I Is quantum weirdness really weird?

[vanhees71]

I've seen lots of posts on these forums from (I assume) interested non-professionals trying to get a grasp of what QM means. Ultimately we can't satisfy them; the only answer we give essentially boils down to "learn the maths and the formalism". I wish we could do better :-)

Maybe all I'm saying is that the formalism isn't 'enough' for me. I get the feeling that I'm in the minority here

Well, the danger of investigating nature in realms what are not immediately available to our everyday experience is that you have to change your view of the world and your intuition about it. Indeed, with QT we "zoom" into the smallest structures of matter our senses are unable to register without the help of technology. It's not surprising that this reveals a totally different structure than what we are used to in the macroscopic world that is directly senseable without technical aids. What you describe concerning your experience is precisely what the natural sciences are after: You observe nature and try to find a mathematical description of it, and this leads to intuition. The amazing thing is, how far this concept to comprehend the "inner workings" of nature leads in a pretty successful way. We are able to describe structures down to some femto meters with one pretty clear concept called quantum theory, and there's no hint that this theory is invalid at any point. The ability to formulate such a comprehensive mathematical theory and to apply the abstract findings within this theory to the real world, use it to construct all kinds of further technology and invent new experiments to investigate it even further shows that we've built already a pretty good understanding, if not intuition, for what goes on at a 15 orders of magnitude smaller scale than what is graspable by our bare senses!

That's amazing enough, and there's no need to invoke additional weirdness to make it interesting to the lay man. Of course, you cannot explain Hilbert space formalism and group-representation theory to everybody, but you can, in a qualitative way explain the results of this abstract thinking and the resulting practical experiments, observation, and finally technology forming more and more our daily life. E.g., the laptop I'm using right now to type this posting is just a very real thing originating from such abstract theories as electrodynamics (Maxwell 1865) and quantum theory applies to semiconductor materials (Born, Jordan, Heisenberg; Schrödinger; Dirac 1925/26)! The very possibility to do this, would have been very "weird" even to a 19th century engineer, but nowadays any kid can use it. I think that's the way one should think about the results of fundamental research in the natural sciences and not present it as some weird magic!

 

[zonde]

Bell's inequality is the result of Bell's theorem. Experiments show Bell's inequality is not valid. Thus there must be some part of the hypothesis of Bell's theorem that is not valid. What do you think it is?

It's locality assumption that does not hold.

 

[zonde]

That only works in models whose predictions are computed by functions that are defined on some state space. This is exactly not the case in QM.

Assume we have a model whose predictions are computed by functions $O_\xi :\Lambda\rightarrow\mathbb R$. Then we can add these functions and multiply them as follows: $(O_\xi + O_\zeta)(\lambda) := O_\xi(\lambda) + O_\zeta(\lambda)$ and $(O_\xi O_\zeta)(\lambda) := O_\xi(\lambda) O_\zeta(\lambda)$
Given some element $\lambda\in\Lambda$, we can define define the evaluation map $v_\lambda$ that takes a function $O_\xi$ and evaluates it at $\lambda$: $v_\lambda(O_\xi) := O_\xi(\lambda)$
It is now easy to prove that $v_\lambda(O_\xi + O_\zeta) = v_\lambda(O_\xi) + v_\lambda(O_\zeta)$ and $v_\lambda(O_\xi O_\zeta) = v_\lambda(O_\xi) v_\lambda(O_\zeta)$. We take these identities as the definining identities for an evaluation map.

In quantum mechanics, observables aren't functions $O_\xi : \Lambda\rightarrow\mathbb R$, but rather operators $\hat O_\xi$ that are defined on a Hilbert space. We can now ask ourselves whether this is just an artifact of the formulation. It turns out that it is impossible to reformulate the theory in the previous language. If it were possible to map the operators $\hat O_\xi$ to ordinary functions $O_\xi$ on some state space $\Lambda$, then there would be evaluation maps $v$ such that at least for commuting $\hat O_\xi$, the defining identities of such evaluation maps would be satisfied, i.e. for commuting $\hat O_\xi$, $\hat O_\zeta$, we would have $v(\hat O_\xi + \hat O_\zeta) = v(\hat O_\xi) + v(\hat O_\zeta)$ and $v(\hat O_\xi \hat O_\zeta) = v(\hat O_\xi) v(\hat O_\zeta)$. The Kochen-Specker theorem tells us that no such evaluation map $v$ exists. However, if the $\hat O_\xi$ could be mapped to ordinary functions on some state space $\Lambda$, there would be plenty of these evaluation maps: One for every $\lambda\in\Lambda$. Thus, not all quantum mechanical observables $\hat O_\xi$ can be represented as ordinary functions $O_\xi:\Lambda\rightarrow \mathbb R$ on some state space $\Lambda$. Hence, QM violates counterfactual definiteness.

