Is Bell's Theorem a Valid Solution to the Locality Versus Nonlocality Issue?

[harrylin]

[..]
Bell's inequality derivation relies on the fact that they originate from THE SAME ensemble, therefore you cannot use three different ensembles and expect them to just work. If you think three different ensembles should work, you should start out from that assumption and derive the inequalities and show that you can still obtain them. However, many authors have done that, and obtained different inequalities which no experiment or QM has ever violated. See the articles I mentioned earlier.

Ok then you appear to understand (and agree) with De Raedt et al's most recent paper on this issue. If so, please confirm it in the discussion thread about their paper. As I'm trying to understand the strengths and weaknesses of that argument, elaborations of that argument as formulated by them will be welcome.

Cheers,
Harald

 

[billschnieder]

Well, you can if you have a single ensemble of particle pairs, and then for each pair you choose which combination of properties to measure using a rule that is statistically uncorrelated with the hidden properties of each pair--the "no-conspiracy assumption" which you seem to have forgotten about. Read my post #11 on this thread for more on this point, and consider the following part in particular:

I simply ignore this argument which you keep repeating, because if far off the mark.
1) In any Bell test experiment, you are not dealing with a single ensemble but at least 3 different ensembles! Your argument here seems to be similar to saying, if the average price of a stock for the first 100 days of the year is different from the second 100 days of the year, then there must be conspiracy because we are dealing with a single stock (cf what you call single ensemble above).
2) You are confused if you think it is possible to randomly select based on hidden properties you know nothing about. Read up on Bertrands paradox for more on this point.
Try to understand my argument and you will see that this line of argument you are suggesting as rebuttal is definitely not. If you think you've understood my argument, summarize it in your own words. Then respond to this very simple analogy:

I suppose you know about the triangle inequality which says for any triangle with sides labeled x, y, z where x, y, z represents the lengths of the sides

z <= x + y

Note that this inequality applies to a single triangle. What if you could only measure one side at a time. Assume that for each measurement you set the label of the side your instrument should measure and it measured the length destroying the triangle in the process. So you performed a large number of measurements on different triangles. Measuring <z> for the first run, <x> for the next and <y> for the next.

Do you believe the inequality
<z> <= <x> + <y>

Is valid? In other words, you believe it is legitimate to use those averages in your inequality to verify its validity?

 

[billschnieder]

Ok then you appear to understand (and agree) with De Raedt et al's most recent paper on this issue. If so, please confirm it in the discussion thread about their paper. As I'm trying to understand the strengths and weaknesses of that argument, elaborations of that argument as formulated by them will be welcome.

Cheers,
Harald

You are right, my argument is exactly the same as theirs and the same as that of the other authors in the articles I mentioned earlier. I will summarize it in the thread you mention when I get some time.

 

[JesseM]

I simply ignore this argument which you keep repeating, because if far off the mark.
1) In any Bell test experiment, you are not dealing with a single ensemble but at least 3 different ensembles!

Probably it would be better to avoid the word "ensemble", since a statistical ensemble normally refers to a hypothetical collection of possible outcomes which may be much larger than the number actual sampled in any experiment. But we can consider a single list of particle pairs each of which is hypothetically assumed to be associated with a particular (unknown to us) set of three predetermined results for our three measurement settings (if we're dealing with the case where measurements at the same setting are guaranteed to yield identical results). Then it's true that your measurements divide this single list into multiple sub-lists like this list where you measured a,b or the list where you measured b,c, but as long as the no-conspiracy condition holds than the probability that the collection of sub-lists will violate an inequality not violated by the original list gets smaller and smaller as the number of entries on the list gets larger and larger.

2) You are confused if you think it is possible to randomly select based on hidden properties you know nothing about. Read up on Bertrands paradox for more on this point.

Bertrand's paradox has to do with the ambiguity in the phrase "random chord" since this phrase does not define any particular probability distribution on chords, what that has to do with this situation I don't know since it's assumed there is some specific physical procedure (say, a random number generator) for making the choice of what detector setting to use on each trial. Anyway, does your answer mean you accept that if the no-conspiracy condition holds (i.e. P(λ|a,b) = P(λ|b,c) and so on for other combinations of detector settings) that would mean the inequality becomes more and more likely to be true (under local realism) as the number of trials grows, but that you just don't think the no-conspiracy condition is one there are any good physical arguments for believing in? If so I would say you probably haven't thought it through very carefully.

For example, suppose Alice and Bob are a great distance apart, and suppose on each successive day they are each measuring one member of a particle pair that was emitted from a source between them more than a year ago. And suppose we are defining the "hidden variables" for each particle to be restricted to a cross-section of the past light cone of each measurement (like "region 3" in fig. 2 on p. 3 of this paper), at a time one year before the measurement was actually performed but after the time the past light cones of the two measurements had ceased to overlap (again see the diagram in that paper). Now suppose that one day before measurement, Alice and Bob make their decision about which measurement setting to use based on the behavior of some very chaotic system, like the weather that day or a chaotic pendulum, which according to the butterfly effect might be in a completely different state at that time (leading to a different choice of detector settings) if even one tiny condition were different anywhere in the past light cone of that moment one year earlier. In this case, it would be a physically bizarre situation indeed if over a hypothetical infinite set of trials of this type, there were some consistent correlation between the complete set of all physical conditions throughout the past-light-cone-cross-section (which all contribute to the behavior of the chaotic system) and the three predetermined results of the particle, which we would normally assume to depend only on some small subset of conditions in the same past-light-cone-cross-section (perhaps just variables associated with the spatial location of the particle itself, or its immediate neighborhood). I suppose you could imagine that the particle's behavior is itself deterministically chaotic so that even if it has predetermined results they, too, depend on the complete set of all physical conditions throughout the past light cone, but in this case the fact that the two particles always have the same result whenever the three experimenters pick the same measurement setting would itself be physically bizarre.

Try to understand my argument and you will see that this line of argument you are suggesting as rebuttal is definitely not. If you think you've understood my argument, summarize it in your own words.

Are you not arguing that the probability of getting a given value of lambda (or a given set of predetermined results) should not be assumed to be the same on trials where we pick one combination of measurement settings (say, a and b) as it is on trials where we pick a different combination (say, b and c)?

 

[Gordon Watson]

Because this line of argument leads inevitably to Bell inequalities, as I and others have been trying to explain to you since you started posting here. Suppose we have some large number of particle pairs, from the above you should agree that in each pair, the two particles should have some definite set of predetermined results like [+ on a, - on b, + on c] or [- on a, +on b, + on c] etc.? And for any collection of things (like particle pairs) where each member of the collection either does or doesn't have each of three possible properties A, B, and C (say A=+ on angle a, B=+ on angle b, C=+ on angle c, so "not A"=- on angle a, "not B"=- on angle b, and "not C"=- on angle c), simple arithmetic shows the whole collection must satisfy this inequality:

Number(A, not B) + Number(B, not C) ≥ Number(A, not C)

There's a proof on this page, but I think their proof is not as simple as it could be, the simplest way of seeing it is this:

Number(A, not B) = Number(A, not B, C) + Number(A, not B, not C) [since any member of the group satisfying A, not B must either have or not have property C]
Number(B, not C) = Number(A, B, not C) + Number(not A, B, not C)
Number(A, not C) = Number(A, B, not C) + Number(A, not B, not C)

And plugging this into the above inequality and cancelling like terms from both sides gives:

Number(A, not B, C) + Number(not A, B, not C) ≥ 0

Which obviously must be true since the number with any given set of properties must be ≥ 0!

Anyway, whether you like my proof or the one on the page I linked to better, hopefully you agree that if we knew the complete set of three predetermined properties for a collection of particle pairs, the inequality Number(A, not B) + Number(B, not C) ≥ Number(A, not C) would be satisfied? If so, it's a short step from there to the statement that if you measure two properties for a large number of particle pairs, P(A, not B|measured a and b) + P(B, not C|measured b and c) ≥ P(A, not C|measured a and c) (basically the only extra assumption needed is that the probability the experimenters will pick a given pair of axes to measure is uncorrelated with the triplet of predetermined results prior to measurement). I discuss this more on post #11 here, but we can also discuss it here if you agree with the inequality Number(A, not B) + Number(B, not C) ≥ Number(A, not C) for all particle pairs but don't agree that for measurements this implies P(A, not B|measured a and b) + P(B, not C|measured b and c) ≥ P(A, not C|measured a and c).

