EPR Argument

The Einstein–Podolsky–Rosen argument challenges the completeness of the quantum state by combining perfect correlations with a criterion for physical reality and a locality premise. It is not Bell’s theorem and does not establish that a local completion actually exists. The historical EPR page treats the 1935 context; here the emphasis is the argument’s logical structure.

Predicting a distant result with certainty

A convenient modern version uses the spin singlet

$$|\psi^-\rangle=\frac{|01\rangle-|10\rangle}{\sqrt2}.$$

If Alice and Bob measure spin along the same axis, their outcomes are perfectly anticorrelated. Alice can therefore predict Bob’s result with certainty once she knows her own outcome. She may choose, for example, a $Z$ or an $X$ measurement. Bob’s reduced state before any communicated result remains $I/2$.

This spin formulation is due to Bohm, not the original EPR paper, which used idealized position–momentum correlations. Ideal sharp continuous-variable correlations also involve nonnormalizable limits; a realistic finite-squeezing experiment uses quantified uncertainty criteria rather than literally infinite precision.

From prediction to an element of reality

EPR’s sufficient reality criterion says that if a physical quantity can be predicted with certainty without disturbing the system, there is an element of reality corresponding to it. Applying it requires a locality premise: Alice’s choice and measurement should not change Bob’s physical situation when the systems are suitably separated.

Alice could predict Bob’s $Z$ value by measuring $Z$, or his $X$ value by measuring $X$. If Bob’s physical properties cannot depend on which distant choice she makes, the reasoning assigns both corresponding elements of reality. Yet no qubit state has sharp values of both $X$ and $Z$. Under the proposed completeness criterion, the quantum state then fails to represent all the supposed physical properties.

The counterfactual step is central. Alice does not measure both incompatible observables on one particle. The argument compares alternative possible local choices while claiming that the distant physical reality is unchanged. Removing that premise changes the inference.

What the conclusion does not supply

EPR infer incompleteness under their premises. They do not construct a hidden-variable dynamics, derive a Bell inequality or show that the additional properties can reproduce all quantum correlations. Bell’s theorem later demonstrates that a setting-independent Bell-local completion cannot reproduce the full set of relevant quantum predictions.

The debate is also not settled by observing that Alice’s conditional state assignment changes. Conditional predictions can change with information while Bob’s marginal remains the same. A claim about Bob’s physical ontology requires more than a statement about Alice’s updated probabilities. Bohr’s reply challenges how the experimental context enters the reality criterion.

Exercise: certainty conditional on a record

For the singlet, find $p(B_Z=+1)$ before Alice’s record is received and $p(B_Z=+1\mid A_Z=-1)$. Does the difference imply a controllable message?

Solution

The marginal is $1/2$; the conditional probability is one. Alice cannot choose her measurement outcome, so Bob cannot distinguish her uncommunicated result from his local data. Sorting by Alice’s record requires classical communication. The distinction is compatible with no-signaling regardless of how one assesses EPR’s reality premise.

References

• Bohm, David, Quantum Theory, Prentice-Hall (1951), chapter 22 — spin version of EPR.
• Einstein, Albert, Boris Podolsky, and Nathan Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,” Physical Review 47, 777–780 (1935), doi:10.1103/PhysRev.47.777.