Scholarpedia article on Bell's Theorem

[DevilsAvocado]

It is only when Alice has collected all her pluses and minuses and Bob has done the same that they start comparing time-tags to see coincidences and then they can figure out what the angular difference was at the moment of detection!

Not the angular difference, but the relative angle (a-b).

[all bolding mine]

The photon gives no rodent's behind whether Alice's angle was chosen randomly or not! All it meets is a polarizer at a given angle resulting in an OUTCOME of (+, -, or non-detection).

Misunderstanding; yes the angle is chosen randomly, but the experimental QM outcome/result is also always random 50/50 (+/-).

I'm afraid you have seriously misunderstood. Nobody is assuming static/predetermined results. The result is random for a given photon, but once Alice has measured and obtained +1 for that photon at 67.5°, it is a mathematical/logical error to say Alice would have obtained -1 had she measure that specific photon at the same specific angle 67.5°! You can't set a realism assumption and them immediately gut it and expect it to stay put.

More misunderstandings; we all know that it’s impossible to measure both +1 and -1 for one photon in one measurement, and most agree that if we repeat the measurement at the same angle – the result is 100% random.

Most also agree that if some dude comes up with a theorem that shows “1 + 1 = 9” we don’t have to prove this theorem to be correct, before proving it to be incorrect. I think it is called common sense.

Huh? I just show you that it is impossible to derive Bell's theorem without using CFD

Your derivation is quite strange since you are claiming that to be able to use CFD in any theorem we are obligated to actually measure these values in experiments, which of course most understand is impossible:

We do not need to sacrifice anything, because there is nothing there there to start with. The terms in Bell's inequality and the CHSH can never be tested experimentally, if reasoning correctly. The inequalities can never be violated if reasoning correctly. So I guess what has to be sacrificed is buffoonery.

We have three terms here C(a,b), C(a,c), C(b,c). Those terms can never be all factual as far as the EPR experiment is concerned. At least two of them MUST be counterfactual! There is no other way. Thinking otherwise is just buffoonery. The inequalities can NOT be derived UNLESS the other two terms are counterfactual. As soon as you see that, you realize immediately that NO EXPERIMENT can ever measure them all! None! You can measure one but not the other two. Is that clear enough?

No comment, speaks for itself.

So then, we are left with a lot of experimentalists who do not know what they are doing, publishing in lofty journals whose editors and reviewers do not know what they are doing, a many who love mysticism regurgitating what they've read without thinking for themselves. No news here.

I could be wrong, but to me it’s actually more mysticism in regarding all these highly educated/skilled peoples rigorously scrutinizing the mathematical and experimental result of Bell’s theorem – and not one see what you see...

Why thousands? If you were wrong, then one would have been enough!

A lot of ignorance out there, including Schrödinger et al., right?

Having eliminated all the experiments, we now have QM left. How come then that QM can violate the inequalities? Because the terms that people calculate from QM and substitute into the inequalities in order to obtain violation, are not the correct terms.

They've calculated and used the following three terms (scenario X):
C(a,b) = QM correlation for what we would get if we measure (a,b)
C(b,c) = QM correlation for what we would get if we measure (b,c)
C(a,c) = QM correlation for what we would get if we measure (a,c)

When Bell's inequalities DEMAND that the correlations should be (scenario Y):
C(a,b) = QM correlation for what we would get if we measure (a,b)
C(a,c) = QM correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = QM correlation for what we would have gotten had we measured (b,c) instead of (a,b)

More counterfactual confusion.

Now what is the probability that we would have obtained + at Alice if Alice and Bob had measured at angles (a,c) instead of (a,b). Note this is counterfactual. If you answer 1/3 you need to learn some probability theory. The correct answer is 1, we already know that measuring the photon at angle a gives +, where is the conspiracy in that?! Knowing what was obtained in the factual experiment, changes the probability we calculate for the counterfactual situation, nothing spooky involved. Now we can carry this all the way and include coincidences and you will see that using scenario X correlations in Bell's inequality is deeply flawed.

Well, to me all the above is good example of flawed conspiracy spookiness. You mix performed experimental outcomes with mathematical counterfactual speculations. I have never seen anything like it. You need to learn some QM theory.

QM gives the correct answer for the experiments performed, but neither QM nor the experiments can provide the right answers for substitution in the inequalities.

So what are the actual measured correlations? I hope you understand the experimental difference between entangled and non-entangled photons?

Are you serious? You must have misunderstood something very fundamental about the EPR experiment. For each photon that leaves the source and heads towards Alice, you have a single outcome, (+, -, or non-detection). Alice's polarizer is set to a specific angle say 67.5° for that photon. Same thing for Bob.

