Scholarpedia article on Bell's Theorem

[stevendaryl]

Let me try to clarify what, exactly, is impossible in a local hidden variables theory.

First, what is the correlation function, which I'm calling $C(\alpha, \beta)$?

If you have a list, or run, of 4-tuples of numbers: $\alpha_n, A_n, \beta_n, B_n$ where for each $n$, $\alpha_n$ and $\beta_n$ are angles, and $A_n$ and $B_n$ are each either $+1$ or $-1$, then you can compute a correlation $C(\alpha, \beta)$ as follows:

$C(\alpha, \beta) = \frac{1}{N_{\alpha \beta}} \sum A_n B_n$
where the sum is over those runs $n$ such that $\alpha_n = \alpha$ and $\beta_n = \beta$, and where $N_{\alpha \beta}$ is the total number of such runs.

Suppose that we generate such a list as follows: We create a sequence of spin-1/2 twin pairs. On run $n$, one particle is detected by Alice using a spin-measurement device aligned in the x-y plane at angle $\alpha_n$ away from the x-axis, and the other is detected by Bob at an angle $\beta_n$. If Alice measures spin-up, then $A_n = +1$. If Alice measures spin-down, then $A_n = -1$. If Bob measures spin-up, then $B_n = +1$. If Bob measures spin-down, then $B_n = -1$.

The prediction of quantum mechanics is that in the limit as the number of trials at each angle goes to infinity, is that

$C(\alpha, \beta) = - cos(\beta - \alpha)$

At this point, let's specialize to specific values for $\alpha$ and $\beta$. Assume that $\alpha$ and $\beta$ are always chosen to be from the set
{ 0°, 120°, 240°}. A way to explain the correlations using deterministic local hidden variables is to assume that corresponding to run number $n$ there is a hidden variable $\lambda_n$, which is (or determines) a triple of values $\lambda_n = \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle$. Then $A_n$ and $B_n$ are deterministic functions of the angles $\alpha$ and $\beta$ and the "hidden variable" $\lambda_n$:

$A_n = A_{\alpha_n\ n}$
$B_n = B_{\beta_n\ n} \equiv - A_{\beta_n\ n}$

Now, here's where the impossibility claim arises: If we have a sequence of triples $\langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle$, then we can compute correlation functions as follows:

$C'(\alpha \beta) = \frac{1}{N} \sum A_{\alpha\ n} B_{\beta\ n}$

where $N$ is the total number of runs.

I'm using a prime to distinguish this correlation function from the previous. The difference between the two is that $C(\alpha \beta)$ is computed using those runs in which Alice happens to choose detector angle $\alpha$, and Bob happens to choose detector angle $\beta$. In contrast, $C'(\alpha \beta)$ is computed using all runs, since by assumption, $A_{\alpha\ n}, B_{\beta\ n}$ determines what Alice and Bob would have gotten on run n had they chosen settings $\alpha$ and $\beta$.

The impossibility claim is that there is no sequence of triples $\langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle$ such that the corresponding $C'(\alpha, \beta)$ agrees with the quantum mechanical prediction for the correlation.

This impossibility claim is NOT contradicted by actual experiments, because an actual experiment cannot measure the triple $\langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle$; it can only measure two of the three values.

 

[billschnieder]

Let me try to clarify what, exactly, is impossible in a local hidden variables theory.
...

You argument supports my point (which apparently you still have not considered carefully) not yours. The reason being -- why would any sane person expect the EPR experiment to provide a list of triples rather than 4 lists of doubles?

Forget about probabilities and separability. We start from the inequalities already derived, OK! We have terms 4 correlation terms in the CHSH inequality.

(1) we calculate the terms from QM and obtain a violation - this is the origin of Bell's theorem
(2) we measure the terms from an experiment and obtain a violation. We also realize that the results match QM.

Do you agree with this? If you do then we can dig deeper into the origin of the violation.

 

[stevendaryl]

You argument supports my point (which apparently you still have not considered carefully) not yours.

Well, I don't understand what your point is. I don't think you've been very clear.

The reason being -- why would any sane person expect the EPR experiment to provide a list of triples rather than 4 lists of doubles?

Because of perfect anti-correlations between the two detectors when they are set at the same detector angle, we can assume, under a local hidden-variables theory, that the outcome is a deterministic function of the hidden variable $\lambda$. That means that there is a function $F(\alpha, \lambda)$ returning +1 or -1 for whether Alice will measure spin-up or spin-down at angle $\alpha$. So if Alice has three possible angles to choose from, then there are three relevant quantities:

$F(a,\lambda), F(b,\lambda), F(c,\lambda)$

The first gives the result if Alice happened to choose angle $a$, the second if Alice happened to choose angle $b$, etc.

Bob's results are anti-correlated with Alice's, so we can get Bob's results by using
$F'(\alpha,\lambda) = 1 - F(\alpha,\lambda)$

Forget about probabilities and separability. We start from the inequalities already derived, OK!

