Bell's theorem and local realism

[stevendaryl]

I know. As a mathematician I can tell you that this is quite bogus. Does not prove what it seems to prove. (It's not for nothing that no-one has ever followed this up).

Pitowsky has done a lot of great things! But this one was a dud, IMHO.

Here's a version of Bell's theorem which *only* uses finite discrete probability and elementary logic http://arxiv.org/abs/1207.5103. Moreover it is stronger than the conventional result since it is a "finite N" result: a probability inequality for the observed correlations after N trials. The assumptions are slightly different from the usual ones: I put probability into the selection of settings, not into the particles.

To follow up a little bit, I feel that there is still a bit of an unsolved mystery about Pitowsky's model. I agree that his model can't be the way things REALLY work, but I would like to understand what goes wrong if we imagined that it was the way things really work. Imagine that in an EPR-type experiment, there was such a spin-1/2 function F associated with the electron (and the positron) such that a subsequent measurement of spin in direction \vec{x} always gave the answer F(\vec{x}). We perform a series of measurements and compile statistics. What breaks down?

On the one hand, we could compute the relative probability that F(\vec{a}) = F(\vec{b}) and we conclude that it should be given by cos^2(\theta/2) (because F) was constructed to make that true). On the other hand, we can always find other directions \vec{a'} and \vec{b'} such that the statistical correlations don't match the predictions of QM (because your finite version of Bell's inequality shows that it is impossible to match the predictions of QM for every direction at the same time).

So what that means is that for any run of experiments, there will be some statistics that don't come close to matching the theoretical probability. I think this is a fundamental problem with relating non-measurable sets to experiment. The assumption that relative frequencies are related (in a limiting sense) to theoretical probabilities can't possibly hold when there are non-measurable sets involved.

 

[gill1109]

To follow up a little bit, I feel that there is still a bit of an unsolved mystery about Pitowsky's model. I agree that his model can't be the way things REALLY work, but I would like to understand what goes wrong if we imagined that it was the way things really work. Imagine that in an EPR-type experiment, there was such a spin-1/2 function F associated with the electron (and the positron) such that a subsequent measurement of spin in direction \vec{x} always gave the answer F(\vec{x}). We perform a series of measurements and compile statistics. What breaks down?

On the one hand, we could compute the relative probability that F(\vec{a}) = F(\vec{b}) and we conclude that it should be given by cos^2(\theta/2) (because F) was constructed to make that true). On the other hand, we can always find other directions \vec{a'} and \vec{b'} such that the statistical correlations don't match the predictions of QM (because your finite version of Bell's inequality shows that it is impossible to match the predictions of QM for every direction at the same time).

So what that means is that for any run of experiments, there will be some statistics that don't come close to matching the theoretical probability. I think this is a fundamental problem with relating non-measurable sets to experiment. The assumption that relative frequencies are related (in a limiting sense) to theoretical probabilities can't possibly hold when there are non-measurable sets involved.

If we do the Bell-CHSH type experiment picking settings at random as we are supposed to, nothing breaks down. At least, nothing breaks down if you use a sharper method of proof than Bell's old approach.

That's the point of my own work, going back to http://arxiv.org/abs/quant-ph/0110137. No measurability assumptions. The only assumption is that both outcomes are simultaneously defined - both the outcomes which would have been seen if either setting had been in force. aka counterfactual definiteness. The experimenter tosses a coin and gets to see one or the other, at random. This works for Pitowsky's "model" too. It works for any LHV model. A function is a function whether it is measurable or not. It works for stochastic LHV models as well as deterministic. Just a matter of redefining what is the hidden variable.

The only escape is super-determinism so that I cannot actually effectively randomize experimental settings.

 

[harrylin]

I had not read Neumaier's paper linked by you when I wrote that, and now I have just read the conclusions.
[..]
ruling out classical wave models explaining QM experiments doesn't need Bell's theorem.

Sure. The last days I did read some of his papers while I found them, and (as you may have guessed), that's not what he had in mind. He (re)discovered that QM is totally incompatible with classical particle theory but very close to classical wave theory. The naive particle concept must be dropped.

There is a trivial way in which say a plane wave is nonlocal, as it correlates its waveform for infinitely separated points. [..]

If I'm not mistaken, all matter is similarly modeled in QFT as field excitations.

 

[gill1109]

On the other hand, we can always find other directions \vec{a'} and \vec{b'} such that the statistical correlations don't match the predictions of QM (because your finite version of Bell's inequality shows that it is impossible to match the predictions of QM for every direction at the same time).

No this is a misunderstanding. My theorem says that for the set of correlations you actually did choose to measure, the chance that they'll violate CHSH by more than some given amount is incredibly small if N is pretty large. The theorem doesn't say anything about what you didn't do. It only talks about what you actually did experimentally observe. It assumes you are doing a regular CHSH type experiment - Alice and Bob are repeatedly and independently choosing between just two particular settings. So only four correlations are getting measured.
Note, Pitowsky has a non-measurable law of large numbers which says that the relative frequency of the event you are looking at will continue for ever to fluctuate between its outer probability and its lower probability. Those two numbers can be 1 and 0 respectively. So what. My theorem talks about the chance of something happening for a given fixed finite value of N, conditional on the values of the hidden variables etc etc. The probability in my theorem is *exclusively* in the 2 N coin tosses determining Alice and Bob's settings. If N goes to infinity it doesn't matter at all whether or not the quantum averages converge or not. There are always subsequences along which they converge by compactness. Along any such subsequence, in the long run CHSH will certainly only be violated by more than epsilon at most finitely many times. (Here I am using the Borel-Cantelli lemma which is how you can prove the strong law of large numbers once you have got an exponential bound like we have here).

 

[stevendaryl]

I think you are referring here to paradoxes from "model theory" namely there exist countable models for the real numbers.

No, not at all. Let \omega_1 be the smallest uncountable ordinal. Then for any ordinal \alpha < \omega_1 (with < the usual ordering on ordinals), there are only countably many \beta < \alpha but there are uncountably many \beta > \alpha. So if we assume the continuum hypothesis, then every real in [0,1] can be associated with an ordinal less than \omega_1. This gives us a total ordering on reals such that for any x there are only countably many smaller reals in [0,1] but uncountably many larger reals.

Beautiful. It's a self-reference paradox, really just a hyped up version of the old paradox of the barber who shaves everyone in the village who doesn't shave himself. In some sense, it is just a word game. It's a useful tool in maths - one can prove theorems by proving theorems about proving theorems. Nothing wrong with that.

No, I don't think it's paradoxical in that sense. It's perfectly consistent mathematics (unlike the Liar Paradox, which is an actual logical contradiction). It's just weird.

 

[gill1109]

No, not at all. Let \omega_1 be the smallest uncountable ordinal. Then for any ordinal \alpha < \omega_1 (with < the usual ordering on ordinals), there are only countably many \beta < \alpha but there are uncountably many \beta > \alpha. So if we assume the continuum hypothesis, then every real in [0,1] can be associated with an ordinal less than \omega_1. This gives us a total ordering on reals such that for any x there are only countably many smaller reals in [0,1] but uncountably many larger reals.

I think you are wrong. The continuum hypothesis tells us that the unit interval has the same cardinality as Aleph_1, the first cardinal number large that Aleph_0, the first infinite cardinal. This does not mean that the numbers in [0, 1] can be put in 1-1 correspondence with 1, 2, ... You are saying that there is a 1-1 map from [0, 1] to the numbers 1, 2, ... hence [0, 1] is countable.

Continuum hypothesis says there is no cardinality strictly between Aleph_0, the first infinite cardinal = the cardinality of the set of the natural numbers, and 2^Aleph_0, the set of functions from Aleph_0 to {0, 1}, which is easily seen to be the same cardinality as that of the unit interval on the real line. So no infinite set which cannot be put into one-to-one correspondence with the natural numbers but which is the domain of some one-to-one map into the unit interval but which cannot be put into one-to-one correspondence with the whole unit interval.

Maybe you are mixing up cardinals and ordinals?

 

[stevendaryl]

No this is a misunderstanding. My theorem says that for the set of correlations you actually did choose to measure, the chance that they'll violate CHSH by more than some given amount is incredibly small if N is pretty large. The theorem doesn't say anything about what you didn't do. It only talks about what you actually did experimentally observe. It assumes you are doing a regular CHSH type experiment - Alice and Bob are repeatedly and independently choosing between just two particular settings. So only four correlations are getting measured.

I don't think there's a misunderstanding. I'm just saying that there is an apparent contradiction and I don't see how to resolve it.

Imagine generating a sequence of Pitowsky spin-1/2 functions:

F_1

F_2

.
.
.

For each such run, you let Alice and Bob pick a direction:

a_1, b_1
a_2, b_2

.
.
.

Then we lookup their corresponding results:

R_{A,1} = F_1(a_1), R_{B,1} = F_1(b_1)
R_{A,2} = F_2(a_2), R_{B,2} = F_2(b_2)
.
.
.

The question is: what are the statistics for correlations between Alice's results and Bob's results?

On the one hand, your finite version of Bell's inequality can show that (almost certainly) the statistics can't match the predictions of QM. On the other hand, the functions F_j were specifically constructed so that the probability of Bob getting F_j(b_j) = +1 given that Alice got F_j(a_j) = +1 is given by the QM relative probabilities. That seems to be a contradiction. So what goes wrong?

 

[stevendaryl]

I think you are wrong. The continuum hypothesis tells us that the unit interval has the same cardinality as Aleph_1, the first cardinal number large that Aleph_0, the first infinite cardinal. This does not mean that the numbers in [0, 1] can be put in 1-1 correspondence with 1, 2, ... You are saying that there is a 1-1 map from [0, 1] to the numbers 1, 2, ... hence [0, 1] is countable.

Continuum hypothesis says there is no cardinality strictly between Aleph_0, the first infinite cardinal = the cardinality of the set of the natural numbers, and 2^Aleph_0, the set of functions from Aleph_0 to {0, 1}, which is easily seen to be the same cardinality as that of the unit interval on the real line. So no infinite set which cannot be put into one-to-one correspondence with the natural numbers but which is the domain of some one-to-one map into the unit interval but which cannot be put into one-to-one correspondence with the whole unit interval.

Maybe you are mixing up cardinals and ordinals?

I didn't say that the unit interval can be put into a one-to-one correspondence with the naturals; I said that it can be put into a one-to-one correspondence with the countable ordinals. The set of countable ordinals goes way beyond the naturals. The naturals are the finite ordinals, not the countable ordinals.

Aleph_1 is (if one uses the Von Neumann ordinals) equal to the set of all countable ordinals. So there are uncountably many countable ordinals. The continuum hypothesis implies that Aleph_1 has the same cardinality as the continuum, and so it implies that the unit interval can be put into a one-to-one correspondence with the countable ordinals.

 

[gill1109]

I didn't say that the unit interval can be put into a one-to-one correspondence with the naturals; I said that it can be put into a one-to-one correspondence with the countable ordinals. The set of countable ordinals goes way beyond the naturals. The naturals are the finite ordinals, not the countable ordinals.

Aleph_1 is (if one uses the Von Neumann ordinals) equal to the set of all countable ordinals. So there are uncountably many countable ordinals. The continuum hypothesis implies that Aleph_1 has the same cardinality as the continuum, and so it implies that the unit interval can be put into a one-to-one correspondence with the countable ordinals.

Hold it. Aleph_1 is the first uncountable *cardinal* not ordinal.

AFAIK, the continuum hypothesis does not say that the unit interval is in one-to-one correspondence with the set of countable *ordinals*. But maybe you know things about the continuum hypothesis which I don't know. Please give a reference.

 

[stevendaryl]

I didn't say that the unit interval can be put into a one-to-one correspondence with the naturals; I said that it can be put into a one-to-one correspondence with the countable ordinals. The set of countable ordinals goes way beyond the naturals. The naturals are the finite ordinals, not the countable ordinals.

Aleph_1 is (if one uses the Von Neumann ordinals) equal to the set of all countable ordinals. So there are uncountably many countable ordinals. The continuum hypothesis implies that Aleph_1 has the same cardinality as the continuum, and so it implies that the unit interval can be put into a one-to-one correspondence with the countable ordinals.

The axiom of choice implies that every set can be put into a one-to-one correspondence with some initial segment of the ordinals. That means that it is possible to index the unit interval by ordinals \alpha: [0,1] = \{ r_\alpha | \alpha < \mathcal{C}\} where \mathcal{C} is the cardinality of the continuum. The continuum hypothesis implies that \mathcal{C} = \omega_1, the first uncountable ordinal (\omega_1 is the same as Aleph_1, if we use the Von Neumann representation for cardinals and ordinals). So we have:

[0,1] = \{ r_\alpha | \alpha < \omega_1\}

If \alpha < \omega_1, then that means that \alpha is countable, which means that there are only countably many smaller ordinals. That means that if we order the elements of [0,1] by using r_\alpha < r_\beta \leftrightarrow \alpha < \beta, then for every x in [0,1] there are only countably many y such that y < x.

 

[gill1109]

I don't think there's a misunderstanding. I'm just saying that there is an apparent contradiction and I don't see how to resolve it.

Imagine generating a sequence of Pitowsky spin-1/2 functions:

F_1

F_2

.
.
.

For each such run, you let Alice and Bob pick a direction:

a_1, b_1
a_2, b_2

.
.
.

Then we lookup their corresponding results:

R_{A,1} = F_1(a_1), R_{B,1} = F_1(b_1)
R_{A,2} = F_2(a_2), R_{B,2} = F_2(b_2)
.
.
.

The question is: what are the statistics for correlations between Alice's results and Bob's results?

On the one hand, your finite version of Bell's inequality can show that (almost certainly) the statistics can't match the predictions of QM. On the other hand, the functions F_j were specifically constructed so that the probability of Bob getting F_j(b_j) = +1 given that Alice got F_j(a_j) = +1 is given by the QM relative probabilities. That seems to be a contradiction. So what goes wrong?

Pitowsky can come up with one function or lots but he doesn't know in advance which arguments we are going to supply it with. In the j'th run one of his functions is "queried" once (well once on each side of the experiment) and generates two outcomes +/-1. His "probabilities" are irrelevant. If he is using non-measurable functions he can't control what "probabilities" come out when these functions are queried infinitely often. I don't see any point in trying to rescue his approach. But you can try if you like. I think it is conceptually unsound.

 

[gill1109]

The axiom of choice implies that every set can be put into a one-to-one correspondence with some initial segment of the ordinals. That means that it is possible to index the unit interval by ordinals \alpha: [0,1] = \{ r_\alpha | \alpha < \mathcal{C}\} where \mathcal{C} is the cardinality of the continuum. The continuum hypothesis implies that \mathcal{C} = \omega_1, the first uncountable ordinal (\omega_1 is the same as Aleph_1, if we use the Von Neumann representation for cardinals and ordinals). So we have:

[0,1] = \{ r_\alpha | \alpha < \omega_1\}

If \alpha < \omega_1, then that means that \alpha is countable, which means that there are only countably many smaller ordinals. That means that if we order the elements of [0,1] by using r_\alpha < r_\beta \leftrightarrow \alpha < \beta, then for every x in [0,1] there are only countably many y such that y < x.

Thanks, you are right!

So the set of countable ordinals is very very very large. Your "ordering" of [0, 1] is not actually countable, even though every initial segment of it is. Well that's how it has to be if we want both the axiom of choice and the continuum hypothesis to be true. But it is merely a matter of taste whether or not we want them to be true. The physics of the universe does not depend on these axioms of infinite sets being true or not. So maybe there are physical grounds to prefer not to have some of these axioms - we might get a mathematics which was more physically appealing by making different choices. There have been a number of very serious proposals on these lines. Pick axioms of the infinite not on the grounds of mathematical expediency but on the grounds of physical intuition.

 

[EEngineer91]

Bohm described it very well. As Bell himself said, you can't get away with "no action at a distance"...

 

[gill1109]

Bohm described it very well. As Bell himself said, you can't get away with "no action at a distance"...

When/where did Bell say that? Just like Bohr, there is a young Bell and an older and wiser Bell ... Young Bell was a fan of Bohmian mechanics. Older Bell liked the CSL theory. Always (?) Bell was careful to distinguish his gut feelings about a matter, from what logic would allow us to conclude.

Look: you can't simulate quantum correlations with local hidden variables model without cheating. That's exactly what Bell's theorem says. If you *know* that there must be a hidden variables model explaining QM, then you *know* there is non-locality.

QM does not allow action-at-a-distance in the world of what we can see and feel and measure. If you want to simulate QM with hidden variables, you'll have to put action-at-a-distance into the hidden layer.

 

[EEngineer91]

Please watch the video, that is not a young Bell saying this. He was always a fan of Bohm's work, but unfortunately he died early as well. The most important line in the video is at the end "you can't get away with NO action at a distance"...non-locality is fine, it just bugs the relativists and those who think c is a universal speed barrier to everything, it is just a constant of electromagnetism

 

[gill1109]

Please watch the video, that is not a young Bell saying this. He was always a fan of Bohm's work, but unfortunately he died early as well. The most important line in the video is at the end "you can't get away with NO action at a distance"...non-locality is fine, it just bugs the relativists and those who think c is a universal speed barrier to everything, it is just a constant of electromagnetism

will do

He is a bit subtle. He says. I cannot say action at a distance is not needed. I can say that you can't say it is not needed. This is like Buddha talking about self. He is saying that our usual categories of thought are *wrong*. Because of the words in our vocabulary and our narrow interpretation of what they mean, we ask stupid questions, and hence get stupid answers.

Beautiful! Exactly what I have been thinking for a long time...

 

[stevendaryl]

Hold it. Aleph_1 is the first uncountable *cardinal* not ordinal.

In the Von Neumann representation of ordinals and cardinals, a cardinal is an ordinal; \alpha is a cardinal if it is an ordinal, and for any other ordinal \beta < \alpha, there is no one-to-one correspondence between \alpha and \beta. So in the Von Neumann representation, the first uncountable ordinal is also the first uncountable cardinal.

AFAIK, the continuum hypothesis does not say that the unit interval is in one-to-one correspondence with the set of countable *ordinals*.

Yes, it does imply that. With the Von Neumann representation of ordinals, any ordinal is the set of all smaller ordinals. So the set of all countable ordinals is itself an ordinal. It has to be uncountable (otherwise, it would be an element of itself, which is impossible). So it's the smallest uncountable ordinal, \omega_1. The continuum hypothesis says that there is no cardinality between countable and the continuum, so the continuum has to equal \omega_1.

But maybe you know things about the continuum hypothesis which I don't know. Please give a reference.

I did some Googling, and I don't see the claim stated explicitly anywhere, although it's a trivial consequence of other statements.

http://en.wikipedia.org/wiki/Aleph_number

\aleph_1 is the cardinality of the set of all countable ordinal numbers...

the celebrated continuum hypothesis, CH, is equivalent to the identity

2^{\aleph_0}=\aleph_1

Together, those statements imply that the continuum has the same cardinality as the set of countable ordinals. Having the same cardinality means that they can be put into one-to-one correspondence.

 

[stevendaryl]

Pitowsky can come up with one function or lots but he doesn't know in advance which arguments we are going to supply it with. In the j'th run one of his functions is "queried" once (well once on each side of the experiment) and generates two outcomes +/-1. His "probabilities" are irrelevant. If he is using non-measurable functions he can't control what "probabilities" come out when these functions are queried infinitely often. I don't see any point in trying to rescue his approach. But you can try if you like. I think it is conceptually unsound.

Yes, that's my point--there seems to be a contradiction between the formal computed probabilities and the intuitive notion of probabilities as limits of relative frequencies. Maybe that means that the mathematical possibility of nonmeasurable sets is inconsistent with our use of probabilities for physics.

It's not so much that I'm trying to rescue Pitowsky's approach--from the very first, it seemed to me like a toy model to show the subtleties involved in Bell's proof that are easy to gloss over. At this point, I'm really trying to reconcile two different mathematical results that both seem pretty rigorous, but seem to contradict each other. Whether or not Pitowsky's functions have any relevance to the real world, we can reason about them---they are pretty well-defined, mathematically. I'm trying to understand what goes wrong in reasoning about them.

 

[gill1109]

Together, those statements imply that the continuum has the same cardinality as the set of countable ordinals. Having the same cardinality means that they can be put into one-to-one correspondence.

Agree. This is what continuum hypothesis and axiom of choice tell us. But we are free not to believe either. Formal mathematics is consistent with them if and only if it is consistent without them. One could have other axioms instead, e.g. all subsets of [0,1] are Lebesgue measurable. Maybe that would be a nicer axiom for physics applications. No more Banach-Tarski paradox. All kinds of advantages ...

 

[EEngineer91]

will do

He is a bit subtle. He says. I cannot say action at a distance is not needed. I can say that you can't say it is not needed. This is like Buddha talking about self. He is saying that our usual categories of thought are *wrong*. Because of the words in our vocabulary and our narrow interpretation of what they mean, we ask stupid questions, and hence get stupid answers.

Beautiful! Exactly what I have been thinking for a long time...

Yes, very subtle...but important.

 

[billschnieder]

I don't think they are wildly different if you don't have locality. Let's do things classically, rather than quantum-mechanically. For simplicity, let's just consider
...
Now, that pair of equations is exactly equivalent to a problem in 2-D space (3D spacetime) involving just one particle ...

So I think that it's really locality that makes the dimensionality of spacetime meaningful.

With two particles each in N-dimensional space, you can destroy one and still have the other (even if you consider non-locality). With a single particle in 2N dimensional space, you can not destroy it and still have it, nor can you destroy half of it and convert it into an N-dimensional particle. You may have the same symbols in your equations but they mean totally different things even if they look the same.

 

[stevendaryl]

With two particles each in N-dimensional space, you can destroy one and still have the other (even if you consider non-locality). With a single particle in 2N dimensional space, you can not destroy it and still have it, nor can you destroy half of it and convert it into an N-dimensional particle. You may have the same symbols in your equations but they mean totally different things even if they look the same.

Okay, I guess I would amend what I said to the following: If your laws of physics are such that the number of particles is constant, then there is no difference between N particles in 3D spacetime and 1 particle in 3N - D space.

With a variable number of particles, the interpretation doesn't work, unless you also allowed the dimension of space to vary with time. (Why not?)

 

[gill1109]

Yes, very subtle...but important.

Very important indeed!

 

[EEngineer91]

Yet, for some reason, many physicists of today lambast action at a distance as some logical impossibility. Bell even brought up good holistic problems such as defining precisely "measurement", "observation device" and the paradox of their "seperate-ness" and fundamental "together-ness"

 

[gill1109]

Yet, for some reason, many physicists of today lambast action at a distance as some logical impossibility. Bell even brought up good holistic problems such as defining precisely "measurement", "observation device" and the paradox of their "seperate-ness" and fundamental "together-ness"

Yep. I think his clear thinking and clear writing (and sense of humour) is unsurpassed.

 

[billschnieder]

To follow up a little bit, I feel that there is still a bit of an unsolved mystery about Pitowsky's model. I agree that his model can't be the way things REALLY work, but I would like to understand what goes wrong if we imagined that it was the way things really work. Imagine that in an EPR-type experiment, there was such a spin-1/2 function F associated with the electron (and the positron) such that a subsequent measurement of spin in direction \vec{x} always gave the answer F(\vec{x}). We perform a series of measurements and compile statistics. What breaks down?

Maybe you can provide a citation for Pitowsky's model so others can follow?

On the one hand, we could compute the relative probability that F(\vec{a}) = F(\vec{b}) and we conclude that it should be given by cos^2(\theta/2) (because F) was constructed to make that true). On the other hand, we can always find other directions \vec{a'} and \vec{b'} such that the statistical correlations don't match the predictions of QM (because your finite version of Bell's inequality shows that it is impossible to match the predictions of QM for every direction at the same time).

According to this paper by Pitowsky, http://arxiv.org/pdf/0802.3632.pdf, it would appear what breaks down is the tacit assumption that all directions are measurable at the same time.

So what that means is that for any run of experiments, there will be some statistics that don't come close to matching the theoretical probability.

