Entanglement, CHSH and Bell tests

Write the most general product of two spin-½ states: (α|↑⟩ + β|↓⟩) ⊗ (γ|↑⟩ + δ|↓⟩) = αγ|↑↑⟩ + αδ|↑↓⟩ + βγ|↓↑⟩ + βδ|↓↓⟩. Matching it to |Ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2 demands αδ = 1/√2 and βγ = −1/√2, so all four coefficients must be non-zero — yet it also demands αγ = 0 and βδ = 0, which forces one of them to vanish. The two requirements cannot both hold, so no product state exists, and that impossibility is the definition of entangled. Now trace out one particle: ρ_A = I/2, a fair coin with no preferred axis, identical in every basis. The pair is completely specified while each half separately carries nothing at all. Everything the state knows is stored in the relation between the two halves, which is exactly why one detector can never reveal it and a comparison of two detectors can.

A run produces a table, not a wavefunction. Fix the two analyser directions, record ±1 at each station, and count the four coincidence types: E = (N₊₊ + N₋₋ − N₊₋ − N₋₊)/N. For the singlet the prediction is E(θ) = −cos θ, equivalently P(opposite) = cos²(θ/2) and P(same) = sin²(θ/2), with θ the angle between the axes. Three cases anchor the curve. Parallel analysers, θ = 0: opposite on every trial, E = −1 — the perfect anticorrelation angular-momentum conservation already demands. Perpendicular, θ = 90°: P(opposite) = cos²45° = 0.5, the outcomes look independent, E = 0. Antiparallel, θ = 180°: identical every time, E = +1. In between, at θ = 30°, E = −cos 30° = −0.866, so 93.3% of pairs disagree. It is in this middle range, not at the anchor points, that the quantum and local predictions come apart — which is why the optimal CHSH settings contain no aligned pair.

Bell's argument assumes three things and nothing else. Each pair carries some state λ, drawn from a distribution that does not depend on which settings will later be chosen; the outcome A(a, λ) = ±1 at one station does not depend on the distant setting b, nor B(b, λ) on a; and the settings are chosen freely. For a single λ, look at the quantity A(a, λ)[B(b, λ) − B(b′, λ)] + A(a′, λ)[B(b, λ) + B(b′, λ)]. Since B(b, λ) and B(b′, λ) are each ±1, they are either equal or opposite: one bracket is 0 and the other is ±2, so the whole expression is exactly ±2 for every λ. Averaging over λ can only shrink a quantity that never exceeds 2 in magnitude, giving |S| ≤ 2. Notice what is absent — no Hilbert space, no dynamics, no assumption about mechanism. The inequality constrains a class of theories, so measuring |S| > 2 does not adjust a model, it evacuates the class.

The four correlations are not equally useful; the settings must be staggered. Take a = 0°, a′ = 90°, b = 45°, b′ = 135°. Three of the pairs are 45° apart and the fourth, a with b′, is 135° apart, so E(a, b) = E(a′, b) = E(a′, b′) = −cos 45° = −0.7071 while E(a, b′) = −cos 135° = +0.7071. The combination subtracts that one term, so instead of cancelling, all four contributions reinforce: S = −0.7071 − 0.7071 − 0.7071 − 0.7071 = −2.8284. Its magnitude, 2√2, is Tsirelson's bound — the largest |S| quantum mechanics permits for any state and any observables, and well short of the algebraic maximum 4. One conversion trips people up. Polarisation correlations run on twice the angle, E = −cos 2Δ, because a polariser turned through 180° makes the same measurement again, whereas a spin analyser turned through 180° has swapped its up and down outcomes and repeats only after 360°. So for entangled photons the same optimum sits at analyser settings 0°, 22.5°, 45° and 67.5°: half the rotation, identical physics.

Settled: no theory in which outcomes are fixed in advance by variables carried locally can produce the data. That is the whole content, and it is a great deal — it removes a kind of explanation rather than promoting one. Untouched: signalling. Compute Alice's marginal from the joint probabilities at θ = 45°. P(+,+) = ½sin²22.5° = 0.0732 and P(+,−) = ½cos²22.5° = 0.4268, which sum to exactly ½ — and the same ½ appears for whatever setting Bob picks, because his choice cannot alter ρ_A = I/2. His analyser never moves her histogram, so no protocol turns the correlation into a message and relativity's prohibition survives untouched. Also untouched: what replaces locality. Some accounts drop pre-existing values, others drop locality; the experiment does not choose between them. What it does end is the idea that the pair was merely correlated at the source, the way two shoes packed in separate boxes are correlated.

An inequality constrains the data you actually collect, so an experiment has to earn its assumptions. The detection loophole: with detection efficiency η, the undetected pairs could be the awkward ones, and unless η exceeds about 82.8% — that is 2(√2 − 1) for a maximally entangled pair under CHSH, or 2/3 using Eberhard's inequality with a deliberately non-maximal state — the conclusion rests on a fair-sampling assumption rather than on the counts. The locality loophole: each measurement, setting choice included, must finish before light could carry news of it to the other station, and across the 1.3 km separating the two labs in the Delft experiment that budget is 4.3 μs. And freedom of choice: the settings must not be correlated with λ, which is why fast physical random generators drive them. The 2015 runs at Delft, NIST and Vienna closed detection and locality in the same experiment for the first time; Delft reported S = 2.42 from 245 event-ready trials, p = 0.039.