Understanding Bell’s inequality

[rede96]

I’m only an interested layman with no background in physics and just basic math. But I find a lot of physics fascinating and read up when I can.

One thing I’ve struggled with for ages is understanding just exactly what a violation of Bell’s theorem means when referring to the tests done on entangled spin states.

I understand that it rules out the assumption that a particle must objectively have a pre-existing value of spin. But I don’t understand why it rules out a particle having a set of properties which are set when created that just mean the outcome of its spin state will be random depending on other factors.

For example, I don’t imagine an electron to have a definite value of spin at any point in time. But what I can imagine is that it’s spin state coalesces when it comes into contact with the magnetic field when being measured based on certain properties of that electron. And those certain properties may also mean when an entangled pair is measured it will always be correlated when measured at the same angle.

So taking the above statement as an example, why does a violation of Bell’s theorem rule that out?

I respectfully ask people not to just quote the math. It’s not the math per se I struggle with. It’s how it applies that I’m not understanding.

Thanks.

 

[DrChinese]

One thing I’ve struggled with for ages is understanding just exactly what a violation of Bell’s theorem means when referring to the tests done on entangled spin states.

I understand that it rules out the assumption that a particle must objectively have a pre-existing value of spin. But I don’t understand why it rules out a particle having a set of properties which are set when created that just mean the outcome of its spin state will be random depending on other factors.

For example, I don’t imagine an electron to have a definite value of spin at any point in time. But what I can imagine is that it’s spin state coalesces when it comes into contact with the magnetic field when being measured based on certain properties of that electron. And those certain properties may also mean when an entangled pair is measured it will always be correlated when measured at the same angle.

So taking the above statement as an example, why does a violation of Bell’s theorem rule that out?

It's a bit hard to explain without referencing the math. After all, that's why there's an Inequality involved. But I think I can get some of the idea over.

You are probably aware that there are so-called "perfect correlations" when a pair of entangled particles are measured at the same angle, right? If Alice measures at 14 degrees, and Bob measures at 14 degrees, they always get the same/different results (depending on whether they have correlated or anti-correlated spins). For simplicity, let's say they are correlated and so are the same at any angle setting. Follow? I think this is precisely the case you are using.

Clearly, so you hypothesize the "spin state coalesces when it comes into contact with the magnetic field when being measured based on certain properties of that electron". Ok, that "coalescing" to either + or -, 1 or 0, H or V or whatever cannot in any way on a random factor associated with the "magnetic field" (your example). Do you see that? Because if it did, they wouldn't have perfect correlations! Sometimes they would be different due to the random factor of the measurement device/field. Let me know if I need to explain this further.

Therefore: the particles themselves must have the "secret sauce" so that when it encounters a similar measurement setting, it always gives a specific answer. So I will describe some elements of the secret sauce. It must be specific enough that as you rotate settings through 360 degrees, both particles give the same answer. Since even small deviations in the angle setting are discernible, let's say there is enough information in that particle for 360 different values. So that way they can disagree if the angle settings are NOT the same, but always agree if they ARE the same.

Particle A, 0 degrees=+, 1 degree=+, 2 degrees=-, ... 359 degrees=+. (or whatever values you want to assign, as long it is to both)
Particle B, 0 degrees=+, 1 degree=+, 2 degrees=-, ... 359 degrees=+.

Now you didn't want them to have pre-existing values like the above. But we are forced into this because no random factor can be introduced by the measurement device. Sure, it can have an impact. But you can just as easily roll that impact - which must be a fixed value when the angle settings are the same - into the chart above.

Do you see the problem here? We are back to where we started when you said: "I understand that it rules out the assumption that a particle must objectively have a pre-existing value of spin." If the measurement device alters the outcome, it must do it the same way for both and therefore need not be split out. You could have a chart like this:

Measurement Device A, 0 to 10 degrees=flip spin, 11 to 20 degrees=don't flip spin, ... (whatever you like)
Measurement Device B, 0 to 10 degrees=flip spin, 11 to 20 degrees=don't flip spin, ...

