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Suppose, Alice has a Billion of photons in random states, while Bob has their entangled pairs. Suppose, Alice can measure the spin of the photons in X axis, while Bob can measure their spin in an axis which is about 60 degrees rotated from the X axis.

If Bob measures his photon spins before Alice measures hers, there is a 50% chance that his photons will point to the up direction. However, if he performs the same test after Alice measures her photons, due to the fact that Alice's measurement affects his, his chance of detecting number of photons in the up direction reduces to 1/4. Reference: https://en.wikiversity.org/wiki/Bell%27s_theorem/Inequality

Now, if, at certain time, Bob decides to measure his photons, he can deduce whether or not Alice performed a measurement. Thus, one information is transmitted faster than light: Alice made a measurement. What's wrong with this argument?

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If Bob measures his photon spins before Alice measures hers, there is a 50% chance that his photons will point to the up direction. However, if he performs the same test after Alice measures her photons, due to the fact that Alice's measurement affects his, his chance of detecting number of photons in the up direction reduces to 1/4.

This is not correct. In QFT spacelike operators commute, meaning that the order does not matter. The results of the measurement are the same regardless of what order they happen in. So in this case the fraction of photons in the up direction is 50%, regardless of who measures first or even whether or not Alice measures anything at all.

What changes is the correlation, which is given by $-\cos(\theta)$. So if they are measuring on the same axis then the correlation is -1, meaning that each up measurement for Alice corresponds to a down measurement for Bob. Bob still measures 50% up and 50% down regardless. In contrast, if they are measuring at 60 degrees then the correlation is -0.5, meaning that for each up measurement for Alice there is a somewhat greater chance of Bob getting a down and for each of Alice’s down measurements there is a somewhat greater chance of Bob getting an up measurement. But overall Bob still measures 50% up and 50% down.

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