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

AI summary

Overview

Research area: Foundations of quantum mechanics — Bell/CHSH nonlocality, local hidden-variable models, and the quantification of loopholes (measurement independence and detection efficiency). The paper sits at the intersection of quantum information theory, convex optimization, and experimental quantum optics.

Technical level: Advanced. The paper is written for specialists: it uses linear programming over local strategy polytopes, exact arithmetic in ℚ(√2), Fourier/Chebyshev analysis, and a proposed optical protocol. The core ideas are explainable, but the full argument requires comfort with Bell-inequality formalism.

Scope (one sentence): The paper computes the exact minimal measurement dependence required for a faithful local model to reach a given CHSH value at a given detector efficiency, derives an observable consequence of its exact detection model (a cos 4(a−b) fringe in the fourfold coincidence sum), and constructs an explicit local mechanism that saturates the resulting bound.

What This Paper Is About

Bell tests rule out local hidden-variable explanations only if you assume measurement independence (settings are free) and fair sampling (detected pairs represent all pairs). This paper asks what happens when you relax both at once: given a detection efficiency η_eff and a target CHSH value S, what is the minimum amount of measurement dependence a local model must "pay" while still reproducing what the experimenter actually sees (singles rates, coincidence rates, unbiased marginals)? Along the way, the author shows that within one natural class of local detection models, exact quantum correlations and angle-independent coincidence rates cannot coexist — leaving a specific, previously unbounded experimental signature.

Key Contributions

1. A closed-form "price surface" ℳ(S, η_eff) = max{0, η_eff((S+2)η_eff − 4)/6} for the minimal measurement dependence of faithful local models, verified against exact linear programming to 3.6×10⁻¹⁶ at 32 grid points spanning S ∈ [2.1, 3.4] and η_eff ∈ [0.85, 1]. Its edges reproduce three known results: Hall's tight line at η_eff = 1, the Garg–Mermin threshold η_eff = 2(√2−1), and the postselection ceiling S = 4/η_eff − 2. The general (S, η_eff) statement is presented as Conjecture 1 pending a parametric dual certificate.

2. Exact certification at the quantum point: ℳ(2√2, 9/10) = (27√2 − 33)/100, proved with primal (68 weights, 36 faithfulness equalities) and dual certificates verified in ℚ(√2) arithmetic, plus a rational ladder of six further exact values.

3. An exact solution of the detection sector and an impossibility result: the unique detection profile D(m) = √m·h(m) (solved to 10⁻¹¹) under which a deterministic sign model reproduces the singlet correlation exactly, with the √m edge law derived analytically. Proposition 1 proves no pure-detection model of this class yields both exact singlet correlations and angle-independent coincidence rates.

4. An explicit local mechanism ("moiré phase locking") with deterministic hidden-phase dynamics and a quenched ensemble of frozen offsets and flight times, which attains the certified floor exactly at η_eff = 1 with 1024 frozen offsets, and reaches zero measurement dependence at η_eff ≈ 0.80 when combined with the solved detection profile.

Main Findings

• The price surface has three known theorem-edges. At η_eff = 1 the formula reduces to Hall's tight line ℳ = (S−2)/6; its zero set is the postselection ceiling S = 4/η_eff − 2; at S = 2√2 the section vanishes exactly at the Garg–Mermin threshold. The inverted form S ≤ (4/η_eff − 2) + 6ℳ/η_eff² shows the measurement-dependence budget is amplified by 1/η_eff², i.e., it acts on the coincidence subensemble.

• Measurement dependence is only implied in the violating region. ℳ = 0 whenever S ≤ 4/η_eff − 2, so ordinary classical statistics implies no measurement dependence at all.

• A natural additive guess is falsified. The naive combination ℳ = (1+√2)/3 − 2/(3η_eff), registered as a hypothesis in advance, shares both endpoints but lies strictly above the true surface in the interior.

• The extremal model is "democratic." At the certified optimum ℳ(2√2, 9/10) = (27√2 − 33)/100, all six pairwise variational distances between the four conditional distributions are equal.

