B Simple proof of Bell's theorem

[jeremyfiennes]

The thread I wanted to post my question on got closed. Recapitulating:

The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester):
Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When both SPOT detectors are aligned, these messages are exactly the same. When the detectors are misaligned, "errors" are generated and the sequences contain a certain number of mismatches. How these "errors" might be generated is the gist of this proof. Step One: Start by aligning both SPOT detectors. No errors are observed. Step Two: Tilt the A detector till errors reach 25%. This occurs at a mutual misalignment of 30 degrees. Step Three: Return A detector to its original position (100% match). Now tilt the B detector in the opposite direction till errors reach 25%. This occurs at a mutual misalignment of -30 degrees. Step Four: Return B detector to its original position (100% match). Now tilt detector A by +30 degrees and detector B by -30 degrees so that the combined angle between them is 60 degrees. What is now the expected mismatch between the two binary code sequences? We assume, following John Bell's lead, that REALITY IS LOCAL. Assuming a local reality means that, for each A photon, whatever hidden mechanism determines the output of Miss A's SPOT detector, the operation of that mechanism cannot depend on the setting of Mr B's distant detector. In other words, in a local world, any changes that occur in Miss A's coded message when she rotates her SPOT detector are caused by her actions alone. And the same goes for Mr B. The locality assumption means that any changes that appear in the coded sequence B when Mr B rotates his SPOT detector are caused only by his actions and have nothing to do with how Miss A decided to rotate her SPOT detector. So with this restriction in place (the assumption that reality is local), let's calculate the expected mismatch at 60 degrees. Starting with two completely identical binary messages, if A's 30 degree turn introduces a 25% mismatch and B's 30 degree turn introduces a 25% mismatch, then the total mismatch (when both are turned) can be at most 50%. In fact the mismatch should be less than 50% because if the two errors happen to occur on the same photon, a mismatch is converted to a match. Thus, simple arithmetic and the assumption that Reality is Local leads one to confidently predict that the code mismatch at 60 degrees must be less than 50%. However both theory and experiment show that the mismatch at 60 degrees is 75%. The code mismatch is 25% greater than can be accounted for by independent error generation in each detector. Therefore the locality assumption is false. Reality must be non-local.

Great. Finally an explanation of Bell's theorem that even I can understand! My question relates to the following part: "Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When both SPOT detectors are aligned, these messages are exactly the same. When the detectors are misaligned, "errors" are generated and the sequences contain a certain number of mismatches." A "mismatch" however would be a mismatch with respect to the code emitted by the other detector, implying a communication between the two. Does not this violate their independence?
Thanks.

 

[Nugatory]

A "mismatch" however would be a mismatch with respect to the code emitted by the other detector, implying a communication between the two.

The hypothesis is that each detector produces its output by randomly flipping bits in the the input that it receives from the central source, without communicating with the other detector. Then, after the fact, we take the output of the two detectors and compare them - only then are we counting the mismatches.

 

[jeremyfiennes]

Thanks. I hadn't got the "central source" bit. But there still seems to be a connection. It step 2 A tilts his detector, and as he does so creates errors by randomly flipping bits. But these are "errors" based on the assumption that detector B is still receiving the same input as he is from the central source. The two detectors are effectively linked via the central source, and are not completely independent.

 

[Nugatory]

Thanks. I hadn't got the "central source" bit. But there still seems to be a connection.

That's the whole point of the exercise. We're showing that:

IF
1) The two detectors receive the same inputs; AND
2) The number of bits flipped by a detector depends only on the angle of that detector relative to the source but not the angle of the other detector;
THEN
there will be a limit on how different their outputs can be.

#2 is the independence requirement.

 

[jeremyfiennes]

Thanks. That seems clear. I was just about to say that had resolved it, but a further point arose in my mind. Local theory predicts a maximum mismatch of 50%. Whereas measurement gives 75%, i.e. a 25% correlation. But shouldn't entanglement cause a higher correlation than expected on a local theory, and not a lower one?

 

[Nugatory]

T But shouldn't entanglement cause a higher correlation than expected on a local theory, and not a lower one?

