Simple proof of Bell's theorem

[Zafa Pi]

Just a note that this reference is not itself accepted, although the jeremyfiennes reference is. The critical piece, although superficially about the cited paper, is actually about something else entirely.

So what is the answer to jeremyfiennes's question in post #27?

 

[Zafa Pi]

You don't need to. The logic is: the mismatch inequality follows from assuming reality and locality; experiment shows that the inequality is violated; therefore, no theory that is both local and realistic can be correct. This argument doesn't require that you be able to actually dream up a local non-realistic theory.

Perhaps nature itself is such a theory?

 

[DrChinese]

So what is the answer to jeremyfiennes's question in post #27?

Nugatory's reply says it. But the question jeremyfiennes raised was: what are some of the intuitive features of realism we might give up?

Look at the interpretations. If you accept MWI, you are accepting multiple worlds. You could alternately reject causality (cause precedes effect).

 

[Zafa Pi]

Following on: from <http://www.nature.com/nature/journal/v446/n7138/abs/nature05677.html>:
"Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned."
Which intuitive features?

There is, so far, no question in my mind that giving up locality allow Alice and Bob to collaborate and create any type of correlations they wish. Thus the quote you chose from the the article is, on its own, false. But before that quote the article also says, "Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations."

There it is! They are not giving up locality in general. The quote you chose is merely highly misleading and the authors shouldn't have said it. Good looking out.

 

[DrChinese]

There is, so far, no question in my mind that giving up locality allow Alice and Bob to collaborate and create any type of correlations they wish. Thus the quote you chose from the the article is, on its own, false. But before that quote the article also says, "Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations."

There it is! They are not giving up locality in general. The quote you chose is merely highly misleading and the authors shouldn't have said it. Good looking out.

They are saying that giving up locality is not, on its own, enough to automatically explain QM. They are not saying that locality must go. There is nothing wrong or misleading with the quotes from the referenced paper. But they must be parsed correctly.

The idea that "intuitive" realism is incompatible with QM goes back a long time, and experiments in the past 25 years have tended to support this idea. There is no single experiment that settles this issue at this time. It still comes back to certain assumptions you are free to make.

 

[Zafa Pi]

They are saying that giving up locality is not, on its own, enough to automatically explain QM.

What they are saying is,
"In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned."
It appears that they are talking about locality in general, but they are not because they also say,
"Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations."

I contend that if the assumption of locality (in general, no FTL communication) is dropped (i.e. FTL communication is permitted) from the Bell business then that permits Alice and Bob to communicate and thus create any correlations at all, with realism intact. This I can prove.
Perhaps there is a semantic problem on what it means to say, "giving up the concept of locality". Like giving up meat still allows animal based B12 tablets.

They are not saying that locality must go.

Neither am I.

The idea that "intuitive" realism is incompatible with QM goes back a long time, and experiments in the past 25 years have tended to support this idea. There is no single experiment that settles this issue at this time. It still comes back to certain assumptions you are free to make.

Is "intuitive" realism incompatible with Bohmian Mechanics? I am not well versed on BM, but it doesn't seem to allow for unfettered FTL communication.

 

[DrChinese]

I contend that if the assumption of locality (in general, no FTL communication) is dropped (i.e. FTL communication is permitted) from the Bell business then that permits Alice and Bob to communicate and thus create any correlations at all, with realism intact. This I can prove.

Is "intuitive" realism incompatible with Bohmian Mechanics? I am not well versed on BM, but it doesn't seem to allow for unfettered FTL communication.

That would be a big proof. Putting forth a non-local realistic theory (such as BM) would not do it.

There is a lot of controversy over the limits that are being accumulated around Bell's Theorem and non-local realistic theories. Generally, the Bohmians deny that those restrictions even apply. But the evidence keeps accumulating that "tends" to encroach on their position. The gist of the argument is that NO realistic theory can mimic QM in all respects. Again, that has not been demonstrated yet; but that is where the experiments are going.

 

[Zafa Pi]

That would be a big proof. Putting forth a non-local realistic theory (such as BM) would not do it.

There is a lot of controversy over the limits that are being accumulated around Bell's Theorem and non-local realistic theories. Generally, the Bohmians deny that those restrictions even apply. But the evidence keeps accumulating that "tends" to encroach on their position. The gist of the argument is that NO realistic theory can mimic QM in all respects. Again, that has not been demonstrated yet; but that is where the experiments are going.

