# Does the Bell theorem assume reality?

I think you are misunderstanding the distinction between what is observed, and a model intended to EXPLAIN those observations.
I'm not. Rather my point is precisely that we have to be careful how we apply models to experiment. You can use a model that contradicts the experiment you are modelling.

Yes, what is observed is that Alice measures the spin on one particle, and Bob measures the spin on another particle. The class of models that Bell is interested in (the class that Einstein, Podolsky and Rose were interested in) are models of the following form:
1. When a twin-pair is produced, there is an associated state variable $\lambda$ describing the pair.
2. When Alice measures her particle, the result depends on (1) facts about her measuring device, and (2) the value of $\lambda$
3. Similarly, when Bob measures his particle, the result depends on facts about his measuring device, and the value of $\lambda$.
So in general, Alice's result $R_A$ can be written as a function $R_A(\lambda, \alpha, \lambda_A)$, where $\alpha$ is the choice of which measurement to perform, and $\lambda_A$ is other facts about Alice's device. Bob's result $R_B$ is similarly a function $R_B(\lambda, \beta, \lambda_B)$ where $\beta$ is his choice of measurement, and $\lambda_B$ is other facts about Bob's device.

So now, we have to take into account a stark fact about these twin-particles: There is PERFECT anti-correlation (or correlation, depending on the exact type of EPR experiment performed). That means that for each $\lambda$, if it happens to be that $\alpha = \beta$ (that is, if Alice and Bob perform the same measurement), they always get opposite result: No matter what $\lambda_A$ and $\lambda_B$ are, we always have:

$R_A(\lambda, \alpha, \lambda_A) = - R_B(\lambda, \alpha, \lambda_B)$
This is all true and irrelevant to the point I'm making.

I'm sure you agree that it is wrong to assume perfect anti-correlation between one particle of a pair and another particle of a different pair even if it is prepared similarly. In the same way as it is wrong to assume perfect anti-correlation between heads of one toss and tails of a different toss, even of the exact same coin.

This implies that $R_A$ and $R_B$ don't actually depend on $\lambda_A$ and $\lambda_B$ at all. If Alice's result is NOT determined by $\lambda$ and $\alpha$, then sometimes she would get a result that would not be anti-correlated with what Bob gets.

So we pick three possible measurements for Alice, $\alpha_1, \alpha_2, \alpha_3$. For each $\lambda$, let $A(\lambda)$ be $R_A(\lambda, \alpha_1)$, let $B(\lambda)$ be $R_A(\lambda, \alpha_2)$ and let $C(\lambda)$ be $R_A(\lambda, \alpha_3)$.

Now, if we create a sequence of twin-pairs, each twin pair is associated with some value of $\lambda$. So we let

$A_n = A(\lambda_n)$
$B_n = B(\lambda_n)$
$C_n = C(\lambda_n)$

where $\lambda_n$ is the value of $\lambda$ for the $n^{th}$ twin pair. These are not measurement results, they are just numbers, unknown functions of $\lambda$ evaluated at $\lambda = \lambda_n$. But we're ASSUMING that the significance of these numbers is that

$A_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_1$.
$B_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_2$.
$C_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_3$.

Under the assumption of perfect anti-correlation, Bob would get $-A_n, -B_n, -C_n$ for the corresponding measurements.

So now, the numbers $A_n, B_n, C_n$ are just three numbers, each are assumed to be $\pm 1$. So we can do manipulations as real numbers to come to the conclusion that:

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

where $\langle A B \rangle = \frac{1}{N} \sum_n A_n B_n$ and $\langle A C \rangle = \frac{1}{N} \sum_n A_n C_n$ and $\langle B C \rangle = \frac{1}{N} \sum_n B_n C_n$, and where $N$ is the number of twin pairs produced.

