Measure entangled particles at different angles and compute CHSH correlations—watch quantum mechanics violate local realism
Bell's theorem is one of the most profound results in quantum mechanics. It proves that no theory based on local hidden variables can reproduce all the predictions of quantum mechanics. The CHSH (Clauser-Horne-Shimony-Holt) inequality provides a practical way to test this: quantum mechanics predicts correlations that violate what any local realistic theory could produce.
Alice and Bob are separated in space, each receiving one particle from an entangled pair in the singlet state |Ψ⁻⟩ = (|01⟩ - |10⟩)/√2. Each observer can choose between two measurement angles (a or a' for Alice, b or b' for Bob). Each measurement returns either +1 or -1. The visualization shows the two measurement stations and plots the correlation E(θ) = -cos(θ) as a function of angle difference.
We compute four correlation values E(a,b), E(a,b'), E(a',b), and E(a',b'), where E(θ,φ) is the correlation between measurements at angles θ and φ. The CHSH parameter is defined as:
S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')|
Any local hidden variable theory must satisfy |S| ≤ 2. Quantum mechanics predicts S can reach 2√2 ≈ 2.828 with optimal angle choices. The simulation shows both the quantum prediction (SQM) and results from simulated measurements (Ssim).
For the singlet state, quantum mechanics predicts E(θ,φ) = -cos(θ-φ), where θ-φ is the angle difference between measurement settings. This stronger-than-classical correlation arises from entanglement. The classical bound S ≤ 2 comes from assuming each particle carries predetermined measurement outcomes (hidden variables) that are independent of the distant measurement choice (locality).
The singlet state |Ψ⁻⟩ = (|01⟩ - |10⟩)/√2 is a maximally entangled two-qubit state. It's antisymmetric: if Alice measures spin-up along any axis, Bob is guaranteed to measure spin-down along the same axis (with probability 1). This perfect anticorrelation at identical angles is the signature of the singlet state.
Suppose each particle pair carried hidden variables λ determining measurement outcomes. If Alice's outcome at angle a is A(a,λ) ∈ {-1,+1} and Bob's is B(b,λ) ∈ {-1,+1}, then correlations average over λ: E(a,b) = ∫ A(a,λ)B(b,λ) dλ. Mathematical constraints on such functions force |S| ≤ 2. But quantum mechanics gives E(a,b) = -a·b = -cos(θ-φ), which can violate this bound. Nature doesn't respect the hidden variable assumption.
The optimal angles for maximal CHSH violation are a = 0°, a' = 90°, b = 45°, b' = 135° (or any configuration with these relative angles). This gives:
This yields S = |-0.707 - 0.707 + (-0.707) + (-0.707)| = 2√2 ≈ 2.828, violating the classical bound by 41%.
Maximal Violation: Click "Maximal Violation" to set optimal angles. Run the simulation and observe SQM ≈ 2.828. The simulated value Ssim should approach this with enough trials. This is the smoking gun: quantum correlations exceed what local realism allows.
Classical Bound: Click "Classical Bound" to set angles that saturate the classical limit (a=0°, a'=90°, b=0°, b'=90°). Here S = 2 exactly—on the boundary between quantum and classical. Local hidden variables could reproduce this, but quantum mechanics does too.
Zero Correlation: Click "Zero Correlation" to set all angles orthogonal (a=0°, a'=0°, b=90°, b'=90°). All correlations vanish, giving S = 0. No violation occurs because measurement axes are chosen poorly.
Scan Alice's First Angle: Set b=22.5°, b'=67.5° (Bob's optimal), and a'=45°. Now slowly sweep Alice's angle a from 0° to 90°. Watch S change. The maximal violation occurs when a and a' are optimally spaced.
Statistical Convergence: Set optimal angles, then vary the number of trials from 100 to 10,000. With few trials, Ssim fluctuates around SQM. With many trials, the agreement tightens. This is how real experiments establish violations with high confidence.
Parallel Measurements: Set a = b = 0° and a' = b' = 45°. When Alice and Bob choose the same angle, E = -1 (perfect anticorrelation for the singlet). When they choose different angles 45° apart, E ≈ -0.707. This asymmetry is the heart of nonlocality.
Design Your Own Test: Try to find angles where S > 2 but not maximal. For example, a=0°, a'=30°, b=15°, b'=45° gives moderate violation. Play with the angles to build intuition: the key is balancing correlations across all four pairs.
Understanding the Correlation Plot: The plot shows E(θ) = -cos(θ) vs angle difference. Local hidden variables can't produce this smooth curve—they're constrained to piecewise linear correlations. Quantum mechanics gives the cosine naturally from the overlap of measurement bases.
Bell's theorem and its experimental confirmation are foundational to modern physics and technology:
Quantum cryptography: BB84 and E91 protocols use entanglement and Bell violations to guarantee eavesdropping detection. Any local hidden variable eavesdropper would reduce correlations below the quantum bound.
Quantum computing: Entanglement is a computational resource. Bell inequality violations certify that qubits are genuinely entangled, not just classically correlated.
Device-independent protocols: Bell tests enable certification of quantum devices without trusting their internal workings. If S > 2, the devices must be exploiting quantum mechanics, regardless of implementation details.
Tests of quantum mechanics: Every Bell test is a test of nature's foundations. Experiments have closed loopholes (locality, detection, freedom-of-choice) to confirm quantum mechanics with exquisite rigor.
Philosophical implications: Bell's theorem rules out local realism—the idea that objects have definite properties independent of measurement and that influences can't propagate faster than light. Quantum mechanics is fundamentally nonlocal (though not superluminal signaling).
When you see S > 2 in this toy, you're witnessing the same phenomenon that has been confirmed in laboratories worldwide: photons, atoms, ions, and superconducting qubits all exhibit these nonlocal correlations. Bell's theorem isn't just theory—it's the cornerstone of the quantum revolution.
In 1964, John Stewart Bell derived his inequalities to test whether quantum mechanics could be completed by hidden variables. Einstein, Podolsky, and Rosen (EPR, 1935) had argued that quantum mechanics was incomplete because it allowed "spooky action at a distance." Bell showed this debate could be settled experimentally. In 1972, Clauser, Horne, Shimony, and Holt formulated the practical CHSH inequality. Aspect's experiments (1981-1982) provided the first convincing violations. Modern loophole-free Bell tests (2015+) have confirmed quantum mechanics beyond any reasonable doubt. The 2022 Nobel Prize in Physics honored Aspect, Clauser, and Zeilinger for this experimental work.