Third Loophole Against Entanglement Eliminated

[Nugatory]

But his theorem goes even beyond that, as it makes a claim about the nature of physical reality itself.

I don't see it that way.

Bell's theorem makes (and proves, to the extent that we can say that any mathematical theorem is proven) a claim about the predictions of a particular class of physical theories; that class is defined by a particular set of common assumptions.

We can ask whether some candidate theory does or does not make those assumptions; and the experimentalists can tell us whether or not these predictions are born out by experiment, and with what level of confidence.

We don't end up with a claim about physical reality until we've brought all three elements together.

 

[DevilsAvocado]

But his theorem goes even beyond that, as it makes a claim about the nature of physical reality itself. IMHO that goes too far, as it suggests that we can draw conclusions about all possible solutions, even ones that we have not considered or can't imagine.

I think lugita15 & Nugatory did a splendid job showing you where you fall short, and frankly I think you go too far in claiming what Bell said and not. In fact he was very precautious not to mention any specific theory and thus making his claim as general as possible. Check it out yourself:

J.S. Bell's Concept of Local Causality
http://arxiv.org/abs/0707.0401

 

[gespex]

Pardon my ignorance, but how did they know they measured 75% of the entangled photons? Even if they can show they measured 75% of the photons, they can't say for sure they'll measure 75% of the *entangled* photons without assuming fair sampling. And the only way to know which photons got entangled, you have to measure them both and count the joint detection rate, right?
So how did they show they measured 75% of the entangled photons?

 

[DevilsAvocado]

Even if they can show they measured 75% of the photons, they can't say for sure they'll measure 75% of the *entangled* photons without assuming fair sampling. And the only way to know which photons got entangled, you have to measure them both and count the joint detection rate, right?

If you could give an example of any physical experiment that can effectively count 100% and report only 75% efficiency, that would be quite nice...

http://arxiv.org/abs/1212.0533]We[/PLAIN] estimate the number of produced pairs to N = 24.2⋅10⁶ per applied setting, yielding a normalized violation of J/N = –0.00524 (± 0.00008).

 

[lugita15]

In fact he was very precautious not to mention any specific theory and thus making his claim as general as possible. Check it out yourself:

J.S. Bell's Concept of Local Causality
http://arxiv.org/abs/0707.0401

I actually happen to think that Travis Norsen (and I suppose Bell, insofar as Travis is quoting him), is making his claim too general. Travis thinks that Bell's theorem allows you to reject locality, while I think that it's only the combination of locality and counterfactual definiteness that can be rejected using Bell. See my post here, in a thread about his views of Bell's theorem. Here is his argument for why counterfactual definiteness is unnecessary for Bell's theorem.

 

[nightlight]

Pardon my ignorance, but how did they know they measured 75% of the entangled photons? Even if they can show they measured 75% of the photons, they can't say for sure they'll measure 75% of the *entangled* photons without assuming fair sampling. And the only way to know which photons got entangled, you have to measure them both and count the joint detection rate, right?
So how did they show they measured 75% of the entangled photons?

That was single channel experiment (from CH inequalities) i.e. they discard half the photons on each side to begin with, before the additional explicit losses (25%) are compounded on the remaining channel on each side. It's a "little bit" misleading to claim you can fix the detection problem of the 2-channel (CHCS inequalities) experiment by simply dropping half of the data upfront, without making additional assumptions about the ignored half. E.g. in a 2-channel experiment, the half of the events which are deliberately dropped in 1-channel experiment, can result in elimination of some events that 1-channel experiment accepts as valid counts (such as double detection events on one or both sides of apparatus which corresponds to 2 PDC pairs; hence the 1-channel experiment assumes that no such multi-pair events would have occurred had they measured the ignored channel).

See preprint (latest V34) for analysis of this and similar recent experiments.

Generally, it was already shown in a series of papers by Marshall, Santos and coworkers (see ref. [22] in the above above preprint) that PDC experiment cannot produce genuine violations of classicality i.e. there is a classical Stochastic ED (SED) model which replicates PDC counts. Basically, PDC pairs, being generated by Poissonian hence classical source, laser, are Poissonian themselves, hence reproducable by classical/positive & non-singular Glauber-Sudarshan distribution. The mentioned assumption in the 1-channel experiments postulates that there are no multiple PDC pairs contradicts the already known (theoretically and experimentally) statistical properties of the PDC sources. The source of their 'enhancement' (resulting in 'apparently' better rate of pair detection than in 2-channel experiment) are precisely these multi-photon events on the remaining channel.