I can see from the discussion that there is in fact some serious confusion about the word outcome:

1. a final product or end result; consequence; issue.
2. a conclusion reached through a process of logical thinking.​

You seem to wobble between 1 & 2 without any specific notion on what you actually mean. Let me give you some examples:

It still does not change the fact that we have outcomes at 4 angles. For each angle there is an outcome.

Are you sure? If you insist, I suppose you can provide a NON-LOCAL dataset of outcomes for three angles a, b, c which violates the inequalities.

The only condition is there are 3 outcomes for 3 angles for each photon measured.

For each photon that leaves the source and heads towards Alice, you have a single outcome, (+, -, or non-detection).

The photon gives no rodent's behind whether Alice's angle was chosen randomly or not! All it meets is a polarizer at a given angle resulting in an OUTCOME of (+, -, or non-detection).

So? Who said any thing about the stream of outcomes appearing at Alice or Bob appearing other than random. That does not change the fact that there is an outcome. I have the files from Weihs' experiment and there is one outcome for each photon detected.

As soon as you see that, you realize immediately that NO EXPERIMENT can ever measure them all! None! You can measure one but not the other two. Is that clear enough?

Consider a pair of photons heading toward Alice and Bob resp, with polarizesr set to the angles (a,b). Let us say the possible outcomes are (+, -, 0 for nondetection) for each side and they are all equaly likely.

is not a conditional probability statement but a statement for the expectation value of the paired product of outcomes at A and B,

The Expectation value for the paired product at two stations is necessarily factorable whether or not the processes generating the outcomes are local or non-local.

is actuall <ab> where a represents the outcomes at angle α and b represents the outcomes at angle β.

Yes, Yes, Yes! For Alice, the outcomes are:

- F(a,λ) if she measures at angle (a)
- F(b,λ) if she had measured at angle (b) instead of at the (a) at which she actually measured
- F(c,λ) if she had measured at angle (c) instead of at the (a) at which she actually measured

Don't you see that those are not the outcomes measured in any real experiment.

Let us start with the first photon pair arriving at Alice and Bob respectively, and assume that the outcome was +1 for Alice and -1 for Bob for the angle pair (a,b).

Therefore the outcomes used to calculate C(a,c), and C(c,b) are not independent of those used to calculate C(a,b).

We are allowed to obtain F(a,λ1)=+1 for Alice for the first outcome used to calculate the C(a,b) correlation and F(a,λ1) = -1 for Alice for the first outcome used to calculate the C(a,c) correlation etc.

Note I'm using the shorthand a,b,c to represent Fa, Fb, Fc which are outcomes not angles. I'm using the 3-term Bell inequality |ab + ac| - bc <= 1 in which each term shares outcomes with the other two terms.

* Note that no violation is ever obtained for any individual pair of outcomes, and consequently no violation is possible for the correlations which are essentially averages of paired products |<ab> + <ac>| - <bc> <= 1
* Note that there are only 8 distinct possible outcome combinations for this scenario each of which always satisfies the inequality

No two terms share the same outcome contrary to scenario Y.

* Note that there are 64 distinct possible outcome combinations in Scenario X as opposed to just 8 in Scenario Y.

Get the point? How can we ever discuss this when you are jumping freely between:

• outcome = experimental result
• outcome = logical derivation
• outcome = calculated expectation
• outcome = probabilities

And on top of this you strangely enough couple counterfactual definiteness + CHSH to the impossibility of experimentally verify the outcome one would have obtained if one had measured a different angle:

• outcome = counterfactual paradox

?

Care to straighten out some question marks?

Counterfactual definiteness is a basic assumption, which, together with locality, leads to Bell inequalities. In their derivation it is explicitly assumed that every possible measurement, even if not performed, can be included in statistical calculations. The calculation involves averaging over sets of outcomes which cannot all be simultaneously factual—if some are assumed to be factual outcomes of an experiment others have to be assumed counterfactual. (Which ones are designated as factual is determined by the experimenter: the outcomes of the measurements he actually performs become factual by virtue of his choice to do so, the outcomes of the measurements he doesn't perform are counterfactual.) Bell's Theorem actually proves that every type of quantum theory must necessarily violate either locality or CFD.

 

[billschnieder]

Not the angular difference, but the relative angle (a-b).

Now I know you are not being serious at all.
Please read this article http://arxiv.org/pdf/quant-ph/9810080v1.pdf and pay attention at figure 2 of the above article, and notice the table of outcomes?

we all know that it’s impossible to measure both +1 and -1 for one photon in one measurement, and most agree that if we repeat the measurement at the same angle – the result is 100% random.

Huh? But it is impossible to repeat the same measurement! This is the whole point. You probably mean if you repeat a similar measurement for many different photons, the results appear random. Of course but Bell's inequalities are derived for the same set of photons (remember realism? remember CFD?) The whole point which you still do not get is that repeating it for different photons does not get you the same result as what you would have gotten, were it possible to repeat for the same photon. While the former may be random, the latter must not be.