..
You will see from my Signature that my local-realism (L*R) is not of the type that you here specify. Indeed, it would be an odd LR that combines Einstein locality with extrinsic properties.

The intrinsic property that I identify as relevant (in my realism) is (simply) the orientation of the total spin.

So then, when I convert your "classical analogy" to the realism of L*R, we find something like:

P(A, ~B) = 1/2 sin^2 (a, b).

P(B, ~C) = 1/2 sin^2 (b, c).

P(A, ~C) = 1/2 sin^2 (a, c).

Thus, your classical construction cannot be maintained, let alone constructed. Or rather: Show me how these facts fit your classical analogy, please.

In short: You convert a possible "measurement" outcome (an extrinsic property) to an intrinsic property of the particle itself. As if I attributed "death" to you today, for (regrettably) you one day will be.

So when you say that "I should agree" -- I cannot!

Does the "death" analogy help you see the error in your mode of thinking; you not being a local-realist?
..

 

[JesseM]

..
You will see from my Signature that my local-realism (L*R) is not of the type that you here specify. Indeed, it would be an odd LR that combines Einstein locality with extrinsic properties.

The intrinsic property that I identify as relevant (in my realism) is (simply) the orientation of the total spin.

"Orientation of the total spin" is not something particles have a definite value for in QM prior to any spin measurement, so this must be some sort of hidden variable. But is it a local hidden variable? Does each particle have its own "orientation" value, or by "total spin" are you talking about both particles at once, and saying "orientation of the total spin" for both particles cannot be defined as a function of local variables associated with each particle? If your model includes nonlocal variables that can't be defined as functions of other local variables (in the way that magnetic flux through an extended surface can be defined as a function of the local magnetic field vector at each point on the surface), then you are not a "local realist" by any physicist's definition!

In short: You convert a possible "measurement" outcome (an extrinsic property) to an intrinsic property of the particle itself. As if I attributed "death" to you today, for (regrettably) you one day will be.

But I already told you that EPR/Bell were not saying the particle must already have a given property prior to measurement, just that it must have local properties that predetermine what property would be observed if that measurement was made (in a local realist model where measuring one particle allows you to predict with certainty what would be observed if the same measurement were performed on the other, that is). That was the point of my whole discussion of Einstein's two-box analogy in post #144 and how instead of the box already containing a ball or not, it might contain a holographic projector which was predetermined to either project or not project an image of a ball in response to the box being opened. Please read that post again if you're not clear on this point! Along the same lines, while it would obviously be incorrect to attribute the property of "death" to me now, it's conceivable that there are properties associated with my body and perhaps some section of the surrounding world at this time that predetermine whether a test for my death at some future date (say Jan 1. 2050) will yield the result "still alive" or "he's dead, Jim", so if someone knew all the relevant properties today they could predict with total certainty what the result of this Jan. 1 2050 test would be.

I asked you specifically about such predetermined results in post #146:

So, do you think a hypothetical omniscient observer with complete knowledge of one particle's local properties (which might include the particle's own "orientation" in your model, but wouldn't include any information about the other particle) at some time prior to measurement (but after the past light cones of the two measurement regions have ceased to overlap, as in "region 3" of fig. 2 at the top of p. 3 in this paper) would be able to predict in advance with total certainty what outcome would be seen if the particle were measured at any of three detector settings? So that the observer could say something like "this particle's local properties ensure it is predetermined to give +1 if measured at angle a, -1 if measured at angle b, and -1 if measured at angle c"? And if both particles are always found to give identical observed results when measured at the same angle, would you agree this implies (under local realism) that for each pair emitted by the source, their local properties must be correlated in such a way as to ensure that one particle must have the same three predetermined results as the other one?

Your short response was "Sure; why not?" Did you actually read my question carefully before responding or were you just being flippant? Do you wish to change your answer now? The comment is asking you whether there are local properties associated with one of the particles that predetermine what result it would give to each of the three possible detector settings, before the experimenter even makes the choice of what detector setting to use. If you agree that each particle has a well-defined set of predetermined answers like this (and that both members of every entangled pair have the same three predetermined answers), I don't see how you can deny that for every single particle, it either does or doesn't satisfy the three combinations of predetermined results (A, not B) and (B, not C) and (A, not C), meaning an observer with magical knowledge of hidden variables could count how many particles satisfy each one and get Number(A, not B) and Number(B, not C) and Number(A, not C) for any series of particle pairs.

 

[SpectraCat]

..

In short: You convert a possible "measurement" outcome (an extrinsic property) to an intrinsic property of the particle itself. As if I attributed "death" to you today, for (regrettably) you one day will be.

..

But measurement outcomes are the only things that can ever be known about quantum systems ... that is one of the postulates of QM. So, it seems to me the "intrinsic property" of the particle itself, if it doesn't correlate to a measurement outcome, must be a hidden variable. The only remaining question after that is whether it is a non-local hidden variable, in which case you have a "weird" interpretation (like Bohmian mechanics), but one that is consistent with Bell's Theorem. If the "intrinsic property" is a local variable, then you are in conflict with Bell's Theorem.

In post #141, you seem to suggest that "pristine lambda" is a local hidden variable which is perturbed by measurement. Correct me if I am wrong, but it seems your chief objection to Bell is the association of measurement outcomes with local hidden variables, is that correct? I would also like to know your answers to the following questions:

1) Does the value of "pristine lambda" uniquely predict the measurement outcome for any possible detector setting?

2) If the answer to the above is "yes", then how is that any different than associating "pristine lambda" with measurement outcomes for detector settings, as done by Bell?

To me, I don't see any difference between what you have put forth and what Bell assumes in his paper, but perhaps if you can answer the above two questions, I will understand your position better.

 

[billschnieder]

1) Does the value of "pristine lambda" uniquely predict the measurement outcome for any possible detector setting?

If by detector setting you mean the sum total of the microscopic state of the detector at the instant of measurement, then the answer is Yes. But if by detector setting you mean the angle which Alice and Bob choose, then the answer is No. Alice and Bob have control over the angle, but not over all the microscopic dynamic properties of the detector assembly which they don't even know about.

2) If the answer to the above is "yes", then how is that any different than associating "pristine lambda" with measurement outcomes for detector settings, as done by Bell?

Because the outcome is caused by both the complete state of the particle and the complete state of the detector. The only parameter which the experimenters have any detailed knowledge of and can control is the detector angle.

That is why I mentioned earlier that the experimenter think they are measuring

|P(a,b) - P(a,c)| <= P(b,c) + 1
But in fact they are measuring

|P(a1,b1) - P(a2,c2)| <= P(b3,c3) + 1

Where the additional numbers indicate that the complete microscopic of the system when P(a1,b1) was measured is not necessarily the same as the state when P(a2,c2) was measured etc. Simiply assuming that it must be the same, without justification is a fatal error. The second inequality above has never been proven by anybody as a valid inequality but this is what Bell proponents are using everyday to proclaim the demise of realism/locality.

I'm sure JesseM will respond that the states must all be the same in the experiment because the experimenters have measured a large number of photons randomly sampling all the possible different states ... etc. This argument is naive for 2 reasons:

1) Bertrand's Paradox shows that "random" does not mean much unless you have specified exactly how you are sampling the variable. In other words, you can only sample a variable randomly if you know in advance the behaviour of the variable. How then are experimenters ever going to be able to randomly sample dynamic microscopic properties which they know nothing of? How are they to know that they have used the correct method to "randomly sample" the hidden properties, if they do not know the exact nature of all the dynamic microscopic properties affecting the outcome?

2) The argument assumes that by averaging a large number of values you obtain a value close to the correct one. JesseM will call it the law of large numbers. What he will not tell you (maybe he doesn't know this), is that the law of large numbers does not apply to non-stationary systems. The stock market for example is local-realistic, deterministic and non-stationary. That is why the average stock price for the first 100 days is not the same as the average stock price for the next 100 days for example. Bell proponents make the additional unsubstantiated assumption that the system being measured in the experiment is stationary.

 

[JesseM]

1) Bertrand's Paradox shows that "random" does not mean much unless you have specified exactly how you are sampling the variable.

Yes, and as soon as you do specify a method then Bertrand's paradox doesn't apply. And I did suggest some possible methods, let me refresh your memory:

Bertrand's paradox has to do with the ambiguity in the phrase "random chord" since this phrase does not define any particular probability distribution on chords, what that has to do with this situation I don't know since it's assumed there is some specific physical procedure (say, a random number generator) for making the choice of what detector setting to use on each trial.

...

Now suppose that one day before measurement, Alice and Bob make their decision about which measurement setting to use based on the behavior of some very chaotic system, like the weather that day or a chaotic pendulum, which according to the butterfly effect might be in a completely different state at that time (leading to a different choice of detector settings) if even one tiny condition were different anywhere in the past light cone of that moment one year earlier.

So there you have it, two perfectly good methods that could only be correlated with the hidden variables denoted by lambda (at a time prior to the decision about what setting to use, as I noted) if there was some extremely weird "conspiracy" between seemingly unrelated events.

2) The argument assumes that by averaging a large number of values you obtain a value close to the correct one. JesseM will call it the law of large numbers. What he will not tell you (maybe he doesn't know this), is that the law of large numbers does not apply to non-stationary systems.

Of course it does, silly. Give me a list of the values of 3 stocks on a large number of successive days, say 1000. On each day, if the triplet has property "A" that means stock #1 is above a certain value (say, its average value over some previous period), if it has property "not A" that means it's below that value; likewise "B" means stock #2 is above some value on that day, "not B" means it's below, and "C" means stock #3 is above some value on that day, "not C" below. If I use a random number generator (or a chaotic pendulum) to decide which pair of stocks to measure that day, do you doubt that as the number of days gets large (again, say 1000), the probability of this inequality being violated would get very low?

Number(A, not B|a day where I measured stock 1 and stock 2) + Number(B, not C|a day where I measured stock 2 and stock 3) ≥ Number(A, not C|a day where I measured stock 1 and stock 3)

For a simple argument for why the time-dependence of stock values (or hidden variables) doesn't matter if my choice of which pair to sample has a time-independent probability that isn't correlated with the stock values/hidden variables, consider this point from [post=3307591]post 11 on the Boole vs. Bell thread[/post]:

Of course Bell does not actually assume that for a finite number of trials, exactly the same values of hidden variables occur on trials where a and b are sampled as on trials where b and c are sampled, only that the probability of a given value of lambda on a trial where the sample was a+b is the same as the probability of that value on a trial where the sample was a+c. And note that this does not exclude the notion that the probability of getting different hidden variable values could vary with time, but in that case if you knew the probability distribution for lambda at the actual times of measurement t1, t2, ... tN then you could construct a total probability distribution for lambda for a randomly selected measurement at one of those N times, and as long as the probability of choosing a+b vs. a+c or b+c was independent of the time of measurement (so for example the measurement at t2 was equally likely to be any of those three), then you can still derive the inequality.

The stock market for example is local-realistic, deterministic and non-stationary. That is why the average stock price for the first 100 days is not the same as the average stock price for the next 100 days for example. Bell proponents make the additional unsubstantiated assumption that the system being measured in the experiment is stationary.

Nah, they don't, that's just you inventing confused strawmen again.

 

[Gordon Watson]

But measurement outcomes are the only things that can ever be known about quantum systems ... that is one of the postulates of QM. So, it seems to me the "intrinsic property" of the particle itself, if it doesn't correlate to a measurement outcome, must be a hidden variable. The only remaining question after that is whether it is a non-local hidden variable, in which case you have a "weird" interpretation (like Bohmian mechanics), but one that is consistent with Bell's Theorem. If the "intrinsic property" is a local variable, then you are in conflict with Bell's Theorem.

In post #141, you seem to suggest that "pristine lambda" is a local hidden variable which is perturbed by measurement. Correct me if I am wrong, but it seems your chief objection to Bell is the association of measurement outcomes with local hidden variables, is that correct? I would also like to know your answers to the following questions:

1) Does the value of "pristine lambda" uniquely predict the measurement outcome for any possible detector setting?

2) If the answer to the above is "yes", then how is that any different than associating "pristine lambda" with measurement outcomes for detector settings, as done by Bell?

To me, I don't see any difference between what you have put forth and what Bell assumes in his paper, but perhaps if you can answer the above two questions, I will understand your position better.

Thanks for your interest in understanding my position.

A-1): Your wording here is not what I would use, but I suspect you mean this. Does the value of "pristine lambda" determine the measurement outcome for any possible detector setting. I say, YES.

A-2): There is no difference IF, by "association", you mean the result of a measurement is determined by "pristine" lambda" and its interaction ["association"] with a detector and its setting.

Hope this helps.

 

[Gordon Watson]

"Orientation of the total spin" is not something particles have a definite value for in QM prior to any spin measurement, so this must be some sort of hidden variable.

Of course.

But is it a local hidden variable?

Of course.

Does each particle have its own "orientation" value,

Of course.

or by "total spin" are you talking about both particles at once, and saying "orientation of the total spin" for both particles cannot be defined as a function of local variables associated with each particle?

I'm talking one particle at a time; with total spin conserved in every pair; no two pairs the same.

If your model includes nonlocal variables that can't be defined as functions of other local variables (in the way that magnetic flux through an extended surface can be defined as a function of the local magnetic field vector at each point on the surface), then you are not a "local realist" by any physicist's definition!

I'm a local realist in the purist sense: My local = Einstein local. My realism = Bell's realism.

But I already told you that EPR/Bell were not saying the particle must already have a given property prior to measurement, just that it must have local properties that predetermine what property would be observed if that measurement was made (in a local realist model where measuring one particle allows you to predict with certainty what would be observed if the same measurement were performed on the other, that is). That was the point of my whole discussion of Einstein's two-box analogy in post #144 and how instead of the box already containing a ball or not, it might contain a holographic projector which was predetermined to either project or not project an image of a ball in response to the box being opened. Please read that post again if you're not clear on this point! Along the same lines, while it would obviously be incorrect to attribute the property of "death" to me now, it's conceivable that there are properties associated with my body and perhaps some section of the surrounding world at this time that predetermine whether a test for my death at some future date (say Jan 1. 2050) will yield the result "still alive" or "he's dead, Jim", so if someone knew all the relevant properties today they could predict with total certainty what the result of this Jan. 1 2050 test would be.

Classical examples are not relevant when discussing the fundamentals of our quantum world; see my earlier comment on this.

I asked you specifically about such predetermined results in post #146:

Your short response was "Sure; why not?" Did you actually read my question carefully before responding or were you just being flippant? Do you wish to change your answer now? The comment is asking you whether there are local properties associated with one of the particles that predetermine what result it would give to each of the three possible detector settings, before the experimenter even makes the choice of what detector setting to use. If you agree that each particle has a well-defined set of predetermined answers like this (and that both members of every entangled pair have the same three predetermined answers), I don't see how you can deny that for every single particle, it either does or doesn't satisfy the three combinations of predetermined results (A, not B) and (B, not C) and (A, not C), meaning an observer with magical knowledge of hidden variables could count how many particles satisfy each one and get Number(A, not B) and Number(B, not C) and Number(A, not C) for any series of particle pairs.

My answer was serious: You defined the omniscient One and I defined Her potential.

Q-1: Is there any reason that you work with Numbers and not normalized distributions?

Also: I gave you HER distributions in post # 155 https://www.physicsforums.com/showpost.php?p=3309843&postcount=155

Q-2: Do you accept them?

 

[JesseM]

Classical examples are not relevant when discussing the fundamentals of our quantum world; see my earlier comment on this.

Then why did you say "My local = Einstein local"? The analogy of the two boxes was Einstein's own to explain what kind of local, objective theory he was looking for, see the quotations from his letter in [post=3270631]this post[/post].

My answer was serious: You defined the omniscient One and I defined Her potential.

Right, but just to clarify, you understood that when I said "omniscient" I just meant total perfect knowledge of all hidden variables at a given moment (without any measurements disturbing anything), along with knowledge of the laws of physics allowing for the most accurate possible prediction of the future given knowledge of that moment? If there is any fundamental indeterminism in the laws of nature, this "omniscient" being may not be able to predict with certainty what will happen in the future given knowledge of hidden variables in the present.

