# Bell Non Locality, Quantum Non Locality, Weak Locality, CDP

• B
Gold Member

## Main Question or Discussion Point

When there are EPR/Bell test inequality violations and no signal faster than light, the terms:
Bell Non Locality, Quantum Non Locality, Weak Locality and the Cluster Decomposition Principle:
Initial state of space like systems can be factorized.
Subsystems remain space like separated.
Then final state can be factorized..
With the following diagram how can the above terms be unified in one comprehensive definition and explanation
for Bell test violations ? Related Quantum Physics News on Phys.org
PeterDonis
Mentor
2019 Award
Initial state of space like systems can be factorized.
If this is the case, then the systems are not entangled and there will be no violation of the Bell inequalities. The QM prediction of violations of the Bell inequalities requires that the systems in question are entangled, which means the state of the system as a whole cannot be factorized.

Gold Member
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stevendaryl
Staff Emeritus
I obtained that definition from post # 7 and thought it was in reference to EPR/Bell inequality violations for entangled particles.
And also that this factorization :
p (abIxy,λ) = p (aIx,λ) p (bIy ,λ) is used to derive the inequality : S = (ab) + (ab') + (a'b) - (a'b') ≤ 2
for entangled particles.
Well, that's the heart of Bell's disproof of local hidden variables: Every local hidden variables theory can be "factorized", but an entangled quantum state cannot be. Therefore, entangled quantum states cannot be explained by local hidden variables.

• PeterDonis and vanhees71
N88
Well, that's the heart of Bell's disproof of local hidden variables: Every local hidden variables theory can be "factorized", but an entangled quantum state cannot be. Therefore, entangled quantum states cannot be explained by local hidden variables.
Please, where are the errors in the following?

Fact 1: Under the product rule for probabilities (here, wrt EPRB), the following formulation can never be false: p(ab|xy,λ) = p(a|x,λ)p(b|xy,λ,a). (A)
Explanation: This factorisation is based on these facts: the inclusion of irrelevant conditionals in a probability function is irrelevant; irrelevant conditionals are best eliminated by experimental facts (not by relying on erroneous opinions, hypotheses, etc).

Fact 2: Bell-test experiments confirm the necessity of x and "a" in p(b|xy,λ,a) .
Explanation: "a" and "b" are correlated by the common hidden-variable λ and the angle (x,y) between x and y.

Fact 3: Bell-test experiments confirm that the correlation is law-like.
Explanation: Under EPRB, the law in (A) is: p(b|xy,λ,a) = sin2 ##\tfrac{1}{2}##(x,y). (B)

Fact 4: Many physicists insist that: p(ab|xy,λ) = p(a|x,λ)p(b|y,λ). (C)
Explanation: (C) is called "Bell's locality hypothesis".

Fact 5: There is no justification for thinking that separated local events cannot be correlated as in (A).
Explanation: My garlic crop is correlated with my separate onion crop; the correlated hidden-variable is the separate but correlated underground water-supplies.

Fact 6: Since λ is Bell's random hidden-variable: p(a|x,λ) = p(b|y,λ) = ##\tfrac{1}{2}.## (D)
So under "Bell's locality hypothesis" (C), we must have this permanent error: p(ab|xy,λ) = p(a|x,λ)p(b|y,λ) = ##\tfrac{1}{4}.## (E)
Explanation: (C) is an erroneous hypothesis in that it overlooks the validity of (A) under local-realism when outputs are correlated by the combination of correlated hidden-variables λ and test-settings x and y are correlated by the angle (x,y) between them.

Thanks; N88

stevendaryl
Staff Emeritus
Please, where are the errors in the following?

Fact 1: Under the product rule for probabilities (here, wrt EPRB), the following formulation can never be false: p(ab|xy,λ) = p(a|x,λ)p(b|xy,λ,a). (A)
Explanation: This factorisation is based on these facts: the inclusion of irrelevant conditionals in a probability function is irrelevant; irrelevant conditionals are best eliminated by experimental facts (not by relying on erroneous opinions, hypotheses, etc).

Fact 2: Bell-test experiments confirm the necessity of x and "a" in p(b|xy,λ,a) .
Explanation: "a" and "b" are correlated by the common hidden-variable λ and the angle (x,y) between x and y.

Fact 3: Bell-test experiments confirm that the correlation is law-like.
Explanation: Under EPRB, the law in (A) is: p(b|xy,λ,a) = sin2 ##\tfrac{1}{2}##(x,y). (B)

Fact 4: Many physicists insist that: p(ab|xy,λ) = p(a|x,λ)p(b|y,λ). (C)
Explanation: (C) is called "Bell's locality hypothesis".

Fact 5: There is no justification for thinking that separated local events cannot be correlated as in (A).
Explanation: My garlic crop is correlated with my separate onion crop; the correlated hidden-variable is the separate but correlated underground water-supplies.

Fact 6: Since λ is Bell's random hidden-variable: p(a|x,λ) = p(b|y,λ) = ##\tfrac{1}{2}.## (D)
So under "Bell's locality hypothesis" (C), we must have this permanent error: p(ab|xy,λ) = p(a|x,λ)p(b|y,λ) = ##\tfrac{1}{4}.## (E)
Explanation: (C) is an erroneous hypothesis in that it overlooks the validity of (A) under local-realism when outputs are correlated by the combination of correlated hidden-variables λ and test-settings x and y are correlated by the angle (x,y) between them.

Thanks; N88
Well, outside of quantum mechanics, there are no counter-examples to the locality condition $P(ab|x,y,\lambda) = P(a|x,\lambda) P(b|y, \lambda)$. That is, outside of QM, whenever two distant variables are correlated, it is because there is a common cause to each, and they are correlated through that common cause. Of course, assumption (C) is "erroneous"---QM is not locally realistic. Bell's theorem clarifies the way in which it differs from a locally realistic theory.

The correlation between your garlic crop and onion crop is not a counter-example. In that case, $\lambda$ represents the water supply, in common to both.

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stevendaryl
Staff Emeritus
Well, outside of quantum mechanics, there are no examples of the locality condition $P(ab|x,y,\lambda) = P(a|x,\lambda) P(b|y, \lambda)$. That is, outside of QM, whenever two distant variables are correlated, it is because there is a common cause to each, and they are correlated through that common cause. Of course, assumption (C) is "erroneous"---QM is not locally realistic. Bell's theorem clarifies the way in which it differs from a locally realistic theory.

The correlation between your garlic crop and onion crop is not a counter-example. In that case, $\lambda$ represents the water supply, in common to both.
The claim is that if $\lambda$ is a complete description of the common causal influences on $a$ and $b$, then the probabilities will factor when conditioned on $\lambda$. It has to be complete for factorization to work.

PeterDonis
Mentor
2019 Award
outside of quantum mechanics, there are no examples of the locality condition
I think you mean that outside of QM there are no examples of violations of the locality condition, correct? QM violates it, but we don't know of any other theory that does.

• stevendaryl
N88
Well, outside of quantum mechanics, there are no examples of the locality condition $P(ab|x,y,\lambda) = P(a|x,\lambda) P(b|y, \lambda)$. That is, outside of QM, whenever two distant variables are correlated, it is because there is a common cause to each, and they are correlated through that common cause. Of course, assumption (C) is "erroneous"---QM is not locally realistic. Bell's theorem clarifies the way in which it differs from a locally realistic theory.

The correlation between your garlic crop and onion crop is not a counter-example. In that case, $\lambda$ represents the water supply, in common to both.
I thought the water-supply, being common to both crops, was an illustrative analogy: equivalent to $\lambda$ being common to both of Bell's factors. No?

stevendaryl
Staff Emeritus
I thought the water-supply, being common to both crops, was an illustrative analogy: equivalent to $\lambda$ being common to both of Bell's factors. No?
Sorry, I think I misunderstood your example. I thought you were arguing that C (the assumption of factorizability) was wrong, and that your example showed it.

