Working Paper · Bell Inequality Analysis · 2025

Bell Violations
Without
Entanglement?

A computational rebuttal of Wang et al. (2025). Post-selection artifacts — not quantum nonlocality — explain the reported CHSH violation.

View on GitHub Read the argument
S ≤ 2 Classical LHV bound No post-selection · proven optimal
2.275 Wang et al. (2025) η ≈ 10⁻¹⁸ · Science Advances ↑ "quantum violation" claimed here
3.716 This work — purely classical Classical LHV + post-selection artifact ↑ exceeds Tsirelson bound classically!
ηc = 82.8% Wang et al. operate at η ≈ 10⁻¹⁸ — twenty orders of magnitude below the fair-sampling threshold.
The Argument

Detection efficiency, not entanglement, determines the result.

A recent paper in Science Advances claims to demonstrate Bell-inequality violation without entanglement, reporting a CHSH value of S = 2.275 ± 0.057 at a detection efficiency of roughly 10−18.

The CHSH inequality holds for any local hidden-variable (LHV) model operating on all detected events. When detection efficiency falls below the critical threshold ηc ≈ 82.8%, post-selection can inflate S above 2 for purely classical systems. The reported S = 2.275 is not evidence of quantum nonlocality; it is evidence of a detection loophole.

We demonstrate this with explicit simulation: a classical LHV model with outcome-dependent post-selection consistently yields S ≈ 3.72 — a value that exceeds both the classical bound (S ≤ 2) and the quantum Tsirelson bound (S ≤ 2√2 ≈ 2.83). Wang et al.'s result lies below what we achieve with no quantum mechanics whatsoever.

The critical efficiency follows from the CHSH analysis itself: ηc = 2/(1 + √2) ≈ 0.828. Any experiment below this threshold requires the fair-sampling assumption, which cannot be verified from the data alone. We provide a straightforward diagnostic protocol for future experiments.

Figures
Horizontal dot chart comparing CHSH parameter S across four models: Classical LHV, Wang et al., Quantum, and Classical with post-selection.
Figure 1. CHSH parameter S across four models. Classical LHV + post-selection (this work) achieves S ≈ 3.72 without any quantum mechanics. Wang et al.'s reported S = 2.275 falls below our classical baseline.
Line plot showing CHSH value S vs detection efficiency η. Analytical LHV bound S = 4/η − 2 in vermilion, simulation sweep in orange with 95% CI.
Figure 2. Analytical LHV bound S = 4/η − 2 (vermilion) with simulation sweep (10 runs × 20,000 trials, 95% CI). The critical threshold ηc ≈ 0.828 separates loophole-free from loophole-vulnerable regimes.
Scatter plot of detection efficiency η vs year for landmark Bell tests from 1972 to 2025. Log scale. Loophole-free experiments shown as filled markers above the 82.8% threshold.
Figure 5. Detection efficiency of landmark Bell tests (1972–2025, log scale). Loophole-free experiments (filled markers, 2015–) exceed ηc ≈ 0.828. Wang et al. (2025) reports η ≈ 10⁻¹⁸.
Explore
LHV Bound Calculator

For a given detection efficiency η, the maximum CHSH value achievable by a classical local hidden-variable model with outcome-dependent post-selection:

Smax(η) = 4 / η − 2
Detection efficiency η 0.83
0.10 ηc = 0.83 ← 1.00
2.82 at Tsirelson bound — this is where the loophole closes.
What's Inside
01
Bell Inequality Simulator

Three physical models — LHV, quantum singlet, and post-selection — with parameter sweeps, reproducible seeds, and CSV output.

02
Formal Proofs

Structured Python proofs for CHSH, Tsirelson bound, and post-selection inflation. Lean 4 formalization included.

03
Research Platform

Universal scaffold: project.yaml schema, figure registry, venue profiles, and validate scripts with full TDD.

04
119 Tests

Bounds verification, no-signaling checks, edge cases, and numerical precision. Run with pytest tests/.

Python NumPy Matplotlib pytest LaTeX / revtex4-2 Lean 4 Jupyter Ruff PyYAML
Citation
@misc{alawein2025bellrebuttal,
  author    = {Meshal Alawein},
  title     = {Bell Violations Without Entanglement?
               A Computational Rebuttal of Wang et al. (2025)},
  year      = {2025},
  note      = {meatheadphysicist research platform},
  url       = {https://github.com/alawein/meatheadphysicist},
}