Are the limits to prediction by QM provable?

Question

Quantum mechanics states that only two aspects of a quantum system can be predicted with certainty: 1) the average and 2) the standard deviation of many measurements of identically prepared quantum systems. (At least, that's the message I took from my Quantum Chemistry textbook).

I believe Einstein did not like the fact that you couldn't predict more than those two things -- that there's a fundamental limit to what science can help us predict about events that do occur. Any other measured information from the system is essentially "random".

What I'm wondering is whether QM has definitively proven that we are unable to obtain any other information about the system?

For example, assume that with each quantum system there is associated a pseudo-random number generator that we don't know about. The "random" information is actually the result of the next iteration of the generator. All statistical tests would say that the information is random, but if we knew what function the PRNG was using, we could predict this data!

I remember reading that experiments have ruled out hidden variable theories. Is a PRNG function an example of a hidden variable?

(By the way, I'm not trying to challenge QM; I learn best by asking contradictory questions like this).

Answer 1

Your textbook is giving quantum mechanics too little credit--- QM predicts the entire probability distribution for any answer to any measurement, not just the mean and standard deviation. There are lots of probability distributions with the same mean and standard deviation, and there are fine probability distributions with infinite mean and infinite standard deviation.

Putting that aside, your randon number generator example is exactly what people mean by hidden variables, and for local hidden variables, these are ruled out by Bell's theorem. For nonlocal hidden variables, you can't rule them out because there is a theory, called Bohmian mechanics, which reproduces quantum mechanics exactly from these, but this theory is enormous and contrived.

The issue is debated ad-nauseum here, you can see the threads where 't Hooft has contributed for more debates where the issues concern the state of the art for hidden variables theories.