Derivation of Correlation Functions

Hidden Variable Theory Correlation

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

Assuming a probability distribution ρ(λ) over hidden variables, the expectation value of the product of measurement results is:

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

In hidden variable models, this function is usually a linear function of the angle θ, such as:

ELHV(a, b) = -αθ

where α is some model-dependent constant. This prediction does not match experimental results.

Quantum Mechanics Correlation

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

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

The correlation function is given by:

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)

This prediction has been experimentally verified, demonstrating that local hidden variable theories cannot fully explain quantum entanglement.