EPR, Bell's Theorem, and Nonlocality

The EPR Argument (1935)

In their landmark 1935 paper, Einstein, Podolsky, and Rosen (EPR) argued that quantum mechanics is an incomplete theory. Their reasoning rests on two assumptions that seem obviously true:

• Realism: If we can predict with certainty the value of a physical quantity without disturbing the system, then there exists an "element of physical reality" corresponding to that quantity.
• Locality: No action on one system can instantaneously affect a distant system.

Consider two particles prepared in the singlet state and sent far apart:

EPR's chain of reasoning:

1. If Alice measures spin along the $z$-axis and gets $\uparrow$, then Bob's spin is definitely $\downarrow$ along $z$. By realism, Bob's $z$-spin was an element of reality.
2. Alternatively, if Alice measures along the $x$-axis, she can predict Bob's $x$-spin with certainty. By realism, Bob's $x$-spin is also an element of reality.
3. By locality, Alice's choice of measurement cannot affect Bob's particle. Therefore both $z$-spin and $x$-spin must simultaneously be elements of reality for Bob's particle.
4. But quantum mechanics says $[\hat{S}_x, \hat{S}_z] \neq 0$ — no quantum state simultaneously assigns definite values to both. Therefore quantum mechanics is incomplete.

Hidden Variable Theories

EPR's argument suggests that quantum mechanics should be supplemented by hidden variables — additional parameters $\lambda$ that, if known, would determine all measurement outcomes with certainty. In such a theory:

The observed correlations would then arise from averaging over the hidden variable distribution$\rho(\lambda)$:

For nearly three decades, the question of whether such hidden variables could reproduce all quantum predictions remained philosophical. Then, in 1964, John Bell transformed it into an experimentally testable question.

Bell's Inequality

Bell proved that any local hidden variable theory must satisfy constraints (inequalities) on measurable correlations — and quantum mechanics predicts violations of these constraints.

Derivation of the Original Bell Inequality

Consider three measurement directions $\hat{a}$, $\hat{b}$, $\hat{c}$. For the singlet state with local hidden variables, perfect anticorrelation requires $B(\hat{a}, \lambda) = -A(\hat{a}, \lambda)$. Then:

Since $A(\hat{b},\lambda)^2 = 1$, we can insert $1 = A(\hat{b},\lambda)^2$:

Taking absolute values and using $|A(\hat{a},\lambda)A(\hat{b},\lambda)| = 1$ and$1 - A(\hat{b},\lambda)A(\hat{c},\lambda) \geq 0$:

CHSH Inequality

A more experimentally practical form was derived by Clauser, Horne, Shimony, and Holt (1969). With four measurement settings (Alice: $\hat{a}, \hat{a}'$; Bob: $\hat{b}, \hat{b}'$), define:

For any local hidden variable theory, since $A, B = \pm 1$:

Averaging over $\lambda$:

Quantum Violation

For the singlet state $|\Psi^-\rangle$, the quantum correlation function is:

Choose measurement angles in a plane: $\hat{a}$ at $0°$, $\hat{a}'$ at $90°$,$\hat{b}$ at $45°$, $\hat{b}'$ at $135°$. Then:

The CHSH parameter evaluates to:

Tsirelson's Bound

Is $2\sqrt{2}$ the maximum quantum violation, or can quantum mechanics do even better? Tsirelson (1980) proved a sharp upper bound:

The hierarchy of bounds reveals deep structure:

• $|S| \leq 2$: Local hidden variable theories (classical)
• $|S| \leq 2\sqrt{2}$: Quantum mechanics (Tsirelson's bound)
• $|S| \leq 4$: No-signaling theories (algebraic maximum)

Why nature saturates $2\sqrt{2}$ rather than 4 remains an active area of research in quantum foundations, connecting to information-theoretic principles.

Experimental Tests

Bell inequality violations have been confirmed with increasing rigor over five decades:

What Bell's Theorem Really Means

Bell's theorem is often misunderstood. Here is what it does and does not say:

The no-signaling condition is key: while the correlations between entangled particles are nonlocal, the marginal statistics of either particle alone are independent of what is measured on the other. Formally: