Consider a series of experiments with each experiment involving an entangled photon pair named A and B respectively. In each experiment, the photons will each be measured by polarized filters which can be set to any angle from 0° to 360°. When a measurement is taken, the photon will either align with the filter and therefore be transmitted or it will not align and therefore be absorbed or reflected.[1] The entangled photons have been prepared so that if they are measured by polarizers set to the same angle then they will either both be transmitted or both absorbed, with a 50% chance of observing either outcome.
The amplitude for configurations where the photons are correlated (i.e., have the same alignment) is the cosine of the angle between the two polarizers. Conversely, the amplitude for configurations where the photons are anti-correlated is the sine of the angle. In our experiments, the polarizers are set to either 0°, 120° or 240°.
For the first experiment, the two polarizers are set to 0°. The configurations (along with amplitudes) that are in superposition prior to measurement are as follows:
0°-1. Photon A is aligned and photon B is aligned (cos 0° = 1)
0°-2. Photon A is aligned and photon B is non-aligned (sin 0° = 0)
0°-3. Photon A is non-aligned and photon B is non-aligned (cos 0° = 1)
0°-4. Photon A is non-aligned and photon B is aligned (sin 0° = 0)
When the photons are measured, different versions of the observer become entangled with each configuration that contains amplitude. The probability that an observer will find themselves entangled with a particular configuration is determined by the squared modulus of the amplitude.[2] So this is 50% each for configurations 1 and 3 and 0% for the other two configurations. That is, there is a 100% probability that the photon pair will have the same alignment (i.e., be correlated).
The same configurations apply if both polarizers are set to 120° or 240° respectively since the angle between the polarizers is still 0°. However what happens if the polarizer for photon A is set to 0° and the polarizer for photon B is set to 120°? In this case, the configurations in superposition are:
120°-1. Photon A is aligned and photon B is aligned (cos 120° = -0.5)
120°-2. Photon A is aligned and photon B is non-aligned (sin 120° = 0.866)
120°-3. Photon A is non-aligned and photon B is non-aligned (cos 120° = 0.5)
120°-4. Photon A is non-aligned and photon B is aligned (sin 120° = 0.866)
The probabilities for configurations 1 and 3 are 12.5% each and 37.5% each for configurations 2 and 4. That is, there is a 25% probability that the photon pair will have the same alignment and a 75% probability that it will not. The same configurations apply if the polarizer for photon A is set to 120° and the polarizer for photon B is set to 240° since the angle between the polarizers is still 120°. Similarly if the polarizers are set to 240° and 0° respectively.
The same probabilities also occur if the polarization angle difference is set to 240°. However the configurations are slightly different, as follows.
240°-1. Photon A is aligned and photon B is aligned (cos 240° = -0.5)
240°-2. Photon A is aligned and photon B is non-aligned (sin 240° = -0.866)
240°-3. Photon A is non-aligned and photon B is non-aligned (cos 240° = 0.5)
240°-4. Photon A is non-aligned and photon B is aligned (sin 240° = -0.866)
Note that if the polarizers are set randomly, then the average probability that the measurements will correlate is 50%. That is, (100% + 25% + 25%) / 3 = 50%. Similarly, for measurement anti-correlation, (0% + 75% + 75%) / 3 = 50%.
Now suppose that the entangled photons are light years apart and the polarizer angles are set randomly just prior to measurement. Observers can only be entangled with one of the configurations listed above, so measurements will continue to conform to the specified probabilities.
So Many Worlds produces the correct predictions. However the above scenario is impossible to replicate by any single-world theory that depends on local hidden variables. Such theories entail that a photon has a definite polarization for each of the three angles 0°, 120° and 240° prior to measurement. For this to be true, either the photon has the same alignment value for all three angles (two options - AAA and NNN) or it has two alignment values the same and one different (six options - AAN, ANA, NAA, NNA, NAN, ANN).
Now if we could measure any pair of polarization values, they would either correlate 100% of the time for the first two options or 33% of the time for the last six options. For example, with AAN, the pairings are AA, AN and AN, with only AA correlating. So overall the measurements would correlate at least 33% of the time. This is called a Bell inequality.
It so happens that this inequality can be tested by measuring a pair of photon values via entanglement. To do this, the first polarizer angle is set to 0° and the second polarizer angle is set to 120°. Since photon A and B have identical attributes (i.e., they always correlate when measured at the same angle) then the value measured for photon B at 120° is the value that would have been the value measured for photon A at 120° if it had been measured. As was shown above, the pair of values for photon A at 0° and 120° should correlate at least 33% of the time and so the actual measured values for photon A and photon B should also correlate at least 33% of the time. However, as shown earlier with the 120° configurations, Quantum Mechanics predicts that they will correlate only 25% of the time (cos2 120° = 0.25). So Quantum Mechanics violates the Bell inequality. Therefore photons cannot have precisely-defined values for angles that have not been measured.
One possible loophole remains and that is to ensure that the entangled photons do not somehow communicate with each other. This loophole can be closed by ensuring that the photons are space-like separated when the polarizer angles are set and the measurements performed (i.e., at a distance that would require superluminal communication between the photons). The results are that experiments support the predictions of quantum mechanics and definitively rule out local hidden variable theories.
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[1] This is how polarized sunglasses work. Sunlight normally radiates in all directions. However sunlight that reflects off flat horizontal surfaces such as lakes and roads will be polarized horizontally resulting in a strong glare that vertically polarized sunglasses can block.
[2] This is the Born rule. Note: In order to keep the tables simple, the configuration amplitudes have not been normalized (i.e., the modulus squared probabilities do not add to 1). To normalize, multiply each amplitude by 1/√2.
