Bell Inequalities

A Bell inequality is a linear bound satisfied by every correlation distribution in a specified local model class. It is a witness for that class, not a synonym for entanglement and not a universal inequality independent of the measurement scenario. CHSH is one prominent member; more settings, outcomes or parties change the space of possible witnesses.

Helpful background. Local Hidden Variables defines the local correlation set; CHSH Inequality supplies the standard two-setting example.

A linear functional on conditional probabilities

For finite settings and outcomes, collect the numbers $p(a,b\mid x,y)$ into a vector. A Bell functional is

$$B[p]=\sum_{a,b,x,y}\beta_{abxy}p(a,b\mid x,y).$$

Its local upper bound is the maximum over all local deterministic response tables $a=f(x)$ and $b=g(y)$. Any stochastic local strategy is a convex mixture of these tables, so the same bound applies. The finite convex hull is the local polytope.

A facet inequality supports a maximal-dimensional face of that polytope. Other valid inequalities can still be useful witnesses without defining facets. Numerical optimization of the local bound is a finite combinatorial problem; a proposed quantum or no-signaling maximum is a different optimization and should be labeled separately.

Witness strength depends on the distribution

A given nonlocal distribution violates some separating inequality, but it may obey a particular chosen inequality. Likewise, an entangled state with unsuitable measurements can produce local statistics. The tested object is the state-plus-measurement distribution, and a failed witness is not a general certificate that a state is local.

For CHSH, the derivation establishes the local bound two. Under a white-noise mixture with a maximally violating ideal distribution, the correlators scale by visibility $v$, so $S=2\sqrt2v$. The witness detects nonlocality only for $v>1/\sqrt2$. This threshold belongs to those settings and that noise model. A different inequality or measurement set can respond differently.

Normalization and losses are part of the definition

Coefficients can be written in probability, correlation or game-score form. Comparing reported numbers requires the same normalization and sign convention. Adding a constant or multiplying the functional changes its quoted bound while leaving the underlying separation equivalent.

Detector no-click events also belong to the declared outcome alphabet unless the protocol justifies another treatment. Computing correlations only on a selected detected subset can change the local set. Clauser–Horne-type probability inequalities and event-ready protocols address losses in explicit ways; there is no universal permission to discard missing events.

Choosing a Bell witness

Specify the parties, settings, outcomes and trust assumptions first. Then choose a functional adapted to the accessible state, detector performance and statistical protocol. State the local bound, the quantum prediction and the null-hypothesis test separately. Bell Loopholes describes how trial selection and timing affect that inference.

For an analytic argument with perfect correlations, see GHZ and Hardy. Their ideal contradictions have robust inequality forms when zeros and unit correlations become imperfect.

Exercise: changing the score convention

If $S\le2$ is a local CHSH bound and a game score is $w=1/2+S/8$, determine the local and maximal quantum scores. Does the rescaling change the certified model class?

Solution

The local score is at most $3/4$. The quantum ceiling is $(2+\sqrt2)/4\approx0.8536$. The affine transformation preserves exactly which distributions exceed the local threshold. It changes reported units, not the assumptions or content of the witness.

References

• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014), doi:10.1103/RevModPhys.86.419.
• Clauser, John F., and Michael A. Horne, “Experimental consequences of objective local theories,” Physical Review D 10, 526–535 (1974), doi:10.1103/PhysRevD.10.526.