This is simply a mathematical fact about ANY sequence of triples of numbers $A_n, B_n, C_n$ where each number is $\pm 1$.
Again this is all trivially true and irrelevant so long as $A_n, B_n, C_n$ arise from the same context $n$. The three averages $\langle A B \rangle, \langle A C \rangle, \langle B C \rangle$ are not simply independent averages without any relationship with each other. Based on the way you derived the expression, the inequality relationship embodies all the assumptions you used in it's derivation, including the fact that they are all based on the same context $n$. Note that by dropping the $n$ subscripts, you are being a little careless and perhaps that is why you are not getting the point. This will be crucial when you apply this relationship to experimental data.

Now, the question is whether it is possible to measurement the quantities $\langle A B \rangle$, $\langle A C \rangle$, $\langle B C \rangle$. We can't, actually, because the definition of (for example) $\langle A B \rangle$ is that it is the average of $A_n B_n$ over all values of $n$. But we don't measure $A_n$ and $B_n$ over all possible values of $n$. We only measure it on for some of the $n$. So to compare theory with experiment, we have to assume that the average of $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.
This is the key. You have not clearly stated the assumption. It is not simply that $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.

The assumption is in fact that the relationship between three averages $\langle A_n B_n \rangle, \langle A_n C_n \rangle, \langle B_n C_n \rangle$ from the same context $n$ is the same relationship as that between $\langle A_i B_i \rangle$ from one context $i$ and $\langle A_j C_j \rangle$ from a different context $j$ and $\langle B_k C_k \rangle$ from a yet another context $k$ with $i, j, k$ disjoint.

At the very least, you have to agree that this assumption is implied. Do you disagree? Bell's realism assumption definitely includes this "sub-assumption" if you will, as soon as the inequality relationship is applied to any experiment in which simultaneous measurement of $\langle A_n B_n \rangle, \langle A_n C_n \rangle, \langle B_n C_n \rangle$ was not performed (eg EPRB).

If this assumption is true. It should be possible to start from the variables $A_i, B_i, A_j, C_j, B_k, C_k$ and derive the same relationship as what you derived for $A_n, B_n, C_n$ and ask the question, what additional assumptions will be required in that case. It turns out it will be required to assume that $A_i = A_j, B_i = B_k, C_j = C_k$ and $B_i B_k = 1$ which is the same as assuming perfect correlation between heads of one coin toss and tails of another coin toss.

DrChinese
Gold Member
I'm sure you agree that it is wrong to assume perfect anti-correlation between one particle of a pair and another particle of a different pair even if it is prepared similarly. In the same way as it is wrong to assume perfect anti-correlation between heads of one toss and tails of a different toss, even of the exact same coin.
The only person saying anything about a correlation between one pair toss and another is... you.

You are getting lost in subscripts, and missing the picture Bell presents. Bell shows us that the relationship between 3 pairs of settings (assuming counterfactual definiteness of outcomes for A, B and C) cannot match the quantum expectation. This has nothing whatsoever to do with an experiment. From the Wiki on Bell's Theorem:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

The next question is whether the entangled particle predictions of QM are correct. Any experiment that is run that shows the usual formula is correct will do it. You do not need to test anything at 3 angles (or 4 such as the CHSH) because the only question is whether QM makes the correct prediction (when a local realistic theory predicts something else entirely). So you could do a series of test sat 0 & 120 degrees ONLY, and that would be enough to confirm QM and reject local realism.

So comments about "how we apply models to experiment" are off target. Bell's Theorem sets up a dividing line between QM and Local Realism, and that is independent of an experiment. Any experiment that tests the predictions of QM will be enough to settle things, and that experiment does not need 3 of anything (i.e. AB/BC/AC) to be convincing.

RUTA
If this assumption is true. It should be possible to start from the variables $A_i, B_i, A_j, C_j, B_k, C_k$ and derive the same relationship as what you derived for $A_n, B_n, C_n$ and ask the question, what additional assumptions will be required in that case. It turns out it will be required to assume that $A_i = A_j, B_i = B_k, C_j = C_k$ and $B_i B_k = 1$ which is the same as assuming perfect correlation between heads of one coin toss and tails of another coin toss.
QM assumes you are using the same rotationally invariant state for every trial and the eigenvalues are +/-1 for all measurements. That does mean there are relationships between the outcomes in different trials.