You can see that both have to operate the same to get perfect correlations! So you can completely throw that out as a factor, it can't explain anything.

 

[Zafa Pi]

One thing I’ve struggled with for ages is understanding just exactly what a violation of Bell’s theorem means when referring to the tests done on entangled spin states.

I agree with what @DrChinese had to say, but I am going to take a different tack on your question.

First off, no theorem is violated by anything done in physics, including Bell's theorem. Bell's theorem has two hypotheses, H1 and H2 and then proves an inequality, say, x<y. That inequality is then found to be violated by tests done on entangled entities. This implies that for those tests one or both of the hypotheses must be false.

Usually H1 is is the assumption that nothing, including information, moves faster than light. This is called locality.
H2 is a bit more subtle (and controversial) and may be what you mean when you say:

For example, I don’t imagine an electron to have a definite value of spin at any point in time. But what I can imagine is that it’s spin state coalesces when it comes into contact with the magnetic field when being measured based on certain properties of that electron. And those certain properties may also mean when an entangled pair is measured it will always be correlated when measured at the same angle.

However, I don't really understand what you're saying here, so I will state how I formulate H2.

If there are two different angles we may choose to measure the spin state, say, α and β, and we measured at angle α and got, say, +, then if we had measured at angle β instead we would have gotten something, either + or -, call it ◊.
This version of H2 is called counterfactual definiteness (CFD). What you were saying may be hidden variables which would work for H2.

Summing up, what a violation of Bell's inequality means is that either locality is false, or CFD is false, or perhaps both are false, or I am false.

 

[rede96]

Clearly, so you hypothesize the "spin state coalesces when it comes into contact with the magnetic field when being measured based on certain properties of that electron". Ok, that "coalescing" to either + or -, 1 or 0, H or V or whatever cannot in any way on a random factor associated with the "magnetic field" (your example). Do you see that? Because if it did, they wouldn't have perfect correlations! Sometimes they would be different due to the random factor of the measurement device/field. Let me know if I need to explain this further.

Now you didn't want them to have pre-existing values like the above. But we are forced into this because no random factor can be introduced by the measurement device. Sure, it can have an impact. But you can just as easily roll that impact - which must be a fixed value when the angle settings are the same - into the chart above.

Thanks for that, I really liked you explanation. And yes it make sense and the logic is easy to follow. So if I understand, what we are left with is that the outcome of spin state between the interaction of the electron and magnetic field can't be random AND the electron can't have a pre-existing value of spin. Yet a pair of entangled electrons always show a perfect correlation when measured at the same angle. And there is the confusion! How can the outcome of a measurement be simultaneously non random and non deterministic?

So as I understand it, in very layman's terms, to get around this we assume some sort of non local interaction between the electron pair where once the outcome of spin state is known for one of the pair, some sort of 'instantaneous communication' happens with the other electron to tell it what spin state it needs to be in.

But that leads to even more confusion for me. For example, at what point does this 'instantaneous communication' happen? If I intentionally measure one of the electron pair's spin state prior to the other, then the second electron has been 'told' what state it needs to be in prior to it being measured. But it's already been shown that an electron can't have a pre-existing value prior to it being measured.

Also what 'instantaneous communication' happens when two different angles are being used to measure an entangled pair? For example if one of the electron's is measured 'spin up' at 0 degrees, what is the expected spin state of the second electron which is being measured at 90 degrees? I'd assume that the information being communicated (again excuse the layman's terms as I know this isn't strictly correct) is 'at 0 degrees be spin up' as they have to be positively correlated (in this example) if measured at the same angle. But we know from experiments done that there is only a 50% probability that the second electron will be measured spin up at 90 degrees and different probabilities at different angles. So the outcome seems random. But again it's already been shown above that the outcome of an entangled pair being measured can't be random.

Now I have no idea if that logic holds or not, but can you see where I am getting so confused!