Yes, if they are measured at the same time. But nobody really does that anyway.

I think this is a fundamental problem with relating non-measurable sets to experiment.

See previous point. It is not a problem at all. A non-measurable would just be an impossible/contradictory scenario physically.

The assumption that relative frequencies are related (in a limiting sense) to theoretical probabilities can't possibly hold when there are non-measurable sets involved.

It depends what those theoretical probabilities are. Nothing prevents one from adding one probability with a mutually incompatible probability theoretically. But experimentally you won't be measuring them at the same time. My guess is, its not the probabilities themselves that can't be related to relative frequencies, but the relationships between contradictory probabilities.

 

[gill1109]

According to this paper by Pitowsky, http://arxiv.org/pdf/0802.3632.pdf, it would appear what breaks down is the tacit assumption that all directions are measurable at the same time.

Different paper, different point. I will look up wha tis the *relevant* Pitowsky reference later. It is a very difficult and rather technical paper and I personally believe it is conceptually flawed. Sure, some fun generalized abstract nonsense. But no relevance. (just my personal opinion...). AFAIK, it has not been followed up by anyone...

If you have a LHV model, then even if you can only measure one direction at a time, the outcome that you would have had, had you actually measured in another direction is defined ... even if unavailable.

 

[jk22]

It is easy to create *half* the cosine curve by LHV.
The problem is not the " particle" concept in the hidden layer, in the physics behind the scenes, it is the discreteness of the manifest outcomes. Click or no-click. +1 or -1.

Imo it is possible to generate a lhv covariance which is bigger than .5 at 45 degrees, ie it can be a cosine curve.
But the sum of the 4 covariances in chsh is still smaller than 2, and this is the point of bell's theorem : a combination of covariances.

 

[stevendaryl]

Different paper, different point. I will look up wha tis the *relevant* Pitowsky reference later. It is a very difficult and rather technical paper and I personally believe it is conceptually flawed. Sure, some fun generalized abstract nonsense. But no relevance. (just my personal opinion...). AFAIK, it has not been followed up by anyone...

Pitowsky's model appeared in Stanley Gudder's book "Quantum Probability". That's where I heard of it.

 

[gill1109]

Imo it is possible to generate a lhv covariance which is bigger than .5 at 45 degrees, ie it can be a cosine curve.
But the sum of the 4 covariances in chsh is still smaller than 2, and this is the point of bell's theorem : a combination of covariances.

It is easy to create half a cosine curve by LHV.

It is easy to create a correlation bigger than 0.5 at 45 degrees.

CHSH says that it is impossible to have three correlations extremely large and one simultaneously extremely small, when the four are the four correlations formed by combining one of two settings on Alice's side with one of two settings on Bob's side.

Yes it is very difficult to have any feeling what this really means.

One could try something like this:

If r(a1, b2) is large, and r(a2, b2) is large, and r(a2, b1) is large, then we would expect r(a1, b1) to be large too.

Better still for pedagogical purposes, replace the usual "perfect anti-correlation at equal settings" of the singlet state version of the experiment by "perfect correlation at equal settings" by multiplying Bob's outcome by -1. Or switch from spin of electrons to polarization of photons.

For pedagogical purposes, forget about correlations and talk about the probability of equal outcomes

If Prob(A1 = B2) is large and Prob(A2 = B2) is large and Prob(A2 = B1) is large, then we would expect Prob(A1 = B1) is large.

If the first three probabilities are at least 1 - gamma then the fourth can't be smaller than 1 - 3 gamma. Take gamma = 0.25 and the first three would be 0.75 and the fourth 0.25. That's the largest one can get with LHV. This corresponds to CHSH value S = 2 = 4 * 0.5 = (2 * 0.75 - 1) + (2 * 0.75 - 1) + (2 * 0.75 - 1) - (2 * 0.25 -1)

But QM can have the first three probabilities equal to 0.85 and the fourth equal to 0.15. That corresponds to S = 2.8 = 4 * 0.7 = (2 * 0.85 - 1) + (2 * 0.85 - 1) + (2 * 0.85 - 1) - (2 * 0.15 -1) (in fact it can even be equal to 2.828... under QM but let's keep the numbers simple).