Gisin Phys Lett A 260, 323 (1999)

[11] J.-A. Larsson J Phys A 47, 424003 (2014)

[12] A.

Improvements for AI systems

The improved system can precisely calculate minimal measurement dependence M for any given CHSH value S and detection efficiency ηeff using Hall’s variational measure, utilizing M(S, ηeff) = max[0, ηeff((S + 2)ηeff −4)/6].
The system can certify the quantum point exactly by computing M(2√2, 9/10) = (27√2 − 33)/100, ensuring high-precision verification against Q(√2) arithmetic.
The AI can solve for the unique detection profile D(m) = √m h(m) under which a deterministic sign model reproduces the singlet correlation exactly, deriving the exact quantum correlations and angle-independent coincidence rates are jointly impossible proof.
The system can identify the moir´e phase locking mechanism, modeling how a hidden phase evolves deterministically through a standing beat pattern of the measurement apparatus, allowing for explicit simulation of this mechanism across various parameter ranges.
The improved system can analyze the residual correlation pattern, which is described as a cos 4(a − b) modulation of the fourfold coincidence sum, of relative amplitude up to 12.4%, providing a falsifiable fingerprint for local models that exhibit this effect.