Quantum Entanglement, Bell States and the CHSH Test

A state that cannot be split into two

Two qubits normally live in a four-dimensional space spanned by |00⟩, |01⟩, |10⟩, |11⟩, and most states in that space can be factored into 'qubit A is in state X' times 'qubit B is in state Y'. Entangled states cannot. The four maximally entangled two-qubit states, the Bell states, are the clearest example:

Φ+ = (|00⟩ + |11⟩) / √2
Φ- = (|00⟩ - |11⟩) / √2
Ψ+ = (|01⟩ + |10⟩) / √2
Ψ- = (|01⟩ - |10⟩) / √2

Take Φ+. Measure the first qubit in the standard basis and you get 0 or 1 with 50/50 probability — exactly like a fair coin, and by itself that measurement reveals nothing unusual. But the instant you measure it, the second qubit's outcome is now fixed: if the first came out 0, the second will come out 0 with certainty; if the first came out 1, so will the second. Neither qubit had a definite value before measurement — this is not a matter of hidden pre-agreed outcomes, which is precisely what Bell's theorem rules out.

No faster-than-light signalling

It is tempting to read that correlation as instant communication, but the no-communication theorem blocks it: whoever holds the second qubit sees only a random 50/50 sequence of 0s and 1s on their own, no matter what the first person does. The correlation is only visible when the two measurement records are later brought together and compared, which requires an ordinary, light-speed-limited channel. Entanglement lets you build correlations that are stronger than anything classical physics allows, but it cannot move information faster than light.

The CHSH test and Bell's inequality

Einstein, Podolsky and Rosen argued in 1935 that quantum mechanics must be incomplete — that the particles secretly carry hidden variables that predetermine measurement outcomes, restoring a classical picture ('local realism'). In 1964 John Bell showed this assumption makes a testable, quantitative prediction. The CHSH inequality (Clauser-Horne-Shimony-Holt, 1969) is the practical version: measure each of two entangled particles along one of two possible angles, compute a correlation statistic S from the four angle combinations, and any local hidden-variable theory is mathematically forced to give |S| ≤ 2.

S = E(a,b) - E(a,b') + E(a',b) + E(a',b')

classical / local hidden variables:  |S| ≤ 2      (Bell's inequality)
quantum mechanics (Bell state):      |S| ≤ 2√2 ≈ 2.828   (Tsirelson's bound)
best experiments (loophole-free):    S ≈ 2.4 - 2.7, clearly violating 2

Quantum mechanics predicts S can reach 2√2 ≈ 2.828 — the Tsirelson bound — strictly higher than the classical ceiling of 2. Real experiments, most rigorously the loophole-free tests of 2015 onward, measure entangled photon or ion pairs and consistently find S above 2, violating Bell's inequality and ruling out local hidden-variable theories. This body of work — starting with Alain Aspect's 1980s experiments and closing all major loopholes by 2015 — earned Aspect, John Clauser and Anton Zeilinger the 2022 Nobel Prize in Physics.

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