Here is crux: Locality generally means no FTL influence or communication. So what does dropping locality mean?
If it means that Alice and Bob can communicate at FTL then they can trivially conspire to violate Bells inequality in even more profound ways than the usual QM correlations. (Do I need to show you how?). If, on the other hand, it means something like Bohmian mechanics, that is totally different and does not allow Alice and Bob to communicate at FTL, in spite of the instantaneous actions of the pilot wave.

Here is a simple analogy: A state has had maximum speed on interstate highways of 70mph and announces it is now dropping that restriction. Does that mean you can now go at 90mph? Maybe and maybe not. Perhaps it only applies to emergency vehicles, or maybe like Germany you can go as fast as you like.

So when someone says they have a non-local theory what does it mean to you?

 

[DrChinese]

Locality generally means no FTL influence or communication. So what does dropping locality mean?
If it means that Alice and Bob can communicate at FTL then they can trivially conspire to violate Bells inequality in even more profound ways than the usual QM correlations. (Do I need to show you how?).

Sorry, it's hardly trivial to formulate a theory that can provide the same predictions as QM. You can show a lot of things with a non-local theory. But you can't just say: this theory is non-local and makes the same predictions. So yes, please show me this trivial exercise. Perhaps your FTL theory will feature the following, in addition to violation of Bell inequalities:

1. Entanglement swapping using independent sources.
2. Spin.
3. Signalling limited to c.
4. GHZ effect.

Good luck!

PS Think of the problem this way: just because the speed of light is c does mean a person can walk at c or a car can drive at c. There is a lot more physics involved, think?

The same is true if c were not a limit on transmitting influences (in a non-local theory). Perhaps you might have noticed there aren't any FTL signals or causal influences known to man. Even in QM there is no FTL causal influence that we know of. That is because the causal direction cannot be definitely ascertained. Is it Alice to Bobo? Or Bob to Alice? No one really knows, just guesses.

 

[Zafa Pi]

Sorry, it's hardly trivial to formulate a theory that can provide the same predictions as QM. You can show a lot of things with a non-local theory. But you can't just say: this theory is non-local and makes the same predictions. So yes, please show me this trivial exercise. Perhaps your FTL theory will feature the following, in addition to violation of Bell inequalities:

1. Entanglement swapping using independent sources.
2. Spin.
3. Signalling limited to c.
4. GHZ effect.

Good luck!

PS Think of the problem this way: just because the speed of light is c does mean a person can walk at c or a car can drive at c. There is a lot more physics involved, think?

The same is true if c were not a limit on transmitting influences (in a non-local theory). Perhaps you might have noticed there aren't any FTL signals or causal influences known to man. Even in QM there is no FTL causal influence that we know of. That is because the causal direction cannot be definitely ascertained. Is it Alice to Bobo? Or Bob to Alice? No one really knows, just guesses.

The usual physical set up for a Bell experiment goes something like:
Alice and Bob are 2 light minutes apart and Eve is half way between and simultaneously sends a light signal to each. When Alice receives her signal she flips a fair coin. If it comes up heads selects either +1 or -1 by some objective procedure (i.e., we can duplicate the procedure) and we call that Ah. If she flips a tail she may do the same thing or something else to get At which also = 1 or -1. This takes less than 30 seconds. Bob goes through the same ritual to get Bh and Bt.

Bell's Theorem: Let Ah, At, Bh, and Bt be four numbers that are either 1 or -1. Assume that Ah = Bh (Ah•Bh = 1),
then we have Bell's Inequality: P(At•Bt = -1) ≤ P(At•Bh = -1) + P(Ah•Bt = -1). (Where P is probability)

Proof: P(At•Bt = -1) = P(At•Bt•Ah•Bh = -1) = P(At•Bh•Bt•Ah = -1) = P({At•Bh = -1 and Bt•Ah = 1} or {At•Bh = 1 and Bt•Ah = -1}) =
P(At•Bh = -1 and Bt•Ah =1) + P(At•Bh = 1 and Bt•Ah = -1) ≤ P(At•Bh = -1) + P(Ah•Bt = -1) QED

Suppose that Alice and Bob select 1 for both Ah, At, and Bh, then she gets on the quikfone (FTL) and tells Bob to let Bt = 1 if she flipped heads and let Bt = -1 if she flipped tails. All this takes less than 30 seconds. Then Ah = Bh, but Pr(At•Bt = -1) = 1, P(At•Bh = -1) = P(Ah•Bt = -1) = 0. So Bell's Inequality is violated (and in a more profound manor than QM does by measuring entangled photons) and realism holds.

