Trying to Understand Bell's reasoning

[madhatter106]

Interesting thread, please consider that I do not have any formal education on this subject I find math fascinating in it's relations and patterns.

If I may ask a question is regards to the ratios used?

In the link from Drchinese, which was an interesting read. It is mentioned that the 25% chance is derived from the square of the cosine of 120*. Ok with that wouldn't you need to also take the inverse relationship?

To me the big picture is if your starting with 3 variables the ratios will always be 1/3. Now with the prediction that QM predicts 25% it is because it was derived from the cosine. Am I wrong in thinking that the cosine of one of the variables is simply the ratio against itself? . I know this is nothing new as it's basic trig, what I'm trying to figure is why is the QM ratio is the being taken against itself for A? and the std. is against the whole? Or did I miss something?

 

[madhatter106]

A bit more reading here, so it's related to Malus' Law right?
If the QM model is based off this and this would be the intensity after polarization, it is also the ratio of light percentage that is polarized already to the chosen theta.

So the QM percentage is working from the ratio between percentage of possible existing state to theta. Yet for some reason the follow thru to A,B,C, doesn't seem to be the correct ratio comparison, I can see the 1:3 option for each state but in the case of the polarized beam isn't that based upon the unknown ratio for each A, B, C, state?

 

[madhatter106]

Okay one more ? before I retire for the night. Is this an accurate view?

in the case of polarized light, to assign it a known state it would be pure and no longer an average. then the probability of any other state is forced by that known state.

The QM probability is based on the complete unknown of infinite theta between 0~360. the A,B,C table is already known values that have no internal ratios. I'm assuming the A,B,C values are based on pure polarization? if not is it an accurate comparison? if it's not pure wouldn't the set comparison then have to include the infinite range between 0~120 or the average ratio -0.5? thus simply taking it back to .25?

 

[DrChinese]

By saying "observables have definite values even when I am not looking at it", you ARE effectively saying "I can see the moon even when I am not looking at it". Surely you must be aware that there are contextual observables which only have defined values in specific contexts of observation, and clearly it is naive to think those observables pre-exist the act of observation.
The EPR "elements of reality" only states that there exists objective ontological entities apart from the act of observation. EPR does not demand that those entities be passively revealed by observation, or even be definite.

This is absolutely incorrect, and is somewhat shocking. It is seriously as if you have completely ignored everything important about EPR and Bell to focus on a few things out of context.

The EPR elements of reality have definite values, which can be predicted with 100% certainty prior to observation. That is an experimental fact and has never been in question from 1935 to now. This completely contradicts everything you are saying above.

The relevant question, as I have said in this thread previously, is whether these elements of reality exist simultaneously. Einstein said they do. Why won't you answer a simple question - do you agree with Einstein, yes or no? If you would, then it would clarify your position for the rest of us. If you are unable or unwilling to address this, then please say so and I will be on my way.

 

[DrChinese]

2) In Bell's treatment, Hidden variables are supposed to be responsible for the correlation between A and B. However, when Bell write the equation as follows:

P(AB|H) = P(A|H) * P(B|H)

This equation means that conditioned on H, there is no correlation between A and B. Now please take a moment and let this sink in because I have the feeling you have not understood this point. Note on the left hand side, we have a conditional probability, not a marginal probability, so it is irrelevant whether there is a correlation between A and B marginally or not. The equation clearly says conditioned on the hidden variables, there is no correlation between A and B.

I don't necessarily think your statistics is faulty, but I think you are mis-modeling the setup. Bell refers at various times to a and A, and b and B. Sometimes these are interchangeable, and sometimes they are not. I think the A and B should refer to the results of tests at measurement angles a and b. I think if you re-examine the setup, you will see that the above should include a and b as well.

As I have said many times, the reason you are going 'round in circles is because you are missing the point. Bell is trying to say: outcome B is independent of setting a, and vice versa. Write that statement however you like, and then proceed from there.

 

[JesseM]

By saying "observables have definite values even when I am not looking at it", you ARE effectively saying "I can see the moon even when I am not looking at it".

No, I'm not saying I can see it. I'm saying the hypothetical omniscient observer can see it.

Surely you must be aware that there are contextual observables which only have defined values in specific contexts of observation, and clearly it is naive to think those observables pre-exist the act of observation.

Sure, I never said that the omniscient observer might not see the values of various hidden variables change in response to interaction with a measuring device, just that the variables would have well-defined values at all times.

The EPR "elements of reality" only states that there exists objective ontological entities apart from the act of observation. EPR does not demand that those entities be passively revealed by observation, or even be definite.

Again, I didn't say they have to be passively revealed by observation. I'm not sure what you mean by "even be definite" though. What would an indefinite local hidden variable be like? Certainly we could imagine that certain variables which only take integer values when measured could have non-integer values between measurements, but they all must have some well-defined value.

In fact, those entities could even be of dynamic nature, and in that case you would not talk of definite values will you.

Sure I would. A variable that changes dynamically with time still has a definite value at any given point in time. So, we could imagine an omniscient observer who knows these values at each moment, even if we don't know them.

Look, the basic logic of Bell's proof is based on doing the following:
1. note the statistics seen on trials where both experimenters choose the same measurement angle (the simplest case would be if they always get identical results on these trials)
2. imagine what possible sets of local hidden variables might produce these statistics, if we (or a hypothetical omniscient observer) could see them
3. Show that for all possible sets of local hidden variables that give the right statistics on trials where the experimenters chose the same measurement angles, these hidden variables also make certain predictions about the statistics seen when the experimenters choose different measurement angles, namely that the statistics should satisfy some Bell inequalities
4. Show that quantum mechanics predicts that these same Bell inequalities are violated

The proof does not require that we actually know anything about the specifics of what local hidden variables are present in nature (so it doesn't require that we know the hidden variables associated with a particle or the moon when we aren't looking), it's making general statements about all possible configurations of hidden variables that are consistent with the observed statistics when both experimenters make the same measurement.

Do you disagree that this is the logic of the proof? If you are confident you understand the proof and disagree that this is the basic logic, can you explain where my summary is wrong, and what you think the logic is?

1) P(B|AH) = P(B|H) is NOT guaranteed to be true for a local realist world in which there is no causal influence between A and B. Although causal influence necessarily implies logical dependence, lack of causal influde is not sufficient to obtain lack of logical dependence.

Again, there is not a lack of logical dependence between A and B, since P(B|A) is different from P(B). The point is that in a local realist world, if there is a correlation (logical dependence) between two variables A and B that have a spacelike separation and therefore can't causally influence one another, there must be some cause(s) in the past light cones of A and B which predetermined this correlation.

Like I asked earlier, do you know what a "past light cone" is? If not it's really something you need to research in order to follow any discussion about causality in the context of relativity. If you do know what it means, then suppose we have some event B and we look at its past light cone, and we take the complete set of all facts about what happened in its past light cone (including facts about hidden variables) to be L. Do you disagree that if we know L, then whatever our estimate of the probability of B based on L is (i.e. P(B|L)), further information about some event A which lies outside the past or future light cone of B cannot alter our estimate of the probability of B (i.e. P(B|L) must be equal to P(B|LA)), assuming a universe with local realist laws?

If that wasn't true, then learning B would give us some information about the probability that A occurred, beyond whatever information we could have learned by looking at all the events L in the past light cone of B. Here's a proof--

Show: P(A|LB) not equal to P(A|L), given that P(B|L) not equal to P(B|LA).

Proof: P(A|LB) = P(ALB)/P(LB), by the formula for conditional probability.

P(ALB) can be rewritten as P(B|LA)*P(LA), and likewise P(LB) can be rewritten as P(B|L)*P(L). So, substituting into the above:

P(A|LB) = P(B|LA)*P(LA) / (P(B|L)*P(L))

The formula for conditional probability also tells us that P(A|L) = P(LA)/P(L). So substituting that into the above equation, we get:

P(A|LB) = P(A|L)*P(B|LA)/P(B|L)

From the above equation, the only way P(A|LB) can be equal to P(A|L) is if P(B|LA)/P(B|L) = 1. But we know P(B|LA) is not equal to P(B|L), so this cannot be the case; therefore, P(A|LB) is not equal to P(A|L).

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer. Say A was the event of a particular alien horse on Alpha Centauri winning a race, and B was the event of a buzzer going off in my room; then I know that if I hear the buzzer go off, I should place a bet that when reports of the race reach Earth by radio transmission 4 years later, that particular horse will be the winner, and that will be a piece of information that no one who didn't have access to the buzzer could deduce by examining events in my past light cone. If you think this type of scenario is consistent with relativistic causality in a local realist universe, then I don't know what else to tell you, the idea that you can't gain any new information about an event A by observing an event B at a spacelike separation from it, if you already know all possible information about events in the past light cone of B (or just in a cross-section of the past light cone taken at some time after the last moment when the past light cones of A and B intersected, as I imagined in my analysis in posts 61/62 on the other thread, and is also the assumption used in this paper which discusses relativistic causality as it applies to Bell's analysis, which you should probably look through if my own arguments don't convince you) can basically be taken as the definition of relativistic causality. If you disagree, can you propose an alternate one that's stated in terms of what kind of information you can gain about distant events based only on local observations? Or do you think relativity and local realism place absolutely no limits on information you can gain about events outside your past light cone, allowing arbitrary forms of FTL communication?

The example I in the first few posts points this out clearly

The example you quoted doesn't contradict my point about past light cones. If you knew about everything in the past light cone of opening your envelope, including facts about which cards were inserted into the envelopes before they were sent and what happened to your envelope on its journey to you, then you would already know what color card you'd find before you opened it, and if your friend later knew what card was found in the other envelope and was watching a video of you opening your envelope (and the friend also had full knowledge of everything in the past light cone of your opening your envelope), then that additional knowledge of what happened when the second envelope was opened wouldn't change their prediction about what would happen when you opened yours.

It is OK to go from conditional independence to the equation P(B|AH) = P(B|H) due to conditional independence, but it is definitely not OK to go from causal independence to P(B|AH) = P(B|H).

If we're in a local realist universe respecting relativity, and H represents complete knowledge of every physical fact in the past light cone of B (or every fact in a cross-section of the past light cone taken at some time after the last moment the past light cones of A and B intersected), then yes it is OK. If you disagree, you don't understand relativistic causality.

By the way I use the term conditional independence because that is exactly what the above equation means. P(B|AH) = P(B|H) means that B is conditionally independent of A with respect to H, or A and B are independent conditioned on H.

