A question about Bell's Inequality and hidden variables

[rede96]

Where do you get those numbers?
My calculator gives:
cos^2(50 deg.)=0.413
cos^2(190 deg.)=0.970
cos^2(140 deg.)=0.587

Opps! Sorry, was in a rush as I'm at work and didn't bracket my formula correctly in excel.

You take relative angles. You don't take each angle individually.

Ok great, thanks. So for electrons which are anti correlated then do I use cos^2((angleA - AngleB)/2) or is it cos^2((angleA/2)-(angleB/2))

 

[zonde]

Ok great, thanks. So for electrons which are anti correlated then do I use cos^2((angleA - AngleB)/2) or is it cos^2((angleA/2)-(angleB/2))

Well, the two formulas are the same.
But it's sine not cosine: sin^2((angleA - angleB)/2) or alternatively 1-cos^2((angleA - angleB)/2) (it's the same). Try to calculate sin^2(0), it's 0 as it should be for anticorrelation. And sin^2(180/2)=1.

 

[rede96]

But it's sine not cosine: sin^2((angleA - angleB)/2) or alternatively 1-cos^2((angleA - angleB)/2) (it's the same). Try to calculate sin^2(0), it's 0 as it should be for anticorrelation. And sin^2(180/2)=1.

Great, thanks very much for your help.

Could I just confirm one other thing. When I compare the total number of matches for a number of tests, I only comparable them against the tests where the angles were not the same. E.g if I do 90 tests on all permutations of the three angles, I'll have 30 on average where the angle A = Angle B. So if I got 15 matches in total, I'm comparing that figure as a percentage against 60 tests. (0.25) and not the full 90. Is that correct?

 

[zonde]

Could I just confirm one other thing. When I compare the total number of matches for a number of tests, I only comparable them against the tests where the angles were not the same. E.g if I do 90 tests on all permutations of the three angles, I'll have 30 on average where the angle A = Angle B. So if I got 15 matches in total, I'm comparing that figure as a percentage against 60 tests. (0.25) and not the full 90. Is that correct?

It seems correct.

 

[rede96]

It seems correct.

Thank you.

 

[DrChinese]

How to calculate the percentage match that would be observed in experiments.

As I understood from your post #19, to calculate the % match for anti-correlated particles, for each angle it would be Match%= 1 - (cos^2(theta/2))

So for angles 0, 120 and 240 that would be:

0 is 1 - (cos^2(0 degrees/2)) = 0
120 is 1 - (cos^2(120 degrees/2) = 0.75
240 is 1 - (cos^2(240 degrees/2)) = 0.75

So the average of those is 0.5 So how does the total match prediction = 0.75? With photons it was (cos^2(angle A - angle B)) So is that not the case here?

This is the one bit I'd really like to understand. How to calculate the total match percentage for any set of 3 angles for both correlated and anti-correlated particles.

Also I am assuming that the total match prediction is a percentage of the total number of tests - any tests where the angles where the same.

As zonde already mentioned, and I think you now understand: Theta is the relative angular difference between 2 angles. 1 - (cos^2(Theta/2)) is the same as sin^2(Theta/2)).

The reason I use theta angles that are the same is because it generates an average that is the same as the individual components. That is not really a requirement, just makes everything pretty simple and easy to see. Again, what I am recommending is that you write down a series of hand generated outcomes (for 3 different measurement settings A/B/C) and then calculate the average of Matches. You will quickly see that there are no value sets that give the quantum mechanical expectation values. For anti-correlated electrons, that would be .75 for the Matches. Photons can be correlated or anti-correlated depending on the setup.

 

[rede96]

Again, what I am recommending is that you write down a series of hand generated outcomes (for 3 different measurement settings A/B/C) and then calculate the average of Matches. You will quickly see that there are no value sets that give the quantum mechanical expectation values.

I don't really dispute the outcome of Bell's theorem, it's more the conclusions drawn I struggle with. For example bell's theorem assumes that classically we think of the spin of a particle as being predetermined, we just don't know what it is until we measure it. This obviously isn't the case as the violation of bell's theorem shows. But for me that doesn't mean that we have to rule out local hidden variables. We just rule out the spin being a set value from the start.

I suppose for me it's a bit like twins. If I ask each twin individually at some point in time where they were born, they will always give the same answer. However if I ask the twins individually where they are at any point in time, the answers will vary depending on when I ask. But if I knew enough about the twins I may be able to predict where they will be at a given time.

This to me is analogous to spin. Spin states aren't predetermined, but there is a component of spin for any moment in time, it just depends on when we measure it as to what value we see. And hence if I knew enough about the particle I may be able to predict what spin state I'd measure at a given point in time. Bell's Theorem, as I understand it, does not rule this out.

 

[PeterDonis]

for me that doesn't mean that we have to rule out local hidden variables. We just rule out the spin being a set value from the start.

These are the same thing: "the spin being a set value from the start" is a local hidden variable model.

Spin states aren't predetermined, but there is a component of spin for any moment in time, it just depends on when we measure it as to what value we see.

Unfortunately, this type of model is ruled out by Bell's Theorem. The key statement is "there is a component of spin for any moment in time". That is a local hidden variable model, and it cannot be true if Bell's inequality is violated--which it is in QM.

Bell's Theorem, as I understand it, does not rule this out.

Yes, it does.

 

[rede96]

Unfortunately, this type of model is ruled out by Bell's Theorem. The key statement is "there is a component of spin for any moment in time". That is a local hidden variable model, and it cannot be true if Bell's inequality is violated--which it is in QM.

And this is where I am getting stuck.

For example, if I take a spinning top, draw an arrow on it and then spin the top, (assume the arrow is in the horizontal plane) and ask: is the arrow pointing northwards or southwards, I have no idea. Thus I have to take a measurement. I do this by stopping the top and seeing which direction the arrow is pointing. While the top is spinning, effectively the arrow has no set direction. This to me is analogous to superposition of spin. We can't say there is a directional component of the spin of the top until we take a measurement.

So if I don't know the direction of the arrow, as it doesn't have one until I measure it and depending on when I measure it I'll get a different answer, then how can Bell's Theorem apply to my scenario?

 

[Mentz114]

And this is where I am getting stuck.

For example, if I take a spinning top, draw an arrow on it and then spin the top, (assume the arrow is in the horizontal plane) and ask: is the arrow pointing northwards or southwards, I have no idea. Thus I have to take a measurement. I do this by stopping the top and seeing which direction the arrow is pointing. While the top is spinning, effectively the arrow has no set direction. This to me is analogous to superposition of spin. We can't say there is a directional component of the spin of the top until we take a measurement.

So if I don't know the direction of the arrow, as it doesn't have one until I measure it and depending on when I measure it I'll get a different answer, then how can Bell's Theorem apply to my scenario?

There are two problems with this

1, spin is an axial vector. Your top is spinning in the direction of the axis, up for clockwise and down for counter-clockwise
2, Quantum spin has nothing to do with things spinning !

 

[rede96]

There are two problems with this

1, spin is an axial vector. Your top is spinning in the direction of the axis, up for clockwise and down for counter-clockwise
2, Quantum spin has nothing to do with things spinning !

I'm not sure what that has to do with my point about Bell being a valid test or not? As I understand it Bell Theorem just deal with predicting the outcome of set of related binary outcomes. (Sorry if my terminology is a bit crap!) The mechanism that leads to those outcomes is irrelevant.

So the simple version of Bell for three binary properties of a related system states:

Number of (A and not B) + number of (B and not C) >= number of (A and not C)

I also understood that for this theorem to work we have to assume that the binary values that we measure for A, B and C are absolute. I.e. I can't have an individual something that is a bit A and a bit not A.

And there's what I'm struggling to understand. How can we use Bell's theorem as a test to rule out local hidden variables when the thing we are measuring doesn't even have a definitive spin state until it's measured? I'm so confused lol

 

[Mentz114]

I'm not sure what that has to do with my point about Bell being a valid test or not? As I understand it Bell Theorem just deal with predicting the outcome of set of related binary outcomes. (Sorry if my terminology is a bit crap!) The mechanism that leads to those outcomes is irrelevant.

So the simple version of Bell for three binary properties of a related system states:

Number of (A and not B) + number of (B and not C) >= number of (A and not C)

I also understood that for this theorem to work we have to assume that the binary values that we measure for A, B and C are absolute. I.e. I can't have an individual something that is a bit A and a bit not A.

And there's what I'm struggling to understand. How can we use Bell's theorem as a test to rule out local hidden variables when the thing we are measuring doesn't even have a definitive spin state until it's measured? I'm so confused lol

I was pointing out that your spinning top analogy is inappropriate. .

Bells theorems are a minefield I do not venture into so I can't help you with that.

 

[rede96]

I was pointing out that your spinning top analogy is inappropriate. .

Ah ok. Yes, sure point taken. Thanks.

Bells theorems are a minefield I do not venture into so I can't help you with that.

It is a minefield but it also really intrigues me!

 

[PeterDonis]

While the top is spinning, effectively the arrow has no set direction.

That's not correct. This is a classical model, so at every instant the arrow points in a definite direction, whether you observe it or not. And if you set up a pair of such tops and built in some sort of correlation between the directions of their arrows, and then separated the tops and measured their arrows at spacelike separated events, the correlations between the measurements would obey the Bell inequalities.

This to me is analogous to superposition of spin.

No, it isn't.

 

[PeterDonis]

How can we use Bell's theorem as a test to rule out local hidden variables when the thing we are measuring doesn't even have a definitive spin state until it's measured?

Remember that you yourself said:

As I understand it Bell Theorem just deal with predicting the outcome of set of related binary outcomes. (Sorry if my terminology is a bit crap!) The mechanism that leads to those outcomes is irrelevant.

All of this about "the thing we are measuring doesn't even have a definitive spin state until it is measured" is part of a hypothetical mechanism; it has nothing to do with the outcomes themselves. So it is irrelevant for understanding Bell's Theorem.

The key assumption Bell's Theorem makes is the locality assumption, which is simply that the function that describes the correlations between the outcomes has to factorize: i.e., instead of having a general function $P = f(a, b)$, where, schematically, $a$ denotes the measurement settings for object A, and $b$ denotes the measurement settings for object B, Bell's Theorem assumes that the correlations are described by a function $P = f(a) g(b)$, where $f(a)$ is a function that only depends on the measurement settings at A, and $g(b)$ is a function that only depends on the measurement settings at B.

Bell's Theorem says that if the function describing the correlations factorizes in this way, then the results must obey the Bell inequalities. Or, conversely, if you obtain results that violate the Bell inequalities, then that means that whatever is producing the correlations cannot be described by a function that factorizes in this way. That's all it says. It says nothing about what mechanism is producing the correlations that don't factorize--or indeed about what mechanism might produce correlations that factorize. It is, as you say, purely about the outcomes and the correlations between them, and what kind of function can or can't describe them. That's it.

 

[DrChinese]

And this is where I am getting stuck.

For example, if I take a spinning top, draw an arrow on it and then spin the top,...

So if I don't know the direction of the arrow, as it doesn't have one until I measure it and depending on when I measure it I'll get a different answer, then how can Bell's Theorem apply to my scenario?

But here's the rub (and why I keep saying to write out some examples): How is it that you get PERFECT correlations (or anti-correlation depending on the setup) at ANY angle as long as they are the same angle for both measurements your scenario? You want to have your cake and eat it too! Obviously, for a local realistic hypothesis to work, you have to be able to measure those "tops" at different times while separated and get the exact predicted answer at, say, 64 degrees. Or 176 degrees, or 313 degrees.etc.

If you use my example, and make that work out, you will see the Quantum expectation values cannot be achieved. This takes no complicated math, just the willpower to write down a few specific examples. Then you will quickly see why it won't work out.

 

[rede96]

All of this about "the thing we are measuring doesn't even have a definitive spin state until it is measured" is part of a hypothetical mechanism; it has nothing to do with the outcomes themselves. So it is irrelevant for understanding Bell's Theorem

Yes I got that thanks. I was just trying to imagine a mechanism to help my understanding but probably let it lead me down a wrong path.