The simplest example of this is the GHZ state. See also http://www.phy.pku.edu.cn/~qiongyihe/content/download/3-2.pdf.

Sorry I don't follow you. How replacing function with operator changes my argument? You take something as an input (be it vector or variable) and get something as output. And your motivation for making this calculation is not an input for the calculation(calculation doesn't care if you want to know "what will happen?" or "what would have happened?").

 

[rubi]

Sorry I don't follow you. How replacing function with operator changes my argument?

I believe I have explained it in a crystal clear way. If you still don't understand it, I'm afraid, I can't help.

You take something as an input (be it vector or variable) and get something as output.

Please be more specific. We are working with quantum mechanics. What do you take as input and how are you going to calculate the predictions? How do you simultaneously calculate the value of spin in $x$ direction and spin in $z$ direction? You will find that the quantum mechanical formalism doesn't allow you to do it. Show your calculation, so we can point out your mistake.

And your motivation for making this calculation is not an input for the calculation(calculation doesn't care if you want to know "what will happen?" or "what would have happened?").

I don't understand this sentence.

 

[Zafa Pi]

It's locality assumption that does not hold.

That is certainly one option. That's what Bohm proposed. However, if one wishes to accept that there is no faster than light communication, i.e. locality holds (which seems to be a majority opinion/interpretation) then one is left with giving up counterfactual definiteness, which for the Bell business is the same as realism, or hidden variables, or determinism. I still have not heard of any experiment that can decide the issue, and I personally vacillate.

 

[MindWalk]

At least some (but possibly not all) of what seems weird in QM is the tendency to think of fundamental particles as though they were tiny billiard balls instead of something very different--as though they were property-carrying objects. Perhaps another way of viewing them would permit quantum entanglement, the double-slit experiment's making dots pile up in what looks for all the world like an interference pattern, and so on.

Please do not ask me what that way is.

 

[Zafa Pi]

It's locality assumption that does not hold.

Oops, I just came across:
https://www.physicsforums.com/threads/recent-noteworthy-physics-papers.127314/page-2#post-1307660
Which may negate my last sentence in #125 and possibly negate your #122.

 

[zonde]

Oops, I just came across:
https://www.physicsforums.com/threads/recent-noteworthy-physics-papers.127314/page-2#post-1307660
Which may negate my last sentence in #125 and possibly negate your #122.

Your link illustrates nicely how some physicist are trying to present QM as weird. Here is arxiv link: https://arxiv.org/abs/0704.2529
And here is criticism of that article: https://arxiv.org/abs/0809.4000

 

[zonde]

I believe I have explained it in a crystal clear way. If you still don't understand it, I'm afraid, I can't help.

Your explanation would make sense if counter-factual definiteness could be understood as simultaneous reality of measurement results with different measurement settings. But this is not the case.

I don't understand this sentence.

I am saying that calculation is exactly the same whether you are calculating "what will happen if I will measure spin along x axis?" or "what would have happened if I would have measured spin along x axis?" as long as starting point (preparation of input state) is described exactly the same way.

 

[secur]

Your link illustrates nicely how some physicist are trying to present QM as weird. Here is arxiv link: https://arxiv.org/abs/0704.2529
And here is criticism of that article: https://arxiv.org/abs/0809.4000

The paper by Tausk is good. He's right, "An experimental test of non-local realism" is flawed in a number of ways. Tausk mentions:

"Below I will present a brief analysis of some possible meanings for hypothesis (1), but let me emphasize that my main point here is that hypothesis (1) is simply not used in the deduction of Leggett’s inequality. It is a bit odd, to say the least, that an article that claims to be doing “an experimental test of non-local realism” is apparently trying to accomplish its goal by verifying the violation of an inequality whose proof does not use the very hypothesis that the authors call “realism”!"

Hypothesis 1 is:

"(1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism)"

Tausk missed the reason that hypothesis is included. It's found in "APPENDIX I: AN EXPLICIT NON-LOCAL HIDDEN-VARIABLE MODEL" of the Groblacher ... Zeilinger paper:

"We construct an explicit non-local model compliant with the introduced assumptions (1)-(3). It perfectly simulates all quantum mechanical predictions for measurements in a plane of the Poincare sphere. In particular, the violation of any CHSH-type inequality can be explained within the model and, in addition, all perfect correlations state can be recovered."