That depends on whether you've initially entangled the particles in such a way that measurements on the same axis are expected to produce the same result or opposite results. In the simplified model used in this "proof" (it's not really a proof, it's an pedagogical example) that would correspond to whether the two messages are exactly identical or exact inverses ("one's complement", in computer science terms) of one another, and it's easier to explain if we choose the two messages being identical.

In actual experiments, we most often use photons entangled in such a way that both photons will always pass through filters that are 90 degrees apart, only one will ever pass through filters that are 0 degrees or 180 degrees apart, and it's 50-50 random at both filters when they are 45 degrees apart. For spin-entangled particles (which are seldom used because they're harder to work with than photons) they are usually entangled in such a way that measurements along the same axis always disagree, and if the detectors are 90 degrees apart they will be be 50-50 random. We have observed near-infinite confusion when people switch from one model to the another in mid-explanation

 

[DrChinese]

Thanks. That seems clear. I was just about to say that had resolved it, but a further point arose in my mind. Local theory predicts a maximum mismatch of 50%. Whereas measurement gives 75%, i.e. a 25% correlation. But shouldn't entanglement cause a higher correlation than expected on a local theory, and not a lower one?

It doesn't work that way. The entangled statistics can predict either higher OR lower correlations than a local realistic theory might. At some points they can even be equal. It is strictly a function of the difference [theta] in the choice of measurement angles, the formula being cos^2(theta). That formula won't work for a local realistic theory.

 

[jeremyfiennes]

Thanks both of you. That fills up my brain capacity for the moment. I need time to brood on it. I like the maxim of not changing models in mid explanation.

 

[zonde]

Local theory predicts a maximum mismatch of 50%. Whereas measurement gives 75%, i.e. a 25% correlation. But shouldn't entanglement cause a higher correlation than expected on a local theory, and not a lower one?

Mismatch of 50% is no correlation at all i.e. half of pairs are the same and the other half are different. 75% mismatch is 50% anticorrelation i.e. half of pairs can be viewed as random and half as being opposite. So 75% mismatch gives more certainty than 50% mismatch.
Let me illustrate this with binary strings:
50% mismatch:
A: 10101100
B: 11001010
C: sxxssxxs (the same number of matches "s" as mismatches "x")

75% mismatch:
A: 1010 0101
B: 1100 1010
C: sxxs xxxx (the same number of matches "s" as mismatches "x" in first half and only mismatches "x" in second half)

 

[jeremyfiennes]

Ok, so on this simplified Bell model, at the +/- 30^o position each detector sends out a signal with 25% random mismatches generated by it, and receives a signal with 25% random mismatches generated by the other. Giving an expected overall mismatch at each detector of at the most 50%. In fact 75% is measured, which the local realist theory cannot explain. How does a non-realist non-local theory explain it?

 

[Nugatory]

In fact 75% is measured, which the local realist theory cannot explain. How does a non-realist non-local theory explain it?

We derived the 50% limit by making two assumptions.
1) The two detectors receive the same inputs; AND
2) The number of bits flipped by a detector depends only on the angle of that detector relative to the source but not the angle of the other detector.

When we observe a 75% mismatch, we know that at least one of those two assumptions must be false. #1 is (under the conditions of this toy model) realism, and #2 is locality, so we know that any theory that accurately describes this situation must be either non-local or non-realistic (or both).

Bell's theorem is not intended to explain the results. It's not giving us a theory that explains the experimental observations, it's telling us that any theory that correctly predicts these results cannot be local and realistic.

 

[zonde]

How does a non-realist non-local theory explain it?

QM formalism describes entangled pairs with single mathematical object but it does not give predictions for individual detections.
So if you want to speak about individual detections then you have to examine interpretations of QM. For non-local realist interpretation you can look at Bohmian mechanics.
I am unsure about non-realist explanations. Such explanation would have to take measurements as not being factual (rather extreme for me). I would suggest looking at quantum decoherence (it is not considered interpretation however).

 

[jeremyfiennes]

Thanks. A small one to be going on with while I read up: are counterfactual definiteness and realism the same thing?

 

[jeremyfiennes]

And in the simple Bell model, the common source for both detectors corresponds to locality; and the fixed relation between the detector angle and the mismatches generated corresponds to realism?

