Bell's test: why local recipes stop at 2 and quantum reaches 2.83

hypothesess.com, AI-assisted writing · 2026-11-01

"Entanglement breaks the rules" is said so often that it has stopped meaning anything. Which rule, and by how much? Bell's test gives one number, S. If every measurement result is settled in advance by facts carried along with each particle, S cannot pass 2, as we check by counting below; the inequality in this form is due to Clauser, Horne, Shimony and Holt [3]. Quantum mechanics predicts 2.828, and no more [4]. This notebook entry computes both numbers from scratch, then asks how many measurements it takes to tell the two worlds apart.

What the test measures

Alice and Bob each receive one particle from a shared source. Alice measures hers with setting a or a', Bob with b or b'. Every measurement gives +1 or -1. For each pair of settings, the correlation E is the average of the product of the two results: +1 if they always agree, -1 if they always disagree, 0 if there is no relation.

The test adds four of these correlations with one minus sign:

S = E(a, b) - E(a, b') + E(a', b) + E(a', b').

That is all. The whole argument is about how large S can be.

Every private rule, listed

Suppose each particle carries a rule that fixes its answer for each setting before anyone measures. Alice's rule says what she gets for a and for a'; Bob's says what he gets for b and for b'. Each of those four answers is +1 or -1, so there are 2 x 2 x 2 x 2 = 16 possible rules. We wrote them all down and computed S for each one [1].

Eleven of the sixteen give S = 2 or -2. The other five give 0. None gives anything else. The reason is visible in the formula: S = a(b + b') + a'(b - b'), and when b and b' are each +1 or -1, one bracket is 0 and the other is +2 or -2 [1]. Letting the particles agree on a rule at random before they leave the source only averages these values, so no mixture passes 2 either. This is the content of the inequality that Bell found and that Clauser, Horne, Shimony and Holt put in this four-term form [2][3]. It is a counting fact, not a law of physics.

What quantum mechanics predicts

For two spin-1/2 particles in the singlet state, quantum mechanics gives E(a, b) = -cos(a - b), where a and b are the angles of the two measuring directions. Choose a = 0 degrees, a' = 90, b = 45, b' = 135; the four cosines then line up and S comes to 2.828427, which is 2 times the square root of 2 [1].

Could other angles do better? We tried 200,000 random sets of four angles and never passed 2.82744 [1]. That matches Tsirelson's theorem: quantum mechanics itself tops out at 2 times the square root of 2 [4]. So the quantum world beats the private-rule bound by a factor of 1.414, and stops there [1].

How many pairs before the gap is unmistakable?

A real experiment measures a finite number of pairs, and a finite sample is noisy. We simulated the quantum statistics with N pairs per setting, and beside them a private-rule world sitting exactly on its bound, 400 runs of each [1].

Pairs per setting	Quantum: mean S	Spread	Private rule: mean S	Spread	Private-rule runs above 2.5	Quantum runs below 2
10	2.822	0.450	1.988	0.533	17.0%	3.7%
100	2.839	0.140	2.000	0.172	0.0%	0.0%
1,000	2.826	0.044	2.001	0.054	0.0%	0.0%
10,000	2.829	0.015	1.999	0.018	0.0%	0.0%

With 10 pairs per setting, a private-rule world lands above 2.5 by luck about one time in six, so a tiny run proves nothing [1]. With 100 pairs per setting, 400 pairs in all, the two worlds did not overlap once in 400 runs: the gap of 0.83 is about six times the quantum spread of 0.14 [1]. Each tenfold increase in pairs shrinks the spread by about 3.2, the square root of ten, which is the usual law for averages of independent measurements [1].

What real experiments found

Real detectors are imperfect and real entanglement is never pure, so measured values fall short of 2.828. They still clear 2. Aspect, Grangier and Roger measured entangled photons in 1982 and called their result "the greatest violation of generalized Bell's inequalities ever achieved" [5]; later that year, with the settings switched while the photons were in flight, Aspect, Dalibard and Roger found a violation "by 5 standard deviations" [6]. In 2015 a Delft team closed the remaining loopholes with electron spins 1.3 km apart: 245 trials gave S = 2.42 plus or minus 0.20 [1][7]. The 2022 Nobel Prize in Physics went to Aspect, Clauser and Zeilinger for these experiments [8].

The Delft number is worth a second look against the table above. The sample was small, 245 trials across four settings, so the quoted uncertainty of 0.20 is large; the table above shows how wide the scatter is at 10 pairs per setting [1][7]. Our simulation says the same: at that sample size, a clean separation needs a value well above 2, and 2.42 is about two spreads above it.

Nerd corner

The spread of S comes from the four sample correlations. Each is an average of N products that are +1 or -1, so its variance is (1 - E^2) / N. For the quantum angles every E^2 is 0.5, so the variance of S is 4 x 0.5 / N = 2 / N and the spread is the square root of 2 / N, which reproduces the simulated spreads of 0.450, 0.140, 0.044 and 0.015 [1]. For the private-rule world at the bound, E is plus or minus 0.5, so the variance is 4 x 0.75 / N = 3 / N. The 16-rule enumeration and the random-angle search are in the lab page's script [1].

What it does and does not show

The test rules out one family of explanations: results fixed locally in advance. It does not show that anything travels faster than light. Alice's own results, looked at alone, are 50/50 whatever Bob does; only the comparison of the two lists shows the correlation, and the lists have to be brought together at light speed or slower. Nor does it kill determinism outright: theories where the fixing is non-local, or where the choice of settings is itself part of the plan, survive at a price most physicists find too high. The careful phrase is "local hidden variables", and 2.828 is the exact size of what they cannot do.

Check yourself

Someone runs 10 pairs per setting and gets, say, S = 2.6. Have they shown quantum mechanics beats the bound? No. At that sample size a private-rule world lands above 2.5 about one time in six [1].

Could a cleverer private rule reach 3? No. All 16 rules are listed; the best gives 2, and mixing them can only average [1].

Why do real experiments fall short of the quantum maximum? Imperfect detectors and impure entanglement pull S down; the point is that it stays above 2 with the loopholes closed [7].

This is a lab notebook entry for hypothesess.com, written with AI and checked by hand; every figure in it was computed, not estimated, and the box above recomputes the ones that depend on your numbers.