 

[DrChinese]

Thanks for that, I really liked you explanation. And yes it make sense and the logic is easy to follow. So if I understand, what we are left with is that the outcome of spin state between the interaction of the electron and magnetic field can't be random AND the electron can't have a pre-existing value of spin. Yet a pair of entangled electrons always show a perfect correlation when measured at the same angle. And there is the confusion! How can the outcome of a measurement be simultaneously non random and non deterministic?

So as I understand it, in very layman's terms, to get around this we assume some sort of non local interaction between the electron pair where once the outcome of spin state is known for one of the pair, some sort of 'instantaneous communication' happens with the other electron to tell it what spin state it needs to be in.

But that leads to even more confusion for me. For example, at what point does this 'instantaneous communication' happen? If I intentionally measure one of the electron pair's spin state prior to the other, then the second electron has been 'told' what state it needs to be in prior to it being measured. But it's already been shown that an electron can't have a pre-existing value prior to it being measured.

Also what 'instantaneous communication' happens when two different angles are being used to measure an entangled pair? For example if one of the electron's is measured 'spin up' at 0 degrees, what is the expected spin state of the second electron which is being measured at 90 degrees? I'd assume that the information being communicated (again excuse the layman's terms as I know this isn't strictly correct) is 'at 0 degrees be spin up' as they have to be positively correlated (in this example) if measured at the same angle. But we know from experiments done that there is only a 50% probability that the second electron will be measured spin up at 90 degrees and different probabilities at different angles. So the outcome seems random. But again it's already been shown above that the outcome of an entangled pair being measured can't be random.

Now I have no idea if that logic holds or not, but can you see where I am getting so confused!

No one can explain the mechanism other than via guessing. There are possibilities much as you describe. But there is no single interpretation that fits everything together neatly. The best I can describe is as follows, and take with a grain of salt:

QM predicts correctly the statistical outcome for a specified context. The context includes both the source and the subsequent measurement apparati (which may or may not be local to each other). Outcomes appear completely random when properties are in superposition within the context.

 

[microsansfil]

hi all,

Just for information. David Mermin described a very simple device that allows everyone to realize the enigma we are faced with EPR experiments : A gedanken demonstration http://www.theory.caltech.edu/classes/ph125a/istmt.pdf

Best regards
Patrick
1/
http://inspirehep.net/record/31657/files/vol1p195-200_001.pdf
The basic inequality (the original) is
$$1 + \operatorname{\rho}(B, C) \geq |\operatorname{\rho}(A, B) - \operatorname{\rho}(A, C)|$$
A, B and C: three measured attributes, rho being the "correlation".
2/
Alain Aspect BELL’S THEOREM : THE NAIVE VIEW OF AN EXPERIMENTALIST : https://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf

 

[rede96]

No one can explain the mechanism other than via guessing. There are possibilities much as you describe. But there is no single interpretation that fits everything together neatly. The best I can describe is as follows, and take with a grain of salt:

QM predicts correctly the statistical outcome for a specified context. The context includes both the source and the subsequent measurement apparati (which may or may not be local to each other). Outcomes appear completely random when properties are in superposition within the context.

Ok, thanks for the help. Just one last question if I may. So let's say I was to make an assumption that although an electron (or any particle) doesn't have a pre-exiting value of spin but instead has some intrinsic random property that determines how it's spin state will react when in encounters a magnetic field of a certain property (for example). Is there anything that prohibits that?

 

[Nugatory]

Ok, thanks for the help. Just one last question if I may. So let's say I was to make an assumption that although an electron (or any particle) doesn't have a pre-exiting value of spin but instead has some intrinsic random property that determines how it's spin state will react when in encounters a magnetic field of a certain property (for example). Is there anything that prohibits that?

If that intrinsic property is enough to determine without reference to what's happening with the other particle in the pair how the spin state will react... then the resulting measurements will not display perfect anticorrelation and/or must obey Bell's inequality. So yes, that possibility doesn't work.