 

[zonde]

Thanks. A small one to be going on with while I read up: are counterfactual definiteness and realism the same thing?

It seems that "counterfactual definiteness" and "realism" in Quantum mechanics contexts is used with the same meaning that properties of particles exist independently from measurements.
But outside QM "counterfactual reasoning" generally means asking "what if?" type of questions while "realism" means that there exists reality independent of our observations and models. With that philosophical meaning "realism" justifies scientific method as we can test our models against reality (by performing experiments).

And in the simple Bell model, the common source for both detectors corresponds to locality; and the fixed relation between the detector angle and the mismatches generated corresponds to realism?

No. You can take locality assumption given by Nugatory in post #11: "2) The number of bits flipped by a detector depends only on the angle of that detector relative to the source but not the angle of the other detector." Alternatively we can say that measurements of Alice and Bob are independent.
Common source is more like a given. Basically everything you need to get the right answer for the step one you have to take as given. And without some source of entangled particles you fail at step one.

Returning to assumption that properties of particles exist independently from measurements. I would like to point out that in that simple proof properties of particles do not appear anywhere in the argument. However it relies on "counterfactual reasoning" in it's common sense as it asks "what if?" type of questions (in steps two to four).

 

[jeremyfiennes]

Thanks. I'm realizing that my main problem is to really "get" the meaning of the terms. The worst, "counterfactual definitiveness", has thankfully now gone (the guy who invented it should be shot). So locality in this simplified case is, in Nugatory's words, that "the number of bits flipped by a detector depends only on the angle of that detector relative to the source, but not the angle of the other detector". It is called "locality" because the other detector could be so far away that any dependence effect would have to travel faster than light (?). "Realism" would then be that the total mismatch cannot exceed the sum of the mismatches of the individual detectors?

 

[forcefield]

You can take locality assumption given by Nugatory in post #11: "2) The number of bits flipped by a detector depends only on the angle of that detector relative to the source but not the angle of the other detector."

To argue about locality, don't you need the complementary experiment, where angles are changed while the particles are assumed to be on their way to the detectors ?

 

[DrChinese]

"Realism" would then be that the total mismatch cannot exceed the sum of the mismatches of the individual detectors?

Realism can be considered several different things. In the Bell proof, usually it is the idea that observer Alice, by her choice of measurement setting, does not influence the outcome that Bob sees (and vice versa). Mathematically, that is usually expressed as the independence of the functions that determine the outcomes for Alice and Bob. Therefore you have a Product state with settings a and b. See Bell's (2). If it is independent, then settings of {a,c} or {b,c} would likewise be independent Product states. That allows one to consider combinations of {a,b,c} even though all 3 could not be measured simultaneously. All of that together is realism.

 

[Nugatory]

So locality in this simplified case is, in Nugatory's words, that "the number of bits flipped by a detector depends only on the angle of that detector relative to the source, but not the angle of the other detector".

Yes, but see below.

"Realism" would then be that the total mismatch cannot exceed the sum of the mismatches of the individual detectors?

In the context of this simplified toy model, "realism" is an assumption so basic that it almost goes unstated.

We could reason as if the messages are actual physical pieces of paper, the source is a printer connected to a random number generator (and the the detectors might be slightly buggy character-recognition scanners). The key point is that when the messages leave the source, each bit in the message is established by physical properties (in this case, where on the paper the toner is deposited) of the message; without an assumption of that sort there's no original/unflipped value so no sensible way of interpreting the mismatches as the result of random bit flips. That assumption is (loosely speaking, and within the context of this toy model) "realism"; you need it to derive the conclusion that the total mismatch cannot exceed the sum of the individual mismatches.

But there's only so far you should carry this line of thinking and this toy model. You can drive yourself mad trying to define precisely what you mean by "realism" and "locality", as these are natural language words and they get all squishy under pressure. Bell's actual proof is done with mathematical statements about the probability distributions of the factors affecting the measurement results, and these are much less squishy. No matter how you interpret words like "locality" and "realism", there is an entire class of theories that are excluded by Bell's theorem because they imply probability distributions that cannot produce the observed experimental results.

 

[DrChinese]

To argue about locality, don't you need the complementary experiment, where angles are changed while the particles are assumed to be on their way to the detectors ?

