I QM Assumptions Regarding Entanglement Properties

[Vanadium 50]

So my working hypothesis personal theory is that this is some sort of resonance.

Fixed that for you.

 

[Dadface]

The assumptions are call locality and realism. In his paper, they are the separability condition - Bell's (2) is associated with "locality". And the condition that there is "realism" is expressed around Bell's (14) when he says "let there be another unit vector c" which is to say that there are other elements of reality (in addition to a and b) that cannot be simultaneously be measured, but could have been predicted with certainty a la EPR.

Thank you DrChinese. EPR postulated that there may be "hidden variables". I can only imagine what these hidden variables may be but different possibilities come to mind for example subtle changes in known properties or properties yet to be discovered, or currently accepted assumptions, which are incorrect, about the nature properties. It seems to me that in the realism aspect of Bell's analysis the existence of all hidden variables, whatever they may be, are implied.

I don't see how Bell's work can take into account all possible hidden variables. In fact do we know exactly what any hidden variables are?

Did bell make any assumptions about the properties of the entangled objects. I refer to them as objects for want of a better word but the use of a label for example object or photons or electron implies that the entangled objects are real with real properties. And this pins down more closely where I'm stuck. What is assumed about the properties of the quantum objects referred to in Bells analysis?

Getting tired and I don't think I'm expressing myself very clearly here. Never mind and night night.

 

[DrChinese]

Did bell make any assumptions about the properties of the entangled objects. I refer to them as objects for want of a better word but the use of a label for example object or photons or electron implies that the entangled objects are real with real properties. And this pins down more closely where I'm stuck. What is assumed about the properties of the quantum objects referred to in Bells analysis?

No, not really any particular constraints regarding the nature or number of hidden variables or what form they might take. They could conceivably even be global properties (a hidden variable accessible to all particles). The only real requirement he attempted to enforce was: the setting of a measurement device here NOT influence the outcome of a measurement there. I.e. there could be no influences faster that light.

 

[DrChinese]

I don't see how Bell's work can take into account all possible hidden variables.

You can see the problem yourself if you simply attempt to hand manipulate outcomes. I call it the DrChinese challenge. If you pick certain settings, you can't even HAND PICK results that match QM. You only need about 8 or so examples to see the impossibility. Once you hit that wall, you quickly see why Bell rules out ALL local hidden variable theories. The only way to "win" the challenge is to "cheat". That is, you hand pick knowing what you plan to measure in advance.

 

[jerromyjon]

To my knowledge such a theory is not known of at present.

Sure, there are various interpretations of QM which describe everything observable, and the wavefunction which gives insight into the nature of the quantum world, but I know what you mean from previous posts about a new "more in-depth description", but that theory would still encompass that "spooky action at a distance"! The weirdness of quantum mechanics will always be strange compared to classical mechanics. QM does not follow the macroscopic laws of nature that everyone is used to. Spacelike and timelike separated events coincide with greater probability than is classically possible!

 

[Nugatory]

Now nobody can prove that any future theory is incorrect without knowing exactly what the theory is. I know that may seem very obvious but it can seem that that is what Bell tried to prove.

That is neither what Bell proved, nor what he set out to prove, nor what he said that he proved. What he asserted and then proved is that any theory in which what happens at detector A is independent of the setting of detector B (and vice versa) must disagree with the prediction of quantum mechanics. You don't need to know exactly what that theory is to prove this result, you just have to consider the consequences of having the result at one detector be independent of the other detector and compare them with the quantum mechanical prediction.

He assumed that there are certain features that any attempt at formulating a successful theory should have and then went on to prove that the theory can't be successful at all because it does not conform to the observations.

That's not right either, because the observations in question didn't even exist when Bell came up with his inequality. Bell showed that one class of theories (those in which the results at A are independent of the setting at B) must obey the inequality while quantum mechanics would violate the inequality. Only then did experimentalists go looking for violations (and I consider the most important words in Bell's original paper to be "The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made").

And that's where I'm stuck. What exactly are the assumptions that Bell assumed the theory should have?

He didn't assume that any theory "should have" any particular assumption. Instead, he considered the consequences of one assumption, namely that the results at A are independent of the setting at B. Here we can let Bell speak for himself, from the first paragraph of his paper: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past"

 

[morrobay]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

 

[jerromyjon]

(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Because they are anticorrelated... but, yeah, that doesn't make sense to me either but it's late and I'm tired...

 

[Dadface]

Thank you for your comments everyone. I think (hope) things are starting to make a bit more sense but I need time to try to mull over the comments made.

 

[stevendaryl]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Line 13 explains it. In the anti-correlated case (which I assume is the one that Bell is talking about here), one experimenter (who I always call Alice) measures the spin of one particle, and another experimenter (who I always call Bob) measures the spin of the other particle. Experimentally, they always get the opposite results whenever they measure their spins along the same axis. If Alice gets +1, then Bob gets -1. In the formula

P(a,b) = \int d\lambda \rho(\lambda) A(a, \lambda) B(b, \lambda)

a is the spin-direction chosen by Alice and b is the spin-direction chosen by Bob, and P(a,b) is the correlation, which is the average value of the product of Alice's result and Bob's result. (I really hate the use of the letter P here, because that suggests probability, but nevermind...)

