The EPR Paradox

In 1935, Einstein, Podolsky, and Rosen (EPR) published an argument meant to show that quantum mechanics, as it stood, was incomplete. They never claimed it was wrong — only that the wave function could not be the whole story. Their thought experiment forced a sharp question that took three decades and Bell's theorem to answer.

Two ingredients EPR took for granted

EPR built their case on two assumptions they regarded as self-evident for any reasonable physical theory.

Realism (elements of reality). In their words: if, without disturbing a system in any way, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to that quantity. A property that is predetermined and merely read off by measurement.
Locality. What an experimenter chooses to do here cannot instantaneously influence the real physical state of a system there, if the two are spacelike separated. No faster-than-light influence.

The entangled pair

Consider two spin- $\tfrac12$ particles prepared in the singlet state and then sent in opposite directions to distant observers, Alice and Bob:

|\Psi^-\rangle = \frac{1}{\sqrt{2}}\big(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle\big).

The singlet is rotationally invariant: along any common axis the two spins are perfectly anticorrelated. If Alice measures her spin along $\hat n$ and gets $+\tfrac{\hbar}{2}$ , then a measurement of Bob's spin along the same $\hat n$ is certain to give $-\tfrac{\hbar}{2}$ .

The argument, step by step

Alice measures her spin along $\hat z$ . Without touching Bob's particle, she can now predict with certainty the outcome of Bob's $z$ -measurement (it is the opposite of hers).
By the reality criterion, Bob's $z$ -spin must be an element of reality — a definite value that was "there" all along, since Alice did nothing to Bob's particle (locality).
But Alice could equally have chosen to measure along $\hat x$ . Then Bob's $x$ -spin would be predictable with certainty, so it too must be an element of reality.
Therefore both $S_z$ and $S_x$ of Bob's particle have simultaneous definite values.

Here is the tension. Quantum mechanics says $S_x$ and $S_z$ are incompatible observables — their operators do not commute, $[S_x, S_z] = -i\hbar S_y \ne 0$ — so no state assigns both a sharp value. The wave function $|\Psi^-\rangle$ does not contain those joint values.

The EPR conclusion

EPR concluded: either (a) measuring Alice's particle instantaneously affects Bob's distant particle — which they rejected as "spooky action at a distance" violating locality — or (b) the two quantities really do have definite values that quantum mechanics fails to list. Choosing locality, they argued the quantum description is incomplete. The missing definite values are what later became known as hidden variables.

What EPR got right and what they missed

EPR correctly identified entanglement as the strange heart of the theory; Schrödinger coined the word entanglement in direct response to their paper. What they missed is that "complete the theory with local hidden variables" is not a free move. The next lessons make local realism precise and then show, via Bell's theorem, that its predictions disagree with quantum mechanics — and experiment sides with quantum mechanics. EPR framed exactly the right question; the answer turned out to be the one they least expected.