 

[rede96]

If that intrinsic property is enough to determine without reference to what's happening with the other particle in the pair how the spin state will react... then the resulting measurements will not display perfect anticorrelation and/or must obey Bell's inequality. So yes, that possibility doesn't work.

Thanks for the reply. I’m not sure I fully understand why that’s the case but take it on face value of course.

I guess what I’d really like to understand is just what the math in bells theorem specifically applies to?

I sort of understand, at least in part, that it is testing the assumption that particles have a real value of spin at any point in time, even if we don’t know it. But does it only apply to systems that have binary outcomes for example?

So I’m just struggling to understand how it should be interpreted when thinking about other measurable values / attributes of a system? Is it saying nothing has a real value until measured?

 

[Stephen Tashi]

An explanation of Bell's inequality without reference to quantum mechanics is given in http://quantumuniverse.eu/Tom/Bells Inequalities.pdf

That explanation at least shows that data about measured attributes such as "Owns own home" should obey Bell's inequality. So if there is non-fudged data statistical data for some attributes that violates Bell's inequality then this is evidence that measuring those attributes has a different behavior that measuring the attributes we define in the macroscopic world.

 

[DrChinese]

I sort of understand, at least in part, that it is testing the assumption that particles have a real value of spin at any point in time, even if we don’t know it. But does it only apply to systems that have binary outcomes for example?

No, there is no restriction to spin or other observables with binary outcomes. There are many many forms of entanglement that have been studied and documented. As a rule, some form of a Bell Inequality is used as a yardstick to verify entanglement is present.

 

[Lord Jestocost]

QM predicts correctly the statistical outcome for a specified context...

Thanks for this trenchant statement. To my mind, there is nothing more to say.

 

[rede96]

An explanation of Bell's inequality without reference to quantum mechanics is given in http://quantumuniverse.eu/Tom/Bells Inequalities.pdf

Thanks for the link, although I'm still not sure I fully understand how the math applies to particle spin (See below).

No, there is no restriction to spin or other observables with binary outcomes.

I think this may be where I am getting confused, I didn't understand spin to be a preexisting binary property of a particle.

If I take two playing cards for example, the ace of spades and ace of hearts as my system. The system has two preexisting values. If I select the ace of spades It will always be measured as the ace spades, regardless of who or when it is selected. The measurement process has no bearing on the outcome of the card selected. And in that regard I can understand how Bell's theorem applies. But even without doing a bell test we know particle spin is different.

I can take my selected card, I can turn it over, measure some other properties... but I will always measure it as the ace of spades. But if I take a particle and measure it's spin in the 0 degree angle and get spin UP, then measure it's spin at 90 degrees, then measure it's spin again at 0 degrees, there is only a 50/50 probability I will get spin UP. (EDIT: Depending on how the experiment is set up.)

So how do the same statistics apply to a situation where the measurement has a bearing on the outcome? Because what ever it's 'natural' state was, the measurement must have affected it.

And if I follow that logic (albeit at the risk of being completely wrong!) then in my mind I see the correlations being a factor of spin and measurement, which isn't a binary property of the particle. If that makes sense?

 

[Mentz114]

[]

So how do the same statistics apply to a situation where the measurement has a bearing on the outcome? Because what ever it's 'natural' state was, the measurement must have affected it.

And if I follow that logic (albeit at the risk of being completely wrong!) then in my mind I see the correlations being a factor of spin and measurement, which isn't a binary property of the particle. If that makes sense?

You are right. Spin and polarization measurements project the object into a certain state depending on the alignments of the apparatus. For fermionic quantum spin one can prepare a thermal state where the spin vector(s) can be pointing any which way. A magnetic field aligns the vectors into up or down. in a certain direction. Now the spin in that direction is binary. In the SG case the up and down beams separate spatially.

If we measure a $Z\uparrow$ sample in the $X$-direction we are equally likely to get $X\uparrow$ or $X\downarrow$. This is very basic and usually described in textbooks or lecture notes.