 

[DevilsAvocado]

while I think that it's only the combination of locality and counterfactual definiteness that can be rejected using Bell. See my post here, in a thread about his views of Bell's theorem.

Okay, I posted the reply in that thread.

 

[harrylin]

harrylin, do you agree that if quantum mechanics is experimentally correct, then physical reality cannot possibly obey both counterfactual definiteness and locality (excluding superdeterminism)? If so, how is that not a statement with philosophical significance?

Different people give different meaning to those concepts, which is perhaps why they are often held to be philosophical concepts. That's what I meant, I disagree with the suggestion that the underlying topic is not philosophical.

I'm not sure what you mean by "all possible solutions." Do you mean "all possible experimental situations" or "all predictions of quantum mechanics"?

Neither. Bell made a claim about all possible hypothetical solutions. I don't think that such a sweeping claim is warranted, as it includes anything that he could not imagine.

 

[DevilsAvocado]

Bell made a claim about all possible hypothetical solutions.

Reference please [because this is just dead wrong].

 

[harrylin]

Reference please [because this is just dead wrong].

Sure - it's always good to re-read those texts. For example "Bertlman's socks and the nature of reality". His theorem is a sweeping claim of "cannot be explained" about a class of possible models, without limiting its application to known models:

"the following argument will not mention particles, nor indeed fields, nor any particular picture of what goes on at the microscopic level. [..] The difficulty is not created by any such picture or terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups.
[..]
"certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

 

[DevilsAvocado]

Bell made a claim about all possible hypothetical solutions.

Sure - it's always good to re-read those texts. For example "Bertlman's socks and the nature of reality". His theorem is a sweeping claim of "cannot be explained" about a class of possible models, without limiting its application to known models:

"the following argument will not mention particles, nor indeed fields, nor any particular picture of what goes on at the microscopic level. [..] The difficulty is not created by any such picture or terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups.
[..]
"certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

[my bolding]

If you don’t see your problem here my friend, I’m afraid I can’t help you.

Healthy scientists normally wish their claims to be as generally valid as possible. Very few publish theorems that start: - This idea only works on Mondays, between 4 & 5 p.m., and only if you look the other way. Otherwise it’s perfectly rock-solid!

The only thing you’ve got is the last sentence “cannot be explained, that is to say, without action at a distance”. It can be explained by non-separability, but this doesn’t change the mathematical theorem one single bit.

Words are words and mathematics is mathematics – the old Local Realism has gone fishing in the Norwegian fjords together with the Blue Parrot. Period.

 

[lugita15]

Different people give different meaning to those concepts, which is perhaps why they are often held to be philosophical concepts. That's what I meant, I disagree with the suggestion that the underlying topic is not philosophical.

Well, let me tell you exactly what I mean by these concepts:
1. Locality: An event can only influence things in its future light cone.
2. Counterfactual definiteness: If you make a given measurement, it is always meaningful to ask "What result would you have gotten if you had made this measurement instead?", and there exist a definite (although possibly unknowable) answer to this question.
3. No-Conspiracy Condition: The answer to the question "What result would you get if you make this measurement" is independent of what measurement you actually choose to make.

According to Bell's theorem, if Quantum Mechanics is always right in its experimental predictions, then physical reality cannot possibly obey all 3 conditions. Do you agree or disagree with this statement? And do you agree that it makes a firm claim about physical reality, and not just empirical observations?

Neither. Bell made a claim about all possible hypothetical solutions. I don't think that such a sweeping claim is warranted, as it includes anything that he could not imagine.

By "hypothetical solutions" do you mean hypothetical explanations of observed quantum mechanical phenomena? In that case, yes, Bell did indeed make a claim about all possible explanations, including ones that he presumably could not conceive of; see my statement above.

But there is nothing wrong with such a sweeping claim. Are you familiar with Cantor's proof that there are more real numbers than natural numbers? It starts with the assumption, suppose you had some method, however complicated or unimaginable, to make a one-to-one correspondence between the real numbers and the natural numbers. Then Cantor showed that there would exist a real number which did not map to any natural number, so that method would be unable to make to make such a 1-to-1 correspondence. How was Cantor able to reason about really clever potential methods of counting the real numbers, methods that he never even thought of or imagined? That's the power of proof by contradiction.

