CHSH Bell Test & Tsirelson Bound

What this research found

A three-stage computational study of the Clauser-Horne-Shimony-Holt (CHSH) inequality for two spin-one-half particles in the singlet state, which is the sharpest experimentally testable form of Bell's theorem. Computer algebra proved the singlet correlation exactly, a Monte Carlo simulator drew four million individual measurement outcomes from the Born-rule distribution, and an explicit local hidden-variable model was benchmarked against both. The simulation recovered a CHSH value of 2.827874 ± 0.001414, matching the quantum ceiling of 2√2 (about 2.8284) to within 0.39 standard errors while exceeding the classical limit of 2 by 585.

• Symbolic computation proved that the singlet correlation is exactly the negative inner product of the two measurement directions, for arbitrary directions in three dimensions rather than only coplanar ones, and that the canonical settings of 0, 90, 45, and 135 degrees are a genuine stationary point of the CHSH landscape.
• A global search over the four-angle space with 300 gradient-based restarts recovered the Tsirelson bound of 2.8284271247 to machine precision, agreeing to 4.44 × 10⁻¹⁶, with every restart reaching the same extremal magnitude.
• Sampling a million dichotomic outcomes per correlator, four million state preparations in total, gave a CHSH magnitude of 2.827874 ± 0.001414: 0.39 standard errors from the Tsirelson bound and 585 standard errors above the classical bound of 2. Closed-form and bootstrap uncertainties agreed to about 1%, at 0.0014145 against 0.0013997.
• Bell's 1964 sign model never breached the classical bound across 10,000 random measurement configurations, with an analytic maximum of 1.999998, a mean of 0.7212, and zero violations. Its piecewise-linear correlation meets the quantum cosine only at 0, 90, and 180 degrees, and the gap between them peaks at 45 and 135 degrees, exactly the separations the optimal settings use.
• At the quantum-optimal settings the local model saturates the classical bound at exactly 2.0 rather than falling short of it, which corrects a common informal claim that a hidden-variable prediction there is around 1.41. That isolates the margin of 2√2 minus 2, roughly 0.83, as the purely quantum contribution.

How it was done

The first stage used computer algebra to build the singlet state and the Pauli-matrix observables, evaluate the joint correlation symbolically, and differentiate the CHSH functional at the canonical settings, with a 300-restart gradient-based search over the four-angle space as an independent global check. The second stage implemented an exact quantum Monte Carlo estimator that draws one of four joint outcomes per shot by inverse-CDF sampling from the Born-rule probabilities, so the only error source is finite sample size; uncertainties were computed both in closed form and by a 1,000-replicate nonparametric bootstrap, alongside a convergence study from a hundred to a million shots per correlator. The third stage coded Bell's 1964 sign model with a hidden variable drawn uniformly on the sphere, validated its closed-form correlation against simulation at 61 angles, and swept 10,000 random four-vector configurations both analytically and by Monte Carlo. Everything was seeded for reproducibility and written up as a 39-page report.

Data sources

• Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1:195 (1964) - source of the sign hidden-variable model
• Clauser, Horne, Shimony and Holt, Physical Review Letters 23:880 (1969) - the CHSH inequality
• Cirel'son, Quantum generalizations of Bell's inequality, Letters in Mathematical Physics 4:93 (1980) - the 2√2 quantum bound
• Einstein, Podolsky and Rosen, Physical Review 47:777 (1935)