If we are in agreement on this, do you also agree that for each particle pair, this omniscient being's knowledge of each particle's hidden "orientation" at some time before the experimenters chose their detector settings would allow her to predict with certainty what result the particle would give for each of the three measurement settings? So if we define:

A=predetermined to give spin-up if measured at angle a in the future
not-A=predetermined to give spin-down if measured at angle a in the future
B=predetermined to give spin-up if measured at angle b in the future
not-B=predetermined to give spin-down if measured at angle b in the future
C=predetermined to give spin-up if measured at angle c in the future
not-C=predetermined to give spin-down if measured at angle c in the future

Then at this time before the detector setting was chosen, this omniscient being could use her knowledge of a particle's orientation to put it in some definite category like [A, not-B, C] or [not-A, B, C]? Please give me a clear yes-or-no answer to whether you agree that the "omniscient" being (as defined above) would be able to do this.

And if your answer to that is "yes", then I don't understand why you disagreed with any part of my [post=3308440]post 149[/post]! If you do answer "yes" to the above, then please go back to that earlier post and tell me specifically which statement of mine is the first you disagree with there (perhaps you misunderstood something I was saying in that post).

Q-1: Is there any reason that you work with Numbers and not normalized distributions?

Yes, because first considering numbers of particles with various possible predetermined results like [not-A, not-B, C] is an integral part of how I derive the conclusion that any local realist model where such predetermined results could be known by this type of omniscient being would be one where the probability distributions on measurement outcomes would respect Bell inequalities (so that in turn shows that if QM's probability distributions violate these inequalities, that proves QM's probabilities cannot be derived from such a local realist model).

Also: I gave you HER distributions in post # 155 https://www.physicsforums.com/showpost.php?p=3309843&postcount=155

Q-2: Do you accept them?

I don't accept that those probability distributions could possibly be consistent with "any local realist model where such predetermined results could be known by this type of omniscient being". You can have one but not the other. And I'm asking you to try to follow along the argument as to why a local realist model implies those probability distributions are impossible (given the assumption of perfect correlations whenever both are measured at the same angle), and carefully identify the first point in the argument that you disagree with.

 

[billschnieder]

Yes, and as soon as you do specify a method then Bertrand's paradox doesn't apply.

Wrong, Without know the properties by which you intent to randomize and how they behave, it is impossible to "specify a method". In other words, you can not perform an experiment in which a variable is screened-off, if you know nothing about the properties and behaviour of the variable. There's no conspiracy in that.

Take an example in which the price of a companies stock was dependent on the demand for it's product. But you had no idea that the variable "product demand" was a factor, in other words, it was hidden to you, and you had no idea about how "product demand" changed over time. Now tell me a method you will chose to sample the stock price such that "product demand" was screened-off.

The "methods" you claim to have provided is a joke.

2) The argument assumes that by averaging a large number of values you obtain a value close to the correct one. JesseM will call it the law of large numbers. What he will not tell you (maybe he doesn't know this), is that the law of large numbers does not apply to non-stationary systems.

Of course it does, silly.

No it doesn't. Why don't you tell me what the expectation value for Apple's stock price (AAPL), einstein. You have ~30 years of minute by minute stock price data you can work from. You have no clue what you are talking about.

Read this paper http://arxiv.org/pdf/0804.0902, you will learn a thing or two from it.
Or this one: http://arxiv.org/abs/1001.4103v1

 

[Rap]

I have been following this thread but not contributing because I have to admit, I am having some trouble following the arguments. I would like to ask some questions about a particular scenario, however, to help me understand some things. I wonder what would happen if we enclosed Alice, Bob, the detectors, and the entangled particle generator inside a Schroedinger catbox. Replace Alice and Bob each with a piece of radioactive material connected to a geiger counter, which then was rigged to choose one of the three detector orientations with, let's say, equal probability. In the Schroedinger cat experiment, it was easily rigged to yield a 50-50 chance of breaking the poison vial or not breaking the vial. I expect we could come up with a slightly more complicated mechanism to give a 1/3-1/3-1/3 probability of choosing detector A, B, or C. Suppose we know the wave function of the entire setup when we close the box, entangled particles not yet emitted. (I know, practically impossible). Now we propagate the wave function forward using an appropriate relativistic wave equation. After a time interval sufficient to be sure that the train of entangled particles have impinged on the detectors, what does the wave function look like?I'm quite sure that the wave function would have strong components which favored the expected QM statistical results in the "unboxed" scenario, violations much less so. I also think that, for these strong components, there would be an equal probability for each possible way that those statistics could be realized. In other words, if, in the subspace where the Alice-machine selects A and the Bob-machine selects B, and a hundred particles are measured, and the angle is such that we expect 99 percent (anti)coincidence, then the trains with a single coincidence will be most highly favored, and for each of these 100-particle trains with one coincidence, the probability that the coincidence will occur for the third particle is the same as the probability it will occur for the 37th particle, etc. In other words, the particular details of the well-designed detector will not be an issue, causing a particular particle to be more likely to be the odd one out.

I have two questions - 1) What do you think about the idea that these probabilities are equal and 2) what has happened to the concepts of local realism, superluminal effects, counterfactual definiteness, etc. in this scenario?

 

[SpectraCat]

... inside a Schroedinger catbox ...

The best part about a Schrodinger catbox is the poop is only there half the time!

 

[billschnieder]

I have been following this thread but not contributing because I have to admit, I am having some trouble following the arguments. I would like to ask some questions about a particular scenario, however, to help me understand some things. I wonder what would happen if we enclosed Alice, Bob, the detectors, and the entangled particle generator inside a Schroedinger catbox. Replace Alice and Bob each with a piece of radioactive material connected to a geiger counter, ...

Making the system more complicated will not help you to understand better.

Let me try to explain it another way:

For each *specific particle pair* in the experiment, God knows exactly what it will be obtained *if* Alice and Bob measured at angles (a,b). God also knows exactly what will be obtained *if* Alice and Bob instead measure at angles (b,c), *or* at angles (a,c). The real physical situation (hidden variables) of the particle pair exists and determines together with the chosen settings, the outcome of any of those measurements. All three *alternative results* at (a,b), (b,c), (a,c) will be generated by the same actual existing physical situation (hidden variables) of the particles. In other words, they are all *possible* outcomes from the deterministic interaction of the *hidden variables* with the detectors.

God then writes down an inequality using the three *possibilities* which MUST be obeyed.

|P(a,b) - P(a,c)| <= P(b,c) + 1

Note that of the three *possibilities*, only one can be actualized since *one pair* of photons can only be measured ONCE by Alice and Bob (ie, *if* Alice and Bob choose settings (a,b) and measure *the* photon pair, it becomes impossible for them to choose different settings and measure *the same* photon pair!). Remember that it is God who wrote the inequality, and he does not need to perform any experiments to obtain his *possibilities* in his inequality. The poor experimenter on the other hand is given an impossible task: "Perform three mutually exclusive measurements (a,b), (b,c), (a,c)". So he naively thinks to himself, let me measure (a,b) on particle pair 1 ie, (a1,b1), (b,c) on particle pair 2 ie (b2,c2), and (a,c) on particle pair 3 ie (a3,c3), hopefully all three particles are exactly the same and I should be able to substitute those results in God's inequality. Unfortunately for him, he obtains a violation. But being so foolish, he concludes that God's assumption that particles have pre-existing properties which determine the outcome, is false. He forgets to question his own faulty assumption that results from *different particle pairs* can be used in God's inequality.

 

[Rap]

Making the system more complicated will not help you to understand better.

Yes, its more complicated, but the thing is, if it is a valid system to consider, and you do understand it, then I think all of the questions and their answers are contained in it.

Let me try to explain it another way:

(Explanation)

I understand your explanation, I understand that only one measurement is made so that only one statement is testably true and the other statements are rendered untestable by that measurement. I think it is premature to assume that hidden variables are controlling the situation.

If you consider the whole system inside a box with Alice and Bob replaced by the simple "random event generator" I mentioned, then the wave function at a sufficient time later will consist of a superposition of all possible outcomes. I am assuming that the wave function generated will give correct probabilities, with no hidden variables assumed, no superluminal effects, etc. This wave function contains all of "God's information" that is availiable to us about every possible outcome. When we open the box, the wavefunction collapses to an eigenstate, one of those outcomes. I think that by pondering that wave function, maybe some answers may be found.