I highly recommend these notes "Spin and Quantum Measurement"

 

[DrChinese]

...So how do the same statistics apply to a situation where the measurement has a bearing on the outcome? Because what ever it's 'natural' state was, the measurement must have affected it.

And if I follow that logic (albeit at the risk of being completely wrong!) then in my mind I see the correlations being a factor of spin and measurement, which isn't a binary property of the particle. If that makes sense?

The contradiction is as follows: let's assume that the choice of measurement IS a factor. Yet we get perfect correlations when the same measurement basis is applied to widely separated, spin entangled particles. So you should be able to essentially eliminate that as a factor, again implying that it WAS a purely preexisting condition. (Of course Bell throws a wrinkle in that.)

Of course you can't really eliminate it, but in the quantum formalism ALL factors must be part of the complete measurement context. So when we think of one card existing separate from the other card - we have already made a questionable assumption.

 

[DrChinese]

I think this may be where I am getting confused, I didn't understand spin to be a preexisting binary property of a particle.

When I see the word "binary" in this context, I contrast it with "continuous" measurement values such as velocity, energy, position, etc.

And calling it pre-existing varies with the setup. If you start with particles that have known spin, a subsequent measurement of that observable should give the same result. But if it is in a superposition, it's hard to call it pre-existing in any normal sense.

 

[PeterDonis]

When I see the word "binary" in this context, I contrast it with "continuous" measurement values such as velocity, energy, position, etc.

And calling it pre-existing varies with the setup.

There is another point which might be worth mentioning in this connection. Spin-1/2 measurements are binary because there are only two possible results. But spin-1/2 states can vary continuously, in the sense that for any spin state there is a definite direction in space for which a spin measurement of that state will always give an "up" result, and directions in space can vary continuously. A common term for this description of spin-1/2 states is called the Bloch sphere:

https://en.wikipedia.org/wiki/Bloch_sphere

So for a single spin-1/2 particle, one can indeed view its spin as "pre-existing" since it corresponds to a definite direction in space. However, this model breaks down as soon as we have two entangled spin-1/2 particles.

 

[rede96]

I highly recommend these notes "Spin and Quantum Measurement"

Thanks for the recommendation. I had a quick look, there seems enough there to keep me quiet for a while! :-)

 

[jk22]

What if we considered a change of variables : it is known that classically

$$2/\pi*acos(\cos(a-b))=\int A(a,x)A(b,x)dx$$ with $$A(a,x)=sign(a-x)$$

If we want to get the cos function of quantum mechanics through $$a-b=f(a'-b')$$ then

$$2/\pi*acos(cos(f(a'-b')))=cos(a'-b')\Rightarrow f(a'-b')=acos(cos(\pi/2*cos(a'-b')))$$

then comes a trial : it is supposed that the transformation of the angles acts on both separately hence : $$A(a,x)=A(f(a'),x)=A'(a',x)$$

this can be tried numerically and gives the perfect correlations $$cov(0,b)=cos(b)$$ but the covariance is no more depending on the relative angle $$a'-b'$$ and the Chsh value is $$\leq 2$$ as it shall.

Hence the problem here is not the perfect correlation or cos shaped curve but merely the relative dependency.

 

[rede96]

The contradiction is as follows: let's assume that the choice of measurement IS a factor. Yet we get perfect correlations when the same measurement basis is applied to widely separated, spin entangled particles. So you should be able to essentially eliminate that as a factor, again implying that it WAS a purely preexisting condition. (Of course Bell throws a w

True, but can't I also assume correlations are at least in part affected by or even a direct result of the choice of measurement. And if I assume this, then as the measurement angles I select are made at random, then for any pair of entangled particles I create and intend to measure, I have to assume they have no current spin state until the measurement angle is selected. Which seems silly but if we have to consider all factors then not until all factors have a value can there be a spin state.

Of course you can't really eliminate it, but in the quantum formalism ALL factors must be part of the complete measurement context. So when we think of one card existing separate from the other card - we have already made a questionable assumption.

