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 / 4 / Solution

In the specific Einstein-Podolsky-Rosen completeness criterion, completeness is not just an ability to predict all experimental probabilities. A complete theory must represent every element of physical reality. The associated sufficient reality criterion says that a quantity has such an element if its value can be predicted with certainty without disturbing the system. The EPR paper distinguishes these demands from correctness of the theory's empirical predictions.^[ref]

Their position–momentum example makes the locality assumption crucial. For two systems with ideal correlations,the two joint observables commute, although $[X_B,P_B]=i\hslash$. Measuring $X_A$ permits a certain inference of $X_B$; choosing instead to measure $P_A$ permits a certain inference of $P_B$. If a spacelike-separated choice at A cannot alter the physical reality at B, the choice should not determine which properties B possesses. Combining that independence with the reality criterion motivates assigning both quantities to B, while an ordinary individual wave function does not supply both sharp values.

This is not a procedure for experimentally measuring both noncommuting quantities on B at once. It is a counterfactual argument comparing alternative remote measurements, supported by a locality claim about the same remote system. The ideal perfectly correlated position–momentum state is also nonnormalizable; normalizable approximations approach its correlations. The philosophical inference is best assessed in the ideal limit with this qualification stated.

Niels Bohr's reply challenges the criterion's application. He distinguishes a direct mechanical disturbance of the remote system from a dependence of the conditions under which a definite prediction can be meaningfully specified. His appeal to quantum complementarity is therefore not simply the claim that the particles mechanically push one another during the final measurement. It disputes the context-independent assignment of the two elements of reality.^[ref] Whether that response answers EPR's locality challenge is itself a substantive philosophical issue; it is not resolved by restating the uncertainty relation.

A qualified assessment is preferable to an unlabelled yes or no. If EPR's locality and reality premises are retained, the wave-function description is incomplete in their sense. But the later Bell theorem shows that adding ordinary local hidden variables cannot simply recover all quantum correlations while keeping the standard independence assumptions. The natural completion EPR hoped for cannot retain that entire package.

Nonlocal Bohmian mechanics provides one kind of completion with definite configurations. Everettian wave-only accounts use another notion of completeness while rejecting the premise of one absolute set of distant outcomes. An instrumental reading can regard the quantum formalism as complete for assigning experimental probabilities without accepting EPR's representational demand at all. Thus predictive completeness, completeness of an interpretation's ontology, and completeness in EPR's particular sense are different judgements. The EPR argument exposes a conflict among commitments; it does not establish a criterion-independent verdict about every interpretation.