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Well, outside of quantum mechanics, there are no counter-examples to the locality condition $P(ab|x,y,\lambda) = P(a|x,\lambda) P(b|y, \lambda)$. That is, outside of QM, whenever two distant variables are correlated, it is because there is a common cause to each, and they are correlated through that common cause.
No, in classical theory it is as well possible that it is the locality condition which fails.

Suspected criminals A and B make the same claims about what has happened at the crime scene. Is this because that really happened, and they have told the truth, or because A has succeeded to tell his friend B about the questions, and his answer?

PeterDonis
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2019 Award
Suspected criminals A and B make the same claims about what has happened at the crime scene. Is this because that really happened, and they have told the truth, or because A has succeeded to tell his friend B about the questions, and his answer?
Neither of these possibilities violate locality. They just postulate different local interactions (at the crime scene, vs. at some other location where A tells B things).

Neither of these possibilities violate locality. They just postulate different local interactions (at the crime scene, vs. at some other location where A tells B things).
The point was that it violates P(ab|x,y,λ)=P(a|x,λ)P(b|y,λ). With λ being what they have above known before the interrogations, x and y being the questions asked by the interrogators, and a and b their answers.
Of course, the information transfer in the analogy is not violating Einstein causality, but it violates the law of the state, with the police doing everything to prevent it, so it is also a hidden information transfer. And the guys will do everything to hide it. (If they use smuggled mobiles, the communication may be even faster than sound.)

And the aim of this hidden communication is, exactly, to suggest the police to apply the EPR criterion of reality: Once the interrogation of A allowed to correctly predict what B would answer, without influencing B's interrogation in any way, the answer should correspond to some element of reality.

PeterDonis
Mentor
2019 Award
The point was that it violates P(ab|x,y,λ)=P(a|x,λ)P(b|y,λ). With λ being what they have above known before the interrogations
But the only way for the factorizability condition to be violated here is if ##\lambda## does not include information from the actual crime scene. But that's not how ##\lambda## is supposed to be defined. ##\lambda## is supposed to include all information that could have gotten to both measurements by local means--in relativistic terms, it should include all information that is in the intersection of the past light cones of the two measurement events. That has to include information from the actual crime scene.

With ##\lambda## defined to include information from the actual crime scene, then the factorizability condition can't be violated, since all of the correlations between A's and B's answers can be explained by their common knowledge of what happened at the crime scene. Whether that common knowledge comes from both of them being present at the crime scene, or A communicating it to B afterwards, doesn't matter for factorizability.

the aim of this hidden communication is, exactly, to suggest the police to apply the EPR criterion of reality
Which is violated by quantum mechanics, but not by your scenario. Your scenario does not violate locality; as you admit in the above quote, it shows why the EPR criterion is a good criterion for any scenario that does not involve quantum mechanics. In other words, it shows why classical physics does not violate locality.

N88
Sorry, I think I misunderstood your example. I thought you were arguing that C (the assumption of factorizability) was wrong, and that your example showed it.
Thanks SD. We need to be careful here because, as I see it, you have responded to my position in two ways and neither is correct (as I understand them). So I trust I'm being sufficiently careful in the 4 points that follow:

1: Re your above reply, I AM arguing that (C) is wrong; ie, experimentally false! And I'm also arguing that my example shows it.

That is, Fact 4 is a fact: Many physicists insist [that "Bell's local realism" can be represented by]: p(ab|xy,λ) = p(a|x,λ)p(b|y,λ). (C)
Explanation: (C) is called "Bell's locality hypothesis".

But (C) = p(ab|xy,λ) = p(a|x,λ)p(b|y,λ) is false by observation: so "Bell's local realism" or "Bell's locality hypothesis" is immediately false under EPRB. That is, we know that that "a" and "b" will be correlated (logically dependent) : ie, we know the particles are pairwise correlated and we know the detector settings are correlated --- by a function of the angle (x,y). But (C) is the expression for logical-independence! Hence wrong here -- by observation alone.

2: Further, your original reply was this: "Yes, and your water supply example does not violate the factorizability condition." I disagree. My garlic and onion outputs are also logically dependent because of the correlated water supply; ie, two great crops when the water supply is well maintained; two poor crops when the water supply is neglected.

