Does the Bell theorem assume reality?

[lodbrok]

$|\langle A B \rangle + \langle A C \rangle| \leq 1 + \langle B C \rangle$

I disagree, you derive the relationship by assuming a single set of triples of numbers. That is the fact.

Then in a real experiment, we measure the averages for measurements: $\langle A B \rangle$, $\langle A B \rangle$ and $\langle A B \rangle$. We find that that inequality is violated.

Then you perform an experiment in which you measure pairs of numbers (never triples). And the relationship is violated.

The undeniable conclusion is that the measured quantities did NOT come from a sequence of triples of numbers (one triple for each twin pair).

Duh! Isn't that obvious, you never measured triples in your experiment so it is not surprising that you arrive at the conclusion that you don't have triples.

The point I've been trying to tell you is that, by applying your "relationship from triples", to your "experiment of pairs", you are making an assumption that the "three averages from one set of triples" is exactly the same as the "three averages from three disjoint sets of pairs". It is this assumption that has failed. By ignoring subscripts it is easy to not see the problem. And I've been explaining that this assumption is equivalent to saying a particle of one entangled pair is correlated with another particle of a separate entangled pair.

But I've said enough on this topic. Thanks.

 

[DrChinese]

We are discussing the nature of the realism assumption in Bell's derivation and I'm pointing out the subtle additional assumptions when applying the relationship derived to the experiments performed in the manner of EPRB. The perfect anti-correlation assumption is crucial in the derivation of the relationship. All I'm doing is pointing out the import of that assumption when you now apply the relation to an experiment. I use the coin toss example to illustrate that the problem is not even specific to the EPRB experiment or QM or local realism or non-locality or any other physical concept. It is a problem of incompatible degrees of freedom that is elementary. The question of whether or not a physical hidden variable theory can reproduce the predictions of quantum mechanics is completely irrelevant to the point I'm making.

All I'm saying is that you have to very very careful when you do simple arithmetic with some numbers, and then you try to apply the result to data obtained in an experiment not performed exactly as you assumed when deriving the relationship -- because it always involves introducing additional assumptions which may not always be true. Surely, you aren't arguing that Bell's mathematics are not applied to experiment are you? Otherwise why should the application be off target?

As I keep saying, there are 2 things here. One has to do with Bell's Theorem, and that has nothing to do with experiment. Bell ASSUMES counterfactual definiteness of A, B and C; and that the choice of measurement by Alice does not affect the outcome of Bob (and vice versa). The first is realism, the second is separability or locality. There is nothing more than that for assumptions on the local realistic side.

For experiments, you only need to show that the quantum mechanical prediction of -cos(theta) - theta being the angle between A and B - is correct. Note that NO ASSUMPTION is required for this demonstration, other than things such as the Fair Sampling assumption.

In actual experiments, they use CHSH inequality or something else. But that is not a requirement, it is just done as a way of emphasizing the result.

 

[stevendaryl]

I disagree, you derive the relationship by assuming a single set of triples of numbers. That is the fact.

Here's a challenge for you: Write a list containing 20 triples, each triple consisting of three numbers, each of which is $\pm 1$. For example

+1 +1 -1
+1 -1 -1
-1 +1 -1

etc.

Now, for whatever list you came up with, let's compute the following values:

$\langle A B \rangle$: This is the average of the product of the first two numbers
$\langle A C \rangle$: This is the average of the product of the first and third numbers
$\langle B C \rangle$: This is the average of the product of the second and third numbers

The claim being made is that you cannot come up with a list that violates $|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

Try it.

 

[DrChinese]

I disagree, you derive the relationship by assuming a single set of triples of numbers.

Bell ASSUMES counterfactual definiteness of A, B and C; and that the choice of measurement by Alice does not affect the outcome of Bob (and vice versa). The conjunction of those means that choosing an A paired B could not affect the outcome of an A paired with C. What you are talking about is already built in.

I just saw stevendaryl's challenge to you, you should review that. You will see that you can hand pick values, and still never violate the inequality. And yet there are values of A/B/C that would violate that inequality for the quantum mechanical prediction, accepting that one (or both) of the assumptions in the preceding paragraph are invalid. QM does not make those assumptions. Only the local realist does.

 

[stevendaryl]

Duh! Isn't that obvious, you never measured triples in your experiment so it is not surprising that you arrive at the conclusion that you don't have triples.

You're saying that if you only measure two values, then there cannot be a third value that was unmeasured? How in the world do you justify such an assumption? You're saying that it's obvious that there cannot be a hidden-variable explanation of EPR correlations?

Here's a picturesque way of thinking about EPR: Instead of a particle, you have a bundle of three envelopes held together with a paper clip. On one envelope is the label "A", on one envelope is the label "B" and on one envelope is the label "C". These envelopes have the peculiar property that if you open one envelope, the other two burst into flames and burn to ash without your ever knowing what was inside. The envelope that you do open has a slip of paper with either the number +1 or -1 in it.

The local realism assumption is that the two envelopes that burst into flames also contained either +1 or -1, even though you never had a chance to check.

 

[lodbrok]

Here's a challenge for you: Write a list containing 20 triples, each triple consisting of three numbers, each of which is $\pm 1$. For example

+1 +1 -1
+1 -1 -1
-1 +1 -1

etc.

