Introduction
Entanglement is not “strong correlation” — it is a structure of joint quantum states that cannot be written as independent states for the parts. A standard source produces the two-photon singlet (in polarization language) \[ \lvert\Psi^-\rangle=\frac{1}{\sqrt2}\left(\lvert H\rangle_A\lvert V\rangle_B-\lvert V\rangle_A\lvert H\rangle_B\right), \] which predicts anti-correlated outcomes when Alice and Bob measure in the same basis. Rotate analyzers to different angles, and a distinct, sinusoidal correlation appears.
John Bell showed that any local hidden-variable (LHV) theory obeys inequalities that quantum mechanics can violate. The most used variant in photon tests is the CHSH inequality.
Quantum Correlations for Polarization
Let Alice choose setting \(a\) and Bob choose \(b\), each being a polarizer angle. Assign outcomes \(A,B\in\{+1,-1\}\) for “transmitted” vs “reflected.” For the singlet state the expectation value is \[ E(a,b)=\langle AB\rangle=-\cos\big(2(a-b)\big). \] Identical angles give perfect anti-correlation \(E=-1\); orthogonal analyzers give \(E=+1\).
These correlations follow from amplitudes on a Bloch sphere for polarization, where rotations by angle \(\theta\) correspond to phase changes \(2\theta\) in the measurement probabilities.
The CHSH Inequality
In an LHV model, predetermined outcomes \(A(a,\lambda), A(a',\lambda)\) and \(B(b,\lambda), B(b',\lambda)\) depend on a shared hidden variable \(\lambda\). Define \[ S = E(a,b)+E(a,b')+E(a',b)-E(a',b'). \] All such models satisfy \(|S|\le 2\). Quantum mechanics predicts \(|S|\le 2\sqrt2\) (Tsirelson’s bound), and the singlet with \(a=0^\circ, a'=45^\circ, b=22.5^\circ, b'=67.5^\circ\) achieves the maximum \(2\sqrt2\).
Importantly, no faster-than-light signaling is possible: each party’s marginal outcomes remain 50/50 regardless of the other’s choice.
How Experiments Create Entangled Photons
Most optical Bell tests use spontaneous parametric down-conversion (SPDC): a nonlinear crystal probabilistically splits a pump photon into signal and idler with orthogonal polarizations. Configurations (type-I & type-II, Sagnac loops, crossed crystals) engineer high-visibility states while keeping spatial and spectral modes well matched.
Photons are sent to distant analyzers implemented with wave plates and polarizing beam splitters. Single-photon detectors record time-stamped clicks; coincidences within a short window identify pairs. By switching analyzer settings rapidly and ensuring space-like separation, modern tests close locality and freedom-of-choice loopholes.
Loopholes & Modern Milestones
Early experiments suffered from the detection loophole (missed photons bias samples) and the locality loophole (settings not space-like separated). Advances in detector efficiency, fast random setting choice, and large separations produced fully loophole-closed violations of CHSH.
Today, entanglement distribution reaches satellite scales and underlies protocols like quantum key distribution and device-independent randomness expansion. The same mathematics applies beyond photons — trapped ions, superconducting qubits, even massive atoms.