Presets
Model that generates the data
Analyzer angles (degrees)
a
a′
b
b′
Alice picks a or a′, Bob picks b or b′, randomly for every pair. You can also drag a polarizer ring in the 3D view.
Run
Result
S = --
| Setting pair | N coinc. | E measured | E quantum | E sawtooth |
|---|
Correlation E(a−b)
quantum V cos 2Δhidden-variable sawtoothmeasured (±1σ)
CHSH value S as pairs accumulate
No-signalling check
| P(click +) | other side uses 1st | other side uses 2nd |
|---|
Each side's own statistics stay at 50/50 whatever the distant analyzer does, so no message can be sent. Only the comparison of both records shows the correlation.
Experiment 1: S against analyzer angle
Settings a=0, a′=2θ, b=θ, b′=3θ with θ swept from 0 to 90°. Quantum prediction S(θ) = V (3 cos 2θ − cos 6θ), maximum 2√2 V at θ=22.5°. Each point is a fresh Monte Carlo run using the visibility V from the Source tab.
quantum theorysawtooth modelsimulated
Experiment 2: statistical error against N
Repeat the optimal-angle experiment many times at each N and measure the scatter of S. Theory: σS = √(Σ (1−E²)/(N/4)), so it falls as 1/√N.
Independent checks
The science
The question: can hidden variables explain entanglement?
Two photons are made together and sent to two distant analyzers, Alice and Bob. Each picks a polarizer angle and records +1 or −1. When both angles are equal the answers always agree, which a "pre-agreed instruction" (a local hidden variable λ carried by each photon) can explain easily. Bell (1964) showed that for other angle pairs such instructions cannot reproduce what quantum mechanics predicts. Clauser, Horne, Shimony and Holt (1969) turned this into the inequality tested here.
Quantum prediction
For the polarization-entangled state |Φ+> = (|HH> + |VV>)/√2 and analyzers at angles a and b:
P(+,+) = P(−,−) = ½ cos²(a−b)
P(+,−) = P(−,+) = ½ sin²(a−b)
E(a,b) = P(++)+P(−−)−P(+−)−P(−+) = cos 2(a−b)
Each side alone is always 50/50 (no signalling). With white noise mixed in (visibility V), E = V cos 2(a−b). (For the spin singlet the sign is E = −cos(angle between axes); the photon-polarization form has period 180°.)
The CHSH bound
S = E(a,b) − E(a,b') + E(a',b) + E(a',b')
If outcomes are fixed in advance by λ (A(a,λ), B(b,λ) = ±1) then for every λ
A(B−B') + A'(B+B') = ±2 (one bracket is 0, the other ±2)
and averaging over λ gives |S| ≤ 2. The Experiments tab enumerates all 16 deterministic assignments to confirm this. Quantum mechanics gives S = 2√2 ≈ 2.828 at a=0°, a′=45°, b=22.5°, b′=67.5° (four terms of 1/√2 each). Tsirelson (1980) proved 2√2 is the largest value any quantum state can give; the Experiments tab finds it as the largest eigenvalue of the CHSH operator. Algebraically S could reach 4 (no-signalling "PR boxes"), but nature stops at 2√2.
The hidden-variable model in this lab
Each pair gets a random angle λ (uniform over 180°); a detector answers +1 if cos 2(x−λ) > 0, else −1. This is local and deterministic. Its correlation is the sawtooth
E(Δ) = 1 − |Δ|/45° (|Δ| ≤ 90°)
It matches quantum theory at 0°, 45° and 90° but lies below it in between. At the optimal angles it reaches S = 2 exactly: the best a local model can do with these angles (the bound is a bound, not a failure of this model).
Noise, finite statistics and the coincidence window
Visibility V scales every correlation, so S = 2√2 V: the violation disappears below V = 0.707. With N pairs spread over four settings each E has standard error √((1−E²)/(N/4)), so σS ≈ 2.83/√N at the optimum and about 290 pairs are needed to reach a 5σ violation (V=1). Accidental coincidences within the window τ are uncorrelated and dilute V by a factor (1−f), f = Racc/(Racc+Rtrue), Racc = 2τSASB.
The detection loophole
If detectors miss pairs and only coincidences are analysed, a local model may decide which pairs get detected, depending on the setting. The "LHV + dead zones" model does that: a detector stays silent when λ is far from both of its two outcome directions (total efficiency 85%). Counting only coincidences it mimics quantum mechanics at the optimal angles (S ≈ 2.84). Counting every emitted pair (no click = 0) the same data give S ≤ 2 again. Genuine quantum data fail that stricter test too if detectors are inefficient: for maximally entangled photons the all-pairs CHSH test needs η > 2/(1+√2) ≈ 82.8% (Garg and Mermin 1987); smarter tests (Eberhard 1993) work down to 2/3. Detector efficiency here multiplies both sides independently. The loophole-free experiments of 2015 (Hensen et al., Delft, electron spins; Giustina et al., Vienna, and Shalm et al., NIST, photons) closed it together with the locality loophole.
Experiments and the 2022 Nobel Prize
Freedman and Clauser (1972), Aspect and colleagues (1981-82, with analyzers switched while photons were in flight) and Weihs, Zeilinger and colleagues (1998, random fast switching over 400 m) tested Bell inequalities with entangled photons. The 2022 Nobel Prize in Physics was awarded to Alain Aspect, John F. Clauser and Anton Zeilinger for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science. The result rules out local hidden-variable theories (given the usual assumptions, including free choice of settings); it does not allow faster-than-light messages.
What is simplified
- The source is ideal: exactly one photon pair per event, perfect Bell state times visibility V. Real spontaneous parametric down-conversion also makes multi-pair events and has a basis-dependent visibility.
- The bench is schematic: the BBO crystal, the two-channel analyzers (rotating polarizer plus polarizing beam splitter; real set-ups rotate a half-wave plate by θ/2) and the fibre links are generic, and photons travel in straight lines at an arbitrary visual speed. Timing and the locality loophole are not simulated.
- Setting choices come from a seeded pseudo-random generator, not from a quantum random number generator; a hidden-variable model could in principle exploit a deterministic generator.
- The accidental-coincidence rates, 20 kHz pairs and 100 kHz background, are invented but plausible numbers.
- The "LHV + dead zones" model is tuned to match quantum mechanics at the optimal angles only; other angles do not match.
- Statistical errors treat the four correlations as independent binomial estimates.
Pipeline: 3D models built in Blender 5.2 (Cycles-baked optical-table texture), simulation and rendering in Three.js. Key references: Bell, Physics 1, 195 (1964); Clauser, Horne, Shimony and Holt, Phys. Rev. Lett. 23, 880 (1969); Tsirelson, Lett. Math. Phys. 4, 93 (1980).