Perfect anti-correlation means that when b=a, you always get -1. That means that

P(a,a) = \int d\lambda rho(\lambda) A(a, \lambda) B(a, \lambda) = -1

It's just a mathematical fact that that's impossible unless B(a,\lambda) = -A(a,\lambda).

 

[DrChinese]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Well, the Realism condition is the part just after (14) where it says: "It follows that c is another unit vector [in addition to a and b already referenced]". That's when Bell sets up the relationship between 3 observables. Those 3 can't simultaneously exhibit the Quantum Mechanical expectation value.

 

[Dadface]

That is neither what Bell proved, nor what he set out to prove, nor what he said that he proved. What he asserted and then proved is that any theory in which what happens at detector A is independent of the setting of detector B (and vice versa) must disagree with the prediction of quantum mechanics. You don't need to know exactly what that theory is to prove this result, you just have to consider the consequences of having the result at one detector be independent of the other detector and compare them with the quantum mechanical prediction.

That's not right either, because the observations in question didn't even exist when Bell came up with his inequality. Bell showed that one class of theories (those in which the results at A are independent of the setting at B) must obey the inequality while quantum mechanics would violate the inequality. Only then did experimentalists go looking for violations (and I consider the most important words in Bell's original paper to be "The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made").

He didn't assume that any theory "should have" any particular assumption. Instead, he considered the consequences of one assumption, namely that the results at A are independent of the setting at B. Here we can let Bell speak for himself, from the first paragraph of his paper: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past"

I'm still not getting the conclusions that seem to be drawn from EPR and Bell and I think its down to the meaning of "hidden variable theories". In my opinion a hidden variable theory would be, at the very least, just as powerful as any currently accepted quantum theory in that, for example, it would make exactly the same predictions. If it doesn't conform to the observations it shouldn't be called a theory. But I think if any hidden variable theories are developed they would have some extra feature(s), the hidden variables, which quantum theory doesn't have at the present time. Those hidden variables will account for any perceived apparent discrepancies or weirdness in quantum theories.

I think Einstein said that QM is not complete and I think that is as true today as when he first said it. I think quantum theorists are still hard at work. And there is a possibility that a hidden variable theory will be developed. But until that happens we have no knowledge of what that theory is. And that illustrates one of my sticking points because the impression is often given that Bell, and the subsequent testing of his theory ruled out the possibility of a certain type of hidden variable theory. But to rule out a theory you've got to know the full details of that theory not just one assumption you think is made in developing that theory.

 

[stevendaryl]

I think Einstein said that QM is not complete and I think that is as true today as when he first said it. I think quantum theorists are still hard at work. And there is a possibility that a hidden variable theory will be developed. But until that happens we have no knowledge of what that theory is. And that illustrates one of my sticking points because the impression is often given that Bell, and the subsequent testing of his theory ruled out the possibility of a certain type of hidden variable theory. But to rule out a theory you've got to know the full details of that theory not just one assumption you think is made in developing that theory.

Well, that's the power of mathematics. If you have a real number in mind, I can tell you that its square is not equal to -1. I don't need to know all the decimal places of your number to reach that conclusion. In Newtonian mechanics, if there is a force F(\vec{x}) acting on a particle that is constant in time and independent of the velocity of the particle, then I can tell you that the combination of potential energy and kinetic energy is constant. I don't need to know the details of the force.

Mathematics allows you to prove facts about a huge class of situations. You don't need to know precise details about your situation---it's enough to know that it's of a particular type.

 

[Dadface]

Well, that's the power of mathematics. If you have a real number in mind, I can tell you that its square is not equal to -1. I don't need to know all the decimal places of your number to reach that conclusion. In Newtonian mechanics, if there is a force F(\vec{x}) acting on a particle that is constant in time and independent of the velocity of the particle, then I can tell you that the combination of potential energy and kinetic energy is constant. I don't need to know the details of the force.

Mathematics allows you to prove facts about a huge class of situations. You don't need to know precise details about your situation---it's enough to know that it's of a particular type.

Your analogies might be OK for the situations you describe but I don't think they're necessarily relevant to the point I'm trying to make.

A hidden variables theory, should one be developed, will accept all experimental results and will be just as good, if not better, than existing quantum theories at predicting those results. The theory will encompass the findings that have been made during the experimental testing of Bells paper and will encompass all other relevant experimental findings. In fact I can't imagine Einstein et al envisioning a theory that did not conform to observations because such a theory would be a failure.

I think hidden variables is an aspect of a theory, which has not yet been realized and which accounts for any perceived apparent weirdness or discrepancies in the currently known theories. It may be something complicated or it may be something very simple, perhaps an additional sentence or two pointing out something that has been overlooked. Exactly what the hidden variable are remains to be seen. Or perhaps will never be seen. I like to keep an open mind.

