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A theory is complete in the EPR sense only if its description represents every element of physical reality. Their sufficient reality criterion uses certain prediction without disturbance. Applying it to alternative measurements on a remote partner also requires an independence premise concerning the unmeasured system's physical reality; that premise must not be suppressed when presenting the argument.^[ref]

The Einstein–Podolsky–Rosen paradox begins with correlations enabling certain predictions about one system from alternative measurements on a separated partner. Its EPR criterion of reality promotes certainty without disturbance to a sufficient condition for an element of reality. Together with a locality premise, the alternative position and momentum predictions motivate simultaneously real quantities not represented by one standard quantum state. The intended conclusion is incompleteness, not a mathematical inconsistency of the predictions.^[ref]

Niels Bohr challenges the inference from absence of mechanical disturbance to a context-independent assignment of reality. He accepts that there is no mechanical disturbance of the distant system at the relevant stage. His response invokes the experimental conditions defining the possible predictions: arrangements establishing a spatial reference and arrangements enabling a momentum balance are complementary. On this view, changing the local measurement changes the whole phenomenon to which a definite property attribution belongs. It does not remotely kick the other particle. His diaphragm discussion illustrates why specifying the apparatus and its reference frame is part of the physical claim.^[ref]

A spin singlet state gives a clean reconstruction. Its state isHere $|0_x⟩=|+⟩$ and $|1_x⟩=|-⟩$; the minus sign is an irrelevant overall quantum phase. Measuring the first spin along either axis predicts the other's same-axis outcome with certainty. But the two alternative local quantum measurements define different conditional ensembles. Averaging over the unread outcome leaves the distant reduced density matrix $I/2$ in both cases. The contrast between conditional prediction and the unchanged distant marginal is quantum steering together with quantum no-signalling.

This explains how consistent quantum predictions can coexist with the absence of an operational remote disturbance. It also identifies what a Bohrian rejection of the EPR inference amounts to: certainty in alternative contexts does not license combining the corresponding property attributions into one context-independent account. Merely invoking the Heisenberg uncertainty principle would not suffice, since EPR specifically argued that locality permits the attribution before any incompatible distant measurements are jointly attempted.

The reply is satisfactory as an articulation of quantum complementarity, but not a demonstration that the EPR locality premise and completeness are jointly sustainable. The critic can distinguish a change in our permissible predictions from a change in the distant physical condition. If the latter really is unchanged, why must the alternative predictions not reveal pre-existing properties? Bohr's contextual standard rejects that demand rather than deriving its impossibility from neutral premises. Later Bell inequalities sharpen the issue: quantum no-signalling does not imply Bell local causality. A satisfactory contemporary account should acknowledge this difference, rather than crediting Bohr with a proof of Bell-local completeness or interpreting his reply as faster-than-light mechanical influence.

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.

Bell's theorem makes precise an obstruction to reproducing quantum correlations with a local causal completion. Let measurement settings be $a,b$, outcomes $A,B=\pm 1$ and a more complete state $\lambda$. Bell locality factorizes $P(A,B|a,b,\lambda )=P(A|a,\lambda )P(B|b,\lambda )$; setting independence takes the distribution of $\lambda$ to be independent of the freely chosen settings. These assumptions imply a Bell inequality. For example deterministic assignments satisfy $|A(B+B^{\prime })+A^{\prime }(B-B^{\prime })|\le 2$, and averaging gives the CHSH bound $|S|\le 2$. The stochastic version follows by averaging local response probabilities too.

An entangled spin singlet predicts $E(a,b)=-a\cdot b$ and suitable four directions give $|S|=2\sqrt{2}$. Thus the quantum predictions are incompatible with the stated local model, not merely with an unnecessarily specific mechanical hidden-variable proposal. Perfect anticorrelation in the EPR argument had motivated inferring pre-existing properties from remote predictions under locality and a completeness criterion. Bell shows that retaining such a local completion cannot recover the broader quantum correlation pattern.

It would overstate the result to say that it proves the impossibility of every hidden-variable theory. Bohmian mechanics supplies a nonlocal example, and rejecting setting independence or changing the causal model changes the theorem's premises. It would also be wrong to equate Bell locality with operational no-signalling. Quantum outcome probabilities at either wing can remain independent of the distant setting after averaging, even though the joint correlations have no local factorized explanation.

The theorem therefore sharpens the EPR dilemma: one cannot preserve all of the local causal assumptions and the quantum predictions. Empirical Bell violations bear on that conjunction. They do not alone choose one interpretation, make controllable superluminal communication available, or settle every philosophical sense of “realism.”^[ref][ref]