Bell's proof works in the same way. It says, assume that reality obeys certain properties. Then you can show that reality must also obey this other property, and this other property implies that quantum mechanics is not always experimentally correct.

 

[harrylin]

[my bolding]

If you don’t see your problem here my friend, I’m afraid I can’t help you. [..]

I agree with your bolding... so, the same to you my friend.

 

[harrylin]

The discussion here was increasingly becoming like the thread on scholarpedia, and I think that it's better to stop drifting away here; I may rejoin the discussion there, which is indeed more general. However, still a few clarifications:

Well, let me tell you exactly what I mean by these concepts:
1. Locality: An event can only influence things in its future light cone.
2. Counterfactual definiteness: If you make a given measurement, it is always meaningful to ask "What result would you have gotten if you had made this measurement instead?", and there exist a definite (although possibly unknowable) answer to this question.
3. No-Conspiracy Condition: The answer to the question "What result would you get if you make this measurement" is independent of what measurement you actually choose to make.

According to Bell's theorem, if Quantum Mechanics is always right in its experimental predictions, then physical reality cannot possibly obey all 3 conditions. Do you agree or disagree with this statement?

Localty implies even more than you indicate, but that's not important IMHO. However the requirement called counterfactual definiteness appears to be stronger than how you present it here; this was discussed in an earlier thread and it appears that devilsavocado now brings it up again in the thread on scholarpedia ( https://www.physicsforums.com/showthread.php?t=592086&page=28 ). I have problems with understanding all the implications of that requirement, and thus I don't know what to conclude form Bell's theorem.

And do you agree that it makes a firm claim about physical reality, and not just empirical observations?

That was the issue that I brought up!

By "hypothetical solutions" do you mean hypothetical explanations of observed quantum mechanical phenomena? In that case, yes, Bell did indeed make a claim about all possible explanations, including ones that he presumably could not conceive of; see my statement above.

OK, at least we agree on that (contrary to Devilsavocado).

But there is nothing wrong with such a sweeping claim. [..] Cantor showed that there would exist a real number which did not map to any natural number, so that method would be unable [..] to make such a 1-to-1 correspondence. How was Cantor able to reason about really clever potential methods of counting the real numbers, methods that he never even thought of or imagined? That's the power of proof by contradiction.

Bell's proof works in the same way. It says, assume that reality obeys certain properties. Then you can show that reality must also obey this other property, and this other property implies that quantum mechanics is not always experimentally correct.

There is a big difference: Cantor did not try to prove a negative, while Bell did. Bell's proof appears indeed to include certain assumptions about models of reality, despite the claim that his argument did not "mention particles, nor indeed fields, nor any particular picture of what goes on at the microscopic level". But in practice, no physical assumption about reality can be made without any models of reality.

 

[lugita15]

Localty implies even more than you indicate, but that's not important IMHO.

What is the more general conception of locality you have in mind? Regardless, it's possible to give a proof of Bell's theorem with this meager definition of locality: events can only influence events within their future light cone.

However the requirement called counterfactual definiteness appears to be stronger than how you present it here; this was discussed in an earlier thread and it appears that devilsavocado now brings it up again in the thread on scholarpedia ( https://www.physicsforums.com/showthread.php?t=592086&page=28 ).

I have problems with understanding all the implications of that requirement, and thus I don't know what to conclude form Bell's theorem.

Yes, I just replied to DevilAvocado. To sum up, in principle the term "counterfactual definiteness" COULD refer to something more general, but for the purposes of Bell's theorem all we need is the meaningfullness of asking what a measurement that you didn't make would yield if you had made it.

OK, at least we agree on that (contrary to Devilsavocado).

I think DevilsAvocado may have been having a semantic disagreement with you. He may have been saying that you were incorrect to call it a "claim" as opposed to a proven theorem.

There is a big difference: Cantor did not try to prove a negative, while Bell did.

Cantor did try to prove a negative. He said that no attempt to make a one-to-one correspondence between the natural numbers and the real numbers can possibly work. Similarly, Bell said that no attempt to make a local realistic (non-superdeterministic) explanation of the experimental predictions of quantum mechanics can possibly work.

