Bell's experiment

Bell’s Theorem proves that no "local hidden variable" theory can reproduce the predictions of quantum mechanics. In simpler terms: The universe is not both local and realistic.

1. The Setup: Mermin’s Machine

Imagine a setup with three distinct parts: a central source and two remote detectors (A and B).

The Source: Fires two particles in opposite directions toward the detectors.
The Detectors: Each has a switch with three settings (1, 2, 3) and two lights (Red and Green).
The Procedure: For every run, we randomly set the switches on A and B. We then record whether the lights flash Red (R) or Green (G).

2. The Observed Data

When we run this experiment many times, we notice two striking "features" in the data:

Feature 1 (Perfect Correlation): Whenever the switches are set to the same number (1-1, 2-2, or 3-3), the detectors always flash the same color. They are either both Green or both Red.
Feature 2 (The Statistical Average): When the switches are set randomly (without regard for what the other is doing), the lights flash the same color exactly 50% of the time.

3. The "Realistic" Explanation (Hidden Variables)

To explain Feature 1, any classical intuition suggests that the particles must be carrying "instructions." Since they are separated and cannot communicate instantly, they must have "agreed" on what to do for each setting before they left the source.

We can represent these instructions as a triplet, like RGR:

If setting is 1, flash Red.
If setting is 2, flash Green.
If setting is 3, flash Red.

Since the detectors always agree when the settings are the same, both particles must carry the exact same instruction set.

4. The Mathematical Contradiction (Bell's Inequality)

Now, we apply the logic of these instruction sets to Feature 2.
Consider any possible instruction set. There are only two types:

All same: (RRR) or (GGG).
Mixed: (RRG), (RGR), (GRR), (GGR), (GRG), (RGG).

Let’s calculate the probability of getting the "Same Color" for a Mixed set (e.g., RRG, without loss of generality) when settings are chosen at random:
There are $3 \times 3 = 9$ possible setting combinations:

(1,1), (2,2), (3,3) $\to$ Same color (R-R, R-R, G-G).
(1,2), (2,1) $\to$ Same color (R-R).
(1,3), (3,1), (2,3), (3,2) $\to$ Different colors (R-G or G-R).
In a mixed set, the lights agree in 5 out of 9 cases.

Probability (Same) = \frac{5}{9} \approx 55.5 %

For the "All same" sets (RRR), the probability of agreeing is 9 out of 9 ( $100 %$ ).

The Inequality: If the particles carry hidden instructions, the total probability of the lights flashing the same color across all runs must be at least 5/9 (roughly 56%).

5. The Verdict

Quantum mechanics predicts (and experiments confirm) that the lights flash the same color only 50% of the time.

This 50% result is a direct consequence of the quantum mechanical prediction for a singlet state when the detectors are set $120^{\circ}$ apart (which is the required angle for three symmetric settings). The probability $P_{s a m e}$ of getting the same result when the detectors are set to angles $a$ and $b$ is given by:

P_{s a m e} (a, b) = \frac{1}{2} (1 + \cos (θ))

where $θ$ is the angle between the settings.

Same Settings (1-1, 2-2, 3-3): $θ = 0^{\circ}$ .
$P_{s a m e} (0^{\circ}) = \frac{1}{2} (1 + \cos (0^{\circ})) = 1 (100%)$
(Matches Feature 1)
Different Settings (1-2, 2-1, etc.): $θ = 120^{\circ}$ .
$P_{s a m e} (120^{\circ}) = \frac{1}{2} (1 + \cos (120^{\circ})) = \frac{1}{2} (1 - 0.5) = 0.25 (25%)$

Since there are 9 equally likely combinations of random settings (3/9 "Same" and 6/9 "Different"), the total average probability of flashing the same color is:

Total P_{s a m e} = \frac{3}{9} \times P_{s a m e} (0^{\circ}) + \frac{6}{9} \times P_{s a m e} (120^{\circ})

Total P_{s a m e} = \frac{1}{3} \times 1 + \frac{2}{3} \times 0.25 = \frac{1}{3} + \frac{1}{6} = \frac{3}{6} = 0.5 (50%)

Since $50 % < 55.5 %$ , the assumption that the particles carry pre-determined instructions (Hidden Variables) is mathematically impossible.

Conclusion

The "Feature 1" (perfect correlation) suggests the particles are linked, but "Feature 2" proves that this link cannot be explained by any local properties they carried with them. This leaves us with two radical options:

Non-locality: The choice of setting at detector A instantaneously affects the outcome at detector B.
Non-realism: The particles do not have definite properties until they are measured.