# Quantum entanglement information

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It's on the other hand clear that Bell's theorem does not imply spooky action at a distance
My point is, from what one reads every day in popular science articles and what one sees being asked on forums, it is not at all clear to the majority that Bell's theorem does not imply spooky action at a distance, quite the contrary, and this confusion was started by Bell.

There are several aspects (no pun intended) of Bell's theorem that the majority fail to understand. Firstly, that a "hidden variable theory" is not merely the supposition that the measurements are determined by hidden unknown properties of the particles, it also includes a requirement that the experimental averages and correlations be recoverable from the set of hidden variable values by means of standard Kolmogorov probability. Unfortunately many physicists think you can just integrate or average over any sort of mathematical structure - you can't. Even in simplified versions of Bell's theorem where counting arguments are used, there is an erroneous assumption that averaging counts over any sort of structure will result in the same averages and correlations obtained in experiment - it doesn't. (For example, arrange the natural numbers as follows 1,3,2,5,7,4,9,11,6,13,15,8.... whoa!! picking an odd number from the set of natural numbers is "obviously" twice as likely as picking an even ... except it isn't.)

Another point is that Bell's theorem supposes the confusingly named counterfactual definiteness, which does not simply mean that an experiment not performed can be assigned a definite outcome, but that such outcomes can be treated equally with actual experimental outcomes when averaging or calculating correlations, something which relates back to the point about the ability to recover experimental averages from hidden parameters - it doesn't work with counterfactual values of incompatible observables.

vanhees71
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(For example, arrange the natural numbers as follows 1,3,2,5,7,4,9,11,6,13,15,8.... whoa!! picking an odd number from the set of natural numbers is "obviously" twice as likely as picking an even ... except it isn't.)
Can you explain to a simple theoretical physicist as me how you come to this strange conclusion by just reordering the natural numbers? I can also reorder them such as to least all odd numbers before all even numbers. Is then drawing an odd number impossible and if so why?

Of course, probabilities depend on the situation for which they are calculated, and whether you measure the calculated probabilities in random experiments on equally prepared ensembles is of course a question whether your experimental setup is correct or not, which itself has to be verified by carefull observations either, but what has this to do with Bell's work?

My conclusion is that it is NOT true that it is twice as likely to pick an odd number than an even .... is this what you are asking me to explain????

vanhees71
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My point is that you clearly have to describe, how you draw numbers to make a hypothesis about the probabilities to draw an even or odd number and what it may have to do with their ordering. It is clear that associations of probabilities to concretely given situations are not governed by the axiomatic setup of probability theory (like the standard Kolmogorov axioms) but you need other input to do so.

QT is the physicists' way to evaluate probabilities for the outcome of experiments from given preparation procedures (aka quantum states). It's obviously not part of Kolomogov's axioms.

The point I was getting at with the natural numbers, take the arrangement I gave and calculate the average ratio of even to odd numbers for a large number of terms. It gives 1:2. Take the standard arrangement and you get 1:1. So what's the probability of getting an even number when randomly selecting a natural number?

vanhees71
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I still don't understand, how you come to different ratios. The relative number of even and odd numbers are the same and it's the same as the number of natural numbers, because there's a one-to-one mapping between all these sets, and how you order the numbers doesn't make any difference, but I think it's anyway not to the point concerning Bell's work on the foundations of QT.

For the arrangement I gave, for each n, form the ratio of evens to odds in the first n terms of the arrangement.

It is the same as having a dice with 3 faces, two green, one red, the probability of getting green is 2/3.

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Simon Phoenix
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it is not at all clear to the majority that Bell's theorem does not imply spooky action at a distance, quite the contrary, and this confusion was started by Bell.
I'm intrigued by your comments. For me, Bell has always been impeccably clear. In fact in these days of mostly terse, impenetrable, technical papers, reading one of Bell's works is rather refreshing

I don't see how Bell can be at all blamed for any confusion - quite the opposite - it is my view that with a stroke he cut through the befuddlement of earlier thinking on this subject. It is my view that he clarified things enormously by showing that some rather reasonable (classical) assumptions about correlated variables leads to those correlations being constrained. I think he pinned things down beautifully and precisely but I guess others might not see it like that.

