Understanding Bell’s inequality

[Mentz114]

Study the Stern-Gerlach experiment and electron spin. I'll stop short of recommending a source, as I learned from Sakurai, which is brilliant but not really suitable unless you want to study QM in depth.

Note: Sakurai begins with Stern-Gerlach, but doesn't cover Bell's inequality until page 229! He could have done it sooner, of course, but it illustrates my point.

The notes I cite ( and link to ) in post #14 are based on the SG experiments which are brought in immediately.

 

[Stephen Tashi]

It seems to me the only important instruction is what to do when each of the pair encounter the same measurement angle. As there has to be a 100% correlation.

In the experiment usually discussed, the pair have exactly opposite spins when measured at the same angle so I'd call that a negative 100% correlation. Nevertheless, you can understand the basic idea by imagining an experiment where the pair must be in agreement. when measured at the same angle.

So I can imagine a case where the pair have exactly the same but independent instructions on how to react at the different angles. For example at 0 degrees, be 'UP' and 1 & 359 be 'UP and 2 and 358 degrees be 'DOWN' ...and so on for all the angles. Each pair of entangled particles produced may have a completely different set of instructions but the instructions are always the same for each member of the pair, hence they will always correlate when measured and they don't need to know what angle the other was measured at.

That's the general idea for hidden variables (as pointed out by @DrChinese in post #2). If we are inventing the instructions and we know the desired results, it's easy to come up with a population of instructions that give the desired answers when both members of pair are always measured at the same angle.

I know this is a silly example, as there would have to be some way for nature to produce the same permutations of instructions on average that match what we see when the pair are measured at different angles.

Yes, the tricky part is finding instructions that would agree with statistics taken when the two members of the pair are sometimes measured a different angles.

But what I was interested in is if Bell's Inequality covers this situation? In other words are the probabilities associated with the case I mentioned above the same as those mentioned in case 1 from your post?

Yes, Bell's inequality assumes the instructions are assigned according to case 1), where each member of the pair has instructions only for itself. Bell's inequality looks at a situation where the two members of the pair are sometimes measured at different angles - or, more abstractly, when there are different types of measurements. The inequality considers the case when there are 3 different types of measurements.