OK, but when the paper you quoted to support your argument said:

X [is independent of] Y if any information received about Y does not alter uncertainty about X;

They weren't talking about X and Y being conditionally independent with respect to some other variable H, they were talking about X and Y being conditionally independent in the absolute sense that P(X and Y) = P(X)*P(Y). If they wanted to talk about conditional independence with respect to some other variable they would have written:

X is independent of Y with respect to H if any information received about Y does not alter uncertainty about X given H

2) In Bell's treatment, Hidden variables are supposed to be responsible for the correlation between A and B. However, when Bell write the equation as follows:

P(AB|H) = P(A|H) * P(B|H)

This equation means that conditioned on H, there is no correlation between A and B. Now please take a moment and let this sink in because I have the feeling you have not understood this point.

Yes, I understand perfectly well that there is no correlation on A and B conditioned on H, given how Bell's theorem defines H in terms of the complete set of information about all physical facts (including facts about hidden variables) in the cross-sections of the the past light cones of A and B, with the cross-sections taken after the last moment that their past light cones intersect. That was the central basis of my argument in posts 61 and 62 on the the other thread, and it's also discussed extensively in the online paper I linked to above.

Nevertheless, there is a correlation between A and B in absolute terms--if you do a large collection of trials and just look at incidences of A and B, the probability that B happens is different in the subset of trials where A happened than it is in the complete set of all trials (i.e. P(B|A) is different than P(B)).

Note on the left hand side, we have a conditional probability, not a marginal probability, so it is irrelevant whether there is a correlation between A and B marginally or not. The equation clearly says conditioned on the hidden variables, there is no correlation between A and B.

And yet, those same hidden variables are supposed to be responsible for the correlation. This is the issue that concerns me.

Huh? The hidden variables are responsible for the correlation which exists in absolute terms--you know, the correlation that is seen by actual experimenters doing experiments with entangled particles! Since hidden variables are by definition "hidden" to actual experimenters, we have no experimental data about whether there is a correlation between measurements conditioned on the hidden variables, and thus the idea that there's an absolute correlation but no correlation when conditioned on the hidden variables is perfectly consistent with all real-world observations. And if you understood the nature of relativistic causality you'd see that A and B cannot possibly be correlated when conditioned on H, if H represents the complete set of physical facts about past light cone cross-sections of A and B taken after the last moment when the past light cones of A and B intersected.

Giving examples in which A and B are marginally dependent but conditionally independent with respect to H as you have given, does not address the issue here at all. Instead it goes to show that in your examples, the correlation is definitely due to something other than the hidden variables! Do you understand this?

What? Suppose A is the event of me opening an envelope and finding a red card, and B is the event of you opening an envelope and finding a white card, with these two events happening at a spacelike separation. Let H1 represent the complete set of physical facts about everything in the past light cone of A at some time t after the last moment that the past light cones of A and B intersect, and H2 represent the complete set of physical facts about everything in the past light cone of B at the same time t. H can represent the combination of facts in H1 and H2. Now, H1 necessarily includes the fact that the envelope traveling towards me had a red card in it at that moment, and H2 includes the fact that the the envelope traveling towards you had a white card in it at that moment, so H includes both of these facts. Are you arguing that knowing H is not sufficient to completely determine the fact that we will find opposite colors when we open our respective envelopes and look at the cards? Isn't it true that if we know H on multiple trials like this and in each case H tells us the hidden card in the envelope on its way to me was the opposite color to the hidden card in the envelope on its way towards you, that is sufficient to determine that we will always find opposite colors on opening our envelopes (i.e. knowing H for each trial fully determines the correlation between our results on each trial), and that the probability you will find a white card is conditionally independent of the probability I will find a red card with respect to H? (i.e. if you already know what hidden cards were in the envelopes at some time t when they were on their path to us, your estimate of the probability that you found a white card is not altered by the knowledge that I found a red card when I actually opened my envelope)

3) If as you admit, the inequalities can not be derived from P(AB|H) = P(A|H) * P(B|AH), even though you claim the equation is equivalent to P(AB|H) = P(A|H) * P(B|H) in Bell's case, can you tell me why substituting one equation with another which is equivalent, should result in different inequalities, unless they are not really equivalent to start with?

This is a totally bizarre question. I mean, have you ever seen a proof of anything in physics before? You always start with some physical assumptions, then derive a series of equations, each one derived from previous ones using rules which follow either from your physical assumptions or from mathematical identities. Eventually you reach some final equation which is the conclusion you wanted to prove. Given the assumptions of the problem, each new equation is "equivalent" to a previous equation, or to some combination of previous equations. What you seem to be asking here is, "if all the equations in the proof are equivalent to previous ones, why can't I reach the final conclusion using only mathematical identities like P(AB|H) = P(A|H)P(B|AH), without being allowed to make substitutions that depend specifically on the physical assumptions of the problem like P(B|AH)=P(B|H)?" I don't really know how to respond except by saying "Uhhh, it doesn't work that way, in a physics proof you can't get from your starting equations to your final equation using only transformations of equations that are based on pure math, you have to make use of some actual, y'know, physics in some of your transformations. After all, no one said the final concluding equation was 'equivalent' to the starting equations in a purely mathematical sense, they are only equivalent given the specific physical assumptions you're using in the proof." Really, find me an example of any other proof/derivation in physics (say, a derivation of E=mc^2 from the more basic assumptions in relativity), and I'm sure there'd be some step where some physical assumption is used to transform equation(s) X into equation Y (i.e. X and Y are 'equivalent' given the physical assumptions of the problem), and yet equation X would not suffice to derive the final conclusion if we weren't allowed to make any further transformations based on physical assumptions.

 

[DrChinese]

A bit more reading here, so it's related to Malus' Law right?
If the QM model is based off this and this would be the intensity after polarization, it is also the ratio of light percentage that is polarized already to the chosen theta.

So the QM percentage is working from the ratio between percentage of possible existing state to theta. Yet for some reason the follow thru to A,B,C, doesn't seem to be the correct ratio comparison, I can see the 1:3 option for each state but in the case of the polarized beam isn't that based upon the unknown ratio for each A, B, C, state?

Malus does enter into it, yes. But it is just a bit tricky, as the same formula - cos^2(theta) - comes into play several different ways. Because of that, they look the same but may not be entirely.

 

[yoda jedi]

reality.
Exists objective ontological entities apart from the act of observation. not passively revealed by observation, or even be definite.

i agree.

reality does not need, counterfactual definiteness or indefiniteness, contextuality or non contextuality, determinism or indeterminism etc...
REALITY is:

"Being Qua Being"

 

[madhatter106]

Malus does enter into it, yes. But it is just a bit tricky, as the same formula - cos^2(theta) - comes into play several different ways. Because of that, they look the same but may not be entirely.

Ok,
I see that Bells' inequality would graph out as a straight line and the QM would be a sine wave due to Malus' cosine function. Is it wrong to assume then that X,Y,Z probabilities should also be functions of ratios so that the straight line would approach the sine of QM. When I read over the setup the X,Y,Z states are strict individually to a single plane and due to the polarization eq the cos^2 function that is used to deal with the spherical nature of luminosity.

the source light is entangled states of polarization that average a specific luminosity. the source starts in a spherical output and the measurement is in a single plane, then isn't the unpolarized luminosity that is absorbed pure light? it's and even ratio of all possible states and thus not visible. With that the source light is now part of the probability of one of those states, if the light source had any polarization it then is affecting the probability. pure unpolarized light wouldn't be visible, and using a "created" unpolarized source then it's already a known state. the EM field is then the balanced unpolarized state and the change in energy creates a polarization since theta moves from 0 to ? depending upon the way it was created.

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

I'm I seeing it wrong?

 

[JesseM]

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

Things are only odd if you want to explain the statistics using a local realist theory (like classical electromagnetism from which Malus' law is derived), then you find that the statistics predicted by QM are incompatible with the assumptions of local realism. Are you having trouble understanding why? If so you might take a look at the lotto card analogy I offered in post #18 (where different boxes on the card stand for different detector angles, and getting a cherry or lemon when a given box is scratched stands for getting spin-up or spin-down with a given detector angle), which starts in the paragraph beginning with "Suppose we have a machine that generates pairs of scratch lotto cards"...

 

[DrChinese]

Ok,
I see that Bells' inequality would graph out as a straight line and the QM would be a sine wave due to Malus' cosine function. Is it wrong to assume then that X,Y,Z probabilities should also be functions of ratios so that the straight line would approach the sine of QM. When I read over the setup the X,Y,Z states are strict individually to a single plane and due to the polarization eq the cos^2 function that is used to deal with the spherical nature of luminosity.

the source light is entangled states of polarization that average a specific luminosity. the source starts in a spherical output and the measurement is in a single plane, then isn't the unpolarized luminosity that is absorbed pure light? it's and even ratio of all possible states and thus not visible. With that the source light is now part of the probability of one of those states, if the light source had any polarization it then is affecting the probability. pure unpolarized light wouldn't be visible, and using a "created" unpolarized source then it's already a known state. the EM field is then the balanced unpolarized state and the change in energy creates a polarization since theta moves from 0 to ? depending upon the way it was created.

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

I'm I seeing it wrong?

Somewhat. It is easier to follow some of the issues if you remember that there are several ways to determine the polarization of light. The "best" way (of course depends on the situation ) involves using a polarizing beam splitter, a PBS. You can orient this at any angle, and it will split the beam into an H component and a V component. Of course, that is relative to its axis. In this manner, you can see than the PBS does not itself change the light in some manner that you consider to be "active".

Not sure all of what you are asking, but you should definitely check out some of the traditional sources on optical physics. With entangled pairs, you will be looking at single photons but much of the same rules apply. But some do not.

 

[billschnieder]

No, I'm not saying I can see it. I'm saying the hypothetical omniscient observer can see it.

Doesn't matter, there are blind people on Earth who will never see the moon. "seeing the moon" is not a variable that belongs to the moon and has a definite outcome. Seeing the moon is contextual, for a blind person it does not exist at all. An omniscient being can not "see the moon" if they are not looking at it, neither can they know that "Tom can see the moon" if Tom is not looking at the moon. Simply being aware that the moon exists is a different observable from "seeing the moon". And the latter, does not have a definite outcome prior to observation. So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

Again, I didn't say they have to be passively revealed by observation. I'm not sure what you mean by "even be definite" though. What would an indefinite local hidden variable be like? Certainly we could imagine that certain variables which only take integer values when measured could have non-integer values between measurements, but they all must have some well-defined value.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you? However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed. The latter is the EPR definition of realism, the former is definitely not. We will have to agree to disagree here.

