Bell–CHSH tests and local hidden variables

In a deterministic hidden-variable model, denote Alice's possible responses by A-zero and A-one, and Bob's by B-zero and B-one. Each response is plus or minus one. For a fixed hidden variable, factor the CHSH combination into A-zero times the sum of Bob's responses plus A-one times their difference. Bob's responses are either equal or opposite, so exactly one bracket vanishes and the other equals plus or minus two. Averaging preserves the absolute bound of two.

Stochastic local models obey the same bound because local randomness can be included in an enlarged hidden variable, or treated by convexity. The physical assumption is conditional factorization of the two outcome probabilities given the hidden variable and local settings, together with a hidden-variable distribution independent of the chosen settings. Relabelling outcomes or swapping settings changes signs but not the bound. A single fixed setting cannot supply the four counterfactual response combinations needed to derive the constraint.

For the two-qubit singlet, measuring spin signs along unit vectors gives correlation minus their scalar product. For example, Alice chooses $\mathbf a_0=\hat z$ and $\mathbf a_1=\hat x$, while Bob chooses $\mathbf b_0=-(\hat z+\hat x)/\sqrt2$ and $\mathbf b_1=-(\hat z-\hat x)/\sqrt2$. These axes yield three correlations equal to inverse square root of two and a fourth with the opposite sign. Their CHSH magnitude is twice square root of two, the quantum maximum for this scenario. The measurement choices on each side must be incompatible to attain it; replacing all measurements by one common commuting basis cannot produce the same pattern.

Bell violation certifies entanglement under the quantum model, but an entangled mixed state need not violate this particular inequality with the selected measurements. Entanglement and CHSH violation are therefore different tests. The singlet's single-party marginal probabilities remain one half independently of the remote setting. Nonlocal correlations do not provide a controllable faster-than-light communication channel. No-signalling alone permits correlations stronger than the quantum maximum, so it is also a distinct constraint.

Estimate each correlation from same-sign counts minus opposite-sign counts divided by all valid trials for that setting pair. Discarding nondetections in a setting-dependent way can manufacture an apparent violation from local data. Either assign outcomes according to a valid protocol or use a suitable inequality and trial scheme that incorporates inefficiency. A fair-sampling assumption is an additional assumption, not a consequence of a large measured CHSH value.

Space-like separation and sufficiently late setting choices address unwanted communication between measurement events. Setting independence addresses correlations between settings and hidden source information; locality alone does not imply it. Finite data also require a stated statistical test, including possible memory effects and any stopping rule. Computing S above two from a small table is an estimate, not by itself a significance claim. Modern experiments combine these controls; a classroom binomial model illustrates counting uncertainty without certifying their complete experimental architecture.