Quantum entanglement and Bell's theorem
An extension of the quantum mechanics topic — beyond the core syllabus, but wave-particle duality and quantum measurement do appear on the exam. Entangled particles show correlations stronger than any local realism allows (the Bell inequality S≤2); the experiments of Aspect, Clauser and Zeilinger reached S up to 2√2, confirming the nonlocality of quantum mechanics (Nobel 2022).
I'll give you detector angles a and b — guess whether the result breaks the classical bound (S > 2) or stays within classical limits (S ≤ 2).
What just happened?
Two entangled photons fly to two distant detectors set at angles a and b. For each pair of settings we measure the correlation of the outcomes. When you pick the right angles, the sum of these correlations exceeds the bound allowed by any model with “local hidden variables” — meaning the world cannot work the way Einstein thought.
Quantum correlation:
E(a,b) tells how strongly the measurement outcomes of the two detectors are correlated, depending on the angle difference a−b. For a=b the correlation is maximal (E=−1), for a 90° difference — zero.
Bell inequality (CHSH test):
We combine four measurements at four angle combinations (a, a′=a+90°, b, b′=b+90°) into a single number S:
What does this mean in practice?
Any theory assuming that particles have predetermined outcomes (hidden variables) and that information does not travel faster than light (locality) predicts S ≤ 2. The experiments of Aspect, Clauser and Zeilinger (Nobel 2022) confirmed S close to 2.83 — the Bell inequality is broken in nature.
Why you need this for the exam:
It's enough to remember the qualitative conclusion: a measurement of S > 2 rules out local realism and is experimental proof of the nonlocality of quantum mechanics — the heart of the 2022 Nobel Prize in Physics.
Remember: classically S ≤ 2, quantum up to 2√2 ≈ 2.83 — exceeding the classical bound is experimental proof that at the quantum level the world is nonlocal.
Theory & formulas
Entanglement
Two entangled particles form one shared state: measuring one instantly correlates with the outcome of measuring the other, regardless of the distance separating them.
Bell inequality
Any theory with hidden variables and local realism imposes a bound on the strength of the correlations:
Quantum result
Quantum mechanics predicts stronger correlations, up to: Experiments (Clauser, Aspect, Zeilinger — Nobel 2022) measured , confirming nonlocality. Even so, entanglement does not allow information to be sent faster than light — the outcomes are random.
Notation: — the quantity (correlation) in the Bell inequality; classical bound , quantum maximum .
Frequently asked questions
›What is quantum entanglement?
A state of two particles in which measuring one instantly correlates with the outcome of measuring the other, regardless of distance.
›What does the Bell inequality say?
It sets the bound on correlations possible for a theory with hidden variables (local realism): S≤2; quantum mechanics predicts more.
›Does entanglement allow information to be sent faster than light?
No — the correlations are random and cannot be used to send controlled information (no breaking of causality).
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