No. I don't want to start there.

 

[billschnieder]

Well, I don't understand what your point is. I don't think you've been very clear.

No. I don't want to start there.

Then you are deliberately refusing to see my point that's why you keep getting confused about what I'm saying. I'm not arguing with you about how the inequalities are derived so you are wasting your time trying to demonstrate separability etc.

You agreed earlier the correlations calculated from experiments is
$\frac{1}{N} \sum S_{A,n} S_{B,n}$
You admit that the above correlation matches the QM prediction for the experiment. Although you admit that the above correlation is separable as can be seen from the equation, you turn around and contradict yourself by saying it is impossible to obtain the QM correlation in separable form. Don't you see your error?! If it is impossible to obtain the QM correlation in separable form, then it is impossible for the experiment to match QM as you claim!

...the outcome is a deterministic function of the hidden variable $\lambda$. That means that there is a function $F(\alpha, \lambda)$ returning +1 or -1 for whether Alice will measure spin-up or spin-down at angle $\alpha$. So if Alice has three possible angles to choose from, then there are three relevant quantities:

$F(a,\lambda), F(b,\lambda), F(c,\lambda)$

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 if Alice in fact measured at (a), then the last two F(b,λ), F(c,λ) MUST be counterfactual! We are not talking about measuring any random photon, we are talking about what she would have obtained were it possible for her to rewind time and measure the same photon at a different angle! Don't you see that those are not the outcomes measured in any real experiment.

 

[stevendaryl]

Then you are deliberately refusing to see my point that's why you keep getting confused about what I'm saying.

No, I just don't think you've been very clear.

You agreed earlier the correlations calculated from experiments is
$\frac{1}{N} \sum S_{A,n} S_{B,n}$
You admit that the above correlation matches the QM prediction for the experiment.

It's not that I admit it, I'm pointing it out.

Although you admit that the above correlation is separable as can be seen from the equation.

No, I don't agree with that. The kind of separability that I'm talking about is separability of the joint probabilities, not correlations. Bell's assumption is that if all interactions are local, and

$P(\alpha \wedge \beta) \neq P(\alpha) P(\beta)$

then there must be some "hidden variable" $\lambda$ such that

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

The correlations predicted by quantum mechanics cannot be generated by factorable probabilities of this form.

Don't you see that if Alice in fact measured at (a), then the last two F(b,λ), F(c,λ) MUST be counterfactual!

Of course. A hidden variables theory implies that counterfactuals have definite values.

 

[billschnieder]

It's not that I admit it, I'm pointing it out.

Huh? You are pointing out but without agreeing with the way the correlations are calculated in experiments? Do you or do you not agree that those correlations as calculated in the experiments are separable? The equation you provided yourself shows that they are!

The kind of separability that I'm talking about is separability of the joint probabilities, not correlations.
...
$P(\alpha \wedge \beta) = \int d\lambda P(\lambda) P(\alpha | \lambda) P(\beta | \lambda)$
The correlations predicted by quantum mechanics cannot be generated by factorable probabilities of this form.

You must be very confused then. Is $P(\alpha \wedge \beta)$ a joint probability or a correlation? Are you saying the correlations are separable but the joint probabilities are not !? When I demonstrate to you that the correlations from the experiment are separable, you argue that you were talking about probabilities not correlations and then turn around and use the two terms synonymously.

 

[Nugatory]

Don't you see that if Alice in fact measured at (a), then the last two F(b,λ), F(c,λ) MUST be counterfactual

Yes.
And if we proceed under the assumption that these counterfactuals have real numerical values, we end up drawing conclusions that are not supported by experimental results.

We then have two possibilities:
1) Because of some flaw in their design or execution, the experiments also do not falsify our conclusion. Appeals to the fair-sampling loophole would fall in this category.
2) The experiments are not fatally flawed so they do falsify our conclusion; therefore the assumption leading to the conclusion must be false.

Are you going somewhere else with this line of argument?

 

[billschnieder]

Yes.
And if we proceed under the assumption that these counterfactuals have real numerical values, we end up drawing conclusions that are not supported by experimental results.

You are jumping the gun. If you proceed under the assumption of counterfactual terms in the inequality, then the terms in the inequality must be interdependent and therefore can never be measured in any experiment. So you don't even get to any experiment because a genuine experiment to test the inequality is impossible to perform.

We then have two possibilities:
1) Because of some flaw in their design or execution, the experiments also do not falsify our conclusion. Appeals to the fair-sampling loophole would fall in this category.
2) The experiments are not fatally flawed so they do falsify our conclusion; therefore the assumption leading to the conclusion must be false.

There no mention of loopholes in my argument. My argument does not rely or use any loopholes. Once you realize that the experiment is impossible, it becomes nonsensical to even suggest that the experimental result disagrees with the inequality because the experiment you are talking about would be testing something else not Bell's inequality.

