Quantum Correlation Functions Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/tag/quantum-correlation-functions/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Wed, 12 Mar 2025 03:36:14 +0000 en-US hourly 1 https://wordpress.org/?v=6.9.4 Derivation of the Classical and Quantum Correlation Functions in EPR , Bell’s Theorem https://stationarystates.com/entanglement/derivation-of-the-classical-and-quantum-correlation-functions-in-epr-bells-theorem/?utm_source=rss&utm_medium=rss&utm_campaign=derivation-of-the-classical-and-quantum-correlation-functions-in-epr-bells-theorem Wed, 12 Mar 2025 03:36:14 +0000 https://stationarystates.com/?p=818 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 […]

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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.

 

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