Bell's proof appears indeed to include certain assumptions about models of reality, despite the claim that his argument did not "mention particles, nor indeed fields, nor any particular picture of what goes on at the microscopic level". But in practice, no physical assumption about reality can be made without any models of reality.

Can you elaborate on why you think this?

Anyway, let me just ask you point blank: do you agree that if Quantum Mechanics is always right in its experimental predictions, then physical reality cannot possibly obey all 3 conditions I specified in post 47, using the definitions I provided for them? If you disagree, I can try to write you a proof.

 

[billschnieder]

According to Bell's theorem, if Quantum Mechanics is always right in its experimental predictions, then physical reality cannot possibly obey all 3 conditions.

Please could you state explicitly what those experimental predictions of QM are that you are referring to. Thanks.

 

[lugita15]

Please could you state explicitly what those experimental predictions of QM are that you are referring to. Thanks.

Here are the predictions I'm referring to:
When both polarizers are set to the same angle, they always yield the same result.
When the polarizers are set 30 degrees apart, the probability that they yield different results is 1/4.
When the polarizers are set 60 degrees apart, the probability that they yield different results is 3/4.

 

[billschnieder]

Just so that there is no confusion, I would appreciate if you could state the predictions in a manner that is comparable to the inequalities.

|C(a,b)−C(a,c)|≤1+C(b,c).

What does QM predict for

C(a,b) = ?
C(b,c) = ?
and C(a,c) = ?Pick one set of parameters (your favorite) a, b, c and give us the predictions from QM for the above three terms.

 

[lugita15]

Just so that there is no confusion, I would appreciate if you could state the predictions in a manner that is comparable to the inequalities.

|C(a,b)−C(a,c)|≤1+C(b,c).

What does QM predict for

C(a,b) = ?
C(b,c) = ?
and C(a,c) = ?


Pick one set of parameters (your favorite) a, b, c and give us the predictions from QM for the above three terms.

billschneider, I'd prefer to use my own notation if you don't mind, since I'm relying on Nick Herbert's writeup rather than Bell's original proof. If p is the probability that polarizers oriented at -30 and 30 will differ, q is the probability that polarizers oriented at -30 and 0 will differ, and r is the probability that polarizers oriented at 0 and 30 will differ, then Bell's inequality states that p is less than or equal to q + r. Quantum mechanics predicts that p=3/4 and q=r=1/4.

I claim that if QM is right that the polarizers will always give the same results when they are oriented at the same angle, then conditions 1, 2, 3 in post 47 imply the Bell inequality stated above.

 

[billschnieder]

billschneider, I'd prefer to use my own notation if you don't mind, since I'm relying on Nick Herbert's writeup rather than Bell's original proof.

Too bad then.

 

[lugita15]

Too bad then.

Why too bad? Didn't I rigorously spell out the logic of Herbert's proof for you in this thread?

 

[billschnieder]

Why too bad? Didn't I rigorously spell out the logic of Herbert's proof for you in this thread?

And didn't I completely debunk that logic in this thread?
https://www.physicsforums.com/showthread.php?p=3970771#post3970771

 

[lugita15]

And didn't I completely debunk that logic in this thread?
https://www.physicsforums.com/showthread.php?p=3970771#post3970771

You claimed that my error was hidden by the use of inequalities, so I switched to equations. And then your criticism seemed to be that I was being selective in my application of no-conspiracy to only certain probabilities. But that's no mystery: the no-conspiracy condition states that if a statement is meaningful in both "scenarios" (to use your terminology), i.e. if it is meaningful whether you restrict yourself to factual statements or whether you also allow counterfactuals, then the probability of the statement being true is the same in both scenarios. If you'd like, I can explain again the justification for this no-conspiracy condition.

 

[StevieTNZ]

Ease up, darlings.

 

[stevendaryl]

Just so that there is no confusion, I would appreciate if you could state the predictions in a manner that is comparable to the inequalities.

|C(a,b)−C(a,c)|≤1+C(b,c).

What does QM predict for

C(a,b) = ?
C(b,c) = ?
and C(a,c) = ?


Pick one set of parameters (your favorite) a, b, c and give us the predictions from QM for the above three terms.

