Understanding Bell’s inequality

[rede96]

The important distinction between case 1) and case 2) is whether the instructions given one member of the pair apply only to itself or whether the instructions specify what it and its partner must do.

Thanks for your post, that helped me to think about it in that way. I can see the distinction between to the two cases however I don't understand how the probabilities work. 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.

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.

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. 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?

 

[PeroK]

It isn't so much that nature may have properties that aren't knowable until measured but more it seems to suggest that those properties don't exist until measured?

Good point. "Exists" is a dangerous word, I think. We could compromise on "don't have defined values" until measured.

Hidden variables requires that they do have defined values and - potentially at least - could be worked out indirectly or by some new measurement process.

For example, an electron's spin state tells you everything about the electron's spin in whatever direction you measure it. Spin in the x-y-z directions and spin about any intermediate axis.

But, what it tells you in general is that the spin about a given axis does not have a well-defined value - until you measure it. It tells you that if you measure the spin about a given axis you will get spin-up and spin-down with well-defined probabilities.

Bell's inequality tells you, in effect, that there is no way to know any more than these probabilties. In other words, it's not a deficiency in what you can calculate using QM. Quite the reverse, you need the QM calculations to get the observed results.

 

[PeroK]

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. 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?

You've still not grasped the central point. If there are well-defined instructions, distributed across a sample of particles with certain probabilities, then those instructions must behave according to classical probability theory. It doesn't matter what the instructions are.

Bell's inequality tests whether the results of an experiment obey classical probability theory. They do not.

Whereas, the results of the experiment are compatible with QM probability calculations, based on probability amplitudes.

What Bell did was to find an experiment where the results of classical probability and quantum probability calculations differed.

I still think you are barking up the wrong tree. It's not about specific sets of instructions, it's about how classical probability theory does not explain the quantum phenomena.

 

[rede96]

I still think you are barking up the wrong tree. It's not about specific sets of instructions, it's about how classical probability theory does not explain the quantum phenomena.

I think you are right. For now I think the best way forward is just to accept...

If there are well-defined instructions, distributed across a sample of particles with certain probabilities, then those instructions must behave according to classical probability theory. It doesn't matter what the instructions are.

...and start to read up more on probability theories as well as some QM basics. I'd like to understand how to come to the conclusion above, but I don't think there is going to be a short cut!

Thanks for your help.

 

[PeroK]

I think you are right. For now I think the best way forward is just to accept...
...and start to read up more on probability theories as well as some QM basics. I'd like to understand how to come to the conclusion above, but I don't think there is going to be a short cut!

Thanks for your help.

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.

 

[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.