Entanglement Witness Simulator: Bell Inequality & CHSH Test

simulator intermediate ~10 min
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S = 2.83 — maximum quantum violation (Tsirelson bound)

With perfect visibility V = 1 and optimal measurement angles, the CHSH parameter reaches S = 2√2 ≈ 2.83, decisively violating the classical bound of 2 and demonstrating quantum entanglement.

Formula

E(θ_A, θ_B) = -V cos(2(θ_A - θ_B))
S_CHSH ≤ 2 (classical), S_max = 2√2 (quantum)
C = max(0, 2|⟨ψ|σ_y⊗σ_y|ψ*⟩| - 1) (concurrence)

Einstein's Discomfort

In 1935, Einstein, Podolsky, and Rosen argued that quantum mechanics must be incomplete because it predicted correlations between distant particles that seemed to require 'spooky action at a distance.' For decades this remained a philosophical debate — until John Bell showed in 1964 that local hidden variable theories make testable predictions that differ from quantum mechanics. This simulation lets you explore exactly where classical and quantum predictions diverge.

The CHSH Test

Clauser, Horne, Shimony, and Holt reformulated Bell's inequality into an experimentally practical form. Alice and Bob each choose between two measurement settings and record ±1 outcomes. The CHSH parameter S combines four correlation measurements; classical physics demands S ≤ 2, while entangled quantum states can reach S = 2√2. Alain Aspect's landmark experiments in 1982 confirmed the quantum prediction.

Visibility and Noise

Real experiments never achieve perfect entanglement. Noise, loss, and detector imperfections reduce the effective visibility V of the two-photon state. Below V = 1/√2, Bell violations vanish — but the state may still be entangled. Entanglement witnesses provide a more sensitive test, detecting entanglement even when Bell inequalities cannot be violated, at the cost of assuming the quantum mechanical framework.

The 2022 Nobel Prize

The 2022 Nobel Prize in Physics was awarded to Aspect, Clauser, and Zeilinger for experiments with entangled photons and the violation of Bell inequalities. Their work established that nature is fundamentally nonlocal (or non-realist), and enabled practical quantum technologies including quantum key distribution, quantum teleportation, and entanglement-based quantum computing.

FAQ

What is quantum entanglement?

Quantum entanglement is a correlation between particles that cannot be explained by any local hidden variable theory. When two photons are entangled, measuring one instantly determines the state of the other, regardless of distance. This 'spooky action at a distance' was confirmed experimentally and led to the 2022 Nobel Prize in Physics.

What is the CHSH Bell inequality?

The CHSH inequality states that for any local hidden variable theory, S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| ≤ 2. Quantum mechanics allows S up to 2√2 ≈ 2.83 (the Tsirelson bound). Experimental violation of S > 2 rules out local realism.

What is an entanglement witness?

An entanglement witness is an observable W such that Tr(Wρ) < 0 for at least some entangled states but Tr(Wρ) ≥ 0 for all separable states. Witnesses can detect entanglement that Bell tests miss, requiring fewer measurements and working even when Bell inequalities are not violated.

What visibility is needed to violate Bell's inequality?

For the CHSH inequality with a two-qubit state, the critical visibility is V > 1/√2 ≈ 0.707. Below this threshold, the correlations admit a local hidden variable model and Bell's inequality is satisfied, even though the state may still be entangled for any V > 1/3.

Sources

Embed

<iframe src="https://homo-deus.com/lab/quantum-optics/entanglement-witness/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub