In 1964 physicist John Bell derived a mathematical inequality that any theory based on "local hidden variables" — the idea that particles carry predetermined answers to measurement questions, and that no influence travels faster than light — must satisfy. Quantum mechanics predicts correlations between entangled particles that violate Bell's inequality, and experiments by Aspect (1982) and Hensen (2015, loophole-free) have confirmed the quantum prediction to many standard deviations. The CHSH (Clauser-Horne-Shimony-Holt) version of the inequality states |S| ≤ 2 for any local realistic theory, while quantum entanglement with the singlet state achieves S = 2√2 ≈ 2.828 at the optimal measurement angles.
This simulation lets you set Alice's and Bob's two measurement angles (a, a' and b, b') and accumulate thousands of trials. You can compare quantum (entangled singlet pair) and local hidden-variable (deterministic hidden spin) models side by side and watch S converge to its predicted value. The four correlation coefficients E(a,b), E(a,b'), E(a',b) and E(a',b') are tracked live, showing exactly which correlations push S above or below the classical bound.
What is quantum entanglement?
Two particles are entangled when their quantum state cannot be written as a product of individual states — measuring one particle instantaneously determines the outcome statistics for the other, regardless of the distance between them. For a spin-singlet pair (|↑↓⟩ – |↓↑⟩)/√2, if Alice measures spin-up along some axis, Bob's particle is guaranteed to be spin-down along that same axis. This correlation is stronger than anything achievable by pre-agreed (hidden-variable) strategies, as Bell's theorem proves.
What is the CHSH inequality?
The CHSH inequality is S = E(a,b) – E(a,b') + E(a',b) + E(a',b') where each E is the correlation between Alice's and Bob's measurement outcomes (each ±1) for a given pair of angles. Any local hidden-variable theory must give |S| ≤ 2. Quantum mechanics with the singlet state gives E(a,b) = –cos(a – b), and at the optimal Bell angles (0°, 45°, 22.5°, 67.5°) this yields S = 2√2 ≈ 2.828, a clear violation.
Does a Bell violation mean information travels faster than light?
No. Even though the measurement outcomes are correlated in a way that cannot be explained by shared classical information, Alice and Bob cannot use entanglement to send a message faster than light. Each local outcome is still random (50/50 for any single measurement), so Bob cannot extract a signal from his own results alone — he can only detect the correlations by comparing notes with Alice through a conventional (slower-than-light) channel after the fact.
A local hidden-variable (LHV) model assumes that each particle carries a hidden internal state (the "variable") set at the moment the pair is created, and that this state determines the outcome of any measurement deterministically. "Local" means Alice's result cannot depend on Bob's measurement setting, and vice versa. Bell proved that any such model must obey |S| ≤ 2, while quantum mechanics predicts S up to 2√2. Experiments consistently find S ≈ 2.8, ruling out all LHV models.
The quantum correlation for the singlet state is E(a,b) = –cos(a – b). Maximising S = E(a,b) – E(a,b') + E(a',b) + E(a',b') requires choosing angles such that the cosines combine constructively. Setting a = 0°, b = 22.5°, a' = 45°, b' = 67.5° gives differences of 22.5°, 67.5°, 22.5°, and 22.5°. Substituting into the cosine formula gives S = –cos(22.5°) + cos(67.5°) – cos(22.5°) – cos(22.5°) = –3cos(22.5°) + cos(67.5°) = 2√2 ≈ 2.828.
Early Bell experiments had loopholes: the "detection loophole" (only a fraction of photon pairs detected, possibly a biased sample), the "locality loophole" (detectors close enough that a subluminal signal could carry Alice's setting to Bob before his measurement), and the "free-will loophole" (settings might be correlated with hidden variables). The 2015 Delft experiment by Hensen et al. closed all major loopholes simultaneously using entangled electrons in nitrogen-vacancy centres 1.3 km apart, achieving S = 2.42 ± 0.20.
The statistical significance of a Bell violation grows as √N, where N is the number of trials. With N = 1,000 trials the standard error on S is roughly 0.06, placing a quantum violation (S ≈ 2.83 vs the classical bound of 2.0) about 13 standard deviations away from the classical limit — extremely convincing. Real experiments use millions of trials and achieve p-values less than 10⁻¹⁵, making the violation astronomically improbable under any local-realistic model.
Yes — photons are the most common choice for Bell tests because polarisation entanglement is relatively easy to create via spontaneous parametric down-conversion (SPDC) and because photons travel at the speed of light, making the locality loophole easier to close. Alice and Bob measure photon polarisation along their chosen axes; a polarising beam splitter and two detectors replace the Stern-Gerlach magnets of the spin model. The correlations obey the same CHSH formula with E(a,b) = –cos(2(a – b)) due to the factor-of-2 angle mapping for polarisation.
Quantum non-locality is the name given to the correlations between entangled particles that violate Bell inequalities. It is distinct from non-locality in the sense of action-at-a-distance: there is no causal influence propagating between the particles, and the correlations appear only in the joint statistics, not in any local observable. Technically, quantum non-locality is a property of quantum correlations that is incompatible with any local realistic model, but compatible with relativistic causality.
Yes — there are many generalisations. The Mermin inequalities apply to three or more parties and can be violated by GHZ (Greenberger-Horne-Zeilinger) states by a factor that grows exponentially with the number of qubits. Many-body Bell inequalities are relevant to quantum cryptography (device-independent quantum key distribution) and quantum computing, where verifying entanglement without trusting the devices requires demonstrating a Bell violation.
The Tsirelson bound (1980) is the maximum value of S achievable by any quantum state and any quantum measurements: |S| ≤ 2√2 ≈ 2.828. It lies strictly between the classical bound (2) and the algebraic maximum (4). The fact that quantum mechanics does not allow |S| = 4 means it is not "maximally non-local" — there exist hypothetical theories (e.g. "Popescu-Rohrlich boxes") that violate 2√2 while still respecting no-signalling, hinting that something deeper than no-signalling constrains the quantum world.