Most also agree that if some dude comes up with a theorem that shows “1 + 1 = 9” we don’t have to prove this theorem to be correct, before proving it to be incorrect. I think it is called common sense.

I doubt that. If that were the case, we won't have wasted more than half a century on Bell's theorem. Since it is equivalent to "1 + 1 = 9". But I bet, it will be another 50 years before the buffoonery stops.

Your derivation is quite strange since you are claiming that to be able to use CFD in any theorem we are obligated to actually measure these values in experiments, which of course most understand is impossible

No. The use of CFD in the inequalities places that requirement on the experiment, not me. See the simulations in my previous post where the reason is clearly illustrated. Violation of this requirement is enough to obtain violation of the inequalities without any non-locality!

 

[billschnieder]

Well, to me all the above is good example of flawed conspiracy spookiness. You mix performed experimental outcomes with mathematical counterfactual speculations. I have never seen anything like it.

You mean you have never read how Bell's theorem is derived. You have made my argument succintly. Bell's theorem is derived (at least the so-called experimental violations of the inequalities) by mixing performed experimental outcomes with mathematical counterfactual speculations. This is exactly my argument. I'm happy you now see it . Please say it one more time and let is sink in:

Bell's theorem and it's so-called experimental confirmation are obtained by mixing performed experimental outcomes with mathematical counterfactual speculations.

If you think I've used outcome inconsistently in those quotes, then it is you who has reading comprehension issues.

 

[stevendaryl]

BTW I hope you realize that for an EPR experiment in which we have coincidence counting the correct probability expression should be

$P(\alpha \wedge \beta) = \int d\lambda P(\lambda) P(\alpha | \lambda) P(\beta | \alpha, \lambda)$

No, absolutely not. Not according to a local realistic model. That's the whole point of Bell's argument, is that the probability of Bob getting a spin-up result cannot depend on Alice's device setting, which can be changed "in flight".

 

[stevendaryl]

Agreed 100%

The reason for bringing in free will in this, is Bell’s statement on superdeterminism, but to me it’s the same dish as all the other “not so very bright” loopholes.

I don't think it's a matter of "loopholes". It's a matter of what is the meaning of the nonlocal correlations in quantum mechanics. I agree that one way of looking at superdeterminism is pretty unappetizing: the quantum correlations come about through a "conspiracy". On the other hand, if superdeterminism arises in an "organic" way (I mentioned, either on this thread or another the possibility that the actual history of the world is forced on us by self-consistency), I think that would be cool...if there's actually some mathematics to play with, as opposed to philosophical speculation.

 

[stevendaryl]

Huh? You are pointing out but without agreeing with the way the correlations are calculated in experiments?

I didn't disagree with the way that correlations are calculated in experiments.

Do you or do you not agree that those correlations as calculated in the experiments are separable?

I didn't bring up the word "separable correlations". I brought up the notion of a "factorable probability distribution", and I gave a definition of that. Bell proves that the correlations predicted by quantum mechanics cannot arise from such a factorable probability distributions. Now, it is the correlations which are measured, not the probability distribution (I actually don't know why the calculations are done in terms of correlations, rather than probabilities), but the theoretical predictions for the correlations are computed from the probability distributions. Bell's inequality is derived from a certain assumed form of the probability distribution, and his proof shows that any probability distribution of that form cannot lead to a violation of that inequality. He didn't prove that the inequality cannot be violated--of course it can. He proved that it isn't violated in any theory that predicts a probability distribution of a certain form, the so-called "local realistic" theories.

 

[billschnieder]

No, absolutely not. Not according to a local realistic model. That's the whole point of Bell's argument, is that the probability of Bob getting a spin-up result cannot depend on Alice's device setting, which can be changed "in flight".

If you have studied any probability theory, you would not say that. You are confused between outcome functions F(a,L) and Probabilities. In the EPR experiment with coincidence counting, you only consider outcomes at Bob for which there was an outcome at Alice.

P(b|a,L) means probability of an outcome at Bob given that an outcome was measured at Alice for the given Lambda.

 

[DevilsAvocado]

Now I know you are not being serious at all.
Please read this article http://arxiv.org/pdf/quant-ph/9810080v1.pdf and pay attention at figure 2 of the above article, and notice the table of outcomes?

I’m glad to see you in a good mood Bill, but maybe instead of rofl:ing and looking at fancy pictures, you should actually read the paper??

[all bolding mine]

Quantum theory predicts a sinusoidal dependence for the coincidence rate $C^{qm}_{++}(\alpha,\beta) \propto sin^{2}(\beta - \alpha)$ on the difference angle of the analyzer directions in Alice’s and Bob’s experiments. The same behavior can also be seen in the correlation function $E^{qm}(\alpha,\beta) = - cos(2(\beta - \alpha))$.

