Quantum Correlations and Hidden Variable Predictions

1. Correlations Predicted by Quantum Mechanics

For two entangled spin-1/2 particles, the quantum state is:

|ψ⟩ = (1/√2) ( |↑⟩₁ |↓⟩₂ – |↓⟩₁ |↑⟩₂ )

If we measure spin projections along directions a and b, the correlation function predicted by quantum mechanics is:

EQM(a, b) = -cos(θ)

where:

  • θ is the angle between measurement directions a and b.
  • A result of +1 or -1 is obtained for each particle’s spin.

2. Correlations Predicted by Hidden Variable Theories

In a local hidden variable (LHV) theory, measurement results depend on pre-existing hidden variables (denoted λ) rather than quantum superposition.

Each particle has a pre-determined spin value along any measurement direction, meaning:

A(a, λ) = ±1, B(b, λ) = ±1

A hidden variable theory assumes a probability distribution ρ(λ) such that:

ELHV(a, b) = ∫ ρ(λ) A(a, λ) B(b, λ) dλ

Bell’s Inequality Constraint:

|E(a, b) – E(a, c)| + |E(b, c)| ≤ 2

Predicted Correlation in Local Hidden Variable Models:

ELHV(a, b) = -αθ

Since actual experiments confirm the -cos(θ) prediction, we conclude that local hidden variables cannot explain quantum entanglement.

Derivation of the Quantum Mechanical Correlation  – cos(theta)

Step 1: Define the Entangled State

The two-particle singlet state is:

|ψ⟩ = (1/√2) ( |↑⟩₁ |↓⟩₂ – |↓⟩₁ |↑⟩₂ )

Step 2: Define the Spin Measurement Operators

The spin projection operator for particle 1 along direction a is:

S₁(a) = (ħ/2) (σ ⋅ a)

For particle 2 along direction b:

S₂(b) = (ħ/2) (σ ⋅ b)

Step 3: Compute the Expectation Value

The correlation function is given by the expectation value:

E(a, b) = ⟨ψ | (σ₁ ⋅ a)(σ₂ ⋅ b) | ψ⟩

From quantum mechanics, the expectation value of the dot product of Pauli matrices satisfies:

⟨ψ | (σ₁ ⋅ a) (σ₂ ⋅ b) | ψ⟩ = – a ⋅ b

Since a ⋅ b = cos(θ), where θ is the angle between a and b, we obtain:

EQM(a, b) = -cos(θ)

Final Result

The quantum correlation function is:

EQM(a, b) = -cos(θ)

This function describes how the measurement outcomes are perfectly anti-correlated when the measurement directions are the same (θ = 0), and uncorrelated when they are perpendicular (θ = 90°).