What are the Implications of Bell's Theorem?
Consider the following game show: two friends <PERSON> and <PERSON> (X and Y) are selected from an audience to compete for a grand prize, a brand new Ferrari.
The game description:
1. The two contestants are space-like separated.
2. Each contestant will be asked one of three questions {A, B, C} and the questions which are asked of the two contestants need not be the same. Each of these questions has two possible answers, which we will quantify as {+1, -1}.
3. Repeat 2 a large number of times.
4. When the two contestants are asked the same question, they must always give the same answer.
5. When the two contestants are asked different questions, they must agree $ \frac 14 $ of the time.
Now, how can <PERSON> and <PERSON> win this game?
It's simple: They create a strategy, whereby they pre decide on a set of answers to give to the three questions {A, B, C}. This will guarantee that they always give the same answer when they are asked the same question. 603
For example, they may decide on {+1, +1, -1} to {A, B, C}. Let us denote these answers as $v_i(\alpha) = \pm1$ for $\mathcal i$ = X, Y, $\alpha$ = A, B, C.
This will not allow <PERSON> and <PERSON> to satisfy 4.
$\mathscr Theorem:$
There do not exist random variables $v_i(\alpha), \mathcal i$ = X, Y, $\alpha = A, B, C$ such that:
$$ 1. v_i(\alpha) = \pm1 $$ $$ 2. v_X(\alpha) = v_Y(\alpha)\forall \alpha $$ $$ 3. Pr(v_X(\alpha) = v_Y(\beta)) = \frac 14 \forall \alpha, \beta, \alpha \neq \beta $$
$\mathscr Proof:$
Assume for contradiction that there do exist random variables $v_i(\alpha), \mathcal i$ = X, Y, $\alpha = A, B, C$ such that (1-3 hold).
Since $ v_i(\alpha)$ can only take on the two values $\pm1$, we must have $Pr(v_X(A) = v_X(B)) + Pr(v_X(A) = v_X(C)) + Pr(v_X(B) = v_X(C)) \geq 1 $
By condition 2, we then have $Pr(v_X(A) = v_Y(B)) + Pr(v_X(A) = v_Y(C)) + Pr(v_X(B) = v_Y(C)) \geq 1$
Now, by condition 3, we have $ \frac 14 + \frac 14 + \frac 14 \geq 1$ a contradiction.
But, if you look at the predictions of quantum mechanics, it is possible to satisfy (1-3). Experiments have validated quantum mechanics, thus the correlations achieved cannot be due to a pre existing strategy. Then we must wonder how could <PERSON> and <PERSON> always assure that property (2) holds, if they can not simply pre decide on a set of answers to give to the three questions {A, B, C}. It must be that when <PERSON> and <PERSON> are asked a question, they communicate with each other what question is being asked, and agree to an answer, for if not, one would have $Pr(v_X(\alpha) = v_Y(\alpha)) = \frac 12$
$\mathscr Bell's\; Theorem\; Implication:$ Quantum mechanical correlations are not due to pre existing properties $\Rightarrow$ there must exist an exchange of information between entangled subsystems about what properties are being measured on them, and what values of these properties are to be taken on. Combining this with rule (1) of the game implies that this must take place faster than light, recent experiments overwhelmingly suggest instantaneously.
My question is why is this salient point so muddled in the literature? <PERSON>'s theorem is often stated as follows.
$\mathscr No\; theory\; of\; local\; hidden\; variables\; can\; produce\; the\; predictions\; of\; quantum\; mechanics$
That's missing the point. 262