Bell’s Theorem

Bell’s theorem says that some quantum joint probability distributions have no measurement-independent Bell-local representation. The canonical Bell’s Theorem page owns the assumptions ledger, proof, examples, and theorem–experiment distinction.

Helpful background. For the assumptions ledger, proof, and CHSH witness behind this lookup card, consult the canonical Bell’s Theorem page.

Bell-local statement

With local settings $x,y$, outcomes $a,b$, and a shared variable $\lambda$, a Bell-local model has

$$p(a,b\mid x,y) = \int_\Lambda \mu(d\lambda)\, p_A(a\mid x,\lambda) p_B(b\mid y,\lambda),$$

together with measurement independence,

$$\mu(d\lambda\mid x,y)=\mu(d\lambda).$$

Such models obey Bell inequalities. In the standard CHSH scenario,

$$\lvert S\rvert\le2.$$

Quantum mechanics predicts $\lvert S\rvert=2\sqrt2$ for a suitable entangled state and local measurements. Therefore that quantum distribution has no representation of the form above.

Hypotheses to state

• The settings and outcomes defining the probability scenario are fixed.
• The hidden-variable distribution is independent of the chosen settings.
• Conditional on $\lambda$, each local response depends only on its local setting.
• Ordinary probability theory combines the hidden-variable responses.

Determinism is not an additional CHSH premise: local stochastic response functions can be refined into deterministic ones by adjoining local random seeds.

What follows—and what does not

The result excludes the stated class of Bell-local models. It does not exclude all hidden-variable theories, prove that an undefined “realism” is false, or permit faster-than-light communication. Quantum violations remain no-signaling. Entanglement is necessary for Bell violation but is not by itself the same property, and not every entangled mixed state violates CHSH.

The theorem, quantum prediction, and experimental inference are separate layers. Experiments additionally require a spacetime arrangement, setting protocol, detection model, and finite-data analysis.

Quick check

Which assumption fails if $\mu(d\lambda\mid x,y)$ is allowed to depend on the settings?

Solution

Measurement independence fails. Bell factorization alone does not require the hidden-variable distribution to be independent of $x$ and $y$; the independence condition must be declared separately.

References

• J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964).
• 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).