Bell's Theorem says that if the function describing the correlations factorizes in this way, then the results must obey the Bell inequalities. Or, conversely, if you obtain results that violate the Bell inequalities, then that means that whatever is producing the correlations cannot be described by a function that factorizes in this way. That's all it says. It says nothing about what mechanism is producing the correlations that don't factorize--or indeed about what mechanism might produce correlations that factorize. It is, as you say, purely about the outcomes and the correlations between them, and what kind of function can or can't describe them. That's it.

There is one loop hole maybe I thought of and that's if the actual process of measurement has an influence over the result. If I make this assumption then I can assume some local hidden variables and make a classical model to predict the quantum mechanical result. At least at the one set of angles I did where normally the classical model would be violated.

 

[rede96]

How is it that you get PERFECT correlations (or anti-correlation depending on the setup) at ANY angle as long as they are the same angle for both measurements your scenario?

Because my model is just contrived hypothetical nonsense :-) But I did wonder if maybe Entanglement could be independent of spin. A sort of second variable if you will that although influenced by spin may also be influenced in a different way by the magnetic field .

 

[atyy]

There is one loop hole maybe I thought of and that's if the actual process of measurement has an influence over the result. If I make this assumption then I can assume some local hidden variables and make a classical model to predict the quantum mechanical result. At least at the one set of angles I did where normally the classical model would be violated.

Bell does show that this is not a loop hole (although this case is not usually addressed in the simplest derivations).

 

[PeterDonis]

There is one loop hole maybe I thought of and that's if the actual process of measurement has an influence over the result. If I make this assumption then I can assume some local hidden variables and make a classical model to predict the quantum mechanical result.

No, you can't. "The actual process of measurement has an influence over the result" is a mechanism, and, as you and I both agree per my previous post, Bell's Theorem has nothing to do with mechanisms. It's just about correlations between outcomes. That's it.

 

[DrChinese]

... may also be influenced in a different way by the magnetic field ...

As PeterDonis pointed out: it's all wrapped up into one, a single outcome. Remember: you get perfect correlations at the same angle, so the measurement device doesn't add a degree of randomness separately (as compared to the other measuring device). If they add randomness, it is the same randomness.

 

[rede96]

No, you can't. "The actual process of measurement has an influence over the result" is a mechanism, and, as you and I both agree per my previous post, Bell's Theorem has nothing to do with mechanisms. It's just about correlations between outcomes. That's it.

As PeterDonis pointed out: it's all wrapped up into one, a single outcome. Remember: you get perfect correlations at the same angle, so the measurement device doesn't add a degree of randomness separately (as compared to the other measuring device). If they add randomness, it is the same randomness.

When I mention measurement I wasn't referring to measurement errors or differences between various apparatus. I meant the actual process of a photon going through a polariser or an electron going through a magnetic field actually changes the particle is some way so as to produce the result. And it could be different for each particle.

If this was the case, then you can model almost any correlations you want just by playing with the variables / probabilities. So I can artificially model 0 deg, 120 deg and 240 deg and produce the expected matches seen by experiment. I'm not sure it means anything in the real world, but I have a few excel spread sheets where I've done this. But I haven't been able to figure out how to do it for all combinations of angles without making a new set of assumptions each time.

So it's still nonsense I know, but it did make wonder if in theory it could be a possibility.

 

[DrChinese]

And it could be different for each particle.

But it couldn't! Else how would the results match exactly at the SAME angle settings??

You see, that's the thing about replicating the quantum predictions using a local realistic hypothesis. On the one hand, to match EPR, the Alice's result must lead to a perfect prediction of Bob's (on the same measurement basis).

Of course, your idea falls short even without the EPR-like side, but you will never see that as long as you skip the step of actually attempting to match the quantum expectation values by writing out a full data set for a handful of examples. If you did that, your questions would end quickly.

 

[PeterDonis]

I meant the actual process of a photon going through a polariser or an electron going through a magnetic field actually changes the particle is some way so as to produce the result. And it could be different for each particle.

It's still a mechanism, and Bell's Theorem has nothing to do with mechanisms.

If this was the case, then you can model almost any correlations you want just by playing with the variables / probabilities.