So my water supply example, with its correlated crops, DOES violate the factorizability condition: which is "Bell's locality hypothesis".

3: Regarding Fact 3: Bell-test experiments confirm that the correlation is law-like.
Explanation: Under EPRB, the law in (A) is: p(b=1)|xy,λ,a=1) = sin2##\tfrac{1}{2}##(x,y). (B)

So, even if our knowledge of λ were complete, our related prediction must still be (B); otherwise it would be experimentally falsified.

4. But, further, I do not understand why anyone might believe that we could ever know "hidden-variables" like Bell's λ completely. [Moreover, I am able to derive the correct experimental results without such knowledge and without nonlocality.] So the fundamental tenet of Bell's (C) -- ie, know λ completely -- can never be satisfied. And QM does not require such: for, again in my view, QM is so well-founded that we are able to encode incomplete-information re "hidden-variables" in probability relations AND derive the correct experimental outcomes.

In conclusion: somewhere in the above there must be a "fact" that we disagree about; maybe one that I am confused about.

HTH. Thanks again, N88

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PeterDonis
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2019 Award
My garlic and onion outputs are also logically dependent because of the correlated water supply
The water supply is part of ##\lambda##, and ##\lambda## appears in both factors in the factorizability condition. So a correlation due to ##\lambda## can hold without the factorizability condition being violated.

To put it in terms of your garlic and onion crops, if we factor out the common cause, the water supply (##\lambda##), there is no residual correlation between the two crops: the garlic crop (##a##) depends only on factors specific to how you grew garlic (##x##), and the onion crop (##b##) depends only on factors specific to how you grew onions (##y##). That is what the factorizability condition expresses, and it will be true for your garlic and onion case. In other words, that case obeys the Bell locality condition.

In QM experiments on entangled particles, however, the above will not be the case: even after you have factored out the common cause (##\lambda##), you still have correlations--the results on the ##a## side of the experiment do not depend solely on ##x##, and the results on the ##b## side of the experiment do not depend solely on ##y##. It's as if, even after factoring out the effect of the water supply (and any other common causes), somehow changing the way you grew garlic changed the yield of your onion crop. That does not happen anywhere outside of QM experiments on entangled particles--those are the only experiments anyone has ever done that violate the Bell locality condition.

N88
The water supply is part of ##\lambda##, and ##\lambda## appears in both factors in the factorizability condition. So a correlation due to ##\lambda## can hold without the factorizability condition being violated.

To put it in terms of your garlic and onion crops, if we factor out the common cause, the water supply (##\lambda##), there is no residual correlation between the two crops: the garlic crop (##a##) depends only on factors specific to how you grew garlic (##x##), and the onion crop (##b##) depends only on factors specific to how you grew onions (##y##). That is what the factorizability condition expresses, and it will be true for your garlic and onion case. In other words, that case obeys the Bell locality condition.

In QM experiments on entangled particles, however, the above will not be the case: even after you have factored out the common cause (##\lambda##), you still have correlations--the results on the ##a## side of the experiment do not depend solely on ##x##, and the results on the ##b## side of the experiment do not depend solely on ##y##. It's as if, even after factoring out the effect of the water supply (and any other common causes), somehow changing the way you grew garlic changed the yield of your onion crop. That does not happen anywhere outside of QM experiments on entangled particles--those are the only experiments anyone has ever done that violate the Bell locality condition.
It's not clear to me why you "factor out" the common cause?

Let me be clear: the garlic [G] and onion [O] example is in no way meant to reproduce EPRB correlations. But it is meant to be a counterpoint to your following statement (a counterpoint that can be experimentally validated):

• "To put it in terms of your garlic and onion crops, if we factor out the common cause, the water supply (##\lambda##), there is no residual correlation between the two crops:"

I've cut off the underground water supply (so the test conditions are now X) and still the separated crops are correlated; ie, under X, I now get a good crop of each [G+, O+] half the time, or a poor crop of each [G-, O-] half the time, with no crossovers. So:

P(G+|X) = P(O+|X) = P(G-|X) = P(O-|X) = ##\tfrac{1}{2}##.