That's done if you want to test whether there is some (currently) unknown signaling or other mechanism in place (often called the locality loophole). This test has been performed a number of times, and there is no sign of such a mechanism. If there is such a mechanism, it must be FTL.

 

[DrChinese]

Realism can be considered several different things. In the Bell proof, usually it is the idea that observer Alice, by her choice of measurement setting, does not influence the outcome that Bob sees (and vice versa). Mathematically, that is usually expressed as the independence of the functions that determine the outcomes for Alice and Bob. Therefore you have a Product state with settings a and b. See Bell's (2). If it is independent, then settings of {a,c} or {b,c} would likewise be independent Product states. That allows one to consider combinations of {a,b,c} even though all 3 could not be measured simultaneously. All of that together is realism.

So in the example in the original post: a=+30, b=-30, c=0 degrees. And you consider the 3 testable combinations {a,c} and {b,c} and {a,b} as if they could each be considered as independent.

 

[jeremyfiennes]

Thanks. The understanding of realism I have got till now is that realistic variables have determined values. If you measure one, and then go back and measure it again (assuming you can), you get the same result each time. That obviously implies no external interference, local or non-local. Quantum variables, where there is only a probability of getting a given result, are therefore non-realistic.
So I am confused to read that "Two important consequences of EPR experiments are that two previously held doctrines of physical reality: causality and local reality, are violated." From the above I understood locality" and "realism" to be two different things. So what is "local reality"? Both together? And if so, what then is "causality", and how does it fit into the above scheme?

 

[Nugatory]

So I am confused to read that "Two important consequences of EPR experiments are that two previously held doctrines of physical reality: causality and local reality, are violated.

Where did you hear that? This is the first time the word "causality" appears in this thread.

 

[jeremyfiennes]

I got the "causality" quote from a 2009 site. But today's <https://en.wikipedia.org/wiki/Bell's_theorem> says "Locality is short for local relativistic causality", so in general the two seem to be taken to be the same (?). Thanks.

Two more questions: 1) Are "local hidden variables" and "local realism" the same thing? And if so, in the case of a high photon energy measurement that gives an electron a definite position and an indefinite velocity, would its "real" velocity be considered a hidden variable?

2) In the 'toy' Bell model, each detector generates and receives signals containing mismatches, and from these derives the overall mismatches. Non-locality could then be each detector generating further mismatches depending on those it receives from the other detector (?). If so, I can see how non-locality can increase the total mismatches beyond those of a local-realist model.

But I can't see the equivalent for non-realism. Here the detector angle would only determine the probability of its generating a certain number of mismatches. But these being random, they could either increase or reduce the original number. And so on average would not increase the overall mismatches beyond those of the local-realist model.

 

[Nugatory]

I got the "causality" quote from a 2009 site. But today's <https://en.wikipedia.org/wiki/Bell's_theorem> says "Locality is short for local relativistic causality", so in general the two seem to be taken to be the same (?).

Maybe not in general, but in this context, yes, "causality" is being used to mean the principle that causes have to happen before effects and cannot propagate at faster than the speed of light. That's pretty much equivalent to "locality". (But see my cautionary note from #19).

1) Are "local hidden variables" and "local realism" the same thing?

I'm going to repeat my cautionary note from #19: There's only so much value you're going to get out of using natural language to categorize these ideas. Clarity is in the mathematical statements. But with said...

And if so, in the case of a high photon energy measurement that gives an electron a definite position and an indefinite velocity, would its "real" velocity be considered a hidden variable?

It would be if you proposed a candidate theory in which the electron has a property that you're going to call "real velocity", and from which the results of some interesting measurements can be predicted. Some other candidate theory might assume the existence of some other hidden variables.

2)...But I can't see the equivalent for non-realism.

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.

 

[jeremyfiennes]

Thanks. Time for reflection required!

 

[jeremyfiennes]

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?

 

[zonde]

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?

You can look at criticism of that article: https://arxiv.org/abs/0809.4000

 

[jeremyfiennes]

Ok. I did. Thanks.

 

[DrChinese]

You can look at criticism of that article: https://arxiv.org/abs/0809.4000

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.

 

[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.