Look, the basic logic of Bell's proof is based on doing the following:
...
Do you disagree that this is the logic of the proof? If you are confident you understand the proof and disagree that this is the basic logic, can you explain where my summary is wrong, and what you think the logic is?

I already explained the logic in the first post, what about that logic which started this thread is unclear or wrong to you? I believe it is clear from that post that if premise (1) fails, the whole logic fails with it. Premise (1) defines how local hidden variable theories consistent with QM and EPR should behave. That premise is my focus and that is why I keep trying to focus the discussion on that point because it is easy to get off-topic without addressing that central issue.

I do not see in your responses so far a convincing reason why we should use
P(AB|H) = P(A|H)*P(B|H) and not P(AB|H) = P(A|H)*P(B|AH)

That is not to say you have not given reasons, just that they are not convincing for reasons I have outlined already.

Again, there is not a lack of logical dependence between A and B, since P(B|A) is different from P(B). The point is that in a local realist world, if there is a correlation (logical dependence) between two variables A and B that have a spacelike separation and therefore can't causally influence one another, there must be some cause(s) in the past light cones of A and B which predetermined this correlation.

Like I asked earlier, do you know what a "past light cone" is?

I ignored that question , because it is an irrelevant distraction from the central issue, and it is so obvious I don't even understand why you bring it up.

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer.

Herein lies the crux of the misunderstanding. In the situation being modeled by Bell, we are not calculating the probability of an event a Alice, we are calculating the probability of a joint event or coincidence between Alice and Bob. Again, note that it is not possible to determine that there is a coincidence unless you jointly consider both outcomes at Alice and Bob. This is the reason why you MUST still use
P(AB|H) = P(A|H)*P(B|AH)

Look at the left hand side, it says the probability of the joint event AB conditioned on H. You have probably heard it asked, "why can't we send information by FTL if it really possible?" The answer comes back to this equation. It is not possible to determine that a coincidence has occurred unless you have access to the results from each side. That is why you need the P(B|AH) because it ensures that the coincidences can be accounted for. However, as I have pointed out already. Therefore by writing the equation as
P(AB|H) = P(A|H)*P(B|H)
Bell has effectively restricted his model to only those situations in which there is no correlation conditioned on H. And in that case, to perform an experiment exactly according to what Bell modeled will require that the experimenters know exactly the nature of H, in order to effectively screen it out.

P(AB|H) = P(A|H)*P(B|H)
Clearly means that conditioned on H, there is no correlation between A and B. It is therefore impossible to for H to cause any correlations whatsoever with this equation. Now can you explain how it is possible for an experimenter to collect data consistent with this equation, without knowing the exact nature of H?

If we're in a local realist universe respecting relativity, and H represents complete knowledge of every physical fact in the past light cone of B (or every fact in a cross-section of the past light cone taken at some time after the last moment the past light cones of A and B intersected), then yes it is OK. If you disagree, you don't understand relativistic causality.

So long as you are trying to link an event about A and B such as coincidences, the simple act of trying to calculate a joint probability forces you to use P(AB|H) = P(A|H)*P(B|AH) and not P(AB|H) = P(A|H)*P(B|H).

It is only possible for A and B to be marginally correlated while at the same time uncorrelated conditioned on H, if H is NOT the cause of the correlation.

Yes, I understand perfectly well that there is no correlation on A and B conditioned on H, given how Bell's theorem defines H in terms of the complete set of information about all physical facts

Are you sure you understand that it? Can you explain how the hidden variables H are supposed to be responsible for the correlation between A and B, and yet conditioned on H there is no correlation between A and B. I do not see anything you have written so far in this thread or the other one answers this question.

Nevertheless, there is a correlation between A and B in absolute terms--if you do a large collection of trials and just look at incidences of A and B, the probability that B happens is different in the subset of trials where A happened than it is in the complete set of all trials (i.e. P(B|A) is different than P(B)).

Huh? The hidden variables are responsible for the correlation which exists in absolute terms--you know, the correlation that is seen by actual experimenters doing experiments with entangled particles! Since hidden variables are by definition "hidden" to actual experimenters, we have no experimental data about whether there is a correlation between measurements conditioned on the hidden variables, ...

In case you are not sure about the terminology, in probability theory, P(AB) is the joint marginal probability of A and B which is the probability of A and B regardless of whether anything else is true or not. P(AB|H) is the joint conditional probability of A and B conditioned on H, which is the probability of A and B given that H is true. There is no such thing as the absolute probability.

... and thus the idea that there's an absolute correlation but no correlation when conditioned on the hidden variables is perfectly consistent with all real-world observations.

I agree, there are cases in which a correlation may exist between A and B marginally, but will not exist when conditioned on another variable, like in some of the example you have give. However, the EPR case being modeled by Bell is not one of such, precisely because Bell is trying to introduce hidden variables H which should be responsible for the correlation between A and B. Do you understand this?

This is a totally bizarre question. I mean, have you ever seen a proof of anything in physics before? You always start with some physical assumptions, then derive a series of equations, each one derived from previous ones using rules which follow either from your physical assumptions or from mathematical identities.

The question is very clear. Let me put it to you in point form and you can give specific answers to which points you disagree with.

1) You say in the specific example treated by Bell, P(B|AH) = P(B|H). It is not me saying it. Do you disagree?
2) The above statement (1) implies that in the specific example treated by Bell, where the symbols A, B and H have identical meaning, P(B|AH) and P(B|H) are mathematical identities. Do you disagree?
3) The above statement (2) implies that in the specific example treated by Bell, where the symbols A, B and H have identical meaning, P(A|H)*P(B|AH) and P(A|H)*P(B|H) are mathematical identities. Do you disagree?
4) The above statement (3), implies that if using P(A|H)*P(B|H) results in one set of inequalities, the mathematically identical statement P(A|H)*P(B|AH) should result in the same set of inequalities where the symbols A, B and H have identical meaning. Do you disagree?
5) Given the above (1-4). Explain to me why it is not possible to obtain the same inequalities by using either P(A|H)*P(B|AH) or P(A|H)*P(B|H).

Eventually you reach some final equation which is the conclusion you wanted to prove. Given the assumptions of the problem, each new equation is "equivalent" to a previous equation, or to some combination of previous equations. What you seem to be asking here is, "if all the equations in the proof are equivalent to previous ones, why can't I reach the final conclusion using only mathematical identities like P(AB|H) = P(A|H)P(B|AH), without being allowed to make substitutions that depend specifically on the physical assumptions of the problem like P(B|AH)=P(B|H)?" I don't really know how to respond except by saying "Uhhh, it doesn't work that way

Bell himself says in his original paper that it is not difficult to reproduce the QM correlations using an equation like P(A|H)*P(B|AH). If two equations are mathematically equivalent, they should give the same numerical result, no?

 

[DrChinese]

So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you? However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed. The latter is the EPR definition of realism, the former is definitely not.

You haven't given such an example. And you keep ignoring the EPR definition: if the sun's color can be predicted with 100% certainty WITHOUT first observing it or disturbing it in any way... then there is an element of reality to its color.

If that is not your definition - and it sounds like it isn't because of the first paragraph: then really, why would anyone care? Why should people care about Schneider's* definition? Wouldn't we want to discuss something with a shared meaning here at a public forum? Bell is all about demonstrating that QM is inconsistent with observer independent elements of reality. That is the shared vision of Bell. So if you want to understand his reasoning, try looking there. It has nothing to do with statistical notation.

*A cryptic play on words :tongue:

 

[billschnieder]

You haven't given such an example. And you keep ignoring the EPR definition: if the sun's color can be predicted with 100% certainty WITHOUT first observing it or disturbing it in any way... then there is an element of reality to its color

Consistent with EPR, I can predict the observed color in a specific context if I know everything about all the elements of reality that are part of the specific context. Yet I can not say the color of the sun exists prior to realization of the specific context. Therefore observables having definite values prior to observation is definitely not the EPR definition. The EPR definition, is "existence of elements of reality which deterministically result in the observables" such that it is possible to predict in advance, what would obtain given all the parameters of a specific context.

 

[JesseM]

Doesn't matter, there are blind people on Earth who will never see the moon.

But "hidden" means hidden to the entire community of human experimenters who can share information with one another. Of course in a local realist universe it might be that future experiments could allow us to observe formerly hidden variables, but it's not important to the proof one way or another.

"seeing the moon" is not a variable that belongs to the moon and has a definite outcome.

I don't know what "belongs to the moon" means. In a local hidden variables theory every basic variable should be associated with a particular point in spacetime. If you want to talk about a human "seeing the moon", that would presumably be shorthand for a certain combination of states of variables in the volume of spacetime where the human was making that observation (a macrostate corresponding to a particular 'microstate' involving the exact values of all the local variables in that region), including variables associated with the location of photons arriving at the human's location from the moon.

Seeing the moon is contextual, for a blind person it does not exist at all.

In a local realist theory there is an objective truth about which variables are associated with a given point in spacetime (and the values of those variables). This would include any variables associated with the region of spacetime occupied by the moon, and any associated with the region of spacetime occupied by a human. The variables associated with some humans might correspond to a state that we could label "observing the moon", and the variables associated with other humans might correspond to a state we could label "not observing the moon", but the variables themselves are all assumed to have an objective state that does not depend on whether anyone knows about them.

A "contextual" hidden variables theory is one where knowledge of H is not sufficient to predetermine what results the particle will give for any possible measurement of a quantum-mechanical variable like position or momentum, the conditions at the moment of measurement (like the exact state of the measuring device at the time of measurement) can also influence the outcome--see p. 39 here on google books, for example. This doesn't mean that all fundamental variables (hidden or not) associated with individual points in spacetime don't have definite values at all times, it just means that knowing all variables associated with points in the past light cone of the measurement at some time t does not uniquely determine the values of variables in the region of spacetime where the measurement is made (which tell you the outcome of the measurement).

An omniscient being can not "see the moon" if they are not looking at it,

As I said before, "omniscient being" is just a cute way of describing what it would be like if all hidden variables were known. You're taking the metaphor way too seriously if you imagine an omniscient being who only knows the values of hidden variables if he is "looking at" them; the only reason for invoking such a being is so we can talk about the objective states of all hidden variables that might influence observable experimental results.

neither can they know that "Tom can see the moon" if Tom is not looking at the moon.