Of course the same thing can be pulled off for GHZ, it just takes a conference call on the quikfone.

I await you objection with bated breath.

 

[jeremyfiennes]

On a more simplistic level, a standard formulation of Bell's Theorem (e.g. #35) is that "No physical theory of Local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics". Local Hidden Variables theories are however realistic, and give uniquely defined values. Whereas Quantum Mechanics' predictions are probabilistic. Does it not go without saying that no realistic theory can ever reproduce probabilistic results, and vice versa?

 

[DrChinese]

The usual physical set up for a Bell experiment goes something like:
... it just takes a conference call on the quikfone.

I await you objection with bated breath.

LOL.

Seriously: what you have presented has no connection whatsoever to theoretical quantum mechanics, no connection to the referenced paper, and completely ignores the criteria I mention. Which is probably fine, as we are drifting further and further from anything relevant to this thread. You might want to read the paper and note that the Leggett inequalities are the ones that were being tested for certain non-local theories - and those theories were excluded by experiment.

 

[DrChinese]

On a more simplistic level, a standard formulation of Bell's Theorem (e.g. #35) is that "No physical theory of Local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics". Local Hidden Variables theories are however realistic, and give uniquely defined values. Whereas Quantum Mechanics' predictions are probabilistic. Does it not go without saying that no realistic theory can ever reproduce probabilistic results, and vice versa?

There is a lot of controversy around the idea that "no realistic theory can ever reproduce probabilistic results". Bohmians think theirs does, for example. Further, that is not a strict deduction from Bell (as well said above in bold ). On the other hand, it's a reasonable supposition and certainly the stuff of many leading edge experiments.

 

[zonde]

You might want to read the paper and note that the Leggett inequalities are the ones that were being tested for certain non-local theories - and those theories were excluded by experiment.

As I understand the paper claimed to test the theories restricted by these assumptions:
"(1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism); (2) physical states are statistical mixtures of subensembles with definite polarization, where (3) polarization is defined such that expectation values taken for each subensemble obey Malus’ law (that is, the well-known cosine dependence of the intensity of a polarized beam after an ideal polarizer)."
Now considering assumption (1) what is supposed role of non-locality in these theories? I'm not sure where it may enter and make any difference to predictions. Unless of course measurement of one particle from entangled pair can instantaneously change "pre-existing" property of the other particle from the pair. But in that case Zafa Pi counterexample is relevant.

 

[zonde]

Does it not go without saying that no realistic theory can ever reproduce probabilistic results, and vice versa?

Certainly no. Look at Chaos theory
Not sure about "vice versa" part however (what would be non-realistic theory?).

 

[DrChinese]

As I understand the paper claimed to test the theories restricted by these assumptions:
"(1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism); (2) physical states are statistical mixtures of subensembles with definite polarization, where (3) polarization is defined such that expectation values taken for each subensemble obey Malus’ law (that is, the well-known cosine dependence of the intensity of a polarized beam after an ideal polarizer)."
Now considering assumption (1) what is supposed role of non-locality in these theories? I'm not sure where it may enter and make any difference to predictions. Unless of course measurement of one particle from entangled pair can instantaneously change "pre-existing" property of the other particle from the pair. But in that case Zafa Pi counterexample is relevant.

Their result is that all such theories (which include their "reasonable" definition of realism), including ALL non-local ones, fail. You (and Zafa Pi) keep assuming precisely that which the paper proves is false.

Non-locality - as a feature of a quantum theory candidate - is not a magic bullet to explain violations of certain inequalities. This is the point you are skipping over.

 

[zonde]

Their result is that all such theories (which include their "reasonable" definition of realism), including ALL non-local ones, fail.

But please explain how you understand assumption (1). Does this assumption include theories where measurement of one particle from entangled pair can instantaneously change "pre-existing" property of the other particle from the pair?