• Exact detection profile: D(m) = √m·h(m), with maximum residual 1.5×10⁻¹², correlation matching −cos 2Δ to 4×10⁻¹¹ (independently checked by Monte Carlo). h is smooth and monotone with h(0)/h(1) = 1.17450854538894; no closed form identified (3π/8 excluded at 10⁻¹²). Maximal-scale efficiency is 80.1%.

• Impossibility (Proposition 1): within the independent-click class, exact singlet correlation forces a non-constant detection function, hence an angle-dependent coincidence rate. Super-quantum postselected values in this class are bought with angle-dependent rates (a hard edge-band model reaching S = 4 shows 48% rate modulation). A flat-rate faithful local model exists only for η_eff ≤ 4/(2+π) ≈ 0.778, and lies outside this class.

• The fringe: the total coincidence rate carries a cos 4Δ-dominated modulation of relative peak-to-peak amplitude 12.4%, with harmonic content R_d/R̄_d = 1 + 0.0577cos 4Δ + 0.0098cos 8Δ + 0.0031cos 12Δ + ⋯ (leading harmonic to first overtone 5.87:1).

• Guilt trade-off: within the interpolating family, a bound δ_f < 2% on the fringe forces ℳ ≥ 45% of Hall's floor; δ_f < 1% forces 68%; δ_f < 0.5% forces 80%. These are statements about the family, not about all local models.

• Hiding theorem (Theorem 2): the fringe takes identical values at the two CHSH relative angles, because cos(2k·45°) = cos(2k·135°) for every harmonic k, so R_d(22.5°) = R_d(67.5°) exactly, term by term. Standard four-point CHSH experiments are therefore blind to it.

• The observational record does not reach the relevant regime. Aspect, Grangier and Roger report constancy of the fourfold sum; Aspect's thesis 26-run series shows total true-coincidence rates 78.8–81.2 s⁻¹ against a per-run standard deviation of 1.3 s⁻¹, a routine invariance test at roughly ±1.5%, hard to reconcile with the full-amplitude profile (which predicts R_d(0°)/R_d(45°) = 1.13, about 86 vs 76 s⁻¹). Brida, Genovese and Piacentini scanned eight relative angles in steps of π/8 with visibility 0.978, reporting D(η)−Δ_min = 0.0073 ± 0.0022 (Santos: Δ_exp = 0.0074) where the profile gives 0.053. Combined, existing data exclude the full amplitude and bound mixtures with flat detection to a detection share of order 15% (about 20% at two standard deviations); the sub-percent region is unmeasured.

• Moiré phase-locking mechanism: 23 active beat terms selected by greedy search, with the census ρ over frozen offsets optimized by linear programming (16×16 grid, or 1024 random/stratified offsets), five flight times, and three coupling scales. On 256 atoms the prices at ε = 0.10, 0.05, 0.02, 0.01 are 1.000, 1.015, 1.235, 1.375; at 1024 offsets the mechanism attains ℳ/ℳ_min = 1.000 at every tolerance down to ε = 0.01, with ℳ_min = (√2−1)/3 Hall's floor. At the 161-pin register, ℳ = 0.1380711874577208 against (√2−1)/3 = 0.1380711874576983 (twelve significant digits), with support spreading to 656 active atoms.

• The 256-atom premiums are coverage artifacts. Densifying the register raises the price (1.375, 1.511, 1.635 × the floor at 41, 81, 161 pins at ε = 0.01), and at 1024 offsets both effects resolve together. Between-pin residuals fall to ≤ 0.026 (worst window) and ≤ 0.017 (full range).

• The floor-attaining census shares a low-order Fourier skeleton. The dominant term is cos(4(ξ_A+ξ_B) − φ) with amplitude 0.35–0.38 against a permutation-null 95th percentile of 0.17–0.18, with φ agreeing across four independent solutions to within 5° (156.5°, 161.1°, 157.1°, 158.0°; chance below 10⁻⁵). A weaker cos(4(ξ_A−ξ_B) − ψ) term appears at 0.12–0.15. Station-local censuses of the form q(τ,s)f(ξ_A)g(ξ_B) fail to reproduce the correlators at ε = 0.01 from all 14 starts (missing by 3–13%), and pay 2.2–3.1× the floor where feasible at ε = 0.05.