Huh? But it is impossible to repeat the same measurement! This is the whole point.

I did say “if we repeat the measurement at the same angle”, that is of course not the same measurement. But I think these words is the key to all this confusion – you require for all counterfactual values to be realized in the real experiment [which you also know is impossible] – if not, you think you have proved something wrong, which of course is very wrong.

You probably mean if you repeat a similar measurement for many different photons, the results appear random.

Lot of very strange statements lately... I sure hope you’re not claiming that QM is now a deterministic theory??

I doubt that. If that were the case, we won't have wasted more than half a century on Bell's theorem. Since it is equivalent to "1 + 1 = 9". But I bet, it will be another 50 years before the buffoonery stops.

Really?? ...okay, a completely new definition of scientific refutability... well... let’s see, if you in this brand new epic light want to refute The Flat Earth Society... eh... you must (did I really get that right?? ) first prove that Earth is flat??



(and of course you also have to prove Bell’ theorem correct, before proving it wrong!)

No. The use of CFD in the inequalities places that requirement on the experiment, not me. See the simulations in my previous post where the reason is clearly illustrated. Violation of this requirement is enough to obtain violation of the inequalities without any non-locality!

Okay, you have a working model of Local Realism that violates Bell’s inequalities?? I guess we’re going to help two guys getting to Stockholm, collecting their rightfully reward of two gold medals and $1.4 million.

Quantum Randi Challenge: Help Perimeter Physicist Joy Christian To Collect The Nobel Prize

Bell's theorem and it's so-called experimental confirmation are obtained by mixing performed experimental outcomes with mathematical counterfactual speculations.

Papers, reference, names please, to anyone but you making this hair-raising claim.

If you think I've used outcome inconsistently in those quotes, then it is you who has reading comprehension issues.

Now I know you are not being serious at all:

It still does not change the fact that we have outcomes at 4 angles. For each angle there is an outcome.

The only condition is there are 3 outcomes for 3 angles for each photon measured.

For each photon that leaves the source and heads towards Alice, you have a single outcome, (+, -, or non-detection).

Don't you see that those are not the outcomes measured in any real experiment.

As soon as you see that, you realize immediately that NO EXPERIMENT can ever measure them all! None! You can measure one but not the other two. Is that clear enough?

Totally unclear = contradictory = impossible to discuss

Sorry Bill, you need to state your claims clearly.

 

[DevilsAvocado]

I think that would be cool...if there's actually some mathematics to play with, as opposed to philosophical speculation.

Nooo Steven... you’re way too smart for superdeterminism... everything is a waste, including this discussion (well... some other stuff in this thread actually could be... ):

We always implicitly assume the freedom of the experimentalist... This fundamental assumption is essential to doing science. If this were not true, then, I suggest, it would make no sense at all to ask nature questions in an experiment, since then nature could determine what our questions are, and that could guide our questions such that we arrive at a false picture of nature.

 

[stevendaryl]

If you have studied any probability theory, you would not say that. You are confused between outcome functions F(a,L) and Probabilities.

That distinction has nothing to do with anything I've said. It might have something to do with what you're talking about, but I have no idea--you're not being very clear.

Let me try to make a couple of claims that I believe are true, and you can say definitively whether you agree or disagree with those claims.

1. Bell proved that for all theories of a certain type, the correlations predicted by those theories obey a certain inequality.

2. The correlations predicted by quantum mechanics do not obey that inequality.

3. Therefore, the correlations predicted by quantum mechanics cannot be explained by such a theory.

4. Experimentally, the correlations confirm the predictions of quantum mechanics.

Do you agree with statements 1-4? If so, then what are we arguing about? If not, which one? There is one simplification that is made in the analysis, which is that the quantum mechanical prediction is most easily made in terms of unachievable perfect detections: That is, the assumption that for every pair produced, Alice detects one particle and Bob detects the other. That's an oversimplification, and your point is that this oversimplification makes Bell's result supect, then I don't have much to argue against you. I would have to spend more time thinking about what the implications of non-detection are for Bell's argument.

 

[stevendaryl]

Nooo Steven... you’re way too smart for superdeterminism... everything is a waste, including this discussion (well... some other stuff in this thread actually could be... ):

I view theoretical physics as ultimately entertainment. It has some practical consequences, but those don't really depend on any of the outstanding questions in theoretical physics: the meaning of quantum mechanics, quantum gravity, the origin of time asymmetry, the information paradox of black holes, etc. For all those questions, I just consider it to be a puzzle to be solved for our amusement. It's a matter of taste which solutions seem like cheats. But that isn't important; not everyone laughs at the same jokes, either.