In a local realist universe there is an objective truth about all local variables, and descriptions of macroscopic facts like "Tom seeing the moon" are just shorthand for certain combinations of local variables, much like macrostates vs. microstates in statistical mechanics. So there's an objective truth about whether "Tom sees the moon" is true or false in some particular region of spacetime containing Tom, and the omniscient observer knows whether it's true or false.

Simply being aware that the moon exists is a different observable from "seeing the moon". And the latter, does not have a definite outcome prior to observation. So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

I never once said that observables have definite values prior to observation. I said that all the fundamental physical variables, hidden or otherwise, have definite values at all times. But an "observable" is the outcome of a particular measurement, and it's certainly possible that fundamental physical variables associated with points on the particle's worldline don't uniquely determine this, that fundamental physical variables associated with the measuring device also influence the outcome.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you?

Not unless "color" is one of the fundamental physical variables associated with particular points in spacetime. But whatever these variables are, they do have objective values at every single point.

However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed.

Yes, local "elements of reality" associated with particular points in spacetime, such that all macroscopic facts can be reduced to combinations of facts about these fundamental facts, are what I have been talking about all along.

Look, the basic logic of Bell's proof is based on doing the following:
1. note the statistics seen on trials where both experimenters choose the same measurement angle (the simplest case would be if they always get identical results on these trials)
2. imagine what possible sets of local hidden variables might produce these statistics, if we (or a hypothetical omniscient observer) could see them
3. Show that for all possible sets of local hidden variables that give the right statistics on trials where the experimenters chose the same measurement angles, these hidden variables also make certain predictions about the statistics seen when the experimenters choose different measurement angles, namely that the statistics should satisfy some Bell inequalities
4. Show that quantum mechanics predicts that these same Bell inequalities are violated

The proof does not require that we actually know anything about the specifics of what local hidden variables are present in nature (so it doesn't require that we know the hidden variables associated with a particle or the moon when we aren't looking), it's making general statements about all possible configurations of hidden variables that are consistent with the observed statistics when both experimenters make the same measurement.

Do you disagree that this is the logic of the proof?

I already explained the logic in the first post, what about that logic which started this thread is unclear or wrong to you?

Obviously any short description of "the logic" leaves some stuff out. I am not saying there was anything incorrect about your points 1-3 summarizing Bell's logic, I'm just saying it leaves out any discussion of how Bell arrived at his equation which you mention in 1. My point above is that he does this by imagining we (or a hypothetical omniscient observer) know what the hidden-variables state is, and considering all possible hidden-variables states that could lead to the observed statistics when the experimenters choose to make the same measurement. Do you think this is an incorrect characterization of what Bell is doing?

I do not see in your responses so far a convincing reason why we should use
P(AB|H) = P(A|H)*P(B|H) and not P(AB|H) = P(A|H)*P(B|AH)

That is not to say you have not given reasons, just that they are not convincing for reasons I have outlined already.

But you have not explained whether you disagree with my statements about complete knowledge of all physical variables in the past light cone of some measurement-event, or if so, why. Perhaps this is because you were misunderstanding me and thinking I was talking about "observable", even though I never suggested this. Now that you (hopefully) understand that I am talking about the fundamental local physical variables that must completely determine all macroscopic physical facts in a universe obeying local realist laws, I will re-ask the question, and if you are actually making a good-faith argument here rather than just trying to rhetorically discredit me, I hope you will give me a straight answer:

suppose we have some event B and we look at its past light cone, and we take the complete set of all facts about what happened in its past light cone (including facts about hidden variables) to be L. Do you disagree that if we know L, then whatever our estimate of the probability of B based on L is (i.e. P(B|L)), further information about some event A which lies outside the past or future light cone of B cannot alter our estimate of the probability of B (i.e. P(B|L) must be equal to P(B|LA)), assuming a universe with local realist laws?

Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer.

Herein lies the crux of the misunderstanding. In the situation being modeled by Bell, we are not calculating the probability of an event a Alice, we are calculating the probability of a joint event or coincidence between Alice and Bob.

We are calculating multiple things. In particular, when we go from the mathematical identity P(AB|H) = P(A|H)*P(B|AH) to the equation P(AB|H) = P(A|H)*P(B|H), we are doing the substitution P(B|AH) = P(B|H), and P(B|H) clearly refers to the conditional probability of the single event B at Bob's location given the hidden-variables state H, not to a joint event between Alice and Bob. And you may notice that the equation P(B|H) = P(B|AH) looks a lot like the equation in my question about the light cone above, P(B|L) = P(B|LA). And the section of my post you quoted above was simply about the fact that if P(B|L) was not equal to P(B|LA), this would imply FTL information transmission which is inconsistent with locality, which is trying to show why in a local realist universe it must be true that P(B|L) = P(B|LA). So again, I am hoping for a simple answer for you on the question of whether you agree that local realism does imply that equation must be true in the scenario I described (if you do, then we can go on to examine how well Bell's assumptions about the physical meaning of H resemble my assumptions about the physical meaning of L).

Again, note that it is not possible to determine that there is a coincidence unless you jointly consider both outcomes at Alice and Bob. This is the reason why you MUST still use
P(AB|H) = P(A|H)*P(B|AH)

Look at the left hand side, it says the probability of the joint event AB conditioned on H. You have probably heard it asked, "why can't we send information by FTL if it really possible?"

When I talked about FTL information transmission I wasn't talking about a joint event. Again, I was saying that if P(B|L) was different from P(B|LA), that would imply the possibility of FTL transmission, so we can be confident that in a local realist universe P(B|L) must be equal to P(B|LA). Again, I need an answer to this simple question before I can proceed with the argument.

The answer comes back to this equation. It is not possible to determine that a coincidence has occurred unless you have access to the results from each side. That is why you need the P(B|AH) because it ensures that the coincidences can be accounted for. However, as I have pointed out already. Therefore by writing the equation as
P(AB|H) = P(A|H)*P(B|H)
Bell has effectively restricted his model to only those situations in which there is no correlation conditioned on H. And in that case, to perform an experiment exactly according to what Bell modeled will require that the experimenters know exactly the nature of H, in order to effectively screen it out.

No, the experimenter doesn't know H, H represents variables that can be hidden from the experimenter. Again, see my statement above about the logic of the proof involving imagining we (or an omniscient observer) could know the state of all fundamental physical variables, and derive some statements that logically would have to be true in a local realist universe for any possible state of those fundamental variables which are consistent with the observed results when the experimenters choose the same angles. Please tell me whether you agree or disagree with this statement about Bell's logic in the proof.

Have to go now, will respond to the rest later...

 

[DrChinese]

Consistent with EPR, I can predict the observed color in a specific context if I know everything about all the elements of reality that are part of the specific context. Yet I can not say the color of the sun exists prior to realization of the specific context. Therefore observables having definite values prior to observation is definitely not the EPR definition. The EPR definition, is "existence of elements of reality which deterministically result in the observables" such that it is possible to predict in advance, what would obtain given all the parameters of a specific context.

I don't know if you are talking about semantics or substance.

According to EPR, if I can predict with certainty the result of a measurement of Bob without first observing or disturbing Bob, then there is an element of reality in Bob's observable. There need be no determinism involved regarding Bob, and the outcome could be completely random with no apparent cause. It only needs to be predictable in advance. I would say, by most standards, that means it has a specific value. I don't need to know anything about the context other than what it takes to predict, either. Now, how do you read EPR differently? My reading is about as exact as can be short of quoting EPR, and I assume you have read it. What is there to question here?

 

[DrChinese]

I never once said that observables have definite values prior to observation...

This was the EPR conclusion. And where Bell started from. So it is relevant.

 

[madhatter106]

Tell me if I got this right, from what I understand this isn't about the probability of permutations from an unknown group but the permutations of a 'known' group.

I take it like this, I have a ball that is 1/2 black 1/2 white and when split it will form 1 black and 1 white ball. according to classical physics you are always to going to end up with that arrangement measured or not, whereas QM states that until measured you could have 2 of the same color and that by measuring the one it automatically sets the other.

I'm a bit confused as to why this is problem? by having the 'envelope' of the experiment a known contained value it doesn't remove the probability that you'll need to measure at least one variable to know the other.

Is the confusing bit that there is a possibility that when measuring one value that it still does not mean that what you measured will determine the other, so that if let's say you measure a white ball and assume the other is black but upon receiving the information that the other is white as well it changes your previously measured value since you've become aware of the other state? and that change would have to alter the measurement in 'negative' values.

strange as it seems that makes sense to me, I though probably do not have a conventional view of photons, which allows me to accept that possibility however odd. I think in the classical form that assumption can be made but not from looking at the individual eq. but the whole picture to infer the possibility. I probably sound crazy, I'm just going on how my meager view of physics is to look at the entire range together instead of separately.

 

[DrChinese]

Tell me if I got this right, from what I understand this isn't about the probability of permutations from an unknown group but the permutations of a 'known' group.

I take it like this, I have a ball that is 1/2 black 1/2 white and when split it will form 1 black and 1 white ball. according to classical physics you are always to going to end up with that arrangement measured or not, whereas QM states that until measured you could have 2 of the same color and that by measuring the one it automatically sets the other.

If Alice is one color, Bob is always the expected color. That is not what is in question. And as long as you look at the issue that way - as EPR did - there is the possibility of a classical solution.

The issue has to do with when you look at shades, i.e. angles that are not 90 or 180 degrees apart. At various settings, 0/120/240 being a great one to study, things stop making sense. You must look at that example in detail (or one like it) to understand anything. Or go to the "DrChinese Easy Math" page (just google that) and it lays it out. You already follow the 1/3 bit, so the next part is to realize that is an upper limit and that QM (and experiment) give a value of 1/4. As a result, the EPR logic (elements of reality) is refuted.

 

[madhatter106]

If Alice is one color, Bob is always the expected color. That is not what is in question. And as long as you look at the issue that way - as EPR did - there is the possibility of a classical solution.

The issue has to do with when you look at shades, i.e. angles that are not 90 or 180 degrees apart. At various settings, 0/120/240 being a great one to study, things stop making sense. You must look at that example in detail (or one like it) to understand anything. Or go to the "DrChinese Easy Math" page (just google that) and it lays it out. You already follow the 1/3 bit, so the next part is to realize that is an upper limit and that QM (and experiment) give a value of 1/4. As a result, the EPR logic (elements of reality) is refuted.