• A falsifiable fingerprint: the mechanism softens the correlation extremes — positive residuals where E_QM → −1 and a negative plateau over 76°–116° where E_QM → +1 — visible at ≤ 1% at register angles.

• Composition of the two channels: reweighting coincidences by D_α = (1−α) + αD* gives a one-parameter dial; at α = 1 the LP returns ℳ = 0 at η_eff = 0.797, consistent with the certified detection-only ceiling η_eff ≈ 0.801.

• Proposed protocol: a cross-ratio design at relative angles 0° and 45° (Design A), X = R_d(0,0)R_d(45°,45°)/[R_d(0,45°)R_d(45°,0)], cancels factorizable apparatus responses and equals 1.278 for the solved profile, with a full-torus log-Fourier variant confining apparatus artifacts to the Fourier axes. Stated sensitivity is 5σ at 0.1% peak-to-peak within hours on standard entangled-photon hardware; a flat 0.1% result excludes the solved detection family and its mixtures to a detection share of order one percent.

Methodology in Plain English

The author treats "how much must a local model cheat on measurement independence?" as an optimization problem. Any deterministic local strategy assigns each party an outcome (+1, −1, or no-click) for each of its two settings, giving 3² × 3² = 81 joint strategies; a model is four probability distributions over those 81 strategies, one per setting pair. Requiring the model to reproduce what the experimenter observes — singles rates η_eff, coincidence rate η_eff², vanishing marginals, and a target CHSH value S — adds linear constraints. Measurement dependence is itself linear in those distributions (Hall's variational measure, in half-integral convention M = 2ℳ), so minimizing it is a linear program in 4 × 81 variables plus variational auxiliaries. A floating-point solve identifies the optimal support; every nonzero primal weight and dual multiplier is then reconstructed in exact arithmetic (ℚ(√2) or ℚ) and verified symbolically, so the numbers are theorems rather than numerical estimates.

For the detection sector, the author restricts to a specific class: deterministic sign outcomes sgn cos 2(λ−a) with a shared uniform phase λ, click probability D(m) with m = |cos 2(λ−a)|, and clicks independent given λ. Reproducing the singlet exactly becomes a functional equation between circular autocorrelations, which is solved numerically with the √m edge law built in via a Chebyshev series and Gauss–Jacobi quadrature.

The moiré mechanism is a constructive existence proof: a hidden phase evolves deterministically through the standing beat pattern between the pair and the analyzers, with all randomness pushed into a quenched (frozen once, at pair creation) ensemble of station offsets and flight times. The terrain is chosen by greedy search; the census over offsets is optimized by linear programming over a discrete ensemble.

The experimental proposal exploits symmetry: because the fringe is a function of |a−b| with period 90°, it takes identical values at the two CHSH angles. Moving to relative angles 0° and 45° (and using cross-ratios, plus a two-dimensional settings torus whose Fourier analysis separates factorizable apparatus artifacts from the physical anti-diagonal signal) makes it visible while cancelling detector imbalances.

Why This Matters

The paper converts a philosophical question — "how local could nature be, given what we actually measure?" — into an exact, certified number, and then into a specific experimental target. That is unusual: most loophole analyses treat measurement independence and detection efficiency as separate axes, and this work gives the joint surface. It also sharpens a long-standing complaint about realistic detection models (Pearle noted the setting dependence of the pair-detection rate; Gill called it the model's "major defect") into a proved impossibility for a well-defined class.