But isn't that only if we assume that the outcome of card A can be somehow affected by the second card B? If we assume it doesn't then can't I think of Card A existing separate from card B?

 

[jk22]

What about if we consider the other question : Is quantum mechanics nonlocal ?

Nonlocal would mean that only global properties can be measured, for example we have in the Eprb covariance $$p(+-)=p(-+)$$

but In quantum mechanics could we find states which give a cosine covariance but for which $$p(+-)\neq p(-+)$$
The global quantity AB were here decomposed into distinguishable sides.

Hence where nonlocality in fact only a non valid generalization of a peculiar case ?

 

[PeroK]

I have to assume they have no current spin state until the measurement angle is selected

This is the problem with analysing Bell's inequality without understanding QM! The system has well-defined state. But, that state does not imply specific measurement results for all possible measurements.

For example, if you consider a single unentangled electron. You may know its state. Let's say that is spin-up in the z-direction. The result of a measurement of spin in the z-direction will definitely return spin-up.

But, a measurement of spin in the x or y directions will return spin-up or spin-down, each with 50% probability.

This begs the question: when the electron is z-spin up, does it also have a definite spin in the x and y directions that we simply don't know (hidden variables)? Or, does only a measurement of spin in the x or y directions "cause" the electron to take a definite up or down spin in the measured direction.

On the face of it, there was no way to tell. Bell's inequality was an ingenious way to exploit the difference between normal probability addition (that you would get with hidden variables) and QM probability addition (that is based on the squares of probability amplitudes) to settle the issue.

The details of how Bell's inequality actually does this are not so easy to explain.

But isn't that only if we assume that the outcome of card A can be somehow affected by the second card B? If we assume it doesn't then can't I think of Card A existing separate from card B?

Elementary particles are not like playing cards. Playing cards are distinguishable; electrons, say, are not. So, if you have two electrons in an experiment, you cannot label one electron A and one electron B. You have a system of two electrons. This affects the way probabilities and statistics work when dealing with identical particles.

In particular, there is no electron A and electron B.

Again, you need a better grounding in QM to analyse these questions.

 

[DrChinese]

True, but can't I also assume correlations are at least in part affected by or even a direct result of the choice of measurement. And if I assume this, then as the measurement angles I select are made at random, then for any pair of entangled particles I create and intend to measure, I have to assume they have no current spin state until the measurement angle is selected. Which seems silly but if we have to consider all factors then not until all factors have a value can there be a spin state.

But isn't that only if we assume that the outcome of card A can be somehow affected by the second card B? If we assume it doesn't then can't I think of Card A existing separate from card B?

No argument with me really on these points. You need the entire measurement context when making predictions about an entangled system, and the component parts cannot be considered as separate. How or why that is necessary, I don't know.

 

[rede96]

No argument with me really on these points. You need the entire measurement context when making predictions about an entangled system, and the component parts cannot be considered as separate. How or why that is necessary, I don't know.

I guess that's what I'm trying to figure out. It's a fascinating subject, just not easy for a novice to get his head around! Still thanks very much for your help.

 

[rede96]

The details of how Bell's inequality actually does this are not so easy to explain.

And this seems to be where I'm stuck. I'd really like to understand how Bell's inequality leads to the conclusions it does. But for the life in me I just can't grasp it. I mean if the Bell test simply disproved that entangled particles can't have a pre-exiting component of spin that leads to correlations that are measured then I'd get that. But it seems to be much deeper than that, as I understand it the conclusions are that entangled particles can't have ANY pre-exiting property that would yield the correlations measured.

Also, when I first started reading up on this I imagined the entangled pair to have some 'random' like variables which when they came into contact with the measuring device (in the case of the electrons the magnetic field) would then arrange themselves to yield the direction of spin that was detected. And I thought correlations were simply due to the fact that the entangled pair were exact copies of each other. Or at least the variables that influenced the spin outcome were. But it seems that's not allowed either.