 

[stevendaryl]

We always implicitly assume the freedom of the experimentalist... This fundamental assumption is essential to doing science. If this were not true, then, I suggest, it would make no sense at all to ask nature questions in an experiment, since then nature could determine what our questions are, and that could guide our questions such that we arrive at a false picture of nature.

That's a philosophical issue that doesn't really worry me. If nature can answer the questions that I actually think of asking, then that's good enough for me. To worry about whether there are questions that I could have asked, but didn't is too meta for me.

I don't agree that superdeterminism makes science not worth doing. It's worth doing if we enjoy doing it.

 

[Nugatory]

...
4. Experimentally, the correlations confirm the predictions of quantum mechanics.

Do you agree with statements 1-4? If so, then what are we arguing about? If not, which one? There is one simplification that is made in the analysis, which is that the quantum mechanical prediction is most easily made in terms of unachievable perfect detections: That is, the assumption that for every pair produced, Alice detects one particle and Bob detects the other. That's an oversimplification, and your point is that this oversimplification makes Bell's result suspect, then I don't have much to argue against you.

Based on the exchange in #517, #518, #520, #521, I believe that BillSchneider is indeed rejecting #4, but for a different reason - there's another assumption built into the experiments, one that's not just a simplification but necessary for them to actually falsify the Bell inequality.

As an aside, it's not Bell's result that is suspect in any case. It's a theorem, and if the conclusion follows from the premises it's a valid theorem regardless of the truth of the premises: "If A then B" can be true even if A is false.

 

[stevendaryl]

Based on the exchange in #517, #518, #520, #521, I believe that BillSchneider is indeed rejecting #4, but for a different reason - there's another assumption built into the experiments, one that's not just a simplification but necessary for them to actually falsify the Bell inequality.

Well, don't leave me hanging---what's the assumption? I didn't really get it from Bill's posts.

 

[billschnieder]

I’m glad to see you in a good mood Bill, ...

DA, you do understand the difference between a QM prediction for an experiment and the outcomes actually measured in an experiment don't you, because you seem utterly confused by the difference. Pay attention:

1) QM makes a prediction for the expectation value for measuring a large number of photon pairs at angles α and β for Alice and Bob respectively. It is this expectation value that is -cos 2(α-β). QM says absolutely nothing, and makes absolutely no prediction about the outcome of measuring a single photon pair.

2) In experiments, each photon has an outcome $\ni (+1, -1)$. Hundreds of thousands of photons are measured and long tables are recorded with one entry for each photon measured. Each entry is time-tagged, the time tags are compared, and only coincident entries are considered togeter to calculate <FαF'β> which is then found to match the QM correlation. The table of outcomes at Bob's end is a separate list from the table of outcomes at Alice's end, Bob does not know or case about any "relative angle" nor does Alice. Get it?

I did say “if we repeat the measurement at the same angle”, that is of course not the same measurement. But I think these words is the key to all this confusion – you require for all counterfactual values to be realized in the real experiment [which you also know is impossible]

It is not me who derived the inequalities, it is Bell. It is not me who requires them to be measured in the same experiment. It is sound logic and the use of CFD in the inequalities which require them to be measured in the same experiment if anyone claims they are trying to test the inequalities. I said from the beginning that a genuine Bell test experiment is impossible. It is up to anyone who disagrees with that to make sure they are measured in the same experiment, otherwise they not testing Bell's inequalities or the CHSH no matter what they claim. This is common sense.

 

[billschnieder]

That distinction has nothing to do with anything I've said.

On the contrary, it is at the root of your misunderstanding. When I suggested
$P(\alpha \wedge \beta) = \int d\lambda P(\lambda) P(\alpha | \lambda) P(\beta | \alpha, \lambda)$

You said

No, absolutely not. Not according to a local realistic model. That's the whole point of Bell's argument, is that the probability of Bob getting a spin-up result cannot depend on Alice's device setting

Apparently unaware that if you are right that the Probability at Bob does not depend on the setting at Alice (not that you are), then

$P(\beta | \lambda) = P(\beta | \alpha, \lambda)$
The only time when those two are not equal is when Bob's probability is dependent of Alice's setting. In other words, the equation I gave is ALWAYS CORRECT, but yours in ONLY CORRECT WHEN THERE IS INDEPENDENCE.

But you are thinking that the outcome at Bob does not depend on Alice's setting. In other words, the outcome at Bob is a function of β and λ only ie F'(β,λ). And then you get confused by assuming that this means the probability of Bob's result is independent of the setting at Alice's detector. As I have explained, just because Bob's outcome does not depend on Alice's setting does not mean the probability calculated for Bob's outcome does not depend on Alice's setting. In fact, it must depend on Alice setting if you rely on any kind of coincidence counting.

