Understanding bell's theorem: why hidden variables imply a linear relationship?

[lugita15]

DrC mentioned in a few threads that there are PDC setups where photons not entangled in polarization give identical results when the analyzing polarizers are aligned, but at other settings the results differ from entanglement results.

As DrChinese said, he was talking about identical behavior when both polarizers are oriented at a particular special angle. That's very different from identical behavior at identical angles for all angles.

I just tried to present a simpler situation where, I thought, you'd also get identical results at identical polarizer settings even though the disturbances incident on the analyzing polarizers aren't entangled in polarization.

Well, your scenario doesn't seem to work, and it seems like any such scenario cannot work even in principle.

We're only talking about the situation where x and x1 are the same (where, equivalently, x-x1 = 0, where Theta = 0, where polarizers a and b are aligned).

I think you're misunderstanding what x and x1 are. I'm defining x1 to be the (common) orientation of polarizers a1 and b1, and x to be the (common) orientation of a and b.

So, what you wrote seems to be in agreement with the prediction that the alternate setup will produce identical results at identical settings of the analyzing polarizers.

No, I'm not in agreement with that, unless by "identical settings" you mean that all four polarizers are aligned, and I doubt you mean that. But I maintain that if polarizers a1 and b1 are oriented at x1, and polarizers a and b are oriented at x, where x and x1 are different, then you no longer have to get identical behavior.

If your step 1 applies to nonentanglement setups where the local deterministic (LD) predictions agree with the QM predictions, but where both of those disagree with your prediction, then it would seem that something is wrong in saying that your proof proves that local determinism is incompatible with QM.

But it seems clear to me that step 1, identical behavior at identical angles for all angles, is a special property of quantum entanglement, so the logic doesn't work for nonentanglement setups.

If you instead state your local determinism assumptions this way:

1. Assumption: Determinism holds. (ie., identical antecedent conditions always produce the same results)
2. Assumption: Locality holds. (ie., events on one side of the experiment don't affect events on the other side)

Then, do these assumptions lead to,

If C, then A or B *?

These assumptions would suffice to get to "If C, then A or B" if we also assume that the two entangled particles have identical antecedent conditions, which I think is an overly restrictive assumption. I think it's cleaner just to assume identical behavior at identical angles, since it's an experimental prediction of QM.

 

[Delta Kilo]

Strangely, I am forced to agree with Bill on that. Step 3 of the 'proof' in #164 is only applicable to case (c), that is hypothetical theoretical result for a single pair which cannot be directly measured. The transition to step 4 of the proof is not made explicit and as such is wide open to the kind of attacks we see here.

The missing ingredient is the probability distribution ρ(λ), independent of measurement angles. Having stable probability distribution which does not change from one experiment to another is what holds the whole thing together. It brings the necessary repeatability and allows one to estimate expectation values for different angles in separate experiments.

Frankly, the math of Bell's theorem is so simple, I don't see a point in trying to avoid it. It only leads to unnecessary confusion.

 

[DrChinese]

Strangely, I am forced to agree with Bill on that. Step 3 of the 'proof' in #164 is only applicable to case (c), that is hypothetical theoretical result for a single pair which cannot be directly measured. The transition to step 4 of the proof is not made explicit and as such is wide open to the kind of attacks we see here.

The missing ingredient is the probability distribution ρ(λ), independent of measurement angles. Having stable probability distribution which does not change from one experiment to another is what holds the whole thing together. It brings the necessary repeatability and allows one to estimate expectation values for different angles in separate experiments.

Frankly, the math of Bell's theorem is so simple, I don't see a point in trying to avoid it. It only leads to unnecessary confusion.

Now he has managed to derail you too. LOL. You have to remember that Bill's real point has nothing to do with lugita15's proof, he is trying to say that counterfactual reasoning does not have a physical basis and also does not have a basis in QM. Both of these go without saying, counterfactual cases are not physical by definition. But they clearly exist in the local realistic world, which is what Bell rules out. lugita15's example is perfect, and Bill is using intentionally convoluted logic to push his desired conclusion.

Obviously, lugita15's example follows Bell and Bell should not be in question here. Look at the title of the thread and you will see what a good job Bill has done with his latest hijack.

