Thoughts about Bell, Bohm, Christian, et al.

[AnotherGuest]

The inequality is therefore theory-agnostic and has nothing whatsoever to do with locality.

Here I believe Minkwe is wrong.

(1) The assumption of mathematical existence of outcomes and in the two wings of the experiment for all possible settings and simultaneously, is exactly what most writers call "local realism". It is the combination of "counterfactual definiteness" and "local causality", see http://en.citizendium.org/wiki/Entanglement_(physics). It states that Bob's outcome when he uses setting "b" would have been the same, whatever setting "a" is used by Alice.

These are good old EPR characterisations of realism and locality. Einstein considered that these assumptions do have everything to do with realism and locality. (OK so maybe Einstein was wrong and Bohr was right - the Copenhagen dogma asserts that there is no hidden layer behind quantum mechanics, there are no "event-based" functions "A" and "B" at all.)

(2) Under these assumptions, Bell arrives at an inequality. The inequality would not in general need to be true - it relies on the assumptions of local realism, and of perfect anti-correlation when the settings are equal (and of course, on +/-1 valued outcomes). So the inequality is not theory-agnostic.

The assumptions of perfect anti-correlation when the settings are equal and of +/-1 valued outcomes come from the standard QM analysis of the singlet state; which of course also leads to the singlet correlations.

[Schmelzer]

Do I need to continue? We now see that we get Bell's inequality without any physics whatsoever. All we need is theory-agnostic outcomes and the expectation value of the product of the paired outcomes , or as Bell would say . The inequality is therefore theory-agnostic and has nothing whatsoever to do with locality.

Wow, this was indeed unexpected for me.

But this, of course, explains the heavy emotional reaction against my rejection to accept his attempts to present formulas (1) and (2) as if they contain no nontrivial information, while, in fact, they contain the first half (and the conceptually difficult half) of the whole theorem, namely the EPR argument.

It is, indeed, quite obvious that the equations (1), (2) are far away from being theory-agnostic descriptions of , which is, as I have explained, is already an expression which requires nontrivial context (namely that is not a function defined for all experiments of this type, which would be simply wrong in QM, but a denotation for a fixed single experiment).

This was, initially, not more than a minor quibble from my side, I simply wanted to avoid inaccurate formulations open to misinterpretations. But it hit the point.

And, even if we ignore the point that the very existence of a function for the description of the measurement results already contains nontrivial hypotheses (the existence of whatever hidden variables, which are questioned by those who want to preserve Einstein locality by giving up realism), all one can get without locality, thus, theory-independent, is only

which does not allow to prove Bell's inequality.

[Heinera]

Thanks Joy, I was hoping Heine will flip a couple of pages to equations (13) to (15) of Bell's paper and verify that plays no role in the derivation of the inequality anyone can can follow along from Bells equation (13):



, Since
and after factorizing out , remembering that we get


The second term on the right is .

Here you use the integers from 1 to N as lambda, and 1/N as rho. Fine.

Do I need to continue? We now see that we get Bell's inequality without any physics whatsoever. All we need is theory-agnostic outcomes and the expectation value of the product of the paired outcomes , or as Bell would say . The inequality is therefore theory-agnostic and has nothing whatsoever to do with locality.

No. You use exactly the same locality assumptions as Bell. The fact that you assign values to both and means that you postulate the possibility of counterfactual outcomes. The fact that you in this expression



use the same values in both sums means the theory assumes locality; the value of is independent of the setting in the other station.

[Heinera]

As far as I know mathematics, if is some function defined on a set , then in the expression you can give the variable any other name you like, as long as it is not simultaneously in use for something else. You can give it the name if you like. So you could also write if you want to. Nothing has changed. What's in a name? The important thing is that the function is the same, and the set over which we integrate, , is the same. One could also just write . It stands for the same thing.

Yes, what's in a name? The point is that is no longer an independent variable, but a function of and . The space you integrate over will thus be different for different values of and . And as you correctly stated earlier, this space must be the same for all for Bell's theorem to hold.

[AnotherGuest]

As far as I know mathematics, if is some function defined on a set , then in the expression you can give the variable any other name you like, as long as it is not simultaneously in use for something else. You can give it the name if you like. So you could also write if you want to. Nothing has changed. What's in a name? The important thing is that the function is the same, and the set over which we integrate, , is the same. One could also just write . It stands for the same thing.

Yes, what's in a name? The point is that is no longer an independent variable, but a function of and . The space you integrate over will thus be different for different values of and . And as you correctly stated earlier, this space must be the same for all for Bell's theorem to hold.