 

[billschnieder]

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

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

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

 

[Nugatory]

Once you realize that the experiment is impossible, it becomes nonsensical to even suggest that the experimental result disagrees with the inequality because the experiment you are talking about would be testing something else not Bell's inequality.

I think that falls under #1 - the experiment that has been done is so flawed in its execution or interpretation as to neither confirm or falsify the conclusion.

If that's what you mean, at least it's a coherent statement whose merits can be discussed.
(It's also a statement that I disagree with, but let's try to identify what we're arguing about before we have the argument )

 

[billschnieder]

I think that falls under #1 - the experiment that has been done is so flawed in its execution or interpretation as to neither confirm or falsify the conclusion.

If that's what you mean, at least it's a coherent statement whose merits can be discussed.
(It's also a statement that I disagree with, but let's try to identify what we're arguing about before we have the argument )

Fair enough. Why do you disagree with it? Here is the argument again, please explain what part of it you disagree with:

QM violates 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) = 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)

When Bell's inequalities DEMAND that the correlations should be (scenario Y):
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)

Experiments violate the inequalities because the correlations measured correspond to scenario X not the correct scenario Y. QM and Experiments agree with each other because they both refer to scenario X not Bell's scenario Y. In other words, if you insist on using the terms from QM and experiment to compare with the inequality, then you are making an extra assumption that the correlations in scenario X and Y are equivalent. Now once you obtain a violation, it is this assumption that should be thrown out. As I have demonstrated already, those two scenarios are different without any non-locality or conspiracy, and such an assumption should never even be introduced if reasoning correctly.

 

[billschnieder]

Now let me explain again why Scenario X is different from Scenario Y, photon by photon.

As stevendaryl explained the correlation is calculated as:
$C(\alpha, \beta) = \frac{1}{N} \sum F_{\alpha n} F'_{\beta n}$

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). In other words, the first outcome which goes into the C(a,b) calculation is F(a,λ1)=+1 for Alice, and F'(b, λ1)=-1.

For Scenario Y, then, the first outcome which goes into calculating the remaining two terms is immediately restricted by that result to F(a,λ1)=+1, and F'(c, λ1) = ? for calculating C(a,c) and F(c,λ1)= ?, and F'(b, λ1) = -1 for calculating C(c, b). This is obviously the case because, if Alice got F(a,λ1)=+1, she would not have gotten anything other than what she got had she measured at the same angle which she did in fact measure at and if Bob obtained F'(b, λ1) = -1 he wouldn't have gotten anything other than what he got, had he measured at the same angle at which he measured at. 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 can then go to the next photon pair, and the next and the same conditions apply for the whole set used to calculate the correlation.

However for Scenario X, we are not dealing with counter-factuals and there is no restriction to specific photons. We can measure any random photon for any of the correlations. 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. As you can see, Scenario X has many more degrees of freedom than Scenario Y. The terms in scenario X are independent of each other contrary to scenario Y.

 

[billschnieder]

Now let us simulate all the possibilities for each scenario and demonstrate that while scenario Y would never result in a violation, violations can be expected for scenario X.

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.
Scenario Y:

a,b,c = (+1,+1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a,b,c = (+1,+1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a,b,c = (+1,-1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a,b,c = (+1,-1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a,b,c = (-1,+1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a,b,c = (-1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a,b,c = (-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a,b,c = (-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True

* 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

Now for Scenario X, we measure C(a,b) from one set of photons, C(c,b) from a different set of photons and and C(a,c) from yet another set of photons. Really what we are measuring is C(a1,b1), C(a2,c2) and C(c3,b3) and if we substitute in the inequality, we actually have
|a1b1 + a2c2| - b3c3 <= 1 in which there is no dependency between any of the terms. No two terms share the same outcome contrary to scenario Y. Simulating this, we get

(see next post)

 

[billschnieder]

a1,a2,b1,b3,c2,c3 = (+1,-1,+1,+1,-1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,-1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,+1,+1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
...
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,-1,-1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,-1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,-1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,-1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True

(see post 125 for the full simulation results)

* Note that there are 64 distinct possible outcome combinations in Scenario X as opposed to just 8 in Scenario Y.
* Note also that the inequality is violated many times (in 1/4 of the cases).
* Note that when a1=a2 and b1=b3 and c2=c3 as required by scenario X, the inequality is NEVER violated.

 

[DevilsAvocado]

I don't like dragging free will into discussions about science, because I just don't think it has any relevance. "Free" choices that humans make could very well be determined by conditions at a microscopic level, and that wouldn't make much difference, in practice. What I thought was the difference between determinism and superdeterminism is this:

• A theory is deterministic if a past state uniquely singles out one possible future state.
• A theory is superdeterministic if there is only one possible past, as well.

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.

 

[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.