I feel like I'm arguing a two-front war here. On the one hand, I don't think that it's impossible to have a superdeterministic explanation for QM statistics. On the other hand, I think that such a theory would be very bizarre, and nothing like any theory we've seen so far.
Let me go through a stylized description of an EPR-like experiment so that we can see where the superdeterminism loophole comes in.
We have a game with three players, Alice, Bob and Charlie. Alice and Bob are in different rooms, and can't communicate. In each room, there are three light bulbs colored Red, Yellow and Blue, which can be turned off or on.
The game consists of many many rounds, where each round has the following steps:
- Initially, all the lights are off.
- Charlie creates a pair of messages, one to be sent to Alice and one to be sent to Bob.
- After Charlie creates his messages, but before they arrive, Alice and Bob each choose a color, Red, Yellow or Blue. They can use whatever criterion they like for choosing their respective colors.
- When Alice receives her message, she follows the instructions to decide whether to turn on her chosen light, or not. Bob similarly follows his instructions.
After playing the game for many, many rounds, the statistics are:
- When Alice and Bob choose the same color, they always do the opposite: If Alice's light is turned on, Bob's is turned off, and vice-versa.
- When Alice and Bob choose different colors, they do the same thing 3/4 of the time, and do the opposite thing 1/4 of the time.
The question is: What instructions could Charlie have given to Alice and Bob to achieve these results? The answer, proved by Bell's theorem, is that there is no way to guarantee those results, regardless of how clever Charlie is, provided that
- Charlie doesn't know ahead of time what colors Alice and Bob will choose.
- Alice has no way of knowing what's going on in Bob's room, and vice-versa.
The superdeterministic loophole
If Charlie does know what choices Alice and Bob will make, then it's easy for him to achieve the desired statistics:
- Every round, he randomly (50/50 chance) sends either the message to Alice: "turn your light on", or "turn your light off"
- If Alice and Bob are predestined to choose the same color, then Charlie sends the opposite message to Bob.
- If Alice and Bob are predestined to choose different colors, then Charlie will send Bob the same message 3/4 of the time, and the opposite message 1/4 of the time.
Why the superdeterministic loophole is implausible
The reason that the superdeterministic loophole is not possible is because Alice and Bob can choose any mechanism they like to help them decide what color to use. Alice might look up at the sky, and choose the color based on how many shooting stars she sees. Bob might listen to the radio and make his decision based on the scores of the soccer game. For Charlie to be able to predict what Alice and Bob will choose can potentially involving everything that can possibly happen to Alice and Bob during the course of a round of the game. The amount of information that Charlie would have to take into account would be truly astronomical. The processing power would be comparable to the power required to accurately simulate the entire universe.
Why I think the superdeterministic loophole is actually impossible
What the superdeterministic loophole amounts to is that somehow Charlie has information about the initial state (before the game began) of the universe, s_0, and somehow he has a pair of algorithms, \alpha(s_0) and \beta(s_0) that predict the choice of Alice and Bob as a function of the initial state. The problem is that even if there were such algorithms, they computational time for computing the result would be greater than just waiting to see what Alice and Bob choose. So Charlie couldn't possibly know the results in time to choose his instructions to take those results into account.
Why not? Remember, we're allowing Alice and Bob to use whatever mechanism they like to decide what color to pick. So suppose Alice picks the same algorithm, \alpha, and chooses whatever color is NOT returned by \alpha(s_0)? In other words, she runs the program, and if it returns "Red", she picks "Yellow". If it returns "Yellow", she picks "Blue". If it returns "Blue", she picks "Red". She can base her choice on anything, so if there is a computer program \alpha that she can run, then she can base her choice on whatever it returns.
The only way for it to be possible that \alpha(s_0) always gives the right answer for Alice is if it takes so long to run that Alice gives up and makes her choice before the program finishes.
This is actually a fairly standard argument that even if the universe is deterministic, if you tried to construct a computer program that is guaranteed to correctly predict the future, the future would typically arrive before the computer program finished its calculations. No matter how clever the algorithm, no matter how fast the processor, there is no way to guarantee that the prediction algorithm would be faster than just waiting to see what happens.