That hypothesis is necessary not for Leggett inequality but for the task in this appendix, to replicate the CHSH experiment results. It's important to note the model does NOT replicate all QM results, it's carefully tailored for just this class of experiments. As they mention at the bottom of the appendix, "If this relation is not satisfied the model does not recover quantum correlations. ... This is the origin of the incompatibility with general quantum predictions."

What they're trying to do is give the general impression that this non-local model seems to match existing experimental results (but it doesn't) and they've disproven that "myth" with the Leggett inequality. Thus devaluing the fact that non-locality is important in Bell-type experiments.

Finally, I'd appreciate it if someone could explain this sentence from appendix 1:

"Therefore, in the next step, one must find the conditions for which both x1 and x2 take values from [0, 1] and x1 and x2 take values from [0, 1]."

Probably just a typo but I can't figure out what they meant to say.

 

[rubi]

Your explanation would make sense if counter-factual definiteness could be understood as simultaneous reality of measurement results with different measurement settings. But this is not the case.

I gave a mathematical definition of counterfactual definiteness in my post #97. My post #120 then proves that it is violated by QM. It's irrelevant what vague words we choose to describe the definition. What matters is the mathematical definition, which appears in Bell's proof and is not satisfied by QM.

I am saying that calculation is exactly the same whether you are calculating "what will happen if I will measure spin along x axis?" or "what would have happened if I would have measured spin along x axis?" as long as starting point (preparation of input state) is described exactly the same way.

In QM, you neither calculate "what will happen" nor "what would have happened". The theory allows only probabilistic predictions. Nevertheless, we can show that these probabilistic predictions must be incompatible with counterfactual definiteness.

Your link illustrates nicely how some physicist are trying to present QM as weird. Here is arxiv link: https://arxiv.org/abs/0704.2529
And here is criticism of that article: https://arxiv.org/abs/0809.4000

Among the list of authors of the first paper are some of the most highly respected living scientists like Anton Zeilinger and Markus Aspelmeyer. The second paper doesn't seem to be accepted for publication and it's author isn't even affiliated with an university.

 

[secur]

Your explanation would make sense if counter-factual definiteness could be understood as simultaneous reality of measurement results with different measurement settings. But this is not the case.

It's not important how we define "counter-factual definiteness", but "simultaneous reality of measurement results with different measurement settings" is, I think, exactly the point. It's necessary to assume this, to prove Bell-type inequalities like CHSH. Of course, they're violated in reality, because this assumption is not true. Therefore, BTW, it appears you've understood rubi's explanation correctly - if we ignore the sentence "But this is not the case".

Among the list of authors of the first paper are some of the most highly respected living scientists like Anton Zeilinger and Markus Aspelmeyer. The second paper doesn't seem to be accepted for publication and it's author isn't even affiliated with an university.

What matters is how good a paper is, not how good its authors are.

 

[zonde]

It's not important how we define "counter-factual definiteness", but "simultaneous reality of measurement results with different measurement settings" is, I think, exactly the point. It's necessary to assume this, to prove Bell-type inequalities like CHSH.

Bell does not assume this. He assumes that there is a model that is local (remote measurements are independent) and it can reproduce QM prediction of perfect anticorrelations. From that it follows that in such a model there exist functions $A_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ and $B_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ such that $A_{EPR}(\alpha)=-B_{EPR}(\beta)$ when $\alpha=\beta$.

So if we say that additional assumption is needed we have to disagree with Bell on that "it follows" part.

 

[rubi]

Bell does not assume this. He assumes that there is a model that is local (remote measurements are independent) and it can reproduce QM prediction of perfect anticorrelations. From that it follows that in such a model there exist functions $A_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ and $B_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ such that $A_{EPR}(\alpha)=-B_{EPR}(\beta)$ when $\alpha=\beta$.

So if we say that additional assumption is needed we have to disagree with Bell on that "it follows" part.

No, that doesn't follow. The locality assumption is $A(\alpha,\beta,\lambda) = A(\alpha,\lambda)$. The EPR argument cannot be used to prove anything, because it is objectively invalid. The GHZ experiment shows that EPR's criterion for elements of reality must be rejected. See http://www.phy.pku.edu.cn/~qiongyihe/content/download/3-2.pdf that I have already posted earlier. The elements of reality that EPR think exist, just do not exist, no matter how intuitive the EPR argument might seem to you.