Practical implications:

• Fair-sampling diagnostics: the fourfold coincidence sum is a standard check in Bell tests. The paper shows the diagnostic is blind at the CHSH angles and specifies which angle choices are not, plus what a sub-percent null result would actually bound.
• Detector and source calibration: the cross-ratio design is calibrated against two-channel efficiency imbalance and designed so that factorizable apparatus responses cancel, making it usable for characterizing real entangled-photon sources rather than only for foundational tests.
• Benchmarking local-realist simulators: the price surface gives a quantitative figure of merit for any classical model claiming to reproduce singlet statistics — how much setting information it must consume at a given efficiency.
• Loophole accounting in device-independent protocols: knowing the exact joint cost of relaxing measurement independence and detection efficiency tells practitioners how much of each assumption a security argument can afford to give up.

Industry relevance: the work bears on quantum communication security proofs (device-independent and semi-device-independent QKD), on standardization of Bell-test hardware verification, and on quantum random number generation based on Bell violation, wherever assumptions about detector efficiency and setting independence must be audited rather than assumed.

Future Directions

• Prove the price surface in general. Conjecture 1 is verified numerically at 32 grid points and certified exactly at the quantum point and six rational points; a parametric dual certificate for all (S, η_eff) remains open. The observed symmetry — all six pairwise variational distances equal at the optimum — is proposed as the organizing principle.
• Derive the mechanism's two found ingredients. Both the terrain U (selected by greedy search) and the census ρ (chosen by linear programming) are found rather than derived. Any formation process must act at pair creation and reproduce the low-order Fourier skeleton relative to the apparatus beat structure.
• Close the sub-percent experimental gap. The proposed protocol targets 5σ at 0.1% peak-to-peak; the paper notes the truncated coincidence-count requirement implies a substantial data set. A flat result bounds the detection share of the solved family to order one percent, while a surviving fringe would be a joint (a,b) dependence no single-pair, fixed-imperfection apparatus produces.
• Settle the model's classification. Whether the moiré mechanism counts as "supermeasured" in the Hance–Hossenfelder–Palmer sense, or as a causal form of measurement dependence, is explicitly left open, as is the continuum fidelity limit off the pinned register angles.

Target Audience

Best suited to researchers in quantum foundations and Bell-inequality theory, mathematical physicists working on hidden-variable models and loophole quantification, and experimental quantum-optics groups running entanglement-based Bell tests who want a concrete, calibrated null measurement to perform. The optimization and certification portions will also interest specialists in convex geometry and exact-arithmetic verification. Graduate students with a working knowledge of CHSH will find the introduction and Sections 5–6 accessible; the full paper rewards readers comfortable with linear programming duality and Fourier analysis.

Authors’ abstract

For local hidden-variable accounts of the CHSH experiment that are faithful -- reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency $η_{\rm eff}$ -- I determine the minimal measurement dependence $M$ (Hall's measure) needed to reach a CHSH value $S$. Linear programming over all local strategies yields $M(S,η_{\rm eff})=\max\{0,η_{\rm eff}((S+2)η_{\rm eff}-4)/6\}$, whose edges reproduce Hall's bound, the Garg-Mermin threshold, and the postselection ceiling. The quantum point is certified exactly: $M(2\sqrt{2},9/10)=(27\sqrt{2}-33)/100$, by primal and dual certificates in $\mathbb{Q}(\sqrt{2})$. I solve the detection profile under which a deterministic sign model reproduces the singlet exactly, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible in this class. Mixtures of this profile with measurement dependence trade $M$ against a $\cos 4(a-b)$ modulation of the fourfold coincidence sum, up to 12.4% in relative amplitude, identical at the two CHSH angles; existing data exclude its full amplitude and bound its mixtures with flat detection to a detection share of order 15%; its sub-percent content is unbounded. A two-setting cross-ratio and a settings-torus log-Fourier protocol, calibrated against channel-efficiency imbalance, reach $5σ$ at 0.1% within hours; a flat result excludes the solved family to a detection share of order 1%. A local mechanism, moiré phase locking, with deterministic phase evolution and quenched randomness, attains the certified floor exactly at $η_{\rm eff}=1$ and, with the solved profile, reaches $M=0$ at $η_{\rm eff}\approx 0.80$. Its floor-attaining census shares a low-order Fourier skeleton across independent solutions and is not realizable by station-local offsets on the terrains tested.