If is a supposed to stand for a function instead of a variable, you cannot integrate with respect to it. If it is meant to stand for a variable, then its name is irrelevant. In the first case, the expression is meaningless. In the second case, it means exactly the same as

[minkwe]

Here you use the integers from 1 to N as lambda, and 1/N as rho. Fine.

I absolutely do not. is a summation index representing particles, not lambda.

No. You use exactly the same locality assumptions as Bell. The fact that you assign values to both and means that you postulate the possibility of counterfactual outcomes. The fact that you in this expression

Just because you proclaim it doesn't make it true. There is not locality assumption in it. Please review my previous posts which I have explicity defined as non-local, and yet the derivation will proceed exactly the same, even using Bell's notation, simply replace with

use the same values in both sums means the theory assumes locality; the value of is independent of the setting in the other station.

False. There is no locality in that. There may be counterfactual definiteness, but absolutely no locality assumption.

[minkwe]

(1) The assumption of mathematical existence of outcomes and in the two wings of the experiment for all possible settings and simultaneously, is exactly what most writers call "local realism".

Most writers are wrong here. There is no locality in that assumption. It is counterfactual definiteness which is a different thing completely from locality. As anyone can see by considering and . Until now I haven't said anything about counterfactual definiteness. But I believe I've shown convincingly that there is absolutely no locality assumption required to obtain the inequalities.

(2) Under these assumptions, Bell arrives at an inequality. The inequality would not in general need to be true - it relies on the assumptions of local realism, and of perfect anti-correlation when the settings are equal (and of course, on +/-1 valued outcomes). So the inequality is not theory-agnostic.

You guys keep saying "local realism" which I understand is common verbiage, but the derivation is devoid of locality. Perhaps you think counterfactual definiteness applies only to certain theories. We shall find out next, but there is absolutely no locality in my derivation so far.

[Heinera]

use the same values in both sums means the theory assumes locality; the value of is independent of the setting in the other station.

False. There is no locality in that. There may be counterfactual definiteness, but absolutely no locality assumption.

What I wrote is the very definition of locality.

[AnotherGuest]

(1) The assumption of mathematical existence of outcomes and in the two wings of the experiment for all possible settings and simultaneously, is exactly what most writers call "local realism".

Most writers are wrong here. There is no locality in that assumption. It is counterfactual definiteness which is a different thing completely from locality. As anyone can see by considering and . Until now I haven't said anything about counterfactual definiteness. But I believe I've shown convincingly that there is absolutely no locality assumption required to obtain the inequalities.

(2) Under these assumptions, Bell arrives at an inequality. The inequality would not in general need to be true - it relies on the assumptions of local realism, and of perfect anti-correlation when the settings are equal (and of course, on +/-1 valued outcomes). So the inequality is not theory-agnostic.

You guys keep saying "local realism" which I understand is common verbiage, but the derivation is devoid of locality. Perhaps you think counterfactual definiteness applies only to certain theories. We shall find out next, but there is absolutely no locality in my derivation so far.

Counterfactual definiteness would give us the existence of and ; relativistic causality adds to this: does not depend on , does not depend on . Thus the two together give us existence of and . Which many, many writers call "local realism". But everyone is free to give it another name. It doesn't matter for the ensuing derivation "where" is thought to be located. If you want to think of as being "all around us, everywhere" then you are also including some models that you would call "non-local" as well. So be it.

You are not allowing arbitrary non-local models: you get Bell's inequality, which puts restrictions on the correlation function.

EPR's motivation for (essentially) coming up with and , where the settings are "measure position" and "measure momentum" and in their minds, is coming from the source of the two particles, were the combination of relativistic causality, counterfactual definiteness, and "perfect anti-correlation when the two settings are the same". Exactly what Bell has assumed so far, too (not surprisingly).

Of course we are also assuming that experimenters can choose settings freely ("no conspiracy").

[minkwe]

What I wrote is the very definition of locality.

It absolutely is not, despite your claims. The definition of locality is:
- Locality means the outcome depends only on information inside the light-cone of the station.
- Non-locality means the outcome depends on information outside the light-cone of the station.

I have shown that if with a represents information outside of the light-cone of the station, then the same equations apply as if you use , which is restricted to information only inside the light-cone of the station. In fact, I have shown that you can derive the inequality without invoking any hidden variables at all.

I could have started by saying:

Assume that at given moment in time, 3 outcomes spontaneously appear at at three distant stations where settings "a", "b", and "c" are in effect. Let successive measurements of these outcomes be made Note that no hidden variables are involved, yet, I can derive the inequalities in exactly the same way, without any reference to locality, or even any hidden variables.



