Research-level question in a solvable setting
How can four ordinary two-outcome measurements reveal a bound violated by quantum theory?
A two-qubit singlet predicts correlations that cannot all arise from a local hidden-variable model with predefined ±1 outcomes. Rotate four independent analyzer axes, add adjustable white noise, and compute the complete CHSH combination alongside the actual four joint probabilities.
Prerequisites: quantum-state normalization, Schrödinger time evolution, matrix spectral problems, and the corresponding section of the English lecture notes. The mathematical explanation below is integral to the interactive exercise.
Quantitative predictions
Computed observables
CHSH as a function of symmetric Bob axes
Hold Alice axes fixed at their current settings, then choose Bob's two directions ±β. Compare with local ±2 and the Tsirelson bounds ±2√2.
Predicted CHSH value versus Bob symmetric analyzer angle
The current independently selected B and B′ angles are reported above; the sweep studies the special symmetric family B=β, B′=−β.
Correlations versus relative analyzer angle
A Werner singlet with visibility v has unbiased local outcomes but angle-dependent cross-correlations.
Joint singlet spin correlation E as a function of relative axis angle
For spin-½ Pauli axes, E(a,b)=−v cos(a−b), and its magnitude never exceeds one.
Four joint outcome probabilities
One analyzer pair displays probabilities (++,+−,−+,−−), so correlations can be reconstructed from actual Born probabilities.
Four joint ±1 outcome probabilities for the selected Alice a and Bob b pair
p(st)=[1−st v cos(a−b)]/4, with s,t∈{±1}. Local marginals always equal 1/2.
Derive and compare
Mathematical model
Consider the singlet |Ψ−〉=(|↑↓〉−|↓↑〉)/√2. Spin measurements along axes a and b return random values A,B∈{−1,+1}, with correlator E(a,b)=〈(a·σ)⊗(b·σ)〉=−a·b. If the axes lie in a common plane and angles are measured in degrees, E(a,b)=−cos(a−b). The Werner mixture ρ=v|Ψ−〉〈Ψ−|+(1−v)I₄/4 multiplies this correlator by visibility v.
Combine four independently chosen settings into S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′). A local-hidden-variable theory assigning outcomes A(a,λ), A(a′,λ), B(b,λ), B(b′,λ) to each λ gives an expression A(B+B′)+A′(B−B′), whose magnitude equals 2 for every deterministic λ. Averaging any distribution of such λ therefore yields |S|≤2.
Quantum mechanics permits |S|≤2√2: the Tsirelson bound. For a=0°, a′=90°, b=45°, b′=−45°, the singlet predicts S=−2√2v. Thus this ideal geometry violates the local bound exactly when v>1/√2. This threshold refers to the Werner family, ideal projective detection, and perfect independent preparation; real experimental loophole analysis requires further controls.
The joint-outcome chart allows a second route to calculating E=P++ +P−−−P+−−P−+. Equal marginal probabilities alone never imply classical correlations: the nonlocal signature is the inconsistency of all four conditional correlations with a shared local model.
Test an exactly soluble physical limit
For v=0 all E and S vanish and all four joint outcomes are equally likely. At v=1 with a=b, measurement results are perfectly anticorrelated. At the symmetric quantum-optimal angles |S|=2√2. At precisely v=1/√2 the ideal maximum reaches the local bound but does not exceed it.
Assumptions, uncertainties, and reliability
Ideal spatially separated qubit singlet/Werner mixtures, coplanar spin axes, binary projective outcomes, fair sampling, and no detection or communication loopholes. Correlation plots are theoretical expectations, not a loophole-free experimental Bell test.
Explore independently
Three worked investigations
- Derive a bound. Write the deterministic local expression A(B+B′)+A′(B−B′) and explain why at least one parenthesis is zero, giving |S|=2.
- Find the quantum optimum. Fix the 0°,90° Alice axes and sweep symmetric Bob settings. Identify β≈45° and compare the reported S with 2√2.
- Add noise. Decrease v with optimal geometry. Locate the threshold 1/√2; then reconstruct the correlator from the four actual joint probabilities.
Self-check: What is the maximal CHSH magnitude predicted by quantum theory for two dichotomic observables per party?
Extended exercise: derive one prediction independently, identify the model approximation or physical symmetry that supports it, and test its numerical convergence. AI guidance is useful for generating further test cases after the derivation has been checked.
Interpretation
What survives beyond the model
Quantum correlations depend on the joint choice of incompatible observables, and four complementary spin settings reveal an experimentally meaningful difference from local hidden-variable predictions. The model separates local marginals, bipartite probabilities, noise sensitivity and the global CHSH constraint.
Lecture notes: 07_bell_inequality.tex ↗.