Now, for whatever list you came up with, let's compute the following values:

$\langle A B \rangle$: This is the average of the product of the first two numbers
$\langle A C \rangle$: This is the average of the product of the first and third numbers
$\langle B C \rangle$: This is the average of the product of the second and third numbers

The claim being made is that you cannot come up with a list that violates $|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

Try it.

The fact that you would suggest this tells me you understood nothing of what I said. What you say above is all trivially true and irrelevant since it is what you assume to arrive at the relationship. Your error is that you do not appreciate the difference between what you've outlined above and what actually happens in experiments. Your three averages above are not disjoint but those from the experiment are. You are using the same $A$ column data in calculating both the $\langle A B \rangle$ and the $\langle A C \rangle$ averages.

If a single particle pair has $f$ degrees of freedom, for a set of triples from N particle pairs, you have $N f$ degrees of freedom. However, from 3 disjoint sets of N particle pairs (like in the EPRB experiment) you have $3 N f$ degrees of freedom. That is, there is no common column of data in any of the three averages. In other words, each of the 6 columns from experiments is free to vary independently of the other 5. That is, your averages from experiment are actually calculated from 6 random variables. This is not the case in your averages used for the derivation where there are only 3 random variables. If you want to argue that all this is unimportant, that's your choice. But keep in mind that doing statistics with correlated variables without taking degrees of freedom into consideration is very unwise to put it mildly.

At the very least, you are assuming that "degrees of freedom does not matter". You have to admit at least that, to be consistent.

I've made my point and I don't intend to reply any further.

 

[stevendaryl]

The fact that you would suggest this tells me you understood nothing of what I said.

That's true. What you've said makes no sense to me. I think it's because you're just confused.

But in any case, you agree with the impossibility, right?

Your error is that you do not appreciate the difference between what you've outlined above and what actually happens in experiments.

It's not an error.

I understand that in an experiment, you don't measure three values for each twin-pair. You only measure two. So there is an assumption that the average of $A_n B_n$ over all values of $n$ is equal to the average over those values of $n$ for which $A$ and $B$ are measured. But that is part of the model that Bell's inequality shows cannot explain the EPR correlations. If you generate a sequence of triples and someone else (Alice and Bob) picks two out of three values for each triple in the sequence, then if the choice was not known ahead of time, and Alice and Bob make their choices without knowledge of the values of the triples, then the statistics should be the same for the partial set as they were for the complete set.

This is really a basic assumption of sampling theory. If there is a probability of $P$ that a person is left-handed, and I randomly pick $N$ people and $N_L$ of them are left-handed, then it is expected that $P \approx \frac{N_L}{N}$. The assumption is that my choice of which people to check does not affect the relative frequencies.

Now, there can certainly be challenges to such fairness assumptions. Maybe for whatever reason, left-handed people are less likely to selected by whatever process I was using to select people. That's a possibility. And I suppose that's a loophole in Bell's argument. But what that amounts to is the assumption that Alice or Bob's choice is influenced by the value of $\lambda$. We can arrange things so that Alice making her choice is at a spacelike separation from the creation of the twin-pair, so that there is no possibility of influence (except faster-than-light influences).

I've made my point and I don't intend to reply any further.

I don't think you've made a point.

 

[DarMM]

The fact that you would suggest this tells me you understood nothing of what I said. What you say above is all trivially true and irrelevant since it is what you assume to arrive at the relationship. Your error is that you do not appreciate the difference between what you've outlined above and what actually happens in experiments. Your three averages above are not disjoint but those from the experiment are. You are using the same $A$ column data in calculating both the $\langle A B \rangle$ and the $\langle A C \rangle$ averages.

This is really no different from what @stevendaryl is saying, but consider that list he gave and then consider probability distributions over his list, i.e. assigning chances to each triple which is the probability it is the underlying list of values in that round of the experiment. Even acknowledging that it's not the same $A$ value in each round, over millions of experiments the average over all lists should still obey $|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

 

[RUTA]

So you are saying, according to QM, one particle from one pair is perfectly anti-correlated with another particle from a different similarly prepared pair? That is contrary to my understanding but what do I know. My understanding is that there is no correlation between particles from one pair and those of another pair.

How could you possibly infer that from what I said?

 

[stevendaryl]

This is something people should keep in mind for Physics Forums. The purpose of this website is for people to discuss and ask questions about standard scientific theories, experimental results, theorems, etc. It's educational. This is really not the place for people to show that Einstein or Bell or Cantor whoever were wrong. If you suspect that some standard, excepted result is wrong, this is not really the place to go to convince people. That's something that should be done in a scientific journal.

So it's appropriate to post saying "I don't understand Bell's derivation, could someone explain this step to me?" It's not appropriate to post saying "Bell made a mistake, and nobody noticed until now." Bell could very well have made a mistake that nobody noticed before. If you think that happened, write up a paper and try to get it published. But this forum is not really for publishing original research.

 

[PeterDonis]

I've made my point

Other posters do not appear to agree. But that's moot in any case because...

I don't intend to reply any further.

Which means we can go ahead and just close this thread.