But if the correlations produced by your model factorize in the way I described, then they must obey the Bell inequalities. Or, conversely, if the correlations produced by your model violate the Bell inequalties, then they can't factorize in the way I described. Again, that has nothing whatever to do with any mechanism; it's just a mathematical fact about the correlations between the outcomes, regardless of what mechanism produces them.

That means, if you want to understand what Bell's Theorem is telling you, you should stop trying to make claims about mechanisms and work on trying to understand what the factorization condition means.

 

[atyy]

When I mention measurement I wasn't referring to measurement errors or differences between various apparatus. I meant the actual process of a photon going through a polariser or an electron going through a magnetic field actually changes the particle is some way so as to produce the result. And it could be different for each particle.

If this was the case, then you can model almost any correlations you want just by playing with the variables / probabilities. So I can artificially model 0 deg, 120 deg and 240 deg and produce the expected matches seen by experiment. I'm not sure it means anything in the real world, but I have a few excel spread sheets where I've done this. But I haven't been able to figure out how to do it for all combinations of angles without making a new set of assumptions each time.

So it's still nonsense I know, but it did make wonder if in theory it could be a possibility.

No, you cannot. Bell explicitly considers "hidden variables" associated with the measuring apparatus also, and the possibility that the measuring apparatus interacts with the quantum system. As I said, the simpler derivations may omit this consideration, but if you do the full, most sophisticated version, you come up with the same result. The simpler version is usually given, because it gives the same answer as the more sophisticated derivations.

It is also possible to do the full derivation (ie. include measurement effects and hidden variables associated with the measuring apparatus) using just a few lines if one uses the graphical notation in Fig 19. of https://arxiv.org/abs/1208.4119.

There are loopholes, but the idea you raise is not one of them. There is a quite comprehensive discussion of loopholes in https://arxiv.org/abs/1303.2849.

 

[morrobay]

It's still a mechanism, and Bell's Theorem has nothing to do with mechanisms.
But if the correlations produced by your model factorize in the way I described, then they must obey the Bell inequalities. Or, conversely, if the correlations produced by your model violate the Bell inequalties, then they can't factorize in the way I described. Again, that has nothing whatever to do with any mechanism; it's just a mathematical fact about the correlations between the outcomes, regardless of what mechanism produces them.

That means, if you want to understand what Bell's Theorem is telling you, you should stop trying to make claims about mechanisms and work on trying to understand what the factorization condition means.

Could you show/elaborate on the factorization condition and how correlations predicted by QM and outcomes violating the inequality cannot factorize :
p (ab|xy,λ) = p (a|x,λ) p(b|y,λ)
S = (ab) + (ab') + (a'b) - (a'b') ≤ 2

 

[PeterDonis]

Could you show/elaborate on the factorization condition and how correlations predicted by QM and outcomes violating the inequality cannot factorize

Any paper giving a proof of the Bell inequalities or their equivalents (such as the CHSH inequalities, which it looks like you are quoting) will show this.

 

[rede96]

Of course, your idea falls short even without the EPR-like side, but you will never see that as long as you skip the step of actually attempting to match the quantum expectation values by writing out a full data set for a handful of examples. If you did that, your questions would end quickly.

If what you are advising is to write down all the permutations (e.g. ++-, +-+...etc) and see that these always have to be >= 0.333 I get that. My issue is that as I don't fully understand how a violation of this leads to non locality.

That means, if you want to understand what Bell's Theorem is telling you, you should stop trying to make claims about mechanisms and work on trying to understand what the factorization condition means.

I guess that's the issue, I don't understand what the factorization condition means at all or more importantly how it relates to reality other than it predicts a result we don't get in experiment. Hence why I focus on the mechanisms to help my understanding. But that doesn't mean I dismiss the math. It just means I don't understand how it relates to the real world.

So do you mind if I ask a question to help with my understanding? Hypothetically speaking imagine if there was a particle that when tested individuality through an uneven magnetic field always gave a probability of 50% for 'up' and 50% for 'down' (EDIT: So for all intents and purposes acted just like any spin 1/2 particle.)

But when the particle was entangled, it lost it's probability and one of them always pointed up and one of them only pointed down, regardless of the angle it was tested on (EDIT: e.g. they were now only attracted to the north or south pole of the magnet.)