I did went through most of it last night and thank you, it was a good read. When I see the example of 0/120/240 I instantly go back to trig and the periodic function of those values. the cos^2 value ratio in respect to theta is integral to the outcome.

does the question become why at ratios other than 1:1:sqrt2 do things stop making sense? graphing that ratio will always be a straight line by it's definition and the other a wave with periodic rates. so anything other than right angles will have anomalous results, esp as cosine or sine theta approaches infinity right?

So fundamentally the EM field has some hidden attribute that when the charge is not perpendicular there are strange results. this would be akin to saying that there is another 'variable' between b and e on the EM field that affects those states, yes?

 

[ThomasT]

... Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails.

I agree with this statement, but not necessarily for the reason you gave. (I don't fully understand it yet, having read through the thread quickly.)

Bell's formulation is sufficient to rule out a certain set of lhv theories, but it doesn't imply anything about Nature except that the disparity between Bell's ansatz (and thus Bell inequalities) and the experimental situations does make violations of Bell inequalities useful as indicators of entanglement.

Here's some observations:

1. The hidden variable, H in your notation, is irrelevant in the joint context. Coincidence rate, P(A,B), is solely a function of Theta, the angular difference of the polarizers.

You wrote (replying to another poster):
And yet, those same hidden variables are supposed to be responsible for the correlation. This is the issue that concerns me. Giving examples in which A and B are marginally dependent but conditionally independent with respect to H as you have given, does not address the issue here at all. Instead it goes to show that in your examples, the correlation is definitely due to something other than the hidden variables! Do you understand this?

I think I understand this.

2. The relevant hidden variable is the relationship (wrt some common motional property, usually spin and polarization because of the relative frequency of optical Bell tests) between the entangled entities. This relationship is the physical entanglement, and it is the deep cause of the observed correlations. This relationship, the entanglement, varies so slightly from pair to pair that it's , effectively, a constant, and can only be produced via quantum processes -- and this is accounted for in the QM treatment via an emission model applied to a particular preparation. In other words, QM assumes a local common cause for the entanglement.

3. Bell's locality condition reduces to P(A,B) = P(A)P(B) , which is the definition of statistical independence.

4. The observed statistical dependence is essentially due to three local (c-limited) processes: a) the production of entanglement via emission, b) the filtration of the entangled entities by a global measurement parameter, the angular difference of the crossed polarizers, and c) the data matching process, the final link in a local causal chain that ultimately produces the statistical dependence.

The requirements set forth by Bell for an lhv theory of entanglement seem to be at odds with the reality of the experimental situation(s) that produce the correlations that allow the conclusion that entanglement has been produced. So, it shouldn't be surprising that inequalities based on Bell's formulation are violated by Bell tests as well as the predictions of QM.

However, despite the problematic nature of lhv accounts of entanglement, the foundation of a c-limited, locally causal understanding of entanglement is at hand.

 

[DrChinese]

So fundamentally the EM field has some hidden attribute that when the charge is not perpendicular there are strange results. this would be akin to saying that there is another 'variable' between b and e on the EM field that affects those states, yes?

That would be an attempt to restore local realism, which just won't be possible. Recall that you can entangle particles at other levels as well, such as momentum/position or energy/time. Although it shows up as one thing for spin, you cannot explain it in the manner you mention.

Even for spin, if you look at it long enough, you realize that there is no solution to the mathematical problems. Bell's Inequality is violated because there is no local realistic solution possible.

 

[billschnieder]

JesseM:
Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

The equation
P(AB|L) = P(A|L)P(B|L)
Is NOT NECESSARILY true given the definition of L as encompassing all facts about fundamental physical variables in the past light cones of A and B. Note the emphasis! So don't give me an example in which it is true and claim that it is always true. In case you are not aware, your claim that the above equation is ALWAYS true, for local elements of reality is what is well known as the prinicple of common cause (PCC). There are numerous treatments showing the problems with it and I don't need to go into that here. Look up Simpson's paradox and Bernstein's Paradox.

This is discussed in the Stanford Encyclopedia of Philosophy available online here: http://seop.leeds.ac.uk/entries/physics-Rpcc/

The simple reason the above is not always true is because it is not always possible to specify L such that it "screens off" the correlation as those paradoxes mentioned above indicate.

ThomasT:
The requirements set forth by Bell for an lhv theory of entanglement seem to be at odds with the reality of the experimental situation(s) that produce the correlations that allow the conclusion that entanglement has been produced.

I agree.
P(AB|L) = P(A|L)P(B|L)
Means that there is no longer any correlation between A and B conditioned on L, because it has been screened-off by L. Deriving Bell's inequalities using the above equation implies that the only data (A, B) capable of being compared with the inequalities must be uncorrelated. In order to collect such data will require the experimenters to know exactly the nature of the hidden variables in order to collect it. Therefore Bell's inequalities apply only to independent or uncorrelated data.

 

[JesseM]

JesseM:
Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

The equation
P(AB|L) = P(A|L)P(B|L)
Is NOT NECESSARILY true given the definition of L as encompassing all facts about fundamental physical variables in the past light cones of A and B. Note the emphasis! So don't give me an example in which it is true and claim that it is always true.

Given local realism, yes it is. Do you disagree that if P(B|L) was not equal to P(B|LA), that would imply P(A|L) is not equal to P(A|BL), meaning that learning B gives us some additional information about what happened at A, beyond whatever information we could have learned from anything in the past light cone of B (proof in post #41)? Do you disagree that this is a type of FTL information transmission, since we're learning about an event outside our past lightcone that can't be derived from any information in our past light cone? If you do disagree that this is FTL information transmission, can you explain how you would define FTL information transmission which presumably must be forbidden in a local relativistic theory?

In case you are not aware, your claim that the above equation is ALWAYS true, for local elements of reality is what is well known as the prinicple of common cause (PCC). There are numerous treatments showing the problems with it and I don't need to go into that here. Look up Simpson's paradox and Bernstein's Paradox.

This is discussed in the Stanford Encyclopedia of Philosophy available online here: http://seop.leeds.ac.uk/entries/physics-Rpcc/

Can you find any sources that claim the principle of common cause would be violated in a relativistic universe with local realist laws, where the "cause" can stand for every possible local microscopic fact in the past light cone of one of the two events? Most of the problems discussed in the article you link to above arise from the fact that they are trying to find "causes" that are vague macro-descriptions which don't specify all the precise microscopic details which might influence the correlations. Note in the "conclusions" section where they say:

One should also not be interested in common cause principles which allow any conditions, no matter how microscopic, scattered and unnatural, to count as common causes. For, as we have seen, this would trivialize such principles in deterministic worlds, and would hide from view the remarkable fact that when one has a correlation among fairly natural localized quantities that are not related as cause and effect, almost always one can find a fairly natural, localized prior common cause that screens off the correlation. The explanation of this remarkable fact, which was suggested in the previous section, is that Reichenbach's common cause principle, and the causal Markov condition, must hold if the determinants, other than the causes, are independently distributed for each value of the causes. The fundamental assumptions of statistical mechanics imply that this independence will hold in a large class of cases given a judicious choice of quantities characterizing the causes and effects. In view of this, it is indeed more puzzling why common cause principles fail in cases like those described above, such as the coordinated flights of certain flocks of birds, equilibrium correlations, order arising out of chaos, etc. The answer is that in such cases the interactions between the parts of these systems are so complicated, and there are so many causes acting on the systems, that the only way one can get independence of further determinants is by specifying so many causes as to make this a practical impossibility. This, in any case, would amount to allowing just about any scattered and unnatural set of factors to count as common causes, thereby trivializing common cause principles.

So, the types of problems with the "principle of common cause" when we are restricted to these sorts of macroscopically describable causes don't apply to the "principle of common cause" when we are talking about every microscopic physical fact in the past light cone of a particular event. Something similar seems to be true with Simpson's paradox and Bernstein's paradox--for example, look at the last two pages of http://scistud.umkc.edu/psa98/papers/uffink.pdf (presented at http://scistud.umkc.edu/psa98/papers/abstracts.html#uffink), which says:

Also, in order to evade the Simpson paradox, it seems that one can save the principle by specifying that the cause C is a sufficient causal factor with respect to a class of events. It would be reasonable to take this class to include at least all events in the past of C, perhaps also those outside of C's causal future. However, this means one needs to introduce concepts from the space-time background in the principle.

...

Remarkably, a variant of the principle of the common cause taking explicity account of relativistic space-time has been around for a long time, although it is seldom discussed in the philosophical literature. It is Penrose and Percifal's (1962) principle of conditional independence.

These authors consider two spacelike separated bounded regions A and B in spacetime, and let C be any region which dissects the union of the past-light cones of A and B into two parts, one containing A and the other containing B. The P(A&B|C) = P(A|C)*P(B|C) where A, B, C are the histories of the regions A, B and C, i.e. complete specifications of all events in those regions.

For our discussion, the salient points in which this formulation differs rom other formulations are, first, in this version only non-local correlations are to be explained ... Thirdly, conditional independence is demanded only upon conditionalizing upon the entire history of a region C. This entails that the problems such as Simpson's paradox connected with incomplete specifications of the factors cannot appear.

The paper is titled The Principle of the Common Cause faces the Bernstein Paradox, so presumably when the author says "problems such as Simpson's paradox" this is meant to apply to Bernstein's paradox as well.

Also note that the Stanford Encyclopedia of Philosophy article actually discusses Penrose and Percifal's argument about picking a region C which divides the past light cones of A and B, in section 1.3, 'the law of conditional independence'. They state the conclusion of the "law of conditional independence", namely P(A&B|C) = P(A|C)*P(B|C), without attempting to dispute that it should hold in a classical relativistic universe. But they treat this "law of conditional independence" as a different claim from "Reichenbach's common cause principle" which is the main topic of the article (again seemingly because Reichenbach's principle is based on distinct macroscopically identifiable 'causes'), saying "This is a time asymmetric principle which is clearly closely related to Reichenbach's common cause principle and the causal Markov condition ... one cannot derive anything like Reichenbach's common cause principle or the causal Markov condition from the law of conditional independence, and one therefore would not inherit the richness of applications of these principles, especially the causal Markov condition, even if one were to accept the law of conditional independence."

So again, if you think that my version of the common cause principle could fail in a relativistic universe with local realist laws, even if the "common cause" is defined as the complete set of microscopic physical facts in one measurement's past light cone (or in a region of spacetime which divides the overlap of the two past light cones of each measurement as with the 'law of conditional independence' formulation by Penrose and Percifal above), you need to either find authors who specifically talk about such detailed specifications, or else actually make the argument yourself rather than trying to dismiss it with vague references to the literature.

