Measured at ~10−18 effective acceptance, where post-selection diagnostics remain decisive.
A Reproducible Rebuttal
Is S > 2 Enough Without Entanglement?
A 2025 paper in Science Advances claimed to violate Bell’s inequality without entanglement. The result drew immediate attention, but its interpretation depends on the post-selection rule.
Key claims
The experiment reports a CHSH value of S = 2.275 ± 0.057, obtained from a four-photon frustrated interference setup. The authors claim this constitutes a Bell inequality violation without entanglement — a result that, if correct, would challenge decades of quantum foundations.
However, the protocol retains only fourfold coincidences, discarding the vast majority of detection events. At an effective acceptance of η ≈ 10−18, the post-selection window acts as a hidden filter that preferentially retains correlated outcomes.
Argument in 60 Seconds
Three Numbers Decide the Case
A classical model stays at S = 2.001 without selection, jumps to S = 3.716 under post-selection, and makes the reported S = 2.275 non-diagnostic at vanishing effective acceptance.
No selection, 100% acceptance. The model stays below the Bell bound.
Same source, biased acceptance. The artifact exceeds the Tsirelson bound.
The Claim
What Wang et al. Actually Did
On August 1, 2025, Wang et al. published a striking result: a measured CHSH value of S = 2.275 ± 0.057 from a multiphoton interference experiment. They claimed this constituted a Bell inequality violation — evidence that their optical system exhibited quantum nonlocality.
The apparatus uses four-photon frustrated interference, also described in the follow-up literature as interwoven frustrated down-conversion. Four SPDC pathways are arranged so that a retained event is a fourfold coincidence: one photon detected in each output mode, with local phases α and β controlling the interference pattern.
That is already not a standard Bell trial. Wang et al. keep only fourfold coincidences, infer the −1 outcomes by shifting a phase by π, and normalize the reported joint probabilities with counts taken at four different phase settings. In an ordinary Bell test, each trial yields definite local ±1 outcomes at the chosen settings without that counterfactual renormalization step.
This is why the formal claim on this site stays narrow. The optics are real and interesting, but three independent published critiques now converge on a stronger diagnosis: the reported violation is driven by fourfold post-selection and unconventional normalization rather than by a loophole-free demonstration of nonlocality. Wang et al. themselves acknowledge open locality and sampling loopholes and rely on fair sampling.
Bell test simulation. Select measurement settings and observe CHSH correlations.
CHSH S
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Interactive A Bell test in action. Particle pairs fly from the source to Alice and Bob’s detectors. Select measurement settings and watch the correlation tally build. +1 outcome −1 outcome
The Number That Matters
One in a Quintillion
Wang et al.’s critical number was an effective acceptance or post-selection probability of approximately 10−18. That is not an ordinary per-arm detector specification. It is the fraction of trials that survive the fourfold-coincidence filter. In everyday terms: one retained event for every quintillion opportunities.
In CHSH-style reasoning, the local bound S ≤ 2 transfers cleanly only when the retained sample tracks the emitted ensemble. At ~10−18 effective acceptance, that transfer cannot simply be assumed. Three numbers define the landscape:
S value gauge: Classical zone 0 to 2, Quantum zone 2 to 2.828, Artifact zone 2.828 to 4. Wang et al. reported S equals 2.275.
The S Landscape Classical models stay below 2. Quantum mechanics reaches 2√2. Post-selection artifacts can push all the way to 4. Wang et al. reported S = 2.275, so the relevant question is which evidentiary regime that statistic occupies.
Classical LHV
S ≤ 2
Quantum Tsirelson
S ≤ 2√2 ≈ 2.828
Post-Selection Artifact
S → 4
The Detection Loophole
Classical Counterexamples Under Post-Selection
The answer, as Pearle (1970) and Eberhard (1993) showed decades ago, is yes — if the retained-event rate is low enough. When detectors miss particles, or when post-selection keeps only a tiny subset, the surviving sample is no longer representative. A classical model can exploit these gaps to mimic quantum correlations.
The critical threshold is:
Detection Loophole Threshold
Below this CHSH benchmark, the maximum CHSH value a classical model can achieve is not 2 — it’s 4/η − 2. At Wang et al.’s effective acceptance of ~10−18, that classical ceiling is astronomically high. Drag the slider below to see for yourself.
Classical CHSH Bound
For any local hidden variable theory satisfying Locality, Realism, and Freedom:
View proof sketch
Define X(λ) = A(a,λ)[B(b,λ) − B(b′,λ)] + A(a′,λ)[B(b,λ) + B(b′,λ)]. Since B outcomes are ±1, either [B(b) − B(b′)] = 0 and [B(b) + B(b′)] = ±2, or vice versa. In either case |X(λ)| ≤ 2. Integration over λ preserves the bound.
LHV Maximum S
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Interactive Sweep the CHSH benchmark parameter η to see the local LHV ceiling grow. At η = 70%, classical models can reach S = 3.71 — exceeding even the quantum limit. Wang et al. sits at ~10−18 effective acceptance, deep in the loophole-vulnerable regime.
Post-Selection
When You Discard 99.9999…% of Your Data
The detection loophole is a specific case of a more general problem: post-selection. When you keep only events that satisfy certain criteria, you’re no longer testing Bell’s inequality on the full ensemble. You’re testing it on a biased sample.
Here’s the mechanism. Suppose a classical source produces outcomes deterministically. Now introduce a filter: keep events with probability phi when outcomes match a desired pattern, and plo otherwise. If phi > plo, you’ve biased the sample toward correlations that inflate S.
