Bell’s Theorem and the Two-Particle EPR Example

Bell’s theorem directly refutes the EPR argument by showing that quantum mechanics produces correlations that cannot be explained by any local hidden variable theory.

Step 1: The EPR Assumption

EPR assumed that if quantum mechanics were correct but incomplete, then the properties of particle 2 (such as momentum or position) must have been pre-determined before measurement. This suggests the existence of hidden variables (denoted as λ) that determine measurement outcomes.

According to local realism, measurements at one location should not instantly influence results at another distant location.

Step 2: Bell’s Inequality – A Test for Local Hidden Variables

Bell constructed an inequality that must be satisfied if the world follows local hidden variables.

For two particles, let’s define:

  • A(a, λ) → Measurement result of particle 1 along direction a
  • B(b, λ) → Measurement result of particle 2 along direction b
  • Each measurement result is either +1 or -1.

If hidden variables pre-determine these results, then the correlation function is given by:

E(a, b) = ∫ ρ(λ) A(a, λ) B(b, λ) dλ

Bell derived an inequality that must be satisfied by any local hidden variable theory:

|E(a, b) – E(a, c)| + |E(b, c)| ≤ 2

Step 3: Quantum Mechanics Violates Bell’s Inequality

Quantum mechanics predicts different correlation functions than hidden variable theories.

For a spin-entangled two-particle system, quantum mechanics gives:

E(a, b) = -cos(θ)

Experiments measure this and show violations of Bell’s inequality, proving that no local hidden variable theory can reproduce quantum mechanics’ predictions.

Key Conclusion for the Two-Particle EPR Example

  • EPR claimed that measuring p₁ determines p₂ instantly, meaning p₂ must have existed before measurement.
  • Bell showed that the correlations in quantum mechanics are too strong to be explained by hidden variables.
  • Experiments confirmed quantum mechanics’ prediction, showing nonlocality—meaning measurement at one location does affect the other particle instantaneously.

Thus, EPR’s assumption that momentum and position are pre-determined is incorrect!