 

[Nugatory]

Well, don't leave me hanging---what's the assumption? I didn't really get it from Bill's posts.

coming...

 

[DrChinese]

... I said from the beginning that a genuine Bell test experiment is impossible. It is up to anyone who disagrees with that to make sure they are measured in the same experiment, otherwise they not testing Bell's inequalities or the CHSH no matter what they claim. This is common sense.

Per usual, your ongoing misunderstanding of Bell goes completely against the grain of nearly everything written about the subject. Bell tests are merely experiments to show that the QM prediction is correct for selected angles (or various other observables), nothing more. Hundreds have been performed and published in peer-reviewed journals.

Bell's Theorem, on the other hand, shows that QM's predictions for (the same) selected angles are incompatible with the assumptions of local realism. Therefore Bell tests support QM, which by Bell's Theorem proves one of the local realistic assumptions wrong.

 

[stevendaryl]

On the contrary, it is at the root of your misunderstanding.

No, I think you are confused.

 

[DrChinese]

Bill:

Please publish your pet personal ideas rather than bring them here. We've been through this time and time again. PhysicsForums is for generally accepted science. If you have a suitable published reference for your statement about Bell tests not being valid, provide it. Else I will report your post.

-DrC

 

[billschnieder]

Let me try to make a couple of claims that I believe are true, and you can say definitively whether you agree or disagree with those claims.

1. Bell proved that for all theories of a certain type, the correlations predicted by those theories obey a certain inequality.

I agree that Bell derived certain inequalities. But I do not necessarily agree that the key assumptions required to obtain the inequalities are the ones you think they are. However, for the purpose of the discussions here, I do not care about the derivation, the inequalities are valid and we can start from there as I've told you previously, although I'll be happy to discuss in another thread why those inequalities are more general than you think.

2. The correlations predicted by quantum mechanics do not obey that inequality.

What correlations? This is one of the issues. Please spell out how you have arrived at this conclusion. Write down the inequality and write down the correlations which violate the inequality, term by term.

3. Therefore, the correlations predicted by quantum mechanics cannot be explained by such a theory.

We do not even reach this point yet. We have to address #2.

4. Experimentally, the correlations confirm the predictions of quantum mechanics.

Yes, the correlations from the experiments match QM.

There is one simplification that is made in the analysis, which is that the quantum mechanical prediction is most easily made in terms of unachievable perfect detections

My argument does not depend or rely on any loopholes.

 

[stevendaryl]

It depend on the setting at Alice (not that you are), then

$P(\beta | \lambda) = P(\beta | \alpha, \lambda)$
The only time when those two are not equal is when Bob's probability is dependent of Alice's setting. In other words, the equation I gave is ALWAYS CORRECT, but yours in ONLY CORRECT WHEN THERE IS INDEPENDENCE.

I KNOW that. That's what you get from assuming local realism. Bell's theorem is about locally realistic theories. The assumption that Bell was making is that the correlation between Alice's result and Bob's result is due entirely to the shared hidden variable $\lambda$. That is, the correlation goes away completely once you fix $\lambda$.

So what Bell proved was that

IF the joint probability $P(\alpha \wedge \beta)$ for Alice detecting spin-up at angle $\alpha$ and Bob detecting spin-up at angle $\beta$ has the form

$P(\alpha \wedge \beta) = \int d\lambda P_L(\lambda) P_A(\alpha | \lambda) P_B(\beta | \lambda)$

THEN the correlation between Alice's result and Bob's result will obey a certain inequality. It's certainly possible that the joint probability distribution doesn't have that form. As you point out, the general case is:

$P(\alpha \wedge \beta) = \int d\lambda P_L(\lambda) P_A(\alpha | \lambda) P_B(\beta | \lambda \wedge \alpha)$

Bell isn't talking about the general case. He's talking about the case in which the correlation between Alice's result and Bob's result is completely due to the presence of the hidden variable $\lambda$

 

[billschnieder]

Based on the exchange in #517, #518, #520, #521, I believe that BillSchneider is indeed rejecting #4

This is wrong. The experiments match QM. I do not reject #4.

 

[billschnieder]

No, I think you are confused.

I think you are confused.

Bell isn't talking about the general case. He's talking about the case in which the correlation between Alice's result and Bob's result is completely due to the presence of the hidden variable $\lambda$

Bell isn't talking about joint probability distributions. But using Expectation values for the paired product of outcomes at two stations, where each outcome is a deterministic function of the setting at the station and the lambda in play with possible values +1/-1. I told you to forget about probabilities but you decided to keep getting confused by them.

 

[Nugatory]

Based on the exchange in #517, #518, #520, #521, I believe that BillSchneider is indeed rejecting #4

This is wrong. The experiments match QM. I do not reject #4.