The result is striking: as the bias ratio grows, S can approach 4 — the algebraic maximum — from purely classical data. Watch it happen:
Post-Selection Inflation
Outcome-dependent post-selection on LHV data can achieve:
View proof sketch
Define selection f(A,B) = 1 when AB = +1, f = ρ when AB = −1. As ρ → 0, the post-selected estimator Epost(a,b) → ±1. The resulting CHSH value S → 4.
Event stream visualization. Particles flow through a selection filter.
Events accepted
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Post-selected S
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Interactive Event pairs flow through a filter. Correlated pairs pass; anti-correlated pairs are discarded. Watch how selective filtering inflates S beyond the quantum limit — from purely classical data. Correlated (kept) Anti-correlated (filtered)
Exact post-selected S
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Acceptance Rate
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Verdict
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Interactive Sweep the exact toy filter used in the paper along the slice phi = 1 - plo. The paper point is plo = 0.10, which yields 70% acceptance and S = 26/7 ≈ 3.714.
The Proof
A Constructive Counterexample
We built a classical hidden-variable model. No quantum mechanics. No entanglement. Just deterministic outcomes from a hidden variable λ, sampled uniformly from [0, 2π).
Without post-selection, this model produces S = 2.001 (95% CI: [1.998, 2.004]) — consistent with the classical bound, as it should be.
With outcome-dependent selection (phi = 0.9, plo = 0.1), the same classical source yields S = 3.716 (95% CI: [3.714, 3.717]) at 70% acceptance. That exceeds both the classical bound and the Tsirelson bound.
| Dataset | S | Acceptance |
|---|---|---|
| Full (no selection) | 2.001 ± 0.009 | 100% |
| Post-selected | 3.716 ± 0.004 | 70.0% |
| Wang et al. (2025) | 2.275 ± 0.057 | ~10−18 |
The implication is clear: values above 2.828 are not evidence of stronger-than-quantum correlations. They indicate that the selected ensemble is no longer a fair Bell sample. The apparent violation comes from the filter, not from the physics.
Tsirelson Bound
In quantum mechanics with optimal local observables:
View proof sketch
The CHSH operator Ŝ2 = 4I − [A,A′] ⊗ [B,B′]. Since commutator norms satisfy ‖[A,A′]‖ ≤ 2, we get Ŝ2 ≤ 8I, so max⟨Ŝ⟩ = √8 = 2√2.
Classical LHV
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Quantum
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Interactive Adjust the four measurement angles to explore how CHSH S varies. The optimal quantum settings (0°, 90°, 45°, 135°) produce S = 2√2. No classical model exceeds S = 2 — unless you post-select.
What Would Convince Us
The Standard for Bell Tests
The history of Bell tests is instructive. The first experiments — Freedman & Clauser (1972), Aspect et al. (1982) — had open loopholes. They were brilliant pioneering work, but their authors were careful not to claim more than the data warranted.
The community converged on a stricter standard: close the relevant loopholes directly, or publish diagnostics strong enough to show that selection is not doing the work. For CHSH-style fair-sampling discussions, η > 82.8% is the familiar benchmark. CH and Eberhard analyses can go lower, and the decisive 2015 photon tests were framed that way.
In 2015, three independent groups closed the case by engineering around the loopholes instead of normalizing them away: Hensen et al. (Delft), used an event-ready architecture, while Giustina et al. (Vienna) and Shalm et al. (NIST) used high-efficiency entangled-photon tests analyzed with CH/Eberhard-style logic.
For future experiments with low acceptance rates, we recommend reporting:
- Raw correlations before any selection, if available.
- Acceptance rates broken down by measurement setting.
- No-signaling tests on both selected and unselected data.
- Independence tests between acceptance and measurement settings.
- Sensitivity analysis: quantify the range of selection-induced distortion consistent with the observed result.
If these diagnostics are not available, the appropriate conclusion is not “Bell inequality violated” but rather “cannot rule out classical explanation.”
Timeline ready. Use horizontal scroll, swipe, or the arrow buttons to reveal all seven entries.
Timeline From Bell’s 1964 theorem through loophole-free tests in 2015 to Wang et al.’s 2025 claim. Use horizontal scroll, swipe, or the arrow controls to explore the full sequence.
Testing Dashboard
Live Physics And Algorithm Snapshot
This dashboard stays inside the argument surface. It watches the existing demos, keeps a short in-memory history, and turns the latest interaction into one scannable view of the physics thresholds, post-selection behavior, and shareable rebuttal state.
KPIs
Current diagnostics
Trend Charts
Recent slider history
Current S
No samples yet
LHV max
No samples yet
Acceptance rate
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Breakdown
Per-demo status
| Demo | Current values | Interpretation |
|---|---|---|
| Waiting for demo state... | ||
Evidentiary Caution.
The conclusion is narrower than the strongest claims made in the debate. A local hidden-variable source with outcome-coupled acceptance reproduces the evidentiary regime of a post-selected Bell claim and can exceed the Tsirelson bound without entanglement. The Wang et al. statistic, considered as a selected CHSH number at ultra-low effective acceptance, therefore does not by itself establish quantum nonlocality.
The broader critique literature now goes further: three independent analyses trace the reported violation to fourfold post-selection and unconventional normalization. This site uses that convergence as context while keeping the formal thesis narrower than “Wang et al. are classical.”
The remedy is methodological: report the raw correlations, the setting-resolved acceptance structure, and the diagnostics that constrain fair sampling. Until then, the appropriate interpretation of a low-efficiency post-selected violation is evidentiary caution.
Bibliography