 

[JesseM]

I never once said that observables have definite values prior to observation...

This was the EPR conclusion. And where Bell started from. So it is relevant.

It is relevant, yes. And once you realize why, in a local realist universe, it must be true that P(AB|H)=P(A|H)P(B|H) for the right choice of H, then you can also show that if there is a perfect correlation between measurement results when the experimenters choose the same detector angles, then in a local realist universe the only way to explain this is if H predetermines what measurement results they will get for all possible angles. But the general conclusion that P(AB|H)=P(A|H)P(B|H) for the right choice of H doesn't require us to start from that assumption, so if bill is unconvinced on this point it's best to try to show why the conclusion would be true even if we don't assume identical detector settings = identical results.

 

[ThomasT]

It is relevant, yes. And once you realize why, in a local realist universe, it must be true that P(AB|H)=P(A|H)P(B|H) for the right choice of H, then you can also show that if there is a perfect correlation between measurement results when the experimenters choose the same detector angles, then in a local realist universe the only way to explain this is if H predetermines what measurement results they will get for all possible angles. But the general conclusion that P(AB|H)=P(A|H)P(B|H) for the right choice of H doesn't require us to start from that assumption, so if bill is unconvinced on this point it's best to try to show why the conclusion would be true even if we don't assume identical detector settings = identical results.

You seem to understand that the individual measurements and the joint measurements are dealing with two different hidden parameters.

Then it should be clear why P(AB|H) = P(A|H) P(B|H) is a formal requirement that doesn't fit the experimental situation.

 

[JesseM]

You seem to understand that the individual measurements and the joint measurements are dealing with two different hidden parameters.

Then it should be clear why P(AB|H) = P(A|H) P(B|H) is a formal requirement that doesn't fit the experimental situation.

P(AB|H)=P(A|H)P(B|AH) is a general statistical identity that should hold regardless of the meanings of A, B, and H, agreed? So to get from that to P(AB|H)=P(A|H)P(B|H), you just need to prove that in this physical scenario, P(B|AH)=P(B|H), agreed? If you agree, then just let H represent an exhaustive description of all the local variables (hidden and others) at every point in spacetime which lies in the past light cone of the region where measurement B occurred. If measurement A is at a spacelike separation from B, then isn't it clear that according to local realism, knowledge of A cannot alter your estimate of the probability of B if you were already basing that estimate on H, which encompasses every microscopic physical fact in the past light cone of B? To suggest otherwise would imply FTL information transmission, as I argued in post #41.

 

[JesseM]

(continuing an unfinished reply to billschnieder's post #47)

P(AB|H) = P(A|H)*P(B|H)
Clearly means that conditioned on H, there is no correlation between A and B. It is therefore impossible to for H to cause any correlations whatsoever with this equation. Now can you explain how it is possible for an experimenter to collect data consistent with this equation, without knowing the exact nature of H?

I suppose it depends what you mean by "cause" the correlations, but it is completely consistent with this equation that P(AB) could be different than P(A)*P(B). And "collect data consistent with this equation" is ambiguous since the experimenter can't actually know H--again, the only experimental data is about A and B, H represents some set of objective physical facts that must have definite truth-values in a universe with locally realist laws, but there is no claim that we can actually determine the specific values encompassed by H in practice. That's where the idea of an "omniscient being" comes in. Do you disagree that in a local realist universe, we can make coherent statements about what would have to be true of H if H represents something like "the complete set of fundamental physical variables associated with each point in spacetime that lies within the past light cone of some measurement-event B", even if we don't actually know what the values of all those variables are?

It is only possible for A and B to be marginally correlated while at the same time uncorrelated conditioned on H, if H is NOT the cause of the correlation.

"Cause of" needs some kind of precise definition, it's not a statistical term. But intuitively this claim seems pretty silly. For example, being a smoker increases your risk of dying of lung cancer, and also increases your risk of having yellow teeth, and most people would say that smoking has a causal influence on both. Meanwhile, even if there is a marginal correlation between yellow teeth and lung cancer (people who have yellow teeth are more likely to get lung cancer and vice versa), most people would probably bet that this was a case where "correlation is not causation"--yellow teeth don't have a direct causal influence on lung cancer or vice versa. Suppose we find there is a marginal correlation between yellow teeth and lung cancer, but also that P(lung cancer|smoker & yellow teeth) is not any higher than P(lung cancer|smoker) (this is a little over simplistic since heavy smokers are more likely to get both lung cancer and yellow teeth than light smokers, but imagine we are dealing with a society where all smokers smoke exactly the same amount per day). Would you say this proves that smoking cannot have been the cause of the correlation between lung cancer and yellow teeth?

In any case, Bell's theorem doesn't require any discussion of "causality" beyond the basic notion that in a relativistic local realist theory, there should be no FTL information transmission, i.e. information about events in one region A cannot give you any further information about events in a region B at spacelike separation from B, beyond what you already could have determined about events in B by looking at information about events in B's past light cone.

Are you sure you understand that it? Can you explain how the hidden variables H are supposed to be responsible for the correlation between A and B, and yet conditioned on H there is no correlation between A and B. I do not see anything you have written so far in this thread or the other one answers this question.

Since I don't really understand what your objection to this is in the first place, I can only "explain" by pointing to various examples where this is true. The smoking/yellow teeth/lung cancer one above is a simple intuitive example, but I've also given you numerical examples which you just ignored. For example, in the scratch lotto card example from post #18, we saw the marginal correlation that whenever Alice chose a given box (say, box 2 on her card) to scratch, if Bob also chose the same box (box 2 on his card to scratch), they always found the same fruit; but this correlation could be explained by the fact that the source always sent them a pair of cards that had the same combination of "hidden fruits" under each of the three boxes on each card. And then later in that post I also gave an example where two flashlights had hidden internal mechanisms that determined the probabilities they would turn on, with one sent to Alice and one to Bob; if you don't know which hidden mechanisms are in each flashlight, there is a marginal correlation between the events of each one turning on (I explicitly calculated P(A|B) and showed it was different from P(A)), but if you do have the information H about which hidden mechanism was in each one's flashlight before they tried to turn them on, then conditioned on H there is no correlation between A and B (and I explicitly calculated P(A|BH) and showed it was identical to P(A|H)).

In case you are not sure about the terminology, in probability theory, P(AB) is the joint marginal probability of A and B which is the probability of A and B regardless of whether anything else is true or not. P(AB|H) is the joint conditional probability of A and B conditioned on H, which is the probability of A and B given that H is true. There is no such thing as the absolute probability.

Fair enough. But I think you pretty clearly understood from context what I meant by "absolute probability".

I agree, there are cases in which a correlation may exist between A and B marginally, but will not exist when conditioned on another variable, like in some of the example you have give.

You are saying there may be cases where A and B are marginally correlated, but not correlated when conditioned on H? And yet you also just got through saying that you can't understand how H can be responsible for the marginal correlation between A and B, and yet they are not correlated when conditioned on H? The only real difference I see between the two is that word "responsible for", which isn't any sort of statistical terminology as far as I know. What do you mean by it? Are you talking about some intuitive notion of causality, or of one fact being "the explanation for" another? As I said before, following Bell's theorem does not require introducing such vague notions, it's just about analyzing whether one fact can provide information about the probability of some event beyond what you already knew from other facts.

Still it would help me understand you better if you would explain what "responsible for" means in the context of specific examples like the ones I provided. If Alice and Bob in the lotto card example always find the same fruit on trials where they choose to scratch the same box (a perfect marginal correlation), but I happen to know that on every single trial the source sent them both cards with an identical set of "hidden fruits" under the three boxes, can I say that this fact about the hidden fruits is "responsible for" the marginal correlation they observed?

The question is very clear. Let me put it to you in point form and you can give specific answers to which points you disagree with.

1) You say in the specific example treated by Bell, P(B|AH) = P(B|H). It is not me saying it. Do you disagree?

I agree.

2) The above statement (1) implies that in the specific example treated by Bell, where the symbols A, B and H have identical meaning, P(B|AH) and P(B|H) are mathematical identities. Do you disagree?

Your use of "mathematical identity" is confusing here--if some statement about probabilities can't be proven purely from the axioms of probability theory, but depends upon the specific physical definitions of the variables, I would say that it's not a mathematical identity, by definition. For example P(T)=1-P(H) is not a mathematical identity, but it's a valid equation if T and H represent heads or tails for a fair coin. Do you define "mathematical identity" differently?

Perhaps you are assuming that any relevant physical facts are added as additional axioms to the basic axioms of probability theory so that from then on we are doing a purely mathematical proof with this extended axiomatic system. Still your notion of "mathematical identities" is ambiguous. In a formal proof we have a series of lines containing theorems, each of which are derived from some combination of axioms and previously-proved theorems using rules of inference, until we get to the final line with the theorem that we wanted to prove. If I prove theorem #12 from a combination of theorem #3 and theorem #5 using some ruler of inference, would you say theorem 12 is a "mathematical identity" with 3 and 5? If so, when you say:

4) The above statement (3), implies that if using P(A|H)*P(B|H) results in one set of inequalities, the mathematically identical statement P(A|H)*P(B|AH) should result in the same set of inequalities where the symbols A, B and H have identical meaning. Do you disagree?

If yes to the above, I do disagree. For example, take a look at the various simple logic proofs given in http://marauder.millersville.edu/~bikenaga/mathproof/rules-of-inference/rules-of-inference.pdf . An example from pp. 6-7:

Axioms:
i. P AND Q
ii. P -> ~(Q AND R)
iii. S -> R

Prove: ~S

Proof:

1. P AND Q (axiom i)
2. P (Decomposing a conjunction--1)
3. Q (Decomposing a conjunction--1)
4. P -> ~(Q AND R) (axiom ii)
5. ~(Q AND R) (modus ponens--3,4)
6. ~Q OR ~R (DeMorgan--5)
7. ~R (disjunctive syllogism--3,6)
8. S -> R (axiom iii)
9. ~S (modes tollens--7,8)

Would you say statement 5 above is "mathematically identical" to statements 3 and 4? Even if you are using a definition of "mathematically identical" where that is true, why should it imply that you can reach the final conclusion 9 from statements 3 and 4 without going through the intermediate step of 5? 5 may be an essential step in reaching the conclusion from the axioms, saying that all the statements are "mathematically identical" to previous ones and therefore any given intermediate step should be unnecessary is just playing word games, that's not how mathematical proofs work.