 

[secur]

Bell does not assume this. He assumes that there is a model that is local (remote measurements are independent) and it can reproduce QM prediction of perfect anticorrelations. From that it follows that in such a model there exist functions $A_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ and $B_{EPR} :\Theta\rightarrow\mathbb \{-1;1\}$ such that $A_{EPR}(\alpha)=-B_{EPR}(\beta)$ when $\alpha=\beta$.

So if we say that additional assumption is needed we have to disagree with Bell on that "it follows" part.

I didn't mean to disagree with Bell! To me, he's one of the good guys. So let me just say that with counterfactual definiteness, more or less as you stated it, we can demonstrate CHSH inequality. We could either assume CD, or else derive it from other assumptions, as (you apparently say) Bell did.

 

[zonde]

No, that doesn't follow.

I don't wan to go in long offtopic discussion, so I will end it here.

 

[rubi]

I don't wan to go in long offtopic discussion, so I will end it here.

You don't get away with spreading unscientific claims like this. What you really don't want to do is to refrain from stubbornly denying established scientific facts. If the EPR argument was sound, then it should be straightforward for you to explain why the GHZ experiment didn't invalidate it. Neither would it be offtopic in a thread like this, nor would it take a long discussion. If your claim were a fact, you would be able to provide a mathematical proof and it would take no longer than one post to convince everyone. Apparently, you can't do that.

 

[zonde]

You don't get away with spreading unscientific claims like this. What you really don't want to do is to refrain from stubbornly denying established scientific facts.

I am certainly ready to discuss this topic in separate thread.

If the EPR argument was sound, then it should be straightforward for you to explain why the GHZ experiment didn't invalidate it.

Because measurements of entangled particles are not independent.

If your claim were a fact, you would be able to provide a mathematical proof and it would take no longer than one post to convince everyone.

Which claim? And I don't see how anything in physics can take one post to convince everyone. Unless you mean a post with couple of references to model, prediction and subsequent experiment falsifying that prediction and model. That should convince everyone that particular model is false.

 

[rubi]

I am certainly ready to discuss this topic in separate thread.

You aren't supposed to discuss it. You are just supposed to present evidence in form of a mathematical proof of your claim. That would fit in just one post and you could already have done it if it were possible. Do you realize that this is a completely hopeless endeavour, since there exist several accepted interpretations of QM that are known to be local?

Because measurements of entangled particles are not independent.

Even if that were the case, it would still be completely irrelevant to the EPR argument, since you can only predict the value of the remote spin with certainty after you performed the measurement and you wouldn't disturb it anymore afterwards. Therefore, if the EPR argument was valid, there would have to be a corresponding element of reality, which the GHZ experiment proves to not exist.

Which claim?

The claim that the EPR argument allows you to prove the existence of functions $A,B:\Lambda\rightarrow\mathbb R$.

And I don't see how anything in physics can take one post to convince everyone. Unless you mean a post with couple of references to model, prediction and subsequent experiment falsifying that prediction and model. That should convince everyone that particular model is false.

Mathematical proofs are usually short and can be presented in just a few paragraphs. If the steps are performed correctly, then the proof automatically convinces everyone who understands the involved mathematics. If the proof is longer than just a few paragraphs, then either it is published in a journal or it would be a personal theory, which would violate the rules of PF. So please just show us how you formulate the EPR argument mathematically and how you deduce the existence of the functions $A$ and $B$ from it.

 

[zonde]

Do you realize that this is a completely hopeless endeavour, since there exist several accepted interpretations of QM that are known to be local?

It seems you are referring to this post of mine, right?

It's locality assumption that does not hold.

If that's the case, then you have probably misunderstood me. The question was "What do you think it is?" and I meant to answer "I think, it's locality assumption that does not hold.". I thought that given the context (question) it will be clear. But I should be more careful.

I am certainly aware that it can't be "proved" that it's locality assumption that does not hold. What I think is possible is to critically examine other options and check if they are acceptable from perspective of scientific method. So if you have something to discuss along these lines I might join in.

I will answer other points in your post later.

 

[vanhees71]

What's proven with the violation of Bell's inequality is that nature cannot be described by a local deterministic model. Quantum theory is an example for a local indeterministic model. By construction relativistic QFT is local (and particularly microcausal!). Nevertheless you can have non-local correlations as described by entangled states which describe systems with parts that can be detected at a far distance and which don't have predetermined properties but very strong (sometimes 100%!) correlations. These correlations are stronger than possible in any local deterministic model, and that has been the great achievement of Bell's work: He provided a physically sensible criterion for what's called in a somewhat unsharp way by philosophers of science (including EPR themselves!) "local realism".