The suggestion that locality or even hidden variables (local or non-local) is required is simply indefensible. The only assumption required is the assumption that we are simultaneously considering three numbers from any source whatsoever, Some people call this "counterfactual definiteness", and others call it "realism". But there is absolutely no locality assumption here, whatsoever.

Once we've put this so-called "locality assumption" in its grave, we can proceed to examine the "counterfactual definiteness/realism" assumption.

[AnotherGuest]

Here you use the integers from 1 to N as lambda, and 1/N as rho. Fine.

I absolutely do not. is a summation index representing particles, not lambda.

Just because you proclaim it, doesn't make it true! Of course minkwe intended to be a summation index representing particles. But we can replace the symbol everywhere by and define , and let stand for the (discrete) uniform probability distribution on . Nothing has changed, as far as the mathematics is concerned.

As minkwe himself points out, once we have written down the mathematical assumptions, the mathematical derivation of the inequality does not force any particular interpretation on . If Bell's theorem is more general than people usually think, all the better!

[Heinera]

In the context of Bell's theorem, what is needed for the theorem to hold is counterfactual definiteness (realism), and the property that in the theory, the value is independent of the setting in the other wing, i.e., it always computes to the same value in the theory. This latter property is called locality in the context of Bell's theorem. A theory that use information outside the light-cone of the station would still be called "local" in the Bell sense, as long as this information does not contain the setting in the other wing.

[minkwe]

In the context of Bell's theorem, what is needed for the theorem to hold is counterfactual definiteness (realism), and the property that in the theory, the value is independent of the setting in the other wing, i.e., it always computes to the same value in the theory. This latter property is called locality in the context of Bell's theorem. A theory that use information outside the light-cone of the station would still be called "local" in the Bell sense, as long as this information does not contain the setting in the other wing.

I've just shown you that such a definition of locality is a sham. It is not locality at all, for very clear reasons:

1) By the same definition, if setting "b" is within the light-cone of station A, and the outcome at station A depended on setting "b", then such a manifestly local situation would be characterized as "non-local" according to your definition. A theory that uses information inside the light-cone of the station would still be called "non-local" in the Bell sense, as long as this information contains the setting in the other wing. You see, it is inconsistent.
2) According your definition, if the outcome at station A depends on information outside of its light-cone that is also independent of the setting "b" at the remote station, it would be considered to be "local", despite being manifestly non-local.
3) Such a definition of locality is inconsistent as you can't apply it to single particles where there is only one setting and one station.

As I hope is now obvious, there is no consistent concept of locality required to obtain the inequalities. You are pushing a definition which applies to local setting/outcome dependence but does not apply to non-local setting/outcome independence and trying to claim it as "locality", it simply is not and if Bell believed that, he was woefully mistaken too. No reasonable definition of locality should permit the outcome of measurements at station A to depend on events outside of the light-cone of station A. And no reasonable definition of non-locality should apply to a situation in which all outcomes are determined by events within the light-cone of the station.

All that is required to obtain the inequalities, is the simultaneous consideration of a series of triplets of numbers , without any regard for any physics. Call that "counterfactual definiteness/realism" if you like. But there is absolutely no requirement for any concept of "locality".

[AnotherGuest]

All that is required to obtain the inequalities, is the simultaneous consideration of three numbers , without any regard for any physics. Call that "counterfactual definiteness/realism" if you like. But there is absolutely no requirement for any concept of "locality".

Dear children

Please stop squabbling about words for the time being, and let's look at the logic.

Bell writes "Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another, the orientation of one magnet does not influence the result obtained with the other." It seems that everyone agrees that Bell's physical hypotheses (which are basically EPR's hypotheses) imply (1) and (2). And it seems that everyone agrees that (1) and (2) imply Bell's inequality. And I suppose everyone agrees that Bell's inequality is contradicted by quantum mechanics.

So please, minkwe, tell us your next move. Do you have a problem with quantum mechanics? Or with Bell's physical hypotheses?

[Heinera]

In the context of Bell's theorem, what is needed for the theorem to hold is counterfactual definiteness (realism), and the property that in the theory, the value is independent of the setting in the other wing, i.e., it always computes to the same value in the theory. This latter property is called locality in the context of Bell's theorem. A theory that use information outside the light-cone of the station would still be called "local" in the Bell sense, as long as this information does not contain the setting in the other wing.

I've just shown you that such a definition of locality is a sham.

It is not a sham if one is careful with the definitions, and what one is talking about.