P(G+O+|X) = P(G+|X)P(O+|XG+) = ##\tfrac{1}{2}## ## \neq## P(G+|X)P(O+|X).

Given such a clear difference, I take it that our differences must arise from my misunderstanding something?

PeterDonis
Mentor
2019 Award
I've cut off the underground water supply (so the test conditions are now X) and still the separated crops are correlated
How? Translating this back into Bell's notation, we have eliminated ##\lambda## (the water supply), so the factorizability condition is ##p(ab|xy) = p(a|x) p(b|y)##. If this condition is violated, it means that each crop's yield depends not only on how you grow that crop, but how you grow the other crop. How can that be?

I've cut off the underground water supply (so the test conditions are now X)
No, they aren't. See below.

I take it that our differences must arise from my misunderstanding something?
I think you are misunderstanding Bell's notation. ##x##, in his notation, would correspond to the way you grow your garlic crop: by definition, it only contains factors that are local to the garlic crop. Similarly, ##y## only contains factors that are local to the onion crop. That means that, by hypothesis, and using your binary notation for good (+) or poor (-) crops, we would have

$$P(O^+|X) = P(O^-|X) = P(G^+|Y) = P(G^-|Y) = \frac{1}{2}$$

But we would not have ##P(G^+|X) = P(G^-|X)##, unless you are an incompetent garlic grower, and we would not have ##P(O^+|Y) = P(O^-|Y)##, unless you are an incompetent onion grower. In fact, we would expect ##P(G^+|X) > P(G^-|X)## and ##P(O^+|Y) > P(O^-|Y)##, since you are adapting your growth techniques to each crop individually.

stevendaryl
Staff Emeritus
It's not clear to me why you "factor out" the common cause?
I've cut off the underground water supply (so the test conditions are now X) and still the separated crops are correlated; ie, under X, I now get a good crop of each [G+, O+] half the time, or a poor crop of each [G-, O-] half the time, with no crossovers. So:

P(G+|X) = P(O+|X) = P(G-|X) = P(O-|X) = ##\tfrac{1}{2}##.

P(G+O+|X) = P(G+|X)P(O+|XG+) = ##\tfrac{1}{2}## ## \neq## P(G+|X)P(O+|X).

Given such a clear difference, I take it that our differences must arise from my misunderstanding something?
If even after controlling for water supply, there is a correlation between your garlic crop and your onion crop, then that usually means that there is some other common causal influence other than water supply.

N88
How? Translating this back into Bell's notation, we have eliminated ##\lambda## (the water supply), so the factorizability condition is ##p(ab|xy) = p(a|x) p(b|y)##. If this condition is violated, it means that each crop's yield depends not only on how you grow that crop, but how you grow the other crop. How can that be?
You appear to be confusing logical-implication with causation?? There is no causal influence between the crops; the crop correlations arise from the correlation of the separate local growing conditions (soil, sunlight, rainfall, plough-settings). The outcome correlations in EPRB arise in a similar way; eg, via the correlation of the independent and freely chosen detector settings x and y as a function of their angular difference (x,y).

I think you are misunderstanding Bell's notation. ##x##, in his notation, would correspond to the way you grow your garlic crop: by definition, it only contains factors that are local to the garlic crop. Similarly, ##y## only contains factors that are local to the onion crop. That means that, by hypothesis, and using your binary notation for good (+) or poor (-) crops, we would have

$$P(O^+|X) = P(O^-|X) = P(G^+|Y) = P(G^-|Y) = \frac{1}{2}$$

But we would not have ##P(G^+|X) = P(G^-|X)##, unless you are an incompetent garlic grower, and we would not have ##P(O^+|Y) = P(O^-|Y)##, unless you are an incompetent onion grower. In fact, we would expect ##P(G^+|X) > P(G^-|X)## and ##P(O^+|Y) > P(O^-|Y)##, since you are adapting your growth techniques to each crop individually.
With 5000 acres of garlic and 6000 acres of onions, the crops are very much dependent on the weather! Our competence allows us to make good profits from both good and bad crops!