Fair enough... I am interpreting #4 as "the experiments match QM and therefore falsify Bell's inequality" because I'm pretty sure that's what stevendaryl meant, but yes, that's a slightly different statement than his #4.

If I could ask you how you interpret the outcome of those four posts (#517, #518, #520, #521), suggest a concise statement that does capture the area of disagreement to your satisfaction?

 

[stevendaryl]

I agree that Bell derived certain inequalities. But I do not necessarily agree that the key assumptions required to obtain the inequalities are the ones you think they are. However, for the purpose of the discussions here, I do not care about the derivation, the inequalities are valid and we can start from there as I've told you previously, although I'll be happy to discuss in another thread why those inequalities are more general than you think.

What correlations? This is one of the issues. Please spell out how you have arrived at this conclusion.

Suppose you have a pair of anti-correlated spin-1/2 particles. Then the probability of measuring one particle to have spin-up along an axis $\vec{A}$ is $\frac{1}{2}$. If you then measure the spin of the second particle along axis $\vec{B}$, then the probability of getting spin-up will be either

$sin^2(\frac{1}{2} \theta)$

if the first measurement had result spin-up, or

$cos^2(\frac{1}{2} \theta)$

if the first measurement had result spin-down, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.

Define $S_A$ to be either +1, if the result of the first measurement was spin-up, and -1, if the result was spin-down. Define $S_B$ to be either +1, if the result of the second measurement was spin-up, and -1, if the result was spin-down. Then we can define a "correlation function"

$\langle S_A \cdot S_B \rangle$

to be the expectation value of the product of $S_A$ and $S_B$. From the assumed probabilities, we conclude:

$\langle S_A \cdot S_B \rangle = - cos^2(\frac{1}{2} \theta) + sin^2(\frac{1}{2} \theta) = cos(\theta)$

where I used a trigonometric identity about half-angles.

So that's the quantum prediction for correlation, in the spin-1/2 case.

 

[stevendaryl]

Bell isn't talking about joint probability distributions.

Yes, he certainly is. Maybe you're confused because there is a one step deduction from what I wrote to Bell's starting point that should be made explicit. I've said this a bunch of times, but I will say it again:

We start off by assuming that the joint probability distribution has the form

$P(\alpha, \beta) = \int d\lambda P_L(\lambda) P_A(\alpha | \lambda) P_B(\beta | \lambda)$

Then we note that there is perfect correlation when $\beta - \alpha = 180$ and perfect anti-correlation when $\beta = \alpha$. Such perfect correlation is only possible if the probabilities have the property that: $P_A(\alpha | \lambda) =$ 0 or 1, and $P_B(\beta| \lambda) =$ 0 or 1. More than that, we can show that the quantum predictions imply that

$P_A(\alpha | \lambda) = 1 - P_B(\alpha | \lambda)$

Given that, we can define a function $F(\alpha,\lambda)$ such that

$F(\alpha, \lambda) = +1$ if $P_A(\alpha | \lambda) = 1$

$F(\alpha, \lambda) = -1$ if $P_A(\alpha | \lambda) = 0$

In terms of the function $F$, the predicted correlation between Alice and Bob is given by:

$C(\alpha, \beta) = - \int d\lambda P_L(\lambda)F(\alpha, \lambda) F(\beta, \lambda)$

This is the formula that Bell uses, but it's the same as if you had started with joint probability distributions, and made inferences from the known facts about quantum probability predictions.

 

[stevendaryl]

This is wrong. The experiments match QM. I do not reject #4.

So why, exactly, are you making people guess what your point is, instead of coming out and saying it?

 

[stevendaryl]

Bell isn't talking about joint probability distributions.

I think Bell is the best authority on what Bell was talking about. Here's a scan of a page from his book "Speakable and unspeakable in quantum mechanics":

Equation 11 is the claim that if we knew the causal factors $\lambda$ in common between Alice and Bob, then the probability would factor into a probability for Alice that depends only on $\lambda$ and variables local to Alice (the "a" in the equation) and a probability for Bob that depends only on variables local to Bob(the "b" in the equation).

This is the way that Bell explains his reasoning.

 

[billschnieder]

So why, exactly, are you making people guess what your point is, instead of coming out and saying it?

What are you talking about? What did you think I was doing. I've been explaining what I mean since the beginning of this thread. Long before you even got involved in this thread, and I have done so again in posts #473, #480, #490, #514, #518, #521, #522, #523, #524. But apparently that was too much for you to even read.

 

[billschnieder]

Suppose you have a pair of anti-correlated spin-1/2 particles. Then the probability of measuring one particle to have spin-up along an axis $\vec{A}$ is $\frac{1}{2}$. If you then measure the spin of the second particle along axis $\vec{B}$, then the probability of getting spin-up will be either

$sin^2(\frac{1}{2} \theta)$

if the first measurement had result spin-up, or

$cos^2(\frac{1}{2} \theta)$

if the first measurement had result spin-down, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.