Well, for the spin-1/2 twin pair EPR experiment, the prediction of QM is

C(a,b) = -cos(b-a)

So that inequality becomes:

|-cos(b-a) + cos(c-a)| ≤ 1 - cos(b-c)

For the specific case a=0 degrees, b=30 degrees, c=45 degrees, we have:

|- 0.866 + 0.707| ≤ 1 - 0.966

0.159 ≤ 0.034 FALSE

I don't have the QM predictions for twin photons handy.

 

[billschnieder]

Well, for the spin-1/2 twin pair EPR experiment, the prediction of QM is

C(a,b) = -cos(b-a)

So that inequality becomes:

|-cos(b-a) + cos(c-a)| ≤ 1 - cos(b-c)

But for the twin pair, according to QM, C(a,b) does not commute with C(c,a) nor does it commute with C(b,c). In other words, the three correlations are incompatible. Therefore, although each of the correlations standing alone is a valid QM calculation, combining the three into a single expression gives an invalid QM expression. Unless you introduce the added assumption that commuting observables can be substituted into an inequality based on non-commuting terms without regard for the non-commutatitivity. Another way of see the error is that, you are using three correlations calculated on three different wavefunctions to draw conclusions about three correlations that would have been obtained from a single wavefunction, were it possible to measure them.

This subtle error is the source of the violation, not non-locality or any other spooky business.

That is why I asked the following questions:
1) Do you agree that there are two scenarios involved in this discussion:

Scenario X, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(b,c) = correlation for what we would get if we measure (b,c)
C(a,c) = correlation for what we would get if we measure (a,c)​

Scenario Y, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(a,c) = correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = correlation for what we would have gotten had we measured (b,c) instead of (a,b)​

2) Do you agree that Scenario X is different from Scenario Y?
3) Do you agree that the correlations in Bell's inequalities correspond to Scenario Y NOT Scenario X?
4) Do you agree that correlations calculated from QM correspond to Scenario X not Scenario Y?
5) Do you agree that correlations measured in experiments correspond to Scenario X not Scenario Y?

While the correlations in Scenario X all commute, those in Scenario Y do not all commute.

 

[billschnieder]

You claimed that my error was hidden by the use of inequalities, so I switched to equations. And then your criticism seemed to be that I was being selective in my application of no-conspiracy to only certain probabilities. But that's no mystery: the no-conspiracy condition states that if a statement is meaningful in both "scenarios" (to use your terminology), i.e. if it is meaningful whether you restrict yourself to factual statements or whether you also allow counterfactuals, then the probability of the statement being true is the same in both scenarios. If you'd like, I can explain again the justification for this no-conspiracy condition.

Your no-conspiracy condition is essentially that Scenario X and Scenario Y (from above) are exactly the same, in other words, your no-conspiracy condition is equivalent to saying, the QM result from a single wavefunction must be the same as the QM result from three different wavefunctions.

your argument was:

OK, let me be more explicit in my logic and not use inequalities at all.
1. P(C|w)=P(A|w)+P(B|w)-2P(A & B|w)
2. P(A|x)=.25, P(B|y)=.25, P(C|z)=.75
3. P(A|x)=P(A|w), P(B|y)=P(B|w), P(C|z)=P(C|w) (From no-conspiracy.)
4. P(A|w)=.25, P(B|w)=.25, P(C|w)=.75 (From 2 and 3)
5. .75 = .25 + .25 -2P(A & B|w) (From 1 and 4)
6. P(A & B|w) = -.125 (From 5)

.

And I showed you that step (3) was incomplete, Step (3) What does no-conspiracy say about P(AB|w). According to your logic, no-conspiracy also implies that P(AB|w)=P(AB|x,y). But x and y are two different sets of photons, which means P(AB|x,y) is undefined/meaningless. All you have proven is the triviality that the joint probablity distribution P(ABC|x,y,z) for outcomes from three different sets of photons (x,y,z) is undefined, although the joint probability distribution P(ABC|w) from the single set of photons (w) is well defined. So not unlike what I explained in my previous post, the error is to assume that non-commuting observables can be mixed as if there were commuting observables in the inequality. It is a mathematical error.

 

[DrChinese]

...

