Scholarpedia article on Bell's Theorem

stevendaryl

Staff Emeritus
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

Staff Emeritus
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

Staff Emeritus
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

Staff Emeritus
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?

Last edited:

DrChinese

Gold Member
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

Staff Emeritus
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?

stevendaryl

Staff Emeritus
has been stated "correlations".
you can interpret bell in terms of shareability of correlations and abandon "local cfd" doctrine.
Maybe so, but Bell's original intention was to investigate the possibility of quantum mechanics being explainable in terms of a locally realistic theory, and for such a theory, CFD holds.

stevendaryl

Staff Emeritus
What are you talking about? What did you think I was doing.
I have no idea. That's what I'm trying to find out.

DrChinese

Gold Member
What are you arguing for?
Bill probably wants an admission he is right and everyone else is wrong. In the years I have gone around and around with him, I have never understood where he was driving on this point either. Glad you and Nugatory are doing this with him this time rather than me.

I have an entire list of links to papers "proving" Bell is wrong. If Bill ever wrote one, I could add that to my list. Instead... this.

billschnieder

I have no idea. That's what I'm trying to find out.
No, I don't think you are interested. If you were, you would answer my simple question, which again is this:

stevendaryl said:
2. The correlations predicted by quantum mechanics do not obey that inequality.
Please spell out how you have arrived at this conclusion [your #2]. Write down the inequality and write down the correlations which violate the inequality, term by term.
I'm simply asking you to demonstrate what you claimed yourself in claim #2. What are you afraid of? If you do not understand the question, simply say so and I'll explain again. Bell's inequality has 3 terms. the CHSH has 4 terms. If you claim QM violates the inequality, then you must have 3 terms from QM to substitute in Bell's inequality or 4 terms from QM to substitute in the CHSH in other to demonstrate the violation. This is not rocket science. I'm asking you to provide ALL the three terms you used for Bell's inequality or ALL the 4 terms you used for the CHSH. You can't just write one term and fold your arms. Get it?

Nobody can honestly claim I've not been very clear about what I'm saying, especially after reading post #557. It is one thing to say you disagree with the claims in post #557. It is another thing completely to pretend they are not clear. You haven't even attempted to respond to it. Despite me explaining it clearly multiple times.

billschnieder

Well, don't leave me hanging---what's the assumption? I didn't really get it from Bill's posts.
Because you did not read Bill's posts otherwise you would have seen this in post #521

billschnieder said:
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.

Dale

Mentor
Closed pending moderation.

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