The only person saying anything about a correlation between one pair toss and another is... you.

You are getting lost in subscripts, and missing the picture Bell presents. Bell shows us that the relationship between 3 pairs of settings (assuming counterfactual definiteness of outcomes for A, B and C) cannot match the quantum expectation. This has nothing whatsoever to do with an experiment. From the Wiki on Bell's Theorem:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

The next question is whether the entangled particle predictions of QM are correct. Any experiment that is run that shows the usual formula is correct will do it. You do not need to test anything at 3 angles (or 4 such as the CHSH) because the only question is whether QM makes the correct prediction (when a local realistic theory predicts something else entirely). So you could do a series of test sat 0 & 120 degrees ONLY, and that would be enough to confirm QM and reject local realism.

So comments about "how we apply models to experiment" are off target. Bell's Theorem sets up a dividing line between QM and Local Realism, and that is independent of an experiment. Any experiment that tests the predictions of QM will be enough to settle things, and that experiment does not need 3 of anything (i.e. AB/BC/AC) to be convincing.
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?

stevendaryl
Staff Emeritus
I'm not. Rather my point is precisely that we have to be careful how we apply models to experiment. You can use a model that contradicts the experiment you are modelling.

This is all true and irrelevant to the point I'm making.
No, it is very relevant. The point you are making is wrong. We prove a fact about averages:

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

This is just a mathematical fact about any sequence of triples of numbers (where each number is $\pm 1$). It doesn't have anything to do with any measurements. It's just a 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. The undeniable conclusion is that the measured quantities did NOT come from a sequence of triples of numbers (one triple for each twin pair). The assumption of local realism is that it did come from such a sequence of triples. (Actually, that there is a sequence of triples associated with the sequence of twin pairs. The triples are assumed to be functions of the values of $\lambda$.) So the violation of the inequality disproves local realism.

The futzing around you're doing with indices is just not relevant. The argument as summarized here does not mention indices at all. It's only talking about averages.

stevendaryl
Staff Emeritus
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.
I think you're confused about what assumptions are needed. The stuff you're saying about indices is not correct. Bell's theorem is about averages, not about specific indices.

Now, there is an additional assumption involved, which is that we're assuming that the average of (for example) $A_n B_n$ over all values of $n$ is approximately the same as the average over those values of $n$ for which we actually measured $A$ and $B$. It would be weird if that were not the case, and such a weirdness would require some explanation. The whole point of a local variables theory is to give an explanation to quantum statistics. If it requires an additional unexplainable effect, then that hardly counts as an explanation.

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?
We're saying that Bell's analysis was very very careful, and the results are pretty airtight.

QM assumes you are using the same rotationally invariant state for every trial and the eigenvalues are +/-1 for all measurements. That does mean there are relationships between the outcomes in different trials.
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.

We're saying that Bell's analysis was very very careful, and the results are pretty airtight.
But that is never an argument as opposed to hand-waving. You have to get down to the details. But I've made my point so we can agree to disagree.

stevendaryl
Staff Emeritus
But that is never an argument as opposed to hand-waving. You have to get down to the details. But I've made my point so we can agree to disagree.
I don't think you've made a point. I think you've just shown that you are confused about what Bell's theorem establishes.

stevendaryl
Staff Emeritus
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?
NO. And that is not used in Bell's argument.

$|\langle A B \rangle + \langle A C \rangle| \leq 1 + \langle B C \rangle$
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
Gold Member
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
Staff Emeritus
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.

Last edited:
DrChinese
Gold Member
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
Staff Emeritus
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.

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.

Last edited:
stevendaryl
Staff Emeritus
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
Gold Member
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
Staff Emeritus
This is something people should keep in mind for Physics Forums. The purpose of this web site 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
Mentor
2019 Award