I'm led to believe from previous attempts to understand this that I shouldn't worry over whatever process leads to the result as it's only the result that matters. But I just don't get it! :-(

 

[PeroK]

@rede96 Bell's experiment does not appeal to any previously unknown aspect of QM calculations. So, if you understand QM, Bell has nothing to add. What Bell shows is that those calculations in this cleverly devised scenario are not compatible with hidden variables, which would imply traditional probabilities for those variables.

Those details of the probabilistic calculations are difficult to explain unless you can do QM calculations. A very crude analogy is that Bell arranged the experiment like a triangle where QM calculations say that the squares of the sides add up, whereas hidden variables say that sides add up. The experiment produces a 3-4-5 triangle. Hidden variables would require 3+4 =5 which shows the triangle, hence the experimental result, is not compatible with hidden variables.

So, there is nothing deep about the maths or the calculations, per se. Just very cleverly thought out.

What you are struggling with is how nature may have properties that are not knowable until measured. This applies to all particles, entangled or not. Bell's inequality proves that this is the case, but there is nothing in the experiment that shows why this is the case.

If you did understand the details of Bells inequality that might convince you that hidden variables are not possible, but it would take you little further in understanding how and why nature is like that.

There are simpler examples than Bell - for example, just study spin measurements on a single electron - that suggest strongly that there are no hidden variables. That's why many were convinced before Bells inequality settled the issue.

 

[Stephen Tashi]

Also, when I first started reading up on this I imagined the entangled pair to have some 'random' like variables which when they came into contact with the measuring device (in the case of the electrons the magnetic field) would then arrange themselves to yield the direction of spin that was detected. And I thought correlations were simply due to the fact that the entangled pair were exact copies of each other. Or at least the variables that influenced the spin outcome were. But it seems that's not allowed either.

We can imagine two different general situations

1) Each thing in an entangled pair has instructions that tell it what to do when it encounters various measuring devices. These instructions can be deterministic or they can include probabilistic actions like "For a measuring device of type 2, produce reading..such-and-such... with probability 0.6" etc. The two things in the pair may have the same set of instructions or different sets of instructions, but whatever instructions they have only apply to them individually, not to their partner.

2) Each thing in an entangled pair has the same set of instructions and the instructions specify what it and its partner must do when they encounter various measuring devices. For example, "If object A encounters a measuring device of type 2 and its partner object B encounters a measuring device of type 1 then A must do ...such-and-such and B must do ...so-and-do..."..

In either case 1) or case 2) we can imagine that entangled pairs are being produced so that the instructions vary from pair-to-pair, perhaps in a random manner. 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.The investigation of either case would involve analyzing statistics from entangled pairs being measured by various measuring devices.

It is difficult to imagine how two inanimate objects separated at great distance could coordinate their actions to obey instructions in case 2). Each object would have to know what measurement device the other encountered. It seems natural to formulate a physical theory based on case 1).

The statistics taken from pairs whose instructions are given by case 1) obey Bell's inequality. The instructions given in case 2) can produce a greater variety of statistical results. These need not obey Bell's inequality. Since some experiments produce statistics that violate Bell's inequality, we conclude that physical theories based on case 1) won't work for those phenomena.

I'd say that understanding Bell's inequality as mathematics involves understanding the limitations of statistics that can be produced by case 1) - and it requires knowing some probablity theory.

Understanding the details of a particular experiment that produces statistics that violate Bell's inequality is a topic in physics.

 

[Mentz114]

I guess that's what I'm trying to figure out. It's a fascinating subject, just not easy for a novice to get his head around! Still thanks very much for your help.

Some details can be explained via a model. This might help to see what might be happening and how the data is organised to do the calculation.

 

[rede96]

What you are struggling with is how nature may have properties that are not knowable until measured. This applies to all particles, entangled or not. Bell's inequality proves that this is the case, but there is nothing in the experiment that shows why this is the case.

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?

 

[rede96]

Some details can be explained via a model. This might help to see what might be happening and how the data is organised to do the calculation.

Thanks for the info but I'm afraid I couldn't really follow it. Sorry.

 

[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.