 

[DrChinese]

...Still it would help me understand you better if you would explain what "responsible for" means in the context of specific examples like the ones I provided. If Alice and Bob in the lotto card example always find the same fruit on trials where they choose to scratch the same box (a perfect marginal correlation), but I happen to know that on every single trial the source sent them both cards with an identical set of "hidden fruits" under the three boxes, can I say that this fact about the hidden fruits is "responsible for" the marginal correlation they observed?

... saying that all the statements are "mathematically identical" and therefore to previous ones and therefore any given intermediate step should be unnecessary is just playing word games,...

You are my hero, I can't believe you have stayed in this long.

P.S. I am kicking back and relaxing while you are doing all the heavy lifting.

 

[madhatter106]

That would be an attempt to restore local realism, which just won't be possible. Recall that you can entangle particles at other levels as well, such as momentum/position or energy/time. Although it shows up as one thing for spin, you cannot explain it in the manner you mention.

Even for spin, if you look at it long enough, you realize that there is no solution to the mathematical problems. Bell's Inequality is violated because there is no local realistic solution possible.

Ahh, yes thank you, when I wrote that my intended thought was not to restore realism. I appreciate you mentioning it, helps me word my thoughts better.

As I see it, there is no current known explanation for why this 'action at a distance occurs' right? And I can be wayyyyy off I'm sure, I'm just a layman approaching this. The original thought I didn't write was that photons are from/in another dimension.

 

[DrChinese]

Ahh, yes thank you, when I wrote that my intended thought was not to restore realism. I appreciate you mentioning it, helps me word my thoughts better.

As I see it, there is no current known explanation for why this 'action at a distance occurs' right? And I can be wayyyyy off I'm sure, I'm just a layman approaching this. The original thought I didn't write was that photons are from/in another dimension.

Hey, maybe they are in another dimension. Who knows? What's a dimension here or there among friends?

There is no known mechanism for entanglement, just a formalism. So the formalism is the explanation at this point.

 

[billschnieder]

JesseM:
Since brevity is a virtue, I will not attempt responding to very line of your responses which is very tempting as there is almost always something to challenge in each. Here is a crystallization of my reponse to everything you have posted so far.

1) The principle of common cause used by Bell as P(AB|C) = P(A|C)P(B|C) is not universally valid even if C represents complete information about all possible causes in the past light cones of A and B. This is because
if A and B are marginally correlated but uncorrelated conditioned on C, it implies that C screens off the correlation between A and B. In some cases, it is not possible to define C such that it screens off the correlation between A and B.

Under Conclusions:
If there are fundamental (equal time) laws of physics that rule out certain areas in state-space, which thus imply that there are (equal time) correlations among certain quantities, this is no violation of initial microscopic chaos. But the three common cause principles that we discussed will fail for such correlations. Similarly, quantum mechanics implies that for certain quantum states there will be correlations between the results of measurements that can have no common cause which screens all these correlations off. But this does not violate initial microscopic chaos. Initial microscopic chaos is a principle that tells one how to distribute probabilities over quantum states in certain circumstances; it does not tell one what the probabilities of values of observables given certain quantum states should be. And if they violate common cause principles, so be it. There is no fundamental law of nature that is, or implies, a common cause principle. The extent of the truth of common cause principles is approximate and derivative, not fundamental.

Therefore, Bell's choice of the PCC as a definition for hidden variable theorems by which to suplement QM is not appropriate.

2) Not all correlations necessarily have a common cause and suggesting that they must is not appropriate.

3) Either God is calculating on both sides of the equation or he is not. You can not have God on one side and the experimenters on another. Therefore if God is the one calculating the inequality you can not expect a human experimenter who knows nothing of about H, to collect data consistent with the inequality.

Using P(AB|H) = P(A|H)P(B|H) to derive an inequality means that the context of the inequalities is one in which there is no longer any correlation between A and B, since it has been screened-off by H. Therefore for data to be comparable to the inequalities, it must be screened of with H. Note that P(AB) = P(A|H)P(B|H) is not a valid equation. You can not collect data without screening of (ie P(AB) ) and use it to compare with inequalities derived from screened-off probabilities P(AB|H).

 

[JesseM]

JesseM:
Since brevity is a virtue, I will not attempt responding to very line of your responses which is very tempting as there is almost always something to challenge in each.

In a scientific/mathematical discussion, precision is more of a virtue than brevity. In fact one of the common problems in discussions with cranks who are on a crusade to debunk some mainstream scientific theory is that they typically throw out short and rather broad (and vague) arguments which may sound plausible on the surface, but which require a lot of detailed explanation to show what is wrong with them. This problem is discussed here, for example:

Come to think of it, there’s a certain class of rhetoric I’m going to call the “one way hash” argument. Most modern cryptographic systems in wide use are based on a certain mathematical asymmetry: You can multiply a couple of large prime numbers much (much, much, much, much) more quickly than you can factor the product back into primes. A one-way hash is a kind of “fingerprint” for messages based on the same mathematical idea: It’s really easy to run the algorithm in one direction, but much harder and more time consuming to undo. Certain bad arguments work the same way—skim online debates between biologists and earnest ID afficionados armed with talking points if you want a few examples: The talking point on one side is just complex enough that it’s both intelligible—even somewhat intuitive—to the layman and sounds as though it might qualify as some kind of insight. (If it seems too obvious, perhaps paradoxically, we’ll tend to assume everyone on the other side thought of it themselves and had some good reason to reject it.) The rebuttal, by contrast, may require explaining a whole series of preliminary concepts before it’s really possible to explain why the talking point is wrong. So the setup is “snappy, intuitively appealing argument without obvious problems” vs. “rebuttal I probably don’t have time to read, let alone analyze closely.”

The principle of common cause used by Bell as P(AB|C) = P(A|C)P(B|C) is not universally valid even if C represents complete information about all possible causes in the past light cones of A and B.

Not in general, no. But in a universe with local realist laws, it is universally valid.

This is because
if A and B are marginally correlated but uncorrelated conditioned on C, it implies that C screens off the correlation between A and B. In some cases, it is not possible to define C such that it screens off the correlation between A and B.

It is always possible to define such a C in a relativistic universe with local realist laws, if A and B happen in spacelike-separated regions: if C represents the complete information about all local physical variables in the past light cones of the regions where A and B occurred (or in spacelike slices of these past light cones taken at some time after the last moment the two past light cones intersected, as I suggested in post 61/62 on the other thread and as illustrated in fig. 4 of this paper on Bell's reasoning), then it is guaranteed that C will screen off correlations between A and B. Nothing in the Stanford article contradicts this, so if you disagree with it, please explain why in your own words (preferably addressing my arguments in post #41 about why to suggest otherwise would imply FTL information transmission, like telling me whether you think the example where the results of a race on Alpha Centauri were correlated with a buzzer going off on Earth is compatible with local realism and relativity). If you think the Stanford Encyclopedia article does contradict it, can you tell me specifically which part and why? In your quote from the Stanford Encyclopedia saying why common cause principles can fail, the first part was about "molecular chaos" and assumptions about the exact microscopic state of macroscopic systems:

This explains why the three principles we have discussed sometimes fail. For the demand of initial microscopic chaos is a demand that microscopic conditions are uniformly distributed (in canonical coordinates) in the areas of state-space that are compatible with the fundamental laws of physics. If there are fundamental (equal time) laws of physics that rule out certain areas in state-space, which thus imply that there are (equal time) correlations among certain quantities, this is no violation of initial microscopic chaos. But the three common cause principles that we discussed will fail for such correlations.

Note that of the three common cause principles discussed, none (including the one by Penrose and Parcival) actually allowed C to involve the full details about every microscopic physical fact at some time in the past light cone of A or B. This is why assumptions like "microscopic chaos" are necessary--because you don't know the full microscopic conditions, you have to make assumptions like the one discussed in section 3.3:

Nonetheless such arguments are pretty close to being correct: microscopic chaos does imply that a very large and useful class of microscopic conditions are independently distributed. For instance, assuming a uniform distribution of microscopic states in macroscopic cells, it follows that the microscopic states of two spatially separated regions will be independently distributed, given any macroscopic states in the two regions. Thus microscopic chaos and spatial separation is sufficient to provide independence of microscopic factors.

Also note earlier in the same section where they write:

there will be no screener off of the correlations between D and E other than some incredibly complex and inaccessible microscopic determinant. Thus common cause principles fail if one uses quantities D and E rather than quantities A and B to characterize the later state of the system.

So here common cause principles only fail if you aren't allowed to use the full set of microscopic conditions which might contribute to the likelihood of different observable outcomes, they acknowledge that if you did have such information it could screen off correlations in these outcomes.

The next part of the Stanford article that you quoted dealt with QM:

Similarly, quantum mechanics implies that for certain quantum states there will be correlations between the results of measurements that can have no common cause which screens all these correlations off. But this does not violate initial microscopic chaos. Initial microscopic chaos is a principle that tells one how to distribute probabilities over quantum states in certain circumstances; it does not tell one what the probabilities of values of observables given certain quantum states should be. And if they violate common cause principles, so be it. There is no fundamental law of nature that is, or implies, a common cause principle. The extent of the truth of common cause principles is approximate and derivative, not fundamental.

What you seem to miss here is that the idea that quantum mechanics violates common cause principles is explicitly based on assuming that Bell is correct and that the observed statistics in QM are incompatible with local realism. From section 2.1:

One might think that this violation of common cause principles is a reason to believe that there must then be more to the prior state of the particles than the quantum state; there must be ‘hidden variables’ that screen off such correlations. (And we have seen above that such hidden variables must determine the results of the measurements if they are to screen of the correlations.) However, one can show, given some extremely plausible assumptions, that there can not be any such hidden variables. There do exist hidden variable theories which account for such correlations in terms of instantaneous non-local dependencies. Since such dependencies are instantaneous (in some frame of reference) they violate Reichenbach's common cause principle, which demands a prior common cause which screens off the correlations. (For more detail, see, for instance, van Fraassen 1982, Elby 1992, Grasshoff, Portmann & Wuethrich (2003) [in the Other Internet Resources section], and the entries on Bell's theorem and on Bohmian mechanics in this encyclopedia.)

So, in no way does this suggest they'd dispute that in a universe that did obey local realist laws, it would be possible to find a type of "common cause" involving detailed specification of every microscopic variable in the past light cones of A and B which would screen off correlations between A and B. What they're saying is that the actual statistics seen in QM rule out the possibility that our universe actually obeys such local realist laws.