 

[DrChinese]

But please explain how you understand assumption (1). Does this assumption include theories where measurement of one particle from entangled pair can instantaneously change "pre-existing" property of the other particle from the pair?

Yes, if the following are features as well:

"(1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism); (2) physical states are statistical mixtures of subensembles with definite polarization, where (3) polarization is defined such that expectation values taken for each subensemble obey Malus’ law (that is, the well-known cosine dependence of the intensity of a polarized beam after an ideal polarizer)."

Please keep in mind that we are discussing the result of a paper by a top team in the field. The entire purpose of the paper is to make the point that you keep denying. Which is that just having a non-local element to a theory does not mean it can be formulated to match the predictions of QM. If it has certain realistic elements, it is excluded. There are many many constraints on any candidate non-local theory (or local theory for that matter).

For clarification purposes: Bohmian Mechanics often escapes this result by denying that properties are independent of the measurement (part 1). If so, this paper does not apply.

And just to drive the point home: do you not see that the number of effects we term as "non-local" are limited? They are almost all centered around entangled systems with spatial extent. Spatially separated systems which are not entangled generally do NOT interact. If you say there are non-local effects in a candidate theory, you are compelled to explain how and why those effects are so incredibly limited (why doesn't everything affect everything, for example). Obviously, local theories don't have quite the same problem because there is no action at a distance - but they have other obstacles to overcome.

 

[Nugatory]

Does it not go without saying that no realistic theory can ever reproduce probabilistic results, and vice versa?

No. There's a counterexample in thermodynamics, which makes probabilistic predictions although the hidden variable theory behind it is completely deterministic Newtonian mechanics.

As a more elementary example, I have a really excellent probabilistic theory for describing the behavior of a tossed coin: It comes up heads 50% of the time and tails 50% of the time. Again however, the underlying hidden variable theory is the completely deterministic Newtonian physics, here applied to the motion of the atoms making up the coin and the air around it.

 

[jeremyfiennes]

Ok.

 

[DrChinese]

Ok.

I interpret that to mean the original thread is now at a nice stopping point.

 

[zonde]

But please explain how you understand assumption (1). Does this assumption include theories where measurement of one particle from entangled pair can instantaneously change "pre-existing" property of the other particle from the pair?

Yes, if the following are features as well:

"(1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism); (2) physical states are statistical mixtures of subensembles with definite polarization, where (3) polarization is defined such that expectation values taken for each subensemble obey Malus’ law (that is, the well-known cosine dependence of the intensity of a polarized beam after an ideal polarizer)."

Leggett in his paper "Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem" had assumption 4.:
"4.
$A(a, b, \lambda : B)=A(a, b, \lambda),\;\; B(a, b, \lambda : A)=B(a, b, \lambda),\;\; (2.4)$
i.e., the outcome of the measurement of A is independent of the outcome at the distant station 2 and vice versa ("outcome-independence," cf. Jarrett(5))."

If we allow changing either function $A(a, b, \lambda)$ or $B(a, b, \lambda)$ based on who makes his measurement first we violate Leggett's assumption 4.

 

[zonde]

And just to drive the point home: do you not see that the number of effects we term as "non-local" are limited? They are almost all centered around entangled systems with spatial extent. Spatially separated systems which are not entangled generally do NOT interact. If you say there are non-local effects in a candidate theory, you are compelled to explain how and why those effects are so incredibly limited (why doesn't everything affect everything, for example).

I would like say that this is really good question and it requires good answer. I have thought about it but I would like to hold to myself my speculations.
However I think that it is related to the question how there can be pure states (say coherent polarized beam of light) given entanglement phenomena.

 

[Zafa Pi]

Seriously: what you have presented has no connection whatsoever to theoretical quantum mechanics

What I presented is no more or less than what I claimed in posts #36,38, and 40, i.e. if Alice and Bob can communicate at FTL (they are in effect no longer really separated) they can violate Bell's inequality (get the same correlations as QM) and maintain realism. You denied this, but you are right what Alice and Bob are doing has nothing to do with QM, so what.

If locality means no FTL communication, then if one says they give up locality then an interpretation of that is Alice and Bob can communicate at FTL. If all one means is that there is a non-local element then I agree with you.

 

[Simon Phoenix]

You denied this, but you are right what Alice and Bob are doing has nothing to do with QM, so what.

