Bell Locality and Quantum Correlations

Bell nonlocality is a property of an observed conditional probability distribution relative to a specified class of local explanations. It is not a synonym for entanglement, instantaneous influence, or faster-than-light signaling.

This chapter follows one complete logical chain:

$$\text{Bell-local model} \Longrightarrow \lvert S\rvert\le2 \quad\text{but}\quad \text{quantum theory permits }\lvert S\rvert=2\sqrt2.$$

Required background. Use the Born Rule to compute probabilities for specified quantum states and measurements; use Entangled States to distinguish entanglement from product and separable correlations.

Helpful background. Use Correlations and Covariance to form correlators from joint outcome distributions.

Enter the chapter

Read the three canonical pages in dependency order.

1. Local Hidden Variables defines the model being tested. It separates Bell factorization from measurement independence and proves that every such model is no-signaling.
2. CHSH Inequality derives the local bound, the quantum Tsirelson bound, and the no-signaling algebraic benchmark in one fixed convention.
3. Bell’s Theorem states the existential no-go result, uses CHSH as a witness, and separates the theorem from quantum predictions and experiments.

The order matters. Without the first page, “local” is undefined. Without the second, Bell’s contradiction has no explicit witness. Without the third, an inequality calculation is easily mistaken for a complete interpretation of nature.

The scenario ledger

In the CHSH scenario, two parties receive settings

$$x,y\in\{0,1\}$$

and return normalized outcomes

$$a,b\in\{-1,+1\}.$$

The empirical object is the family $p(a,b\mid x,y)$. Its correlators are

$$E_{xy} = \sum_{a,b=\pm1}ab\,p(a,b\mid x,y),$$

and one CHSH convention is

$$S=E_{00}+E_{01}+E_{10}-E_{11}.$$

Settings, outcomes, and hidden variables are different objects. Reusing one symbol for two roles is a common source of false derivations.

Three nested correlation sets

The benchmark values summarize three model classes:

Model class	CHSH ceiling	Defining feature
Bell-local	$2$	Shared-variable factorization and setting independence
Quantum	$2\sqrt2$	Density operators and local quantum measurements
No-signaling	$4$	Marginals independent of the remote setting

Every Bell-local distribution is no-signaling, but the converse is false. Quantum correlations lie strictly between the two sets in the CHSH scenario. A PR box reaches $4$ while remaining no-signaling, so no-signaling alone does not recover the quantum boundary.

Entanglement, violation, and signaling

These statements must not be conflated:

• An entangled state is not separable across a specified subsystem split.
• A Bell-nonlocal distribution violates some Bell inequality in a specified measurement scenario.
• A signaling distribution lets one party’s marginal depend on the other party’s setting.

Every pure entangled bipartite state displays Bell nonlocality under suitable measurements, but not every entangled mixed state violates CHSH. Quantum Bell violations remain no-signaling. The state, measurement choices, and complete probability distribution must all be specified before any Bell claim is meaningful.

The experimental handoff

The theorem uses ideal conditional probabilities. An experiment estimates them from finite records. A serious Bell-test claim must therefore state:

• how spacelike separation and timing are established;
• how settings are generated and when they are chosen;
• which trials are included and how losses are handled;
• which statistical null model and stopping rule are used;
• which residual assumptions enter the physical interpretation.

These are not repairs to a flawed theorem. They are the bridge from the theorem’s variables to laboratory data. The canonical pages keep the derivation and the experiment conceptually distinct.

Common mistakes

Calling an entangled state “nonlocal” without a measurement model. Bell nonlocality belongs to a distribution. State entanglement is a related but different property.

Equating no-signaling with locality. No-signaling constrains observable marginals; Bell locality demands a stronger hidden-variable factorization.

Treating CHSH as the only Bell inequality. CHSH is the standard $2\times2\times2$ witness, not the definition of Bell nonlocality in every scenario.

Reading $2\sqrt2$ as universal. It is the maximum quantum CHSH magnitude. Particular states and settings can give smaller values.

Exercises

1. A distribution has uniform local marginals and $S=4$. Which of the three sets in the table can contain it?

Solution

Uniform marginals are consistent with no-signaling. The value $4$ exceeds both the Bell-local ceiling $2$ and the quantum ceiling $2\sqrt2$, so the distribution can lie in the no-signaling set but not in the Bell-local or quantum sets. A PR box is the standard example.

2. Why does observing entanglement not by itself establish a Bell violation?

Solution

Entanglement classifies a state relative to a subsystem split. A Bell violation concerns correlations generated by that state together with particular local measurements. Some entangled mixed states do not violate CHSH, and even a state capable of violation produces nonviolating statistics for unsuitable settings.

3. Place each item in the appropriate layer: factorization, $\lvert S\rvert\le2$, singlet correlators, and a finite-sample $p$-value.

Solution

Factorization is a model assumption. The CHSH bound is a theorem derived from it. Singlet correlators are a quantum prediction for a specified state and measurements. A finite-sample $p$-value is part of experimental statistical inference.

References

• J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964).
• N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014).
• J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Physical Review Letters 23, 880–884 (1969).
• A. Fine, “Hidden variables, joint probability, and the Bell inequalities,” Physical Review Letters 48, 291–295 (1982).