2) Not all correlations necessarily have a common cause and suggesting that they must is not appropriate.

I never suggested that all correlations have a common cause, unless "common cause" is defined so broadly as to include the complete set of microscopic conditions in two non-overlapping light cones (two totally disjoint sets of events, in other words). For example, if aliens outside our cosmological horizon (so that their past light cone never overlaps with our past light cone at any moment since the Big Bang) were measuring some fundamental physical constant (say, the fine structure constant) which we were also measuring, the results of our experiments would be correlated due to the same laws of physics governing our experiments, not due to any events in our past which could be described as a "common cause". But it's you who's bringing up the language of "causes", not me; I'm just talking about information which causes you to alter probability estimates, and that's all that's necessary for Bell's proof. In this example, if our universe obeyed local realist laws, and an omniscient being gave us a complete specification of all local physical variables in the past light cone of the alien's measurement (or in some complete spacelike slice of that past light cone) along with a complete specification of the laws of physics and an ultra-powerful computer that we could use to evolve these past conditions forward to make predictions about what will happen in the region of spacetime where the aliens make the measurement, then our estimate of the probabilities of different outcomes in that region would not be altered in the slightest if we learned additional information about events in our own region of spacetime, including the results of an experiment similar to the alien's.

3) Either God is calculating on both sides of the equation or he is not. You can not have God on one side and the experimenters on another.

It is theoretical physicists calculating the equations based on imagining what would have to be true if they had access to certain information H which is impossible to find in practice, given certain assumptions about the way the fundamental laws of physics work. But since they don't actually know the value of H, they may have to sum over all possible values of H that would be compatible with these assumptions about fundamental laws. For example, do you deny that under the assumption of local realism, where H is taken to represent full information about all local physical variables in the past light cones of A and B, the following equation should hold?

P(AB) = sum over all possible values of H: P(AB|H)*P(H)

Note that this is the type of equation that allowed me to reach the final conclusion in the scratch lotto card example from post #18; I assumed that the perfect correlation when Alice and Bob scratched the same box was explained by the fact that they always received a pair of cards with an identical set of "hidden fruits" behind each box, and then I showed that P(Alice and Bob find the same fruit when they scratch different boxes|H) was always greater than or equal to 1/3 (assuming they choose which box to scratch randomly with a 1/3 probability of each on a given trial, and we're just looking at the subset of trials where they happened to choose different boxes) for all possible values of H:

H1: box1: cherry, box2: cherry, box3: cherry
H2: box1: cherry, box2: cherry, box3: lemon
H3: box1: cherry, box2: lemon, box3: cherry
H4: box1: cherry, box2: lemon, box3: lemon
H5: box1: lemon, box2: cherry, box3: cherry
H6: box1: lemon, box2: cherry, box3: lemon
H7: box1: lemon, box2: lemon, box3: cherry
H8: box1: lemon, box2: lemon, box3: lemon

If the probability is greater than or equal to 1/3 for each possible value of H, then obviously regardless of the specific values of P(H1) and P(H2) and so forth, the probability on the left of this equation:

p(Alice and Bob find the same fruit when they scratch different boxes) = sum over all possible values of H: P(Alice and Bob find the same fruit when they scratch different boxes|H)*P(H)

...must end up being greater than or equal to 1/3 as well. Therefore if we find the actual frequency of finding the same fruit when they choose different boxes is 1/4, we have falsified the original theory that they are receiving cards with an identical set of predetermined "hidden fruit" behind each box.

In this example, even if the theory about hidden fruit had been correct, I don't actually know the full set of hidden fruit on each trial (say the cards self-destruct as soon as one box is scratched). So, any part of the equation involving H is imagining what would have to be true from the perspective of a "God" who did have knowledge of all the hidden fruit. And yet you see the final conclusion is about the actual probabilities Alice and Bob observe on trials where they choose different boxes to scratch. Please tell me whether your general arguments about it being illegitimate to have a human perspective on one side of an equation and "God's" perspective on another would apply to the above as well (i.e. whether you disagree with the claim that the premise that each card has an identical set of hidden fruits should imply a probability of 1/3 or more that they'll find the same fruit on trials where they randomly select different boxes).

Therefore if God is the one calculating the inequality you can not expect a human experimenter who knows nothing of about H, to collect data consistent with the inequality.

Using P(AB|H) = P(A|H)P(B|H) to derive an inequality means that the context of the inequalities is one in which there is no longer any correlation between A and B, since it has been screened-off by H. Therefore for data to be comparable to the inequalities, it must be screened of with H. Note that P(AB) = P(A|H)P(B|H) is not a valid equation.

It's true that this is not a valid equation, but if P(AB|H)=P(A|H)P(B|H) applies to the situation we are considering, then P(AB) = sum over all possible values of H: P(A|H)P(B|H)P(H) is a valid equation, and it's directly analogous to equation (14) in http://hexagon.physics.wisc.edu/teaching/2010s%20ph531%20quantum%20mechanics/interesting%20papers/bell%20on%20epr%20paradox%20physics%201%201964.pdf .

 

[billschnieder]

Please tell me whether your general arguments about it being illegitimate to have a human perspective on one side of an equation and "God's" perspective on another would apply to the above as well

The point is that certain assumptions are made about the data when deriving the inequalities, that must be valid in the data-taking process. God is not taking the data, so the human experimenters must take those assumptions into account if their data is to be comparable to the inequalities.

Consider a certain disease that strikes persons in different ways depending on circumstances. Assume that we deal with sets of patients born in Africa, Asia and Europe (denoted a,b,c). Assume further that doctors in three cities Lyon, Paris, and Lille (denoted 1,2,3) are are assembling information about the disease. The doctors perform their investigations on randomly chosen but identical days (n) for all three where n = 1,2,3,...,N for a total of N days. The patients are denoted Alo(n) where l is the city, o is the birthplace and n is the day. Each patient is then given a diagnosis of A = +1/-1 based on presence or absence of the disease. So if a patient from Europe examined in Lille on the 10th day of the study was negative, A3c(10) = -1.

According to the Bell-type Leggett-Garg inequality

Aa(.)Ab(.) + Aa(.)Ac(.) + Ab(.)Ac(.) >= -1

In the case under consideration, our doctors can combine their results as follows

A1a(n)A2b(n) + A1a(n)A3c(n) + A2b(n)A3c(n)

It can easily be verified that by combining any possible diagnosis results, the Legett-Garg inequalitiy will not be violated as the result of the above expression will always be >= -1, so long as the cyclicity (XY+XZ+YZ) is maintained. Therefore the average result will also satisfy that inequality and we can therefore drop the indices and write the inequality only based on place of origin as follows:

<AaAb> + <AaAc> + <AbAc> >= -1

Now consider a variation of the study in which only two doctors perform the investigation. The doctor in Lille examines only patients of type (a) and (b) and the doctor in Lyon examines only patients of type (b) and (c). Note that patients of type (b) are examined twice as much. The doctors not knowing, or having any reason to suspect that the date or location of examinations has any influence decide to designate their patients only based on place of origin.

After numerous examinations they combine their results and find that

<AaAb> + <AaAc> + <AbAc> = -3

They also find that the single outcomes Aa, Ab, Ac, appear randomly distributed around +1/-1 and they are completely baffled. How can single outcomes be completely random while the products are not random. After lengthy discussions they conclude that there must be superluminal influence between the two cities.

But there are other more reasonable reasons. Note that by measuring in only two citites they have removed the cyclicity intended in the original inequality. It can easily be verified that the following scenario will result in what they observed:

- on even dates Aa = +1 and Ac = -1 in both cities while Ab = +1 in Lille and Ab = -1 in Lyon
- on odd days all signs are reversed

In the above case
<A1aA2b> + <A1aA2c> + <A1bA2c> >= -3
which is consistent with what they saw. Note that this equation does NOT maintain the cyclicity (XY+XZ+YZ) of the original inequality for the situation in which only two cities are considered and one group of patients is measured more than once. But by droping the indices for the cities, it gives the false impression that the cyclicity is maintained.

The reason for the discrepancy is that the data is not indexed properly in order to provide a data structure that is consistent with the inequalities as derived.Specifically, the inequalities require cyclicity in the data and since experimenters can not possibly know all the factors in play in order to know how to index the data to preserve the cyclicity, it is unreasonable to expect their data to match the inequalities.

For a fuller treatment of this example, see Hess et al, Possible experience: From Boole to Bell. EPL. 87, No 6, 60007(1-6) (2009)

Note that in deriving Bell's inequalities, Bell used Aa(l), Ab(l) Ac(l), where the hidden variables (l) are the same for all three angles. For this to correspond to the Aspect-type experimental situation, the hidden variables must be exactly the same for all the angles, which is an unreasonable assumption because each particle could have it's own hidden variables with the measurement equipment each having their own hidden variables, and the time of measurement after emission itself a hidden variable. So it is more likely than not that the hidden variables will be different for each measurement. However, in actual experiments the photons are only measured in pairs (a,b), (a,c) and (b,c). The experimenters, not knowing the exact nature of the hidden variables, can not possibly collect the data in a way that ensures the cyclicity is preserved. Therefore, it is not possible to perform an experiment that can be compared with Bell's inequalities.

 

[DrChinese]

Consider a certain disease that strikes persons in different ways depending on circumstances.

...

The reason for the discrepancy is that the data is not indexed properly in order to provide a data structure that is consistent with the inequalities as derived.Specifically, the inequalities require cyclicity in the data and since experimenters can not possibly know all the factors in play in order to know how to index the data to preserve the cyclicity, it is unreasonable to expect their data to match the inequalities.

For a fuller treatment of this example, see Hess et al, Possible experience: From Boole to Bell. EPL. 87, No 6, 60007(1-6) (2009)

Great example, pretty much demonstrates everything I have been saying all along:

a) There is a dataset which is realistic, i.e. you can create a dataset which presents data for properties not actually observed;

But...

b) The sample is NOT representative of the full universe, something which is not a problem with Bell since it assumes the full universe in its thinking; i.e. your example is irrelevant - if you want to attack the Fair Sampling Assumption then you should label your thread as such since one has nothing to do with the other;

c) By way of comparison to Bell, it does not agree with the predictions of QM; i.e. obviously QM does not say anything about doctors and patients in your example; however, you would need something to compare it to and you don't really do that, your example is just playing around with logic.

It would be nice if instead of attempting to attack Bell, you would work on first understanding Bell. Then once you understand it, look for holes.