Probability of getting the same color from both detectors >= 5/9
So I will setup an experiment for which the above inequality makes sense. I will prove that using common sense and locality, the inequality is satisfied. Then I will use some simple Quantum Mechanics and defy locality to prove that the inequality is violated (in fact I will prove the probability to be 1/2). And I will do that using only high school math.
In 1983 Aspect did the corresponding experiment and found that the inequality is indeed violated, confirming that QM is a non local theory (i.e. there are no hidden variables and there is an instant communication between 2 photons).
Rules:
1. We have a source that emits a pair of 2 identical photons. Each pair may have different properties, but the photons inside each pair are identical. Each photons heads towards a different detector.
2. We have 2 detectors that can flash either red or green, depending on the polarization (direction) of the photons.
3. Each detector can be aligned randomly along one of 3 predetermined axes of polarization. Let's assume for example that these axes are at 120 degrees angle with eachother. We name these positions 1, 2 and 3. Each detector can be set independent from the other.
4a. The photon source emits 2 types of pairs of photons, type I and type II. We cannot know which type of pair is emitted at each take of the experiment.
4b. Type I pairs of photons always cause both detectors two flash green (for example, the photons can have polarization that is never opposite to any of the 3 main axes - they will always contain some component in any of the 3 axes; hence it doesn't matter how the detectors are aligned, they will always both flash green). The probability of the detectors flashing the same color is 1.
4c. Type II pairs of photons cause the detectors to flash green only when the detector is oriented at position 2 (the polarization of the photon is along that direction). Then each detector will flash green exactly 1/3 of the time. What is the probability that both detectors will flash the same color? You can prove simply that this is 5/9:
So, whenever any detector is set at position 2 and the other one isn't then they flash a different color (D) ; in any other case they flash the same (S). Take some time to convince yourselves for that since this is the whole proof!
Now, Bell's inequality comes as an answer to the question: What is the probability of getting the same color from both detectors? Answer: the probability is always equal or greater than 5/9. In our case this probability is either 5/9 (if type II photons are emitted) or 1 (if type I photons are emitted).
Probability of getting the same color from both detectors >= 5/9
This is the common sense answer and it seems like there can be nothing else other than that. That is because you implicitly assume that the property of each pair of photons (whether it is type I or II) is pre-determined at the birth of the photons. However in the next part I will prove that QM allows for the probability to be 1/2, which means that the determination of the type happens exactly when the detector flashes, and not beforehand!
To be continued...
