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I am approaching this from an intuitive perspective and I don't speak the language. However, I have been doing a lot of reading about Bell's Theorum and how invalidates either locality or counterfactual definiteness.

From my point of view, locality is the concept that there is no way that two particles can interact with each other "instantaneously" at a distance (i.e. faster than light.). Realism/counterfactual definitiveness is the concept that all particles have some true value for all variables at some instant regardless of whether or not that value was observed. Let's assume for the sake of this question that locality is true, which I believe is the current popular model in QM.

Assuming locality, why do Bell's Inequalities show that conterfactual definiveness must be false?

Edit to respond to below post:

This is crazy. Let me see if I am following you. I'll use your framework.

I'm not really sure what spin is but I'm seeing it as an observable phenomenon that exists as a function of a specific direction in a given particle at an instant of time, and the spin itself at any angle changes (randomly?) over time.

A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and experimentally we would see that they are the same spin This is crazy. Let me see if I am following you. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and they will be the same spin 99% of the time.

Similarly, if we measure the spin at θ in A and at 2θ in B, we will see that these two values are the same spin 99% of the time. Assumingly, this will work for any two angles that are exactly θ apart -- with enough measurements they will be correlated exactly 99% of the time.

However, if we measure the spin at 0 in A and at 2θ in B, they are only correlated 96% of the time. If V0 is at most 1% off from V1, and V1 is at most 1% off from V2, V0 should be anywhere from 0-2% off from V2. But, we find, according to QM and experimentation, it is not! It is less heavily correlated than that.

This is where I lose you. Why does saying no to realism resolve this issue?

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1 Answer

Consider two electrons A and B which have perfectly anticorrelated spins, which means if A's spin is measured in any direction and B's spin is also measured in the same direction, the answer is always opposite.

In this answer, in everything that follows, I will replace "anticorrelated" with "correlated", so I will pretend that in whatever direction, A and B give the same spin answer. This is both theoretically and experimentally wrong, but it is pedagogically correct. You should fix up the minus signs afterwards.

Anyway, if you measure A and B in any direction, they are always the same. So A's spin and B's spin have the same results, so measuring the spin of either one is equally good for revealing what the other would have given you.

In quantum mechanics, if you measure the spin on A in one direction, and measure the spin of B in a direction at an angle $\theta$, the probability of getting the same answer goes as $|\cos(\theta)|^2$ (actually it's $|\cos(\theta/2)|^2$, again pedagogical simplification that doesn't change a thing). So for small angles, the answers to two nearby directions are correlated as $(1-C\theta^2)$ where C is some irrelevant constant.

This means if you measure A and B at two nearby directions tilted by angle $\theta$, and you adust the angle $\theta$ to be appropriately small, you find that the answer is correlated 99%. This means that the counterfactual results to A at zero angle and B at angle $\theta$ are 99% the same, since the actual results for A at zero angle and B at angle $\theta$ are 99% the same.

By rotational symmetry, the counterfactual results to A at angle $\theta$ and B at angle $2\theta$ are also 99% the same, since the relative angle is the same.

But the result of measuring A at angle 0 and B at angle $2\theta$ is $1- C(2\theta)^2 = 1- 4 C\theta^2$, which means the difference of the correlation from 1 is 4 times smaller at double the angle, which means they are only 96% the same.

Consider 3 students in a line, taking a yes/no test. Let's say they are cheating, so that the student in the middle is 99% correlated with the student on the left and on the right, so that out of 1000 questions, only 10 are different between the middle and left, and 10 are different between the middle and right. It follows that at most 20 are different between left and right. This is Bell's inequality, it implies that if something is 99% correlated with two different things, these two things are at least 98% correlated with each other.

But in quantum mechanics, the angle $\theta$ measurement is 99% correlated with each of the angle 0 and angle $2\theta$ results, but the angle 0 and angle $2\theta$ results are only 96% correlated with each other. This is inconsistent with Bell's inequality, so the electrons don't determine the answer ahead of time, counterfactual definiteness is lost, at least if they don't get to rewrite their answers nonlocally in response to what happened to the other.

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Why is it wrong to consider the perfectly correlated case instead the anti-correlated? It means you are preparing them in the state |00>+|11> instead of |01>-|10>, no? –  MBN Aug 21 '12 at 11:02
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@MBN: because that state doesn't stay correlated in any other basis, it doesn't keep the property of full correlation when you rotate by angle $\theta$. There is a theorem somewhere on this site that you can't make full correlation work, only anti-correlation, and it's clearly true, but I didn't work through the proof. –  Ron Maimon Aug 21 '12 at 15:59
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