1) By the same definition, if setting "b" is within the light-cone of station A, and the outcome at station A depended on setting "b", then such a manifestly local situation would be characterized as "non-local" according to your definition. A theory that uses information inside the light-cone of the station would still be called "non-local" in the Bell sense, as long as this information contains the setting in the other wing.

Yes, and among experimentalists, this is called the "locality loophole", i.e. you could have a model that is local in the light-cone sense producing the QM correlations if the setting in one wing is inside the light-cone of the other. But remember that Bell had a hypothetical setup in mind where the events were outside each others light-cones, and in that case the two definitions agree. (Only now are experimentalists close to achieving that setup).

2) According your definition, if the outcome at station A depends on information outside of its light-cone that is also independent of the setting "b" at the remote station, it would be considered to be "local".

Yes, because such information would be irrelevant to Bell's argument.

3) Such a definition of locality is inconsistent as you can't apply it to single particles where there is only one setting and one station.

What do you mean here? I would think the same would apply to any definition of locality.

[minkwe]

Heine,
Even the "locality loophole" is a sham as well since it is often assumed that the inequalities would not be violated if we simply brought the stations within each other's light-cone. But the derivation of inequalities do not care how close together the stations are because no locality assumption is required to obtain the inequalities.

Yes, because such information would be irrelevant to Bell's argument.

It is very relevant, because it tells you clearly that such theories are also forbidden. If Bell's argument is true, then such non-local theories are also forbidden.

What do you mean here? I would think the same would apply to any definition of locality.

If you define locality as "Outcome at station A dependent only on information within the light-cone of Alice", it applies to situations where there is only one particle and one station. If you define it as "outcome at A not dependent on the setting at Station B", then you can't apply that to a single particle where there is no Station B. What about station C, D, E, F, G, ...?

Now that the "locality assumption" is put to rest, let us examine the so-called "counterfactual definiteness" (CFD) or "realism". It is defined variously in the literature as follows:

Wikipedia: In quantum mechanics, Counterfactual definiteness (CFD) is the ability to speak meaningfully of the definiteness of the results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured).
Gill (http://arxiv.org/pdf/1207.5103v6.pdf): Its formulation refers to outcomes of measurements which are not actually performed, so we have to assume their existence, alongside of the outcomes of those actually performed: the principle of realism, or more precisely, counterfactual definiteness.
http://arxiv.org/pdf/1007.4281.pdf: the assumption that a measurement that was not performed had a single definite result.

etc, etc.

Now let us revisit Bell's paragraph 2 of page 1, first three sentences. Assume that you have a single pair of spin-half particles heading in opposite directions to space-like separated stations. Alice orients her magnet along axis "a" and measures +1 (now pay particular attention to the next sentence, remembering the meaning of counterfactual definiteness), if measurement by Alice along axis "a" produces outcome +1 (measurement already performed), then according to quantum mechanics, measurement by Bob along axis "b" must produce outcome -1 and vice versa (measurement yet to be performed).

Why is that not counterfactual definiteness in Quantum Mechanics? Here we are speaking meaningfully of the definiteness of results of measurements which have not been performed. Here QM is assuming that a measurement which has not be performed has a single definite result.

[AnotherGuest]

If Bell's argument is true, then such non-local theories are also forbidden.

Bell's argument is logically correct, it seems to me. So if you buy the initial hypotheses, such non-local theories are also forbidden. But you don't have to buy the initial hypotheses.

Bell has said as much.

Bell doesn't say that he actually believes those initial hypotheses. He just says where those initial hypotheses lead.

[Schmelzer]

Even the "locality loophole" is a sham as well since it is often assumed that the inequalities would not be violated if we simply brought the stations within each other's light-cone.

This assumption would be, of course, nonsensical. The point is that this is irrelevant, because there is no doubt that with communication between A and B allowed one can easily violated Bell's inequalities. So, there is no point in saying QM violates Bell's inequalities A in the future or past lightcone of B, the reaction would be "so what", and to construct local realistic models would be unproblematic.

But the derivation of inequalities do not care how close together the stations are because no locality assumption is required to obtain the inequalities.

Of course, there may be other sets of assumptions which allow to derive the Bell inequalities. Who cares? This would certainly not invalidate Bell's proof which starts with these assumptions. Simply omitting them would invalidate the proof. The freedom to invent other assumptions which allow to prove the inequalities is not at all questioned by Bell. Feel free to prove such theorem.

It is very relevant, because it tells you clearly that such theories are also forbidden. If Bell's argument is true, then such non-local theories are also forbidden.