 

[zonde]

Well, the Realism condition is the part just after (14) where it says: "It follows that c is another unit vector [in addition to a and b already referenced]". That's when Bell sets up the relationship between 3 observables. Those 3 can't simultaneously exhibit the Quantum Mechanical expectation value.

While EPR is talking about reality Bell's argument talks about theories. It says: "This [non-local structure] is charateristic, according to the results to be proved here, of any such theory which reproduces exactly quantum mechanical predictions."
And predictions of a theory we can recalculate as many times as we want. So as long as we can take description of initial state the same for different test parameters (description of initial state does not depend on later test parameters i.e. theory is not superdeterministic) we can redo the calculation for different test parameters and have all these results at our disposal at the same time. And this is exactly what is expected of that hypothetical theory when it says: "It follows that c is another unit vector" i.e. we get prediction for different test parameter but with the same $\lambda$.

 

[stevendaryl]

A hidden variables theory, should one be developed, will accept all experimental results and will be just as good, if not better, than existing quantum theories at predicting those results.

Right. That's the type of theory that Bell was interested in---one that made exactly the same predictions as QM (at least for experiments where QM proved to be correct). That's the type of theory that his proof is about.

 

[zonde]

Your analogies might be OK for the situations you describe but I don't think they're necessarily relevant to the point I'm trying to make.

So you agree that sometimes we can say something can't be true even when we don't know all the details about the situation, right? Say if I claim that I have made many small purchases and the money spent together is more than I had initially, would you say I got it wrong somewhere even without asking what exactly where those small purchases and how much I spent on every purchase?

Then what type of argument would convince you that particular assumptions give us enough information to conclude that any such a theory is impossible? Well, we have to make some inferences from assumptions that we make. How many steps would be acceptable for you? Say we can examine every step for some length so that you can be certain there are no holes in that particular step.

 

[Dadface]

Right. That's the type of theory that Bell was interested in---one that made exactly the same predictions as QM (at least for experiments where QM proved to be correct). That's the type of theory that his proof is about.

Sorry I don't understand this. You say that Bell was interested in a theory that made the same predictions as QM but the theory that was tested had observations that did not make the same predictions as QM. It cannot be described as a theory. I think what Bell and the experimenters did was to show that a theory, as they interpret(ed) it, was a failure because it predicted results that were not borne out by experiment. As I said before, to be classified as a theory a hidden variables theory must, among other things, conform to the observations. Bell and others did not analyse and test a hidden variable theory .They tested what they thought might be a hidden variable theory and then went on to prove it wasn't a theory at all because it did not meet the necessary criteria. None of that means to say that a proper hidden variable theory will not appear in the future.

In a rush and have to go now.

 

[DrChinese]

While EPR is talking about reality Bell's argument talks about theories. It says: "This [non-local structure] is charateristic, according to the results to be proved here, of any such theory which reproduces exactly quantum mechanical predictions."

Bell used EPR's elements of reality as the basis for his paper. No, he did not label it as such except by the title of the paper.

And the statement you quote simple is a restatement of the idea that the only hidden variable theories that are viable are ones in which the setting of Alice influences the outcome for Bob, however remote. Clearly, the realism assumption can be dropped and then that is not an issue. With the current evidence, I am not sure how it makes sense to say the non-commuting observables have definite values at all times. Which was essentially the assertion of EPR (that a more "compete" theory was possible).

 

[zonde]

Bell used EPR's elements of reality as the basis for his paper. No, he did not label it as such except by the title of the paper.

And the statement you quote simple is a restatement of the idea that the only hidden variable theories that are viable are ones in which the setting of Alice influences the outcome for Bob, however remote. Clearly, the realism assumption can be dropped and then that is not an issue. With the current evidence, I am not sure how it makes sense to say the non-commuting observables have definite values at all times. Which was essentially the assertion of EPR (that a more "compete" theory was possible).

Well, yes Bell is using EPR argument to conclude there can be more complete theory if we assume locality.
"Since we can predict in advance [assuming locality] the result of measuring any chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state."
Your objection as I understand is that this "any" is applied to the same initial configuration in the quoted text from Bell paper.
So if we drop "any" from above have we got rid of that EPR assumption? Is this modified inference correct without relaying on that EPR assumption:
"Since we can predict in advance [assuming locality] the result of measuring chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of that measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state for that component of σ2."

 

[zonde]

Sorry I don't understand this. You say that Bell was interested in a theory that made the same predictions as QM but the theory that was tested had observations that did not make the same predictions as QM. It cannot be described as a theory.

Bell considered two predictions of QM:
#1 perfect correlations when measurement angles are the same
#2 imperfect correlations when measurement angles are not the same
He considered all the theories of certain type that satisfied QM prediction #1 and then demonstrated that there is no way how these theories could make QM prediction #2.

 

[PeterDonis]

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[PeterDonis]

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