For example, consider the subspace containing the Alice=A, Bob=B (AB) outcome, Alice is using detector orientation A, Bob is using B. Every possible pair of measurement results will each be represented by an eigenstate with a particular probability, but the ones that yield the answer we expect from QM will be much more likely. Now what about the Alice=A, Bob=C (AC) subspace? Same thing. For the Alice=A, Bob=A (AA) same thing, but more restrictive, only perfect anticorrelation strings have non-zero probability.

Now let's look at the Bell situation. Look at the AB subspace, pick an Alice-string and a Bob-string from the available possibilities. Look for the Bob-string being measured by Alice in the BB subspace, it's there and there and there is only one string that Bob measures and its perfectly anticorrelated with it. Look for the Alice-string being measured by Bob in the AA subspace, it's there, and the string measured by Alice is perfectly anticorrelated. Now look for the Bob-string being measured by Bob in the CB subspace (Alice is set to detector C). Its there, along with a bunch of possible strings that Alice might measure. But if you look for any of those strings that Alice might measure in the CA subspace, they will not be found.

Couldn't it be that this whole scenario is possible, without invoking hidden variables, non-local reality, superluminal speeds, etc., but rather follows from the direct application of unitary propagation to wave functions?

 

[DrChinese]

Making the system more complicated will not help you to understand better.

Let me try to explain it another way:

For each *specific particle pair* in the experiment, God knows exactly what it will be obtained *if* Alice and Bob measured at angles (a,b). God also knows exactly what will be obtained *if* Alice and Bob instead measure at angles (b,c), *or* at angles (a,c). The real physical situation (hidden variables) of the particle pair exists and determines together with the chosen settings, the outcome of any of those measurements. All three *alternative results* at (a,b), (b,c), (a,c) will be generated by the same actual existing physical situation (hidden variables) of the particles. In other words, they are all *possible* outcomes from the deterministic interaction of the *hidden variables* with the detectors.

God then writes down an inequality using the three *possibilities* which MUST be obeyed.

|P(a,b) - P(a,c)| <= P(b,c) + 1

Note that of the three *possibilities*, only one can be actualized since *one pair* of photons can only be measured ONCE by Alice and Bob (ie, *if* Alice and Bob choose settings (a,b) and measure *the* photon pair, it becomes impossible for them to choose different settings and measure *the same* photon pair!). Remember that it is God who wrote the inequality, and he does not need to perform any experiments to obtain his *possibilities* in his inequality. The poor experimenter on the other hand is given an impossible task: "Perform three mutually exclusive measurements (a,b), (b,c), (a,c)". So he naively thinks to himself, let me measure (a,b) on particle pair 1 ie, (a1,b1), (b,c) on particle pair 2 ie (b2,c2), and (a,c) on particle pair 3 ie (a3,c3), hopefully all three particles are exactly the same and I should be able to substitute those results in God's inequality. Unfortunately for him, he obtains a violation. But being so foolish, he concludes that God's assumption that particles have pre-existing properties which determine the outcome, is false. He forgets to question his own faulty assumption that results from *different particle pairs* can be used in God's inequality.

So basically, QM is wrong (since the average value of correlation where theta=120 is 1/3 rather than 1/4) but God presents us with evidence to trick us into believing 1/4 is correct.

Wow, what an interesting and useful scientific hypothesis. Oh, did I leave out that it not falsifiable either?

P.S. I have offered your little gem myself several times here (in jest of course) because it can also be used to "disprove" any and all physical conclusions (General Relativity, all QM, etc.). There is nothing special whatsoever able applying this to Bell. I would hope this is obvious to all.

 

[billschnieder]

I understand your explanation, I understand that only one measurement is made so that only one statement is testably true and the other statements are rendered untestable by that measurement. I think it is premature to assume that hidden variables are controlling the situation.

Nobody is assuming that hidden variables must be controlling the situation, all I am saying that proclamations of the demise of local hidden variables are naive and premature. In other words, the results are consistent with hidden variables. And the apparent paradox is really just due to faulty logic.

If you consider the whole system inside a box with Alice and Bob replaced by the simple "random event generator" I mentioned, then the wave function at a sufficient time later will consist of a superposition of all possible outcomes. I am assuming that the wave function generated will give correct probabilities, with no hidden variables assumed, no superluminal effects, etc. This wave function contains all of "God's information" that is availiable to us about every possible outcome. When we open the box, the wavefunction collapses to an eigenstate, one of those outcomes. I think that by pondering that wave function, maybe some answers may be found.

But that is exactly the answers I have been giving you. You seem to think of QM as an ontological physical theory which it is not. QM is an information theory very similar to Probability theory. QM manipulates information in a consistent way to prevent you from making silly mistakes, just like Probability theory. You run into problems when you start assigning ontological status to the entities within the mathematical framework. The wave function simply encapsulates the information about all the possibilities I mentioned, and what you call "collapse" is what happened when one measurement has been performed such that the others become "impossible". What QM is telling you in you is simply the same thing I told you, just less straight forward.

Couldn't it be that this whole scenario is possible, without invoking hidden variables, non-local reality, superluminal speeds, etc., but rather follows from the direct application of unitary propagation to wave functions?

What scenario, QM does not describe any ontological scenario here. You appear to be assigning ontological status to abstract mathematical constructs such as "unitary propagation" and "wave functions" which might be the real obstacle here. To give you a simple analogy:

Say you are heading out to the shop to buy some groceries, and you have in your mind what you plan to buy, but it is not a simple list, since some of the items will only be bought if you find other items. For example you might have in your mind to buy bananas if you also find peanuts, other wise you will buy apples. This is all in your head, there are different possiblities. It is information, and has no ontological status. This is your "wavefunction". You go to the shop and find peanuts, your wavefunction is now collapsed. What you are suggesting above is like saying there is an existing "thing" which propagates unitarily to give you the result of what you bought in the shop. If this is how you think, you should read this paper for some enlightenment:

http://bayes.wustl.edu/etj/articles/prob.in.qm.pdf
Jaynes, E. T., 1990, `Probability in Quantum Theory,' in Complexity, Entropy, and the Physics of Information, W. H. Zurek (ed.), Addison-Wesley, Redwood City, CA, p. 381;

 

[Rap]

You seem to think of QM as an ontological physical theory which it is not.

I absolutely agree with you, the wave function is not ontological, it is an encapsulation of our knowledge gained from measurements. Unitary propagation tells us the probability of a certain outcome being obtained from a certain measurement made at some point in the future.

But in the scenario I outlined, there was never any implication that it was anything else. We make a (practically impossible) measurement of the wave function before the detectors are chosen by the radioactive material-geiger counter-simple machine (thus removing the vastly more complicated humans Alice and Bob and all implications of "free choice") and before the string of entangled particles are emitted. This wave function encapsulates our knowledge before the box is closed. Then we propagate that wave function. What we end up with after a time long enough for the particles to have been emitted and measured, is a wave function which assigns a probability for every possible outcome we might observe upon opening the box. e.g. "Alice" (the machine) chose detector A, "Bob" chose B, and the strings are [1,1,-1,1,-1...] and [-1,1,1,-1,-1,...]. That's one possibility and the wave function tells us the probability of that happening. If we want to ask "what if Bob chose detector C?", that answer is in there, and I think the answer to all such questions are in there, and they will violate the Bell inequalities. The wave function will give no answer to "what is the probability of simultaneous strings S1, S2, S3" because there is no such measurement. In other words, I think QM will give the correct answer without recourse to hidden variables, superluminal speeds, etc. Assuming that a probability can be assigned to three strings at a time amounts to counterfactual definiteness, and since the wave function is silent on such a probability, I think rejection of CFD is in order. I have read a number of the papers you have referred to, and I guess I am open to the possibility of hidden variables while maintaining the violation of Bell inequalities, but I am not sure how they would fit in with the above scenario.

 

[billschnieder]

The wave function will give no answer to "what is the probability of simultaneous strings S1, S2, S3" because there is no such measurement.

