---
title: "How Bell tests distinguish quantum correlations"
description: "Which assumptions turn a surprising correlation into evidence against a local hidden-variable model?"
canonical_url: "https://sandboxphysics.com/stories/quantum-evidence/bell-tests-correlations-loopholes/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/bell-tests-correlations-loopholes/"
last_modified: "2026-10-24"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# How Bell tests distinguish quantum correlations

Sandbox Physics · 2026-10-15

Which assumptions turn a surprising correlation into evidence against a local hidden-variable model?

## Ask the particles different questions

Two distant detectors keep giving matching answers. The match could have a simple cause: the particles left their source with the same list of instructions. To tell that possibility from a quantum correlation, experimenters change what they ask the particles. A Bell test compares several pairs of measurement settings with a limit that every model in a specified class of local models must obey.

Clauser and Aspect tested this idea with photon polarization, the direction selected by a polarizing filter. They rotated or switched the analyzers and counted how often the answers agreed. The 2022 Nobel Prize recognized their Bell experiments and Zeilinger’s work with entangled photons and quantum information. The early tests, the later improvements and the prize belong to different dates.

Read the evidence: Aspect (2022), Experimental tests of Bell’s inequalities: a first-hand account Nobel Prize in Physics 2022 — popular science background

## The CHSH bound and what it assumes

For the CHSH version, each detector reports one of two outcomes, labeled plus one and minus one. Multiplying the two answers gives plus one when they match and minus one when they differ. The average product, written E, measures their correlation. The letters a and a-prime label the two settings on one side; b and b-prime label the two on the other. The expression combines all four pairs of settings.

The local limit is two. It assumes that each side’s outcome probabilities depend only on its local setting and the shared hidden information, and that the settings are chosen independently of that information. The rule deciding which trials enter the calculation also has to be fixed beforehand.

A violation with an uncertainty that holds up counts against every model that satisfies these assumptions. It does not show that messages can travel faster than light, and it does not rule out every hidden-variable theory one could imagine.

$$
\begin{gathered}\begin{aligned}S={}&E(a,b)+E(a,b\prime)\\&+E(a\prime,b)-E(a\prime,b\prime)\end{aligned}\\|S|\leq 2\end{gathered}
$$

Read the evidence: Larsson (2014), Loopholes in Bell inequality tests of local realism

## What was not detected matters

If the analysis keeps only detected coincidences, the detectors may be picking out an unrepresentative sample. Fair sampling is an assumption and has to be stated as one, not passed off as a result. Fast, random setting choices with the right spacetime separation address a different worry: that the measurement events could signal to each other.

Aspect’s experiment with switching analyzers was an important step, but one early experiment did not close every loophole. Later teams combined better sources, detectors and timing. A 2015 photon experiment by Giustina and collaborators reports closing the most significant loopholes at the same time. Read its inequality and analysis as the paper gives them; they are not a simple CHSH calculation on raw coincidence counts.

Read the evidence: Aspect (2022), Experimental tests of Bell’s inequalities: a first-hand account Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons

## A p-value is not the probability that locality is true

A statistical test asks how surprising the data would be if a null model and its assumptions were true. It does not give the probability that a worldview is correct. How trials are delimited, how losses are handled, how predictable the settings are and when data-taking stops all have to be fixed before the significance is interpreted.

A state reconstruction, called tomography, answers a different question: which quantum state best describes a set of calibrated measurements? Error correction asks how to recover information after noise. Bell tests instead confront a class of local explanations with correlations measured under explicit sampling and timing rules.

Read the evidence: Larsson (2014), Loopholes in Bell inequality tests of local realism Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [See how sampling and detector efficiency affect a Bell test](https://sandboxphysics.com/labs/bell-chsh/)
- [Reconstruct a quantum state from measurements](https://sandboxphysics.com/labs/quantum-state-process-tomography/)
- [See how error correction differs from a Bell test](https://sandboxphysics.com/labs/quantum-error-correction/)

## Sources and reading notes

The linked Bell teaching sample is simulated and does not certify physical randomness, detection or spacetime closure.

1. [Aspect (2022), Experimental tests of Bell’s inequalities: a first-hand account](https://arxiv.org/abs/2212.04737)
2. [Nobel Prize in Physics 2022 — popular science background](https://www.nobelprize.org/prizes/physics/2022/popular-information/)
3. [Larsson (2014), Loopholes in Bell inequality tests of local realism](https://arxiv.org/abs/1407.0363)
4. [Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons](https://arxiv.org/abs/1511.03190)

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