Furthermore, as Vanhees has stressed he elevated a concept that was previously the realm of metaphysical mumblings and obfuscation into something that could be tested in the lab. Bell's work, by any reasonable standard, was a tour de force. It allowed a profound question about the nature of reality to be answered experimentally. Bell's theorem is crystal clear - it states in words that a local, hidden variable theory cannot reproduce all of the results of QM - and in such a way that we can test it. Maybe it's just me - but that's an absolutely stunning result.

How anyone can construe that this implies 'spooky action at a distance' is also quite beyond me, but there's no accounting for what goes on in other folks' heads

Another way of stating Bell's result would be to say that IF there are classical-like variables underlying nature, and discounting super-determinism, THEN these must have non-local dependencies (spooky action at a distance) in order to reproduce all of the results of QM - which is where I would guess the confusion comes from. But it's not right to blame Bell for this confusion - especially not when he wrote with such depth and clarity. He can't be blamed, I feel, for the subsequent confusion of others who misunderstood him.

In my view Bell's work on entanglement is one of the most profound and wonderful pieces of theoretical physics of the 20th century. Sure, according to my understanding, Boole showed, about a century before Bell, that for binary random variables $A,B$ and $C$ the joint marginals $P(A,B), P(A,C)$ and $P(B,C)$ could only be constructed from $P(A,B,C)$ in the usual fashion provided the pairwise joint distributions satisfied something we call the 'Bell inequality'. But how does Bell's rediscovery of this prior result lessen his achievement in any way? Even if Bell had been aware of Boole's earlier work I would still judge that his application of it was a tour de force as far as physics is concerned - but then as Hilbert once remarked "physics is much too difficult to be left to the physicists"

Bell quite explicitly pushed the idea that non-locality is required to explain violation of his inequalities, while at the same time being unaware that there could be anything wrong with his hand wavy integration over lambda. I have been reading his papers and to me its mind blowing how non-rigorous and flawed his arguments are regarding probability.

vanhees71
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No, Bell proved that if you want to reproduce the probabilistic results of QT with a deterministic hidden-variable model it must necessarily be nonlocal. Relativistic QFT is local but not deterministic!

Read Bell, he makes statements like "quantum correlations are not locally explicable". He was not aware that a local theory could be anything other than a hidden variable model where one can integrate willy nilly.

Simon Phoenix
Gold Member
Bell quite explicitly pushed the idea that non-locality is required to explain violation of his inequalities
Not really - Bell said that if we wish to describe nature using variables that have certain seemingly a priori reasonable properties then in a certain experimentally achievable set up these rather general properties imply that the measureable correlation functions are bounded (that is, Bell's inequality, or its more complete version the CHSH inequality)..

QM predicts that this bound can be violated. Therefore QM cannot be described, or replaced by a theory based on these variables possessing those seemingly reasonable properties.

It then becomes a matter of experiment whether the QM prediction holds - if it does, if the inequality is violated, then we have unequivocal proof that nature cannot be described by ANY theory built on variables possessing the reasonable properties our intuition might have demanded.

Colloquially, those reasonable properties are 'realism' and 'locality' - but they have more precisely defined meanings.

So we can sacrifice 'realism' or 'locality' (or both perhaps) and construct a theory that agrees with the QM predictions. Bell did like the 'realism' of the Bohm approach to QM, so in this sense he 'pushed' this interpretation being well aware that (a) it is only one interpretation of QM and (b) it is just a smidgeon on the non-local side of things

But he was also well aware that non-locality is ONLY required if we wish to retain a description of nature using 'realistic' hidden variables. QM tootles along quite merrily without hidden variables - they're not in the slightest bit necessary.

while at the same time being unaware that there could be anything wrong with his hand wavy integration over lambda. I have been reading his papers and to me its mind blowing how non-rigorous and flawed his arguments are regarding probability.
I'm quite prepared to accept that to a mathematician Bell's arguments may well look hopelessly non-rigorous and 'hand-wavy'. Thing is, though, they lead to the correct inequality for the correlation functions - which is rather the essential point, don't you think?