Correct, because it is a nonsensical question. Just like asking "what is a square circle". QM has built-mechanism which prevent you from asking silly questions like that. That is why it works so well.

In other words, I think QM will give the correct answer without recourse to hidden variables, superluminal speeds, etc.

I agree, QM does not need hidden variables to be able to perform its functions as it currently does. But this is not the same as saying the underlying physical processes could not be described using hidden variables. QM in it's current state, is unable to predict single event outcomes. It doesn't mean the physics of single events can not be described by hidden variables. The problem arises when you place QM on a pedestal and worship it as the be-all and end-all theory which it is not, and then conclude from that that anything which is not required in QM, is not permitted in nature.

Assuming that a probability can be assigned to three strings at a time amounts to counterfactual definiteness

This is wrong. As I explained earlier during this thread. CFD does not mean you allow three mutually exclusive statements to be true simultaneously since this will be so nonsensical nobody will ever in his right mind advocate for CFD. CFD simply means you speak definitely of outcomes which are no longer possible. For example, the following two statements:

1) If A is true then X is false.
2) If A is false then X is true.

CFD doesn't mean "X is false, and X is true" -- this is a nonsensical statement. CFD means that we speaking meaningfully and unambiguously about both statements (1) and (2) which can be simultaneously true in their complete states, with their conditioning statements in place, even though only one of them is *actual*. Once you disect them out, you are dealing with nonsense not CFD.

In other words, I am saying CFD is not as nonsensical as you make it look because in that strawman form, nobody with more than a braincell has ever advocated such a thing.

and since the wave function is silent on such a probability, I think rejection of CFD is in order.

I can also say, there is no such thing as a "square circle" therefore a rejection of "square circles" is in order. But I haven't said anything meaningful. Try to understand this instead of looking around for stuff to *reject*.

I have read a number of the papers you have referred to, and I guess I am open to the possibility of hidden variables while maintaining the violation of Bell inequalities, but I am not sure how they would fit in with the above scenario.

There is nothing special about bell inequalities. Boole had derived them 100 years before Bell. Their violation or non-violation should not have some special status. Rather, you should ask what the inequalities represent and what their violation means for the specific case at issue.

In the EPR case, the inequalities are relationships between three simultaneously *actual* variables from the same system since they are derived from the perspective of an omniscient being who is aware of those *actual* variables.

The expectation values from QM and experiments are therefore not compatible since those *actual* variables can not all be measured simultaneously. This is the cause of the violation and it is not clear to me what else you are looking for.

Take any situation in which you have an inequality with *actual* variables and experiments in which one of those *actual* variables can not be measured and you will obtain a violation. And the violation will not mean you have to reject the existence of those *actual* variable. All it will mean is that you can not use *mutually exclusive possibilities* in an expression which expects *simultaneous possibilities* or *actualities*. I have posted one recently concerning the triangle inequality and the OP posted one concerning coin tosses. It really is that simple, if you will get over the yearning need to reject some classical concept.

here is the triangle inequality example again:

I suppose you know about the triangle inequality which says for any triangle with sides labeled X,Y,Z where x, y, z represents the lengths of the sides

z <= x + y

Note that this inequality applies to a single triangle. What if you could only measure one side at a time. Assume that for each measurement you set the label of the side your instrument should measure and it measured the length destroying the triangle in the process. So you performed a large number of measurements on different triangles. Measuring <z> for the first run, <x> for the next and <y> for the next.

Do you believe the inequality
<z> <= <x> + <y>

Is valid? In other words, you believe it is legitimate to use those averages in your inequality to verify its validity?

Please answer this last question, so I know that you understand this issues here.

 

[rlduncan]

Coin Toss Simualtion

It is my intention to provide a simple coin toss simulation to clearly demonstrate that the derivation by Sakurai (http://en.wikipedia.org/wiki/Sakurai%27s_Bell_inequality) is invalid and that Bell’s inequality is pointless. The coin toss experiment consists of n=100 trials of randomly tossing three coins labeled a, b, c in which Alice randomly chooses a coin and records the upper most face and Bob randomly chooses a coin and records the lower most face to guarantee that Alice and Bob will have opposite outcomes 100% of time when choosing the same coin.

Bell’s inequality can be derived by first adding Eq. (3) and Eq. (4) from the above website as follows:

(3) P(a+,c+) =P2 + P4
(4) P(c+,b+) = P3 + P7

The sum gives the following:

P(a+,c+) + P(c+,b+) = P2+ P4 + P3 + P7

But from Sakurai’s Eq. (2): P(a+,b+) = P3 + P4, substituting gives:

P(a+,c+) + P(c+,b+) = P(a+,b+) + P2+ P7 and because probabilities are nonnegative then one can write the inequality as follows: P(a+,c+) + P(c+,b+) ≥ P(a+,b+).

It is easy to test the validity of the above equation that leads to the Bell inequality using the coin tossing experimental data. From the website P2 = abc(++-/--+) and P7 = abc(--+/++-). Let the plus sign represent a head and the negative sign represent a tail and instead of probability use the number or outcomes, n which is more convenient and simpler to analyze. According to Sakurai the following equation should be true for the coin tossing experiment where the outcomes are heads and tails.

nac(HH) + ncb(HH) = nab(HH) + nabc(HHT/TTH) + nabc(TTH/HHT)

The n = 100 trials of the coin tossing is given as a spread sheet and can be viewed as a web page at http://www.atomgeometry.com/EPRB_Coin_Toss.htm . The following is a summary of the relevant data used to test the above equation.

nac(HH) = 3
ncb(HH) = 4
nab(HH) = 1
nabc(HHT/TTH) = 14
nabc(TTH/HHT) = 17

Which suggest that 3 + 4 = 1 + 14 + 17.This is obviously a false statement and remains false no matter how large you make n. It is reasonable to determine from this simple coin toss experiment that any inequality derived from this equation may be true or false depending on the specific outcomes. Regardless, knowing that the equation used to derive the inequality is false renders the inequality meaningless. Furthermore, it must be concluded that Bell’s inequality as derived says nothing about local hidden variables or nonlocality.

 

[rlduncan]

Correct, because it is a nonsensical question. Just like asking "what is a square circle". QM has built-mechanism which prevent you from asking silly questions like that. That is why it works so well.


I agree, QM does not need hidden variables to be able to perform its functions as it currently does. But this is not the same as saying the underlying physical processes could not be described using hidden variables. QM in it's current state, is unable to predict single event outcomes. It doesn't mean the physics of single events can not be described by hidden variables. The problem arises when you place QM on a pedestal and worship it as the be-all and end-all theory which it is not, and then conclude from that that anything which is not required in QM, is not permitted in nature.


This is wrong. As I explained earlier during this thread. CFD does not mean you allow three mutually exclusive statements to be true simultaneously since this will be so nonsensical nobody will ever in his right mind advocate for CFD. CFD simply means you speak definitely of outcomes which are no longer possible. For example, the following two statements:

1) If A is true then X is false.
2) If A is false then X is true.

CFD doesn't mean "X is false, and X is true" -- this is a nonsensical statement. CFD means that we speaking meaningfully and unambiguously about both statements (1) and (2) which can be simultaneously true in their complete states, with their conditioning statements in place, even though only one of them is *actual*. Once you disect them out, you are dealing with nonsense not CFD.

In other words, I am saying CFD is not as nonsensical as you make it look because in that strawman form, nobody with more than a braincell has ever advocated such a thing.



I can also say, there is no such thing as a "square circle" therefore a rejection of "square circles" is in order. But I haven't said anything meaningful. Try to understand this instead of looking around for stuff to *reject*.


There is nothing special about bell inequalities. Boole had derived them 100 years before Bell. Their violation or non-violation should not have some special status. Rather, you should ask what the inequalities represent and what their violation means for the specific case at issue.

In the EPR case, the inequalities are relationships between three simultaneously *actual* variables from the same system since they are derived from the perspective of an omniscient being who is aware of those *actual* variables.

The expectation values from QM and experiments are therefore not compatible since those *actual* variables can not all be measured simultaneously. This is the cause of the violation and it is not clear to me what else you are looking for.

