Physics 1964

On the Einstein Podolsky Rosen Paradox

John Stewart Bell

No theory of local hidden variables can ever reproduce all the predictions of quantum mechanics.

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In depth · the introduction

Two particles can be born so deeply linked that measuring one instantly tells you about the other — and Bell found a way to prove that no hidden agreement, made in advance, could ever explain how linked they really are.

The big idea

Quantum mechanics says two particles can be "entangled" — created together so that their properties stay perfectly matched no matter how far apart they drift. Einstein disliked this. He believed each particle must secretly carry its answers all along, like two travellers handed sealed envelopes at the start; opening one just reveals what was already written inside. No spooky link — just hidden information.

John Bell found a brilliant move: instead of arguing, he calculated. If the particles really do carry secret instructions decided in advance — and if nothing one detector does can reach across to affect the other — then their answers can only agree up to a certain amount. There is a mathematical ceiling. Quantum mechanics predicts the particles agree more than that ceiling allows. So the "sealed envelope" picture cannot be the whole story.

How it came about

In 1935 Einstein, with Boris Podolsky and Nathan Rosen, published a famous challenge: quantum mechanics, they argued, must be incomplete, because it left out the definite "elements of reality" each particle surely carries. Niels Bohr disagreed, and for thirty years the argument looked like philosophy — untestable, a matter of taste. Most physicists shrugged and went on using the equations.

John Bell, an Irish physicist at CERN, could not let it rest. On sabbatical in 1964 he took Einstein's side seriously enough to test it — and discovered, to his own surprise, that Einstein's local, common-sense world makes a different prediction from quantum mechanics, one you can actually measure. His short paper appeared in an obscure new journal that soon folded; for years almost nobody noticed. Then the experimenters came — Clauser, then Aspect, then many more — and ran the test. Quantum mechanics won, every single time.

Why it mattered

Bell turned the deepest question about reality — is the world local? do things carry definite values before we look? — into something a laboratory can answer. The answer is startling: nature really is "non-local" in Bell's precise sense, with particles correlated more tightly than any pre-arranged plan allows. And the strangeness turned useful — it is exactly this that makes uncrackable quantum encryption and quantum computing possible.

A way to picture it

Picture two friends sent to opposite ends of the Earth, each given a coin to flip whenever a referee calls out one of three questions. If they agreed on a plan before parting — "heads to question one, tails to question two…" — then over many rounds there's a strict limit on how often their answers can match when the questions differ. That's the ceiling. Now suppose the real friends match far more often than any plan could permit, no matter how cleverly they schemed beforehand. You'd be forced to conclude they're somehow coordinating in the moment, across the whole planet, instantly. That impossible-seeming coordination is exactly what entangled particles do.

Where it sits

Bell's theorem is the hinge of the quantum story. Behind it stand the founders — Schrödinger, who named entanglement and called it the characteristic trait of quantum mechanics; Heisenberg, whose uncertainty principle first set limits on what we are allowed to know; and the 1935 EPR paper that started the quarrel. Ahead of it lies the whole field of quantum information, and the 2022 Nobel Prize given to the experimenters who carried Bell's test through to its end.

The original document

Original source text

J. S. Bell · Physics Physique Физика 1 (1964): 195–200

I. Introduction

Bell opens by recalling the 1935 argument of Einstein, Podolsky and Rosen: that quantum mechanics is incomplete and ought to be supplemented by additional ("hidden") variables to restore causality and locality. The question of this paper is whether such a completion is possible — and the answer Bell reaches is that it is not, so long as the added variables are local.

The correlation experiment

Following Bohm's spin version of EPR, a pair of spin-½ particles is prepared in the singlet state and flies apart. At distant stations one measures the spin component of one particle along a unit vector a, and of the other along b; each result is +1 or −1. The singlet guarantees that whenever the two settings are equal, a = b, the two results are perfectly anticorrelated.

The locality assumption

A local hidden-variable theory assigns each pair a variable λ, distributed with density ρ(λ). The first result, A(a,λ) = ±1, depends only on the local setting a and on λ — not on the distant setting b — and likewise B(b,λ) = ±1 depends only on b. The measured correlation is the average P(a,b) = ∫ dλ ρ(λ) A(a,λ) B(b,λ). Perfect anticorrelation at equal settings forces A(a,λ) = −B(a,λ).

The inequality (eq. 15)

From these assumptions alone — and no quantum mechanics — a short argument over any three settings yields Bell's inequality: |P(a,b) − P(a,c)| ≤ 1 + P(b,c). It is a bound every local hidden-variable theory must obey. But the quantum-mechanical prediction for the singlet, P(a,b) = −a·b = −cos θ, violates it for suitable angles. The two cannot both be right.

Conclusion

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.

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John S. Bell · 1964