Well I think the point that Dr Chinese is trying to make is that the phenomenon of entanglement goes much deeper, and is more pervasive, than simply being able to violate a Bell inequality.

I've no doubt that one could construct some artificial piece of theory that would be non-local and realistic that would mimic the observations made in a specific Bell inequality experiment. It wouldn't look like physics as we know it (either classical or quantum) but would just be a theory specifically designed to reproduce the features of one experiment. Would the same theory then be able to explain the results from a GHZ state, say? Would the same theory then predict the phenomenon of entanglement swapping? And so on.

The thing is that with QM we have a single coherent and logical theory that explains all of this in one framework - we don't need to introduce all sorts of ad-hoc assumptions for each new experiment - everything follows from the few basic axioms and postulates of QM.

The closest we've got so far (to my knowledge) to a non-local realistic theory that reproduces all the results of QM is Bohmian mechanics - but that, like all of the interpretations of QM, has got its own 'weirdness' (the whole business of interpretation seems to me to be about shifting the awkward bits under the rug we're most comfortable with).

 

[akvadrako]

I have heard of another non-local realistic theory which may not meet everyone's requirements for "realistic" but it does seem to illustrate the possibility.

1. First, every point in space contains a copy of the wavefunction of the entire universe.
2. Each copy updates unitarily.
3. Whenever a measurement is made at one point, that copy is collapsed.
4. Updates are broadcast to all other points.

Some time synchronisation protocols are probably necessary too, but I would think the above could reproduce the expectations of QM.

 

[jeremyfiennes]

I interpret that to mean the original thread is now at a nice stopping point.

No, not just yet please! I now realize I am unclear on the meaning of "hidden variables". I had imagined these to be conceived, but at present immeasurable, properties of the observed object. From Nugatory's reply (#49), however, it seems that they can also be factors in the object's environment. In the analogy of a doctor measuring a patient's blood pressure (measuring the blood pressure of a patient having his blood pressure measured by a doctor), would it be true to say that:
-- unmeasured patient-associated variables (how well he slept, what he had for breakfast, etc) are "hidden"
-- unmeasured environment-associated variables (the temperature, noise level of the ward, etc.) are likewise "hidden"
-- their combined effect is "experimental error", and can be reduced by including the variables in the model
-- the doctor-effect is uncontrollable observer-dependent "uncertainty" -- the patient could react to a male doctor in one way, to a female doctor in another, and so on?

And that:
-- for realists, the patient has a real, doctor-independent blood-pressure, even though it cannot be determined
-- for positivists, it is meaningless to talk of a real blood-pressure, because it cannot be determined
-- for quantum physics, the real blood pressure is what it is measured to be, and before that did not exist?

 

[Nugatory]

-- for realists, the patient has a real, doctor-independent blood-pressure, even though it cannot be determined
-- for positivists, it is meaningless to talk of a real blood-pressure, because it cannot be determined
-- for quantum physics, the real blood pressure is what it is measured to be, and before that did not exist?

This is an unfortunately very confusing example, because all the sources of uncertainty you cite are not problems with the observable we're measuring (there's no problem with the manometer reading), but rather with how good a proxy that measurement is for what the doctor really wants to know, namely what level of treatment for hypertension is indicated. (Or, informally, not only have you not supplied a definition for "real blood pressure", you've made a pretty good case that the phrase is undefined).

For more helpful examples, you might try these three philosophical positions against two phenomena: thermodynamic pressure, for which we have an accepted hidden-variable theory; and quantum spin, for which we do not.

 

[Nugatory]

I had imagined [hidden variables] to be conceived, but at present immeasurable, properties of the observed object.

It's not "hidden variables", it's "hidden variable theory". The hidden variables are just whatever inputs a candidate theory uses to make predictions, so you can't say anything concrete about them except in the context of a particular candidate theory.

Bell's theorem operates, not by proving that there are no hidden variables, but by proving that all candidate theories that have a particular set of mathematical properties will fail. Hidden variable theories only come into the logic because it turns out that all local realistic hidden variable theories (for most generally accepted definitions of "local realistic hidden variable theory") happen to have those properties so are precluded.

(Do note, however, that the previous paragraph is running the history backwards. Bell started with that particular set of mathematical properties because they covered all possible LHV theories - that's what made them interesting)

 

[jeremyfiennes]

So how would a "hidden variable" in general be defined?