Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S28 / 3

The Bell theorem gives a powerful conditional constraint, but does not by itself establish the unrestricted philosophical sentence that every interpretation contains a literal instantaneous action between distant objects. One must specify what locality means and which further assumptions are retained. In particular, Bell local causality is stronger than an inability to send a faster-than-light message.

Let $a,b$ be settings in spacelike-separated laboratories, $A,B\in \{-1,1\}$ their recorded outcomes, and $\lambda$ a sufficient specification of relevant prior physical variables. A Bell-local model factorizes asTogether with measurement independence, $\rho (\lambda \mid a,b)=\rho (\lambda )$, this constrains observable correlations. It allows stochastic response functions: predetermination of all outcomes is not a necessary separate assumption in this general formulation. Bell's 1964 paper demonstrates the incompatibility between a local completion and selected quantum correlations; the two-setting inequality below is a standard later form of the argument, not a claim about the exact notation of that paper.^[ref]

For example, write the local conditional mean outcomes as $A(a,\lambda ),B(b,\lambda )\in [-1,1]$. At each $\lambda$,Averaging over the same setting-independent distribution gives the CHSH inequality, $|S|\le 2$. A suitable spin singlet state instead predicts $|S|=2\sqrt{2}$. Consequently no model retaining the stated factorization and independence assumptions can reproduce all those quantum correlations. Merely replacing deterministic hidden variables with local random noise cannot evade the bound.

However, violation of this bound does not entail operational signalling. For an ordinary entangled state, a remote nonselective measurement leaves the local reduced density matrix unchanged. The local experimenter cannot choose the outcome needed to encode a controllable message. Quantum no-signalling and failure of Bell factorization can therefore coexist; confusing them would misstate both the prediction and the theorem.

Interpretations allocate the explanatory burden differently. In Bohmian mechanics, the guiding dynamics of an entangled configuration is explicitly nonlocal, though quantum equilibrium masks controllable signalling. A collapse interpretation must explain the status of correlations and of any proposed nonlocal state change. Everettian accounts retain the entangled global state and multiple relative records; their defenders can deny a fundamental action that selects one distant outcome. That does not produce a factorizing classical common-cause model of the observed correlations. Whether such an account deserves the word local depends on whether locality means local dynamical interactions, separability, or Bell's single-outcome causal factorization.

One can also reject setting independence or change the assumed causal organization, but such proposals must be developed as physical models, not advertised as refutations of the inequality's proof. The warranted conclusion is that the conjunction of Bell locality, independent settings and the relevant outcome framework is incompatible with the quantum correlations. Calling the failure action-at-a-distance is justified only after an interpretation specifies the physical entities, events and causal commitments to which that phrase applies.