 

[zonde]

Even if that were the case, it would still be completely irrelevant to the EPR argument, since you can only predict the value of the remote spin with certainty after you performed the measurement and you wouldn't disturb it anymore afterwards. Therefore, if the EPR argument was valid, there would have to be a corresponding element of reality, which the GHZ experiment proves to not exist.

And what about pure states? We can predict measurement outcome for pure state when it is eigenvalue of an operator.
And if we model measurement of entangled particle as changing the other particle to pure state I don't see how your argument holds.

The claim that the EPR argument allows you to prove the existence of functions $A,B:\Lambda\rightarrow\mathbb R$.

You will have to refresh my memory. In which post I claimed this?

 

[rubi]

It seems you are referring to this post of mine, right?

No, I was referring to your post #133.

And what about pure states? We can predict measurement outcome for pure state when it is eigenvalue of an operator.
And if we model measurement of entangled particle as changing the other particle to pure state I don't see how your argument holds.

I am arguing that the EPR idea that we can infer the existence of "elements of reality" using the EPR argument must be rejected, since the EPR argument would apply to the GHZ experiment as well, suggesting the existence of certain "elements of reality". However, we can prove that for the GHZ state the assumption of such "elements of reality" is inconsistent with the predictions of QM, which have been checked experimentally. Therefore, the EPR argument cannot be sound. Being able to predict something with certainty (without disturbing the system) does not necessarily imply the existence of some "element of reality", contrary to what the EPR argument would suggest. Of course, given an eigenstate of some operator, we can predict the outcome of certain experiments with certainty, but as I argued, that doesn't mean that we can infer the existence of an "element of reality", because the EPR argument just isn't sound.

We could now analyze why the EPR argument isn't sound and that would probably end up in a long discussion. But given the GHZ experiment, we must at least acknowledge that it isn't sound.

You will have to refresh my memory. In which post I claimed this?

You claimed it in post #133. At least that's how I understand that post.

 

[Zafa Pi]

However, we can prove that for the GHZ state the assumption of such "elements of reality" is inconsistent with the predictions of QM, which have been checked experimentally.

This, of course, is under the assumption of locality.

 

[Zafa Pi]

It's locality assumption that does not hold.

This was your response to my post #104, which asked:
Bell's inequality is the result of Bell's theorem. Experiments show Bell's inequality is not valid. Thus there must be some part of the hypothesis of Bell's theorem that is not valid. What do you think it is?

I agreed this was a valid option, and you also agreed that non-locality can't be proved.
As a theoretical question, suppose we assume locality, then what part of the hypothesis of Bell's theorem do you think is not valid? Do I need to state what I mean by Bell's theorem for you?

 

[zonde]

I am arguing that the EPR idea that we can infer the existence of "elements of reality" using the EPR argument must be rejected, since the EPR argument would apply to the GHZ experiment as well, suggesting the existence of certain "elements of reality". However, we can prove that for the GHZ state the assumption of such "elements of reality" is inconsistent with the predictions of QM, which have been checked experimentally. Therefore, the EPR argument cannot be sound. Being able to predict something with certainty (without disturbing the system) does not necessarily imply the existence of some "element of reality", contrary to what the EPR argument would suggest. Of course, given an eigenstate of some operator, we can predict the outcome of certain experiments with certainty, but as I argued, that doesn't mean that we can infer the existence of an "element of reality", because the EPR argument just isn't sound.

Look, it does not work without locality assumption. That "without in any way disturbing a system" is important part in EPR definition, and it really means without any disturbance. You can't ignore that.
And while "pure" measurements of pure states do not prove "elements of reality" they are consistent with them. So any part of QM that would use only "pure"measurements of pure states can't possibly falsify "elements of reality".
And then of course you can consider Bohmian mechanics as a hint that there is something wrong with your argument.

 

[secur]

Here's a simple proof that QM cannot be "counterfactual definite (CD)". Meaning that one cannot, in general, assign values for different experimental settings run on the same system at the same time, without contradiction. (Note, this is always possible classically, in theory if not in practice). It has nothing to do with locality.

Suppose Alice and Bob can each set their detectors to 0 or 30 degrees. Let the entangled electrons be in positive-parity "twin state", |11> + |00>, where 1 means spin up and 0, spin down. Suppose Bob's setting is opposite to Alice, so when they both set to 30, the total is 60 degrees.

