[ang_cire]

So to make sure I am understanding correctly, the normal distribution of the outcomes is itself evidence that other hidden factors aren't at play, because those factors would produces a less normal distribution?

I.e. if coin toss results skew towards heads, you can conclude some factor is biasing it that way, therefore if the results are (over the course of many tests) 'even', you can conclude the absence of biasing factors?

[Workaccount2]

Basically they get to measure a super position particle twice, by using an entangled pair of it. So two detectors that each measure one of the particle's 3 possible spin directions, which are known to be identical (but usually you only get to make 1 measurement, so now we can essentially measure 2 directions). We then compare how the different spin directions agree or disagree with each other in a chart.

15% of the time they get combination result A, 15% of the time they get combination result B. Logically we would expect a result of A or B 30% of the time, and combination result C 70% of the time (There are only 3 combinatorial output possibilities - A,B,C)

But when we set the detectors to rule out result C (so they must be either A or B), we get a result of 50%.

So it seems like the particle is able to change it's result based on how you deduce it. A local hidden variable almost certainly would be static regardless of how you determine it.

This is simplified and dumbified because I am no expert, but that is the gist of it.

[nathan_compton]

Not really. The shape of the distribution of whatever random numbers you are getting is just a result of the physical situation and nothing to do with the question posed by Bell.

Let me take a crack at this. Quantum Mechanics like this: we write down an expression for the energy of a system using position and momentum (the precise nature of what constitutes a momentum is a little abstract, but the physics 101 intuition of "something that characterizes how a position is changing" is ok). From this definition we develop both a way of describing a wave function and time-evolving this object. The wave function encodes everything we could learn about the physical system if we were to make a measurement and thus is necessarily associated with a probability distribution from which the universe appears to sample when we make a measurement.

It is totally reasonable to ask the question "maybe that probability distribution indicates that we don't know everything about the system in question and thus, were that the case, and we had the extra theory and extra information we could predict the outcome of measurements, not just their distribution."

Totally reasonable idea. But quantum mechanics has certain features that are surprising if we assume that is true (that there are the so-called hidden variables). In quantum mechanical systems (and in reality) when we make a measurement all subsequent measurements of the system agree with the initial measurement (this is wave function collapse - before measurement we do not know what the outcome will be, but after measurement the wave function just indicates one state, which subsequent measurements necessarily produce). However, measurements are local (they happen at one point in spacetime) but in quantum mechanics this update of the wave function from the pre to post measurement state happens all at once for the entire quantum mechanical system, no matter its physical extent.

In the Bell experiment we contrive to produce a system which is extended in space (two particles separated by a large distance) but for which the results of measurement on the two particles will be correlated. So if Alice measures spin up, then the theory predicts (and we see), that Bob will measure spin down.

The question is: if Alice measures spin up at 10am on earth and then Bob measures his particle at 10:01 am earth time on Pluto, do they still get results that agree, even though the wave function would have to collapse faster than the speed of light to get there to make the two measurements agree (since it takes much longer than 1 minute for light to travel to Pluto from earth).

This turns out to be a measureable fact of reality: Alice and Bob always get concordant measurement no matter when the measurement occurs or who does it first (in fact, because of special relativity, there really appears to be no state of affairs whatever about who measures first in this situation - it depends on how fast you are moving when you measure who measures first).

Ok, so we love special relativity and we want to "fix" this problem. We wish to eliminate the idea that the wave function collapse happens faster than the speed of light (indeed, we'd actually just like to have an account of reality where the wave function collapse can be totally dispensed with, because of the issue above) so we instead imagine that when particle B goes flying off to Pluto and A goes flying off to earth for measurement they each carry a little bit of hidden information to the effect of "when you are measured, give this result."

That is to say that we want to resolve the measurement problem by eliminating the measurement's causal role and just pre-determine locally which result will occur for both particles.

This would work for a simple classical system like a coin. Imagine I am on mars and I flip a coin, then neatly cut the coin in half along its thin edge. I mail one side to earth and the other to Pluto. Whether Bob or Alice opens their envelope first and in fact, no matter when they do, the if Alice gets the heads side, Bob will get the tails side.

This simple case fails to capture the quantum mechanical system because Alice and Bob have a choice of not just when to measure, but how (which orientation to use on their detector). So here is the rub: the correlation between Alice and Bob's measurement depends on the relative orientation of their detectors and even though both detectors measure a random result, that correlation is correct even if Alice and Bob, for example, just randomly choose orientations for their measurements, which means Quantum Mechanics describes the system correctly even when the measurement would have had to be totally determined for all possible pairs of measurements ahead of time at the point the particles were separated.

Assuming that Alice and Bob are actually free to choose a random measuring orientation, there is no way to pre-decide the results of all pairs of measurements ahead of time without knowing at the time the particles are created which way Alice and Bob will orient their detectors. That shows up in the Bell Inequality, which basically shows that certain correlations are impossible in a purely classical universe between Alice and Bob's detectors.

Note that in any given single experiment, both Alice and Bob's results are totally random - QM only governs the correlation between the measurements, so neither Alice nor Bob can communicate any information to eachother.