Frontier physics

Quantum entanglement and Bell inequalities

Entanglement produces correlations incompatible with local hidden-variable models; Bell tests distinguish them from quantum predictions.

The two-qubit state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ cannot be factored into separate states. Each measurement is random, yet outcomes are strongly correlated; this does not enable faster-than-light communication.

The CHSH inequality

Two observers independently choose between two settings each. For a singlet and suitable angles, quantum theory predicts $|S|=2\sqrt2$, exceeding the local bound 2. Bell violations are experimentally established; foundational interpretations continue to be studied.

For a two-spin singlet, measurements along unit vectors $\mathbf a,\mathbf b$ have correlation $E(\mathbf a,\mathbf b)=-\mathbf a\cdot\mathbf b$. The CHSH combination $S=E(a,b)+E(a,b’)+E(a’,b)-E(a’,b’)$ obeys $|S|\le2$ for local hidden-variable models, whereas quantum theory permits $2\sqrt2$. Loophole-controlled experiments confirm violations. They rule out that model class under the test assumptions, not enable faster-than-light messaging.

Bell correlations do not permit faster-than-light signaling: each party’s marginal outcome probabilities are independent of the distant measurement choice. Only correlations assembled after classical comparison violate the classical bound. Modern experiments close detection and locality loopholes in the same test. The foundational interpretation remains debated, but the quantum probability structure is precisely confirmed.

For a singlet pair, each ±1 outcome has zero mean, yet jointly measured outcomes depend on the angle between analyzers. Any local hidden-variable model reproducing all such correlations must satisfy CHSH; quantum data exceed that bound. Experiments sample several analyzer settings so the conclusion does not rest on one correlation alone.