Define $S_A$ to be either +1, if the result of the first measurement was spin-up, and -1, if the result was spin-down. Define $S_B$ to be either +1, if the result of the second measurement was spin-up, and -1, if the result was spin-down. Then we can define a "correlation function"

$\langle S_A \cdot S_B \rangle$

to be the expectation value of the product of $S_A$ and $S_B$. From the assumed probabilities, we conclude:

$\langle S_A \cdot S_B \rangle = - cos^2(\frac{1}{2} \theta) + sin^2(\frac{1}{2} \theta) = cos(\theta)$

where I used a trigonometric identity about half-angles.

So that's the quantum prediction for correlation, in the spin-1/2 case.

Sorry, you did not understand the question. You have given me one correlation. Bell's inequality has 3, the CHSH has 4, I want you to simply write down what you claim the QM correlation is for each TERM OF THE INEQUALITY, demonstrating the violation. This is what I said, which you ignored, please read carefully rather than assuming what I'm asking:

Please spell out how you have arrived at this conclusion. Write down the inequality and write down the correlations which violate the inequality, term by term.

That shouldn't be a difficult question now should it?

 

[billschnieder]

Fair enough... I am interpreting #4 as "the experiments match QM and therefore falsify Bell's inequality" because I'm pretty sure that's what stevendaryl meant, but yes, that's a slightly different statement than his #4.

I did not what to guess what he meant, so I responded to what he wrote. You could add a #5 claim that the experiments violate the inequalities, and I will disagree with such a claim for the same reason as I disagree with #2.

If I could ask you how you interpret the outcome of those four posts (#517, #518, #520, #521), suggest a concise statement that does capture the area of disagreement to your satisfaction?

You mean #521 and #522 did not accomplish this clearly enough for you?

1) Do you agree that there are two scenarios involved in this discussion:

Scenario X, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(b,c) = correlation for what we would get if we measure (b,c)
C(a,c) = correlation for what we would get if we measure (a,c)
​

Scenario Y, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(a,c) = correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = correlation for what we would have gotten had we measured (b,c) instead of (a,b)
​

2) Do you agree that Scenario X is different from Scenario Y?
3) Do you agree that the correlations in Bell's inequalities correspond to Scenario Y NOT Scenario X?
4) Do you agree that correlations calculated from QM correspond to Scenario X not Scenario Y?
5) Do you agree that correlations measured in experiments correspond to Scenario X not Scenario Y?

Do you now see the issue?

 

[DrChinese]

1) Do you agree that there are two scenarios involved in this discussion:

Scenario X, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(b,c) = correlation for what we would get if we measure (b,c)
C(a,c) = correlation for what we would get if we measure (a,c)
​

Scenario Y, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(a,c) = correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = correlation for what we would have gotten had we measured (b,c) instead of (a,b)
​

2) Do you agree that Scenario X is different from Scenario Y?
...

The *local realist* DOES NOT believe there is a difference between these scenarios. Most of the rest of us deny the existence of counterfactuals, so your scenario Y makes no sense to us.

That makes you the local realist who does not believe in local realism. And leaves you tilting at windmills. Still.

 

[audioloop]

I think Bell is the best authority on what Bell was talking about. Here's a scan of a page from his book "Speakable and unspeakable in quantum mechanics":

Equation 11 is the claim that if we knew the causal factors $\lambda$ in common between Alice and Bob, then the probability would factor into a probability for Alice that depends only on $\lambda$ and variables local to Alice (the "a" in the equation) and a probability for Bob that depends only on variables local to Bob(the "b" in the equation).

This is the way that Bell explains his reasoning.

has been stated "correlations".
you can interpret bell in terms of shareability of correlations and abandon "local cfd" doctrine.

 

[stevendaryl]

Sorry, you did not understand the question. You have given me one correlation. Bell's inequality has 3, the CHSH has 4, I want you to simply write down what you claim the QM correlation is for each TERM OF THE INEQUALITY, demonstrating the violation. This is what I said, which you ignored, please read carefully rather than assuming what I'm asking:

You can look it up. It would help if you said what your point was, instead of random demands for equations. The claim made by Bell is that the correlation function that I wrote down is not consistent with any locally realistic theory. Are you claiming that the proof is wrong, or what? You've made a dozen or so posts, and I still have absolutely no idea what your point is. Do you think that maybe you're not being clear?

Are you now asking for me to step you through a proof of Bell's theorem? I am not prepared to do that right now, but before I go to the trouble, I would like to know to what end. What are you arguing for?