They're able to run four distinct experiments, with settings (0,0), (0,30), (30,0), (30,30). Assuming CD we can assign values for all, even though we can only do one of them. That is, if we could run each experiment individually at the same time, we would get numbers from these assigned values. Suppose one experimental run generates values for N distinct electron pairs' detections. For convenience, suppose N is as large as we like, so statistical variance is negligible. There will be four sequences of length N: A0, A30, B0, and B30. Then QM says the following must happen.

First, A0 = B0. That is, for each of the N pairs in the two sequences, the values must match. To emphasize this fact let's call the single sequence S0.

S0 and A30 must have 1/4 mismatches, let's call that function M. I.e.,

M(S0, A30) = 1 - {sum from i=1 to n of abs(S0(i) + A30(i) - 1) / N} = sin^2(pi/6) = 1/4.

And the same is true for M(S0, B30).

Finally, we know from QM that M(A30, B30) must be sin^2(pi/3) = 3/4.

Obviously this is impossible. M(S0, A30) + M(S0, B30) = 1/2, setting an upper bound on M(A30, B30), less than 3/4.

QED.

This basically comes from Nick Herbert's 1985 book "Quantum Reality".

 

[Zafa Pi]

Here's a simple proof that QM cannot be "counterfactual definite (CD)". Meaning that one cannot, in general, assign values for different experimental settings run on the same system at the same time, without contradiction. (Note, this is always possible classically, in theory if not in practice). It has nothing to do with locality.

I find this very unfortunate, because it feeds into zonde's objections. Your statement is false, it has everything to do with locality. If we have non-locality then Alice and Bob can conspire to make the measurements come out any way they want. S0 and A30 don't need to have only 25% mismatches they can have 100% mismatches.

Another way to see this is that Bohmian Mechanics is a consistent interpretation of QM, and it allows for the validity of CD (or determinism). However BM is nonlocal.
If one assumes locality then CD is incompatible with QM. That is all that the Bell results show, i.e. local realism (= CD) is ruled out by QM.

For some reason I can't get zonde to agree to my last sentence.

 

[rubi]

This, of course, is under the assumption of locality.

Look, it does not work without locality assumption.

No, the GHZ experiment doesn't need any locality assumption. There is the so called GHZ state and it is incompatible with the idea of "elements of reality", no matter whether you perform the experiment with spacelike distances or not. You will get the same results in both cases. Why don't you first understand the GHZ argument before making such claims? I have given a link to an easily understandable article earlier.

That "without in any way disturbing a system" is important part in EPR definition, and it really means without any disturbance. You can't ignore that.

I don't ignore that. It is however irrelevant for showing the inconsistency of "elements of reality" with the predictions of QM and experiment. Please read the argument again. We can predict with certainty and without disturbing the system the value of Bob's spin after we have measured Alice's spin, both in the EPRB state and in the GHZ state. The EPR argument would then suggest the existence of an "element of reality". The EPR argument must thus be invalid. It's dead simple.

And while "pure" measurements of pure states do not prove "elements of reality" they are consistent with them.

No. The GHZ experiment shows that they aren't.

And then of course you can consider Bohmian mechanics as a hint that there is something wrong with your argument.

Bohmian mechanics is the perfect hint that my argument is completely correct. There are no element of reality for the spin values of the particles in Bohmian mechanics and Bohmians will agree to this. It is impossible. By the way, it is not "my" argument. It's GHZ's argument and I'm only repeating it. So you are in disagreement with highly respectable scientists.--
At this point, the discussion is becoming cumbersome. It makes no sense to discuss this if you don't make sure to have understood the GHZ experiment first.

Anyway, I have asked you earlier for a mathematical proof of the existence of the functions $A,B:\Lambda\rightarrow\mathbb R$, based on a mathematical formulation of the EPR argument. You have ignored this. I'm not wasting any more time on this unless you can present such a proof. It's pointless to keep discussing this with vague language like you do. You claim a mathematical statement, so you have to prove it.

 

[zonde]

It makes no sense to discuss this if you don't make sure to have understood the GHZ experiment first.

It was some time ago but I have read and analyzed GHZ experiments.

As I remember they compared QM prediction with non-contextual local hidden variable prediction.

Before going anywhere further I would like to add that I don't like EPR definition because it seems to exclude contextual hidden variables which certainly are realistic.
Say measurement of relative phase can theoretically give perfectly predictable result. Maybe that is what you are arguing about?