So, to more closely match the Bell-formulation, let's use
$$P(O^+|X) = P(O^-|X) = P(G^+|Y) = P(G^-|Y) = \frac{1}{2}. (J)$$
Then we find agriculturally that
$$P(O^+G^+|XY) = P(O^+|X) P(G^+|XYO^+) = \frac{1}{2} \neq P(O^+|X)P(G^+|Y). (K)$$
So now we have the local farming conditions X and Y separated like the x and y in Bell's locality hypothesis. And the farming conditions X and Y are correlated via a function of the associated growing conditions: just like Bell's detector settings x and y are correlated via a function of the associated settings.

N88
If even after controlling for water supply, there is a correlation between your garlic crop and your onion crop, then that usually means that there is some other common causal influence other than water supply.
Indeed, its called the weather (mostly). Remember, I only removed the underground water-supply. And that was done reluctantly: for reasons that I have yet to understand (but see post above this one). There is apparently some reason to remove λ from Bell's formulation for comparison purposes, but that reasoning is currently beyond me.

stevendaryl
Staff Emeritus
You appear to be confusing logical-implication with causation?? There is no causal influence between the crops; the crop correlations arise from the correlation of the separate local growing conditions (soil, sunlight, rainfall, plough-settings). The outcome correlations in EPRB arise in a similar way; eg, via the correlation of the independent and freely chosen detector settings x and y as a function of their angular difference (x,y).
So the probability of getting a good garlic crop is a function of lots of variables: soil, sunlight, rain, etc. Similarly for an onion crop. Then it's false to say that water supply is the only common causal influence on the two crops. Bell's factorizability condition is only expected to hold if the hidden variable $\lambda$ includes all relevant common causal influences.

stevendaryl
Staff Emeritus
Indeed, its called the weather (mostly).
Then what that means is that $\lambda$ must include both water supply AND weather. Bell's factorizability condition only holds when ALL common causal influences are held constant.

stevendaryl
Staff Emeritus
Indeed, its called the weather (mostly). Remember, I only removed the underground water-supply. And that was done reluctantly: for reasons that I have yet to understand (but see post above this one). There is apparently some reason to remove λ from Bell's formulation for comparison purposes, but that reasoning is currently beyond me.
For simplicity, let's assume that there are two states for the water supply, "good" and "bad. And there are two possible types of weather, "good" and "bad. Then there are four possible values of $\lambda$:
1. $\lambda_{++}$: good water supply and good weather
2. $\lambda_{+-}$ good water supply and bad weather
3. $\lambda_{-+}$ bad water supply and good weather
4. $\lambda_{--}$ bad water supply and bad weather
If water supply and weather are the only variables that are relevant, then

$P(G^+, O^+|\lambda_{++}) = P(G^+|\lambda_{++}) P(O^+ | \lambda_{++})$
(and similarly for all other values of $\lambda$).

If you don't control for weather and only control for water supply, then the probabilities will not factor. If you've controlled for every common causal influence, then the probabilities will factor.

PeterDonis
Mentor
2019 Award
You appear to be confusing logical-implication with causation?
No, I'm just saying that you are confused about how your scenario matches up with Bell's notation and formulas.

the crop correlations arise from the correlation of the separate local growing conditions (soil, sunlight, rainfall, plough-settings)
In other words, the water supply is not the only common factor between the crops, despite what you said earlier. Ok, fine. Then all of these things that correlate between the growing conditions--soil, sunlight, rainfall, plough settings--are all included in ##\lambda##. They aren't included in ##x## and ##y##. That's how Bell defined those variables. If you are using different definitions from his, of course you're going to come up with different formulas. But if you want to compare with Bell's formulas, you need to use his definitions.

With 5000 acres of garlic and 6000 acres of onions, the crops are very much dependent on the weather!
Ok, then weather goes in ##\lambda## as well.

So now we have the local farming conditions X and Y separated like the x and y in Bell's locality hypothesis
No, you don't. See below.

the farming conditions X and Y are correlated
This violates Bell's definitions; by definition, Bell's ##x## and ##y## are uncorrelated.

just like Bell's detector settings x and y are correlated via a function of the associated settings
What "function of the associated settings" are you talking about?