Ok, if you invent some other conditions, which allow to derive the inequalities, non-local theories which fulfill these conditions would be impossible too. Which is quite irrelevant. Except if you want to claim that, say, de Broglie-Bohm theory is impossible. It is a non-local realistic theory. Thus, some non-local theories may be forbidden, other non-local theories theories not.

Now that the "locality assumption" is put to rest, let us examine the so-called "counterfactual definiteness" (CFD) or "realism". It is defined variously in the literature as follows:

Now let us revisit Bell's paragraph 2 of page 1, first three sentences. Assume that you have a single pair of spin-half particles heading in opposite directions to space-like separated stations. Alice orients her magnet along axis "a" and measures +1 (now pay particular attention to the next sentence, remembering the meaning of counterfactual definiteness), if measurement by Alice along axis "a" produces outcome +1 (measurement already performed), then according to quantum mechanics, measurement by Bob along axis "b" must produce outcome -1 and vice versa (measurement yet to be performed).

Why is that not counterfactual definiteness in Quantum Mechanics?

Because there is no counterfactual definiteness in QM. Counterfactual definiteness requires that the outcome of unperformed experiments exists, is well-defined even befor the measurement. But the derivation that of counterfactual definiteness in Bell's argument uses the EPR argument, which requires locality.

In a nonlocal world, the outcomes may not be predefined. If A(a) is "measured", some random process, depending also on hidden variables of the measurement device, may define its result. Then, this result is transferred to B, so that after this the resultat of B(a) is defined as -A(a). But in this case the result B(a) was not predefined, but becomes defined only at the moment of measurement of A(a). All other measurement results B(c), for all c=/=a, remain undefined until the experiment is done, and those of A(c) for all c=/=a, remain undefined forever. This is what happens in dBB theory, where the "measurement result" is, in fact, the result of an interaction, which depends on the hidden variable of the particle as well as of the hidden variable of the measurement device.

[FrediFizzx]

Now let us revisit Bell's paragraph 2 of page 1, first three sentences. Assume that you have a single pair of spin-half particles heading in opposite directions to space-like separated stations. Alice orients her magnet along axis "a" and measures +1 (now pay particular attention to the next sentence, remembering the meaning of counterfactual definiteness), if measurement by Alice along axis "a" produces outcome +1 (measurement already performed), then according to quantum mechanics, measurement by Bob along axis "b" must produce outcome -1 and vice versa (measurement yet to be performed).

Did you mean, "...measurement by Bob along axis "a" must produce outcome -1..."?

[AnotherGuest]

Now let us revisit Bell's paragraph 2 of page 1, first three sentences. Assume that you have a single pair of spin-half particles heading in opposite directions to space-like separated stations. Alice orients her magnet along axis "a" and measures +1 (now pay particular attention to the next sentence, remembering the meaning of counterfactual definiteness), if measurement by Alice along axis "a" produces outcome +1 (measurement already performed), then according to quantum mechanics, measurement by Bob along axis "b" must produce outcome -1 and vice versa (measurement yet to be performed).

Why is that not counterfactual definiteness in Quantum Mechanics? Here we are speaking meaningfully of the definiteness of results of measurements which have not been performed. Here QM is assuming that a measurement which has not be performed has a single definite result.

Very good question, Minkwe.

We can speak meaningfully about what would happen if ... . All physics is about saying what would happen if ....

Counterfactual definiteness goes, I think, beyond that. It is about the question whether all those other things which could have happened if .. are to be considered just as real as the one thing that does actually happen in the circumstances which will actually hold.

Now actually this is still talking about words. Word games for philosophers. But as soon as we start talking about mathematical models, then we do get onto something interesting, I think. We can ask the question: is there a mathematical model which reproduces the predictions of quantum mechanics, but which also goes beyond those predictions and allows, within the mathematics, "existence" of counterfactual outcomes of not performed measurements, alongside of actual outcomes of actually performed measurements? And do those actual and counter-factual outcomes together, satisfy locality, in the sense that whatever measurement Alice makes, does not change which outcome Bob observes?

I think that the whole discussion is not actually about "reality" (whatever that is!) but about mathematical models of reality. And about whether those mathematical models can be extended so as to include also "what would have happened if ...". And about whether that extended model still satisfies natural locality properties.

In other words, local-ness or not local-ness is a property of a model, It depends on what we put in the model. If you put into the model things that needn't be there, like the outcome that Alice would have seen if Bob had done a different measurement from the one that he actually performed, then, it seems, you run into problems with locality.

EPR gave reasons, from QM + locality, why counterfactual measurement outcomes should be thought of as "real".