Take any situation in which you have an inequality with *actual* variables and experiments in which one of those *actual* variables can not be measured and you will obtain a violation. And the violation will not mean you have to reject the existence of those *actual* variable. All it will mean is that you can not use *mutually exclusive possibilities* in an expression which expects *simultaneous possibilities* or *actualities*. I have posted one recently concerning the triangle inequality and the OP posted one concerning coin tosses. It really is that simple, if you will get over the yearning need to reject some classical concept.

here is the triangle inequality example again:

I suppose you know about the triangle inequality which says for any triangle with sides labeled X,Y,Z where x, y, z represents the lengths of the sides

z <= x + y

Note that this inequality applies to a single triangle. What if you could only measure one side at a time. Assume that for each measurement you set the label of the side your instrument should measure and it measured the length destroying the triangle in the process. So you performed a large number of measurements on different triangles. Measuring <z> for the first run, <x> for the next and <y> for the next.

Do you believe the inequality
<z> <= <x> + <y>

Is valid? In other words, you believe it is legitimate to use those averages in your inequality to verify its validity?

Please answer this last question, so I know that you understand this issues here.

Bill, great examples and thorough reasoning.

 

[SpectraCat]

Coin Toss Simualtion

It is my intention to provide a simple coin toss simulation to clearly demonstrate that the derivation by Sakurai (http://en.wikipedia.org/wiki/Sakurai%27s_Bell_inequality) is invalid and that Bell’s inequality is pointless. The coin toss experiment consists of n=100 trials of randomly tossing three coins labeled a, b, c in which Alice randomly chooses a coin and records the upper most face and Bob randomly chooses a coin and records the lower most face to guarantee that Alice and Bob will have opposite outcomes 100% of time when choosing the same coin.

Bell’s inequality can be derived by first adding Eq. (3) and Eq. (4) from the above website as follows:

(3) P(a+,c+) =P2 + P4
(4) P(c+,b+) = P3 + P7

The sum gives the following:

P(a+,c+) + P(c+,b+) = P2+ P4 + P3 + P7

But from Sakurai’s Eq. (2): P(a+,b+) = P3 + P4, substituting gives:

P(a+,c+) + P(c+,b+) = P(a+,b+) + P2+ P7 and because probabilities are nonnegative then one can write the inequality as follows: P(a+,c+) + P(c+,b+) ≥ P(a+,b+).

It is easy to test the validity of the above equation that leads to the Bell inequality using the coin tossing experimental data. From the website P2 = abc(++-/--+) and P7 = abc(--+/++-). Let the plus sign represent a head and the negative sign represent a tail and instead of probability use the number or outcomes, n which is more convenient and simpler to analyze.

All of the above looks fine, however you go off the rails at this point:

According to Sakurai the following equation should be true for the coin tossing experiment where the outcomes are heads and tails.

nac(HH) + ncb(HH) = nab(HH) + nabc(HHT/TTH) + nabc(TTH/HHT)

That is where you start to go wrong .. there is no problem with the equality as written, but you have to make sure you compare apples with apples. ( more below)

[EDIT: Actually, I take that back .. that last equality is simply wrong, if nab, nac and nbc refer to raw coincidence counts, as opposed to probabilities.]

The n = 100 trials of the coin tossing is given as a spread sheet and can be viewed as a web page at http://www.atomgeometry.com/EPRB_Coin_Toss.htm . The following is a summary of the relevant data used to test the above equation.

Here is the problem with your analysis:

The following values are for coincident measurements in that they are ONLY counted for cases where Alice and Bob have made particular choices AND observe particular values:

Note also that the measurements below DO conform to the inequality Sakurai actually wrote (modified for your notation): nab <= nac + nbc ---> 1 <= 3 + 4 (TRUE!)

nac(HH) = 3
ncb(HH) = 4
nab(HH) = 1

The values below are "objective" in that they are counted for EVERY case, whether or not Alice and Bob's choices resulted in them getting a coincident measurement for that case. You should have been clued into the fact that something was up since the values were so much higher than for the coincident measurements above.

nabc(HHT/TTH) = 14
nabc(TTH/HHT) = 17

If you count up P3 and P4 as "objective" measurements, you will find that your equality is exactly preserved ... it has to be, because the numbers on both sides are the same.

What Sakurai was saying was that IF you do enough coincident measurements, you will find that the probability of a particular coincident measurement (say nab) with respect to the pool of successful coincident measurements, will approach the summed probabilities of the particular objective measurements which give possible matches to that coincident measurement. In your case with the coin tosses, the objective probabilities for all of the specific outcomes are equal, so your inequality amounts to 0.25 <= 0.25 +0.25, which is obviously true.

 

[rlduncan]

All of the above looks fine, however you go off the rails at this point:


That is where you start to go wrong .. there is no problem with the equality as written, but you have to make sure you compare apples with apples. ( more below)

[EDIT: Actually, I take that back .. that last equality is simply wrong, if nab, nac and nbc refer to raw coincidence counts, as opposed to probabilities.]

The reason the equation does not sum correctly is because of choosing two coins out of the three coins when tabulating the data. The equation is always valid when analyzing all three coins simultaneously.

For comparison, here is a second spread sheet for n=25 trials for a simultaneous 3-coin toss experiment:

http://www.atomgeometry.com/Simultaneous_Coin_Toss.htm .

Notice that the equation: nab(HT) + nbc(HT) = nac(HT) + nabc(HTH) + nabc(THT) is true, that is, 2 + 8 = 4 + 1 + 5. For this coin tossing experiment the equation is always true and the inequality derived from the equation is always true. This consistency is lacking in the EPRB coin tossing experiments. Which suggest the problem lies in choosing two of the three coin to analyze instead of analyzing all three coins simultaneously (IMHO).

 

[Rap]

The problem arises when you place QM on a pedestal and worship it as the be-all and end-all theory which it is not, and then conclude from that that anything which is not required in QM, is not permitted in nature.

1) You cannot prove that it is not the be-all and end-all theory.
2) To think that science is only about knowledge gained through measurements, and theories which predict the outcomes (or probabilities of outcomes) in the future based on that knowledge, and that questions which implicitly assume unmeasured knowledge are improper, is not the same as putting QM on a pedestal and worshipping it.
3) I agree, putting QM on a pedestal and worshipping it as such is totally anti-scientific.

This is wrong. As I explained earlier during this thread. CFD does not mean you allow three mutually exclusive statements to be true simultaneously since this will be so nonsensical nobody will ever in his right mind advocate for CFD. CFD simply means you speak definitely of outcomes which are no longer possible. For example, the following two statements:

1) If A is true then X is false.
2) If A is false then X is true.

CFD doesn't mean "X is false, and X is true" -- this is a nonsensical statement. CFD means that we speaking meaningfully and unambiguously about both statements (1) and (2) which can be simultaneously true in their complete states, with their conditioning statements in place, even though only one of them is *actual*. Once you disect them out, you are dealing with nonsense not CFD.

We have a prediction from our theory:

1) If A is true then X will be found false.
2) If A is false, then X will be found true.

Once the truth value of A is determined, and X is found to be in accordance with these predictions, we can say that the predictions were both true prior to the measurement. After the measurement, they are counterfactual - they assume that a measurement has not been made when in fact it has. They justify our theory, but they have no other relevance. They must be replaced with, for example:

1a) A was found true, and X was found false.

Counterfactual definiteness is the statement:

1b) If A were found false, then X would have been found true.

To deny CFD is to say that, since 1b is counterfactual, it cannot be assigned a truth value, which is not the same as assigning it a truth value of false. 1b cannot be used as if it were the result of a measurement, since it was in fact not the result of a measurement.

There is nothing special about bell inequalities. Boole had derived them 100 years before Bell. Their violation or non-violation should not have some special status. Rather, you should ask what the inequalities represent and what their violation means for the specific case at issue.

They do not have any special significance to a mathematician, but when applied to physics, I think they take on a special significance.

I have posted one recently concerning the triangle inequality and the OP posted one concerning coin tosses. It really is that simple, if you will get over the yearning need to reject some classical concept.

here is the triangle inequality example again:

I suppose you know about the triangle inequality which says for any triangle with sides labeled X,Y,Z where x, y, z represents the lengths of the sides

z <= x + y

Note that this inequality applies to a single triangle. What if you could only measure one side at a time. Assume that for each measurement you set the label of the side your instrument should measure and it measured the length destroying the triangle in the process. So you performed a large number of measurements on different triangles. Measuring <z> for the first run, <x> for the next and <y> for the next.

