Quantum Entanglement
Entanglement is perhaps the most striking feature of quantum mechanics. When two particles are prepared in an entangled state, their measurement outcomes are correlated in ways that no classical theory can reproduce. Einstein famously objected to this as 'spooky action at a distance,' arguing that quantum mechanics must be incomplete and that hidden variables must exist to explain the correlations locally.
Bell's Theorem
In 1964, John Bell proved that any theory based on local hidden variables must satisfy certain constraints — now called Bell's inequalities — on the correlations between measurements at two distant locations. Quantum mechanics predicts correlations that violate these inequalities. This transforms a philosophical debate into an experimentally testable prediction.
The CHSH Test
The CHSH version of Bell's inequality is the most commonly tested in experiments. Alice and Bob each choose between two detector settings (a, a' and b, b'). For each setting pair, they measure the correlation E between their outcomes. The CHSH parameter S = E(a,b) - E(a,b') + E(a',b) + E(a',b') is bounded by |S| <= 2 for any local hidden variable theory. For entangled singlet states, quantum mechanics predicts E(a,b) = -cos(a-b), which can yield |S| up to 2 sqrt(2) ~ 2.83.
Experimental Confirmation
Starting with Alain Aspect's landmark 1982 experiments, Bell's inequality has been violated in increasingly stringent tests. The 2015 'loophole-free' experiments closed the detection and locality loopholes simultaneously, leaving essentially no room for local hidden variable explanations. In 2022, Aspect, Clauser, and Zeilinger received the Nobel Prize in Physics for their experimental work establishing the violation of Bell's inequalities.
Using the Simulator
Set the four detector angles and observe how the CHSH parameter S changes. The default angles give a clear violation. The S-value meter shows whether you're in the classical (green) or quantum (red) regime. The optimal angles for maximum violation are a=0, a'=45, b=22.5, b'=67.5. Entangled pairs are emitted from the central source toward Alice and Bob's detectors, and measurement results accumulate over time.