That is why I asked the following questions:
1) Do you agree that there are two scenarios involved in this discussion:

Scenario X, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(b,c) = correlation for what we would get if we measure (b,c)
C(a,c) = correlation for what we would get if we measure (a,c)​

Scenario Y, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(a,c) = correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = correlation for what we would have gotten had we measured (b,c) instead of (a,b)​

2) Do you agree that Scenario X is different from Scenario Y?
3) Do you agree that the correlations in Bell's inequalities correspond to Scenario Y NOT Scenario X?
4) Do you agree that correlations calculated from QM correspond to Scenario X not Scenario Y?
5) Do you agree that correlations measured in experiments correspond to Scenario X not Scenario Y?

While the correlations in Scenario X all commute, those in Scenario Y do not all commute.

Bill, you already posted this last line of reasoning in another thread - in fact this is verbatim. That thread is closed for moderation. Please do not continue to post your personal pet arguments that you cannot support with suitable* references. Else you will again be reported. PhysicsForums' Quantum Physics is NOT the place to make arguments that go counter to generally accepted physics. Start your own blog or get one of your papers published. This is an educational** forum first and foremost.

*By PF standards.
** I realize you believe you are educating people by sharing your version of the truth, but that is not how it is defined here.

 

[stevendaryl]

But for the twin pair, according to QM, C(a,b) does not commute with C(c,a) nor does it commute with C(b,c).

C(a,b) is an expectation value, a real number, not an operator. Real numbers always commute.

 

[billschnieder]

C(a,b) is an expectation value, a real number, not an operator. Real numbers always commute.

Huh? it is the result of an operation, the operations do not commute. If the time to put on socks is a real number Ta, and the time it takes to put on shoes is another real number Tb. Won't it be silly to suggest that becase Ta, and Tb are real numbers, then you should say the time to put on Shoes and then put on socks must be Tb + Ta? Don't you realize that because the two operations do not commute you can not simply add the times like that?

 

[stevendaryl]

Huh? it is the result of an operation, the operations do not commute.

Expectation values always commute.

C(a,b) = -cos(b-a)

That is not an operator, it is a real number, for every choice of a and b.

 

[stevendaryl]

Huh? it is the result of an operation, the operations do not commute. If the time to put on socks is a real number Ta, and the time it takes to put on shoes is another real number Tb. Won't it be silly to suggest that becase Ta, and Tb are real numbers, then you should say the time to put on Shoes and then put on socks must be Tb + Ta? Don't you realize that because the two operations do not commute you can not simply add the times like that?

I think you're drifting off into territory where I have nothing to say. You asked me how I would calculate the QM prediction for correlations, and I gave you an answer. I don't think there is anything more for me to say. Correlations are real numbers, and they can be added in any order, and you get the same answers. It's a separate question of what the meaning of an arithmetical expression.

 

[billschnieder]

Correlations are real numbers, and they can be added in any order, and you get the same answers. It's a separate question of what the meaning of an arithmetical expression.

That is where you are mistaken. Do you have a reference for that claim? I just gave you and example with shoes and socks which clearly demonstrates that you are wrong.

On the Problem of Hidden Variables in Quantum Mechanics
John S. Bell. Reviews of Modern Physics, Vol 38, Number 3, (1966)

Page 448:
"A quantum mechanical "system" is supposed to have "observables" represented by Hermitian operators in a complex linear vector space. Every "measurement" of an observable yields one of the eigenvalues of the corresponding operator. Observables with commuting operators can be measured simultaneously."
...
"Any real linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values is the expectation value of the combination".

Page 449
"It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

 

[stevendaryl]

That is where you are mistaken. Do you have a reference for that claim?

The claim that real numbers commute? You want a reference for that?

I just gave you and example with shoes and socks which clearly demonstrates that you are wrong.

No, it doesn't. If T1 = 5 and T2 = 3, then T1 + T2 = 8. There is no issue of whether operators commute.

You've gone off the deep end, here.

 

[DrChinese]

... I just gave you and example with shoes and socks which clearly demonstrates that you are wrong.

Great, Bill. Assuming we have settled some point, none of this has anything to do with the topic of this thread - which is closing another Bell test "loophole".

You don't even seem to object to Bell tests on the grounds of "loopholes" anyway, so I am not sure what you really have to contribute here. Please do not turn this into a discussion of the soundness of Bell's Theorem. If you feel the need to continue, perhaps you could start a thread on the point you have a question about.