Thing is, though, they lead to the correct inequality for the correlation functions - which is rather the essential point, don't you think?
No they don't!!!!!!!!!!!!!! That's the whole point of it. QM leads to the correct inequality, his non-rigorous hand wavy stuff does not. Yet he uses the fact that it does not to claim that non-locality is necessary.

Simon Phoenix
Gold Member
QM leads to the correct inequality
I think you may be missing the point. You're probably best looking at the various derivations of the CHSH inequality (the original Bell inequality is a special case of this). I'm sure you'll find one that satisfies your requirement for rigour. The key thing is that these are all classical in the sense that absolutely no notion from QM is required in the derivation - the CHSH inequality is a bound on classical correlation functions. If we hadn't discovered QM yet we could still derive the CHSH inequality.

But I'm guessing you're one of those few brave souls who believe the CHSH inequality is mathematically incorrect anyway. Good luck with that

Proofs of CHSH use both counterfactual definiteness and locality. As with Bell's theorem the implication is thus again that either counterfactual definiteness is wrong, or locality is wrong or both, so what point are you trying to make?

And let me add that as with Bell, you will find people claiming that it proves locality must be wrong. You see this with Eberhard for example where despite making it explicit that he is appealing to counterfactual definiteness, he does not even consider the possibility that it might be wrong and considers CHSH to be a proof of non-locality.

Simon Phoenix
Gold Member
so what point are you trying to make?
I'm not the one claiming that the Bell/CHSH inequality is wrong

Counterfactual definiteness is a cornerstone of classical physics - in a nutshell it is the claim that objects have properties independent of measurement. Locality is another very reasonable assumption that says that things done 'there' should not affect results 'here' - we can add in a limit on any possible interaction between 'here' and 'there' by appeal to relativity, but the essential principle as far as the experiments are concerned is that the result I obtain in my lab should not depend on the setting chosen in your lab.

These are both eminently reasonable classical assumptions - and from these one can derive the CHSH inequality (with the further assumption that it is possible to make independent random choices for settings). No QM required.

You're the one claiming that there's something mathematically wrong with the derivation of the classical bound on the correlation functions - but so far you haven't explained where the alleged error is. So the point I'm making, if any, is really to try to tease out from you what on earth you're talking about.

I'm not the one claiming that the Bell/CHSH inequality is wrong
Experimental evidence is what claims that they are wrong. The experimental data does not obey Bell's inequality. The experimental data does not obey CHSH. Moreover the experimental data is not the suspect here, it is the assumptions in the proofs of BI and CHSH. One possibility is that the assumption of locality is wrong and the point I am making is that this is what gets pushed as being the only possible thing that is wrong by popular science articles and this is a problem going back to papers by Bell, Eberhard etc who do not consider or are not even aware of the other possibilities.

Counterfactual definiteness is not equivalent to simply claiming particles have properties independent of measurement, there is more to it, but the point of particular importance is the idea that combining counterfactual results with factual results gives the same statistics. It is a mathematical fact that when the counterfactual results are possible alternate results that were not obtained due to an incompatible experiment being performed instead, then when more than 2 variables are involved, the statistics need not be the same as a scenario where the counterfactual results were not obtained simply because no one bothered.

Similarly a "hidden variable theory" is not merely a model where there are hidden parameters present that uniquely determine outcomes, the crucial point is that it is one where one can perform the integrations or averaging required in the proofs of BI and CHSH.

Nugatory
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and the point I am making is that this is what gets pushed as being the only possible thing that is wrong by popular science articles
Yes, but, fixing the world so that popular science articles do not misrepresent QM is beyond the charter of Physics Forums. We do not have the power to accomplish such miraculous results, and if we did we would first take on easier problems like world hunger, global peace, a cure for cancer, and the like.

This thread has moved far beyond its original scope, and it's time to close it.