So if I did the bell test with 3 randomly selected angles, it would still be a 50% chance of detection at each angle, as I don't know which path the particles would take.

I could model this in the same way as the Bell test and would violate bell's inequality as I'd get no matches and would expect >= 0.333. But does that also mean non locality? Couldn't I just say that one particle always pointed up in a magnetic field and one always pointed down. So this was an intrinsic property of the particle we can explain without action at a distance? Or would bell's theorem not apply in that case?

 

[Leo1233783]

Hi All,
sorry for my wording, english is not my language.

the Bell theorem has many possible loopholes. The most serious of them is the fair sample loophole. As Garg & Mermin showed in 86, one can reproduce QM results if he considers that some 'particles' aren't detected. This implies that an experimental strict proof of the theorem needs a detection rate at least better than 75%. Another proof would consist to invalidate the shapes detection predicted by the alternative hidden variables theories. Then, Zellinger showed the inequalities with a detection rate of 92% ( 2013/15) and quantum intrication became a definitive mainstream concept. It is now an inescapable element to dig for new physics ( LQG, ER=EPR, etc ) and philosophy.

Mermin functions exist. Some need hidden variables, other just only one shared value when the simulation starts. One of them, discovered in 2011 , uses trivial classical tunelling and doesn't need a fine tuning at all. Cos² for correlated ( or sin² for uncorrelated ) are perfectly got with a computed and simulated detection rate of 75%. This simulates perfectly what the detectors may measure locally in a 75% experiment. A physical interpretation theory might use the interferences concept and its relation to usable ( local ) information. But to get a perfect cos² the rate is limited to 75% ; thus it is falsified in advance by the 92%'s experiment.

The Mermin paper ( PHYSICAL REVIEW D VOLUME 35, NUMBER 12 15 JUNE 1987, Detector inefficiencies in the Einstein-Podolsky-Rosen experiment by Anupam Garg & N. D. Mermin 1986 ) is not public. This reduces its real impact.

As for today, numerical simulations may produce the nearest outcomes to QM while experiments are probably around 1 or 2% detection rate. This asks the applicability of the mainstream theory to technology.
Some note that Zellinger's experiment is debatable. Other remark that the tapping-detector quantum cryptology is not yet sold by companies. Sometimes I wonder if the quantum computing projects will not end by producing nice specialized analogic computers.

But it is impossible that all well-known physicists are mistaken. Hopefully see a technological product close the debate.

 

[PeterDonis]

My issue is that as I don't fully understand how a violation of this leads to non locality.

"Non locality" is words, and not everyone necessarily agrees that they are the best words to use to describe what violating the Bell inequalities means. But see below.

I don't understand what the factorization condition means

Heuristically, it goes like this: we have some probability, correlation, whatever you want to call it, between two measurement results, call them $a$ and $b$, which are obtained at spacelike separated events. Call the thingie we're interested in (probability, correlation, whatever) $E(a, b)$. We have settings for the two measuring devices, call them $A$ and $B$. So the most general function we can write down to describe how the thingie we're interested in depends on the measurement settings (ignoring any other stuff like "hidden variables"--in all the papers those are integrated over anyway so you always end up with formulas that look like the ones I'm about to write) is $E(a, b) = F(A, B)$, i.e., the thingie were interested in is described by some function $F$ whose arguments are the settings $A$ and $B$.

Now, what the factorization condition means is that the function $F$ can be factored into a function that only depends on $A$ and a function that only depends on $B$, like this: $E(a, b) = f(A) g(B)$. And intuitively, this seems to express what we mean by "locality": that since the measurements are spacelike separated, there can't be any communication between them, so whatever is causing the thingie we're interested in, it should break down into something that only depends on the settings at $A$ (which is what the function $f(A)$ describes) and something that only depends on the settings at $B$ (which is what the function $g(B)$ describes). And Bell and others have shown mathematically that assuming that form for $E(a, b)$--what I'm calling the factorization condition--requires that certain inequalities must hold, which we observe to be violated in actual experiments. So, whatever is causing the thingie we're interested in, it cannot break down into something that only depends on the settings at $A$, and something that only depends on the settings at $B$.