Do you believe the inequality
<z> <= <x> + <y>

Is valid? In other words, you believe it is legitimate to use those averages in your inequality to verify its validity?

Please answer this last question, so I know that you understand this issues here.

Of course, I do not believe that, but I don't understand how that applies to the problem at hand.

 

[rlduncan]

The values below are "objective" in that they are counted for EVERY case, whether or not Alice and Bob's choices resulted in them getting a coincident measurement for that case. You should have been clued into the fact that something was up since the values were so much higher than for the coincident measurements above.


If you count up P3 and P4 as "objective" measurements, you will find that your equality is exactly preserved ... it has to be, because the numbers on both sides are the same.

This is not true as clearly demonstrated in the spread sheet. Look at Trial #4, there is a count for cb(HH), but none for abc(HHT/TTH) or abc(TTH/HHT). There is simply no one-to-one mapping of the counts and the equation will be false because of Trials such as #4. Refer to the Simultaneous 3-Coin Toss experiment (http://www.atomgeometry.com/Simultaneous_Coin_Toss.htm ) and observed the one-to-one mapping for this experiment for all trials.

 

[SpectraCat]

This is not true as clearly demonstrated in the spread sheet. Look at Trial #4, there is a count for cb(HH), but none for abc(HHT/TTH) or abc(TTH/HHT). There is simply no one-to-one mapping of the counts and the equation will be false because of Trials such as #4. Refer to the Simultaneous 3-Coin Toss experiment (http://www.atomgeometry.com/Simultaneous_Coin_Toss.htm ) and observed the one-to-one mapping for this experiment for all trials.

What are you talking about? Your two objective columns correspond to P2 and P7 ... P(c,b) is equal to P3 + P7. So it is obvious why there is no count in those columns for Trial #4 .. trial #4 corresponds to an incidence of P3.

If you had cast your inequalities properly in terms of probabilities rather than raw counts, perhaps you would not have made this error. Writing the objective measurements as probabilities is easy .. just divide by the number of trials. You need to consider the proper normalization of the coincident measurements in order to write them as probabilities (my earlier statement about how to normalized these was incorrect). This means that you need to calculate P(a,b) by dividing nab by the number of trials where Alice and Bob chose (a,b), and do similar normalizations for the P(a,c) and P(b,c).

Your simulations have another issue, which is that you are not doing NEARLY enough trials ... with such small sample sizes you could easily have a selection bias. Right now you are folding in the probabilities that Alice and Bob will pick a particular coin ... those will eventually even out if they are truly random, but for only 100 trials, I wouldn't expect them to be even.

 

[SpectraCat]

The reason the equation does not sum correctly is because of choosing two coins out of the three coins when tabulating the data. The equation is always valid when analyzing all three coins simultaneously.

For comparison, here is a second spread sheet for n=25 trials for a simultaneous 3-coin toss experiment:

http://www.atomgeometry.com/Simultaneous_Coin_Toss.htm .

Notice that the equation: nab(HT) + nbc(HT) = nac(HT) + nabc(HTH) + nabc(THT) is true, that is, 2 + 8 = 4 + 1 + 5. For this coin tossing experiment the equation is always true and the inequality derived from the equation is always true.

Right, because all of those counts are "objective" (in terms of the definition I used in post #174). The problem with your other analysis is that you are comparing raw counts for coincident measurements with those for "objective" measurements. Once you normalize properly, and do enough repeats (10000 should be plenty) those problems will go away.

This consistency is lacking in the EPRB coin tossing experiments. Which suggest the problem lies in choosing two of the three coin to analyze instead of analyzing all three coins simultaneously (IMHO).

It's not a problem .. you just have to normalize your probabilities properly .. see my previous post (#178).

 

[billschnieder]

1) You cannot prove that it is not the be-all and end-all theory.

Experimentally, I can measure with a very high precision the position of a single photon on the screen after going through a double slit. QM can't predict the position of a single photon on the screen, therefore QM is not a be-all and end-all theory. Period. Single events happen all the time, yet QM can not predict any single events. That should be enough to humble QM worshipers.

2) To think that science is only about knowledge gained through measurements, and theories which predict the outcomes (or probabilities of outcomes) in the future based on that knowledge, and that questions which implicitly assume unmeasured knowledge are improper, is not the same as putting QM on a pedestal and worshipping it.

Of course it is not, but claiming that QM is the final physical theory which is good enough and we should stop looking for a better theory which can explain single events is worshipping QM. Claiming that anything in nature which is not explained by QM is not real, is worshiping QM. It is also called the Mind Projection Fallacy. Jaynes explains in the article I cited above:

The failure of quantum theorists to distinguish in calculations between several quite different meanings of 'probability', between expectation values and actual values, makes us do things that don't need to be done; and to fail to do things that do need to be done. We fail to distinguish in our verbiage between prediction and measurement. For example, the famous vague phrases: 'It is impossible to *specify* ...'; or 'It is impossible to *define* ...' can be interpreted equally well as statements about prediction or statements about measurement. Thus the demonstrably correct statement that the present formalism cannot predict something becomes perverted into the logically unjustified -- and almost certainly false -- claim that the experimentalist cannot measure it!
We routinely commit the Mind Projection Fallacy: supposing that creations of our own imagination are real properties of Nature, or that our own ignorance signifies some indecision on the part of Nature. It is then impossible to agree on the proper place of information in physics. This muddying up of the distinction between reality and our knowledge of reality is carried to the point where we find some otherwise rational physicists, on the basis of the Bell inequality experiments, asserting the objective reality of probabilities, while denying the objective reality of atoms! These sloppy habits of language have tricked us into mystical, pre-scientific standards of logic, and leave the meaning of any QM result ambiguous. Yet from decades of trial-and-error we have managed to learn how to calculate with enough art and tact so that we come out with the right numbers!

We have a prediction from our theory:

1) If A is true then X will be found false.
2) If A is false, then X will be found true.

Once the truth value of A is determined, and X is found to be in accordance with these predictions, we can say that the predictions were both true prior to the measurement. After the measurement, they are counterfactual - they assume that a measurement has not been made when in fact it has. They justify our theory, but they have no other relevance. They must be replaced with, for example:

1a) A was found true, and X was found false.

You haven't followed anything I have said about this, you are just repeating an argument I have responded to already:

The statementfs:

*If A is true then X is false.*
*If A is false then X is true.*

Have only one truth value (true or false). They can not be valid at one time and invalid at another time. They can not be true at one time and false at another time. They are statements about the logical relationship between the truth values of two entities (A and X). They are not a statements about X only, or about A only. The above statements are completely different statements from the ones.

*X is false*, *A is true*, *X is true*, *A is false*

The statements will have the same truth value (true or false) regardless of whether or not A is true and whether or not X is true. To suggest that a statement is made *irrelevant* (whatever you mean by that) just because somebody decided to measure X is naive at best. To see this, give the statements to Alice and Bob. Alice enters the room and performs the experiment without Bob knowing that an experiment has been performed.

According to you, Bobs knowledge has instantaneously been changed (made irrelevant) just because Alice made a measurement. So how then is Bob ever supposed to know what part of his knowledge is relevant and what part is not, if he does not have access to all the experiments that have been performed.

If you think such an approach gets you off Bell's hook, you are mistaken as I have explained because:

- Every QM prediction is conditioned on the experimental preparation: *If x is measured you obtain y* etc as I have explained.
- QM predictions for P(a,b), P(b,c), P(a,c) can never be simultaneously measured, therefore, according to you, as soon as Alice and Bob set their devices to (a,b) and do the measurement, P(b,c) and P(a,c) become *irrelevant*.

If we are to go with your CFD definition and approach, we will end up with the conclusion that Bell's inequalities require irrelevant parameters to be used at the same time as relevant ones. So why is that the problem of any local realistic theory rather than Bell's problem? Why is that natures problem rather than Bell's?? (cf. Mind Projection Fallacy)

My main argument in this thread has been to point out to you that it is impossible to test Bell's inequality experimentally because it uses three simultaneous values in its derivation where only two can ever be measured experimentally. Why is this not sufficient reason to you why a violation is obtained. You haven't responded to this argument yet. Nobody has.