 

[ThomasT]

I think the situation you are referring to is as follows:

With a single Type I PDC crystal, you get pairs of photons coming out which have the properties of being entangled on some bases (momentum for example) but are NOT polarization entangled. The input is V> and the output is the pair H>H>. Therefore they match ONLY at the special angles 0/90/etc. They do not match at 45 degrees (for both) or any other angle, instead they follow Product state statistics.

My apologies if something I said implied otherwise.

Ok, thanks. I had misunderstood the situation. Not your fault.

A bit off topic, but I'm curious, wrt this situation how do the results vary for other orientations (the orientations that don't always produce identical results) of the aligned polarizers?

 

[ThomasT]

I think you're misunderstanding what x and x1 are. I'm defining x1 to be the (common) orientation of polarizers a1 and b1, and x to be the (common) orientation of a and b.

Ok. I misunderstood.

I maintain that if polarizers a1 and b1 are oriented at x1, and polarizers a and b are oriented at x, where x and x1 are different, then you no longer have to get identical behavior.

Ok. When x and x1 are aligned, then you'd always get identical results, and as x is offset from x1 then the percentage of identical results would vary as cos²|x-x1|. Is that correct?

Anyway, it appears that my critique of your argument using the alternate setup doesn't work.

But it seems clear to me that step 1, identical behavior at identical angles for all angles, is a special property of quantum entanglement, so the logic doesn't work for nonentanglement setups.

Yes, that seems to be the case. Thanks.

As far as I understand then, your argument does seem valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with the assumption(s) of local determinism.

But I haven't thoroughly considered bill's or DK's objections yet.

 

[Delta Kilo]

But I haven't thoroughly considered bill's or DK's objections yet.

My only objection is to replacing math with handwaving. It makes it all too easy to overlook some small detail and leave the proof open to attacks. Sorry if I didn't make myself clear.

 

[lugita15]

As usual, the error is hidden in the fuzzy description of the experiment. What is the experiment we are talking about?

We are talking about the experiment discussed by Herbert http://quantumtantra.com/bell2.html. Basically, it involves sending two polarization-entangled photons to two different polarizers, set independently by two people.

More specifically, what does "The result at -30° differs from the result at 0°" mean?

You have told me before that you believe in counterfactual definiteness, in the sense that the question "What would happen if you send this photon through a polarizer oriented at angle x?" always has a well-defined answer for all angles x, regardless of whether you happen to actually send the photon through a polarizer oriented at that particular angle. Of course, you can phrase this question in different ways based on whether you did or did not carry out the measurement; if you did carry out it becomes "What result did you get when you sent the photon through a polarizer oriented at angle x", and if you didn't it becomes "What result would you have gotten if you had sent the photon through a polarizer oriented at angle x?". But that is just the fault of linguistics, and the basic question is the same regardless of whether in any particular case it happens to be a factual question or a counterfactual question, since in either case you admitted that the question is valid and has a meaningful answer.

And since you believe in local determinism, for you the fact of identical behavior (of the two photons in entangled pair) at identical angle settings (of the two polarizers) implies that the answer to this question must be determined in advance for all angles x. And not only that, the fact of identical behavior at identical angle settings also implies that the answers to the question for the two photons must be the same for all angles x. Thus for all angles x we can meaningfully talk (even if we don't know what it is) about "the result we would have if we send either photon in this pair through a polarizer oriented at angle x" independent of which of the two photons we're talking about or what angle settings we choose for the two polarizers when we send the two photons through. It is this long thing in quotes that we abbreviate as "the result at angle x".

So when I say "The result at -30° differs from the result at 0°", I really mean "The result you would get if you send either of the photons in the pair through a polarizer oriented at -30° is different from the result you would get if you send either of the photons in the pair through a polarizer oriented at 0°".

Let us examine this more closely.

It appears we are comparing an outcome at -30° with an outcome at 0°, for example we have an entangled pair of photons one heads of to the -30 instrument and another to the 0 instrument and we compare if the outcomes match. This makes sense. But then you talk of "The result at 0° differs from the result at 30°". Where did 30° come from? We had two entangled photons which we measured at -30 and 0, all of a sudden 30 appears which would suggest one of the following possibilities:

a) You were magically able to measure two photons at 3 angles (-30°, 0°, +30°)
b) You measured two different pairs of entangled photons.
c) You are not talking about an actual performable experiment but about a hypothetical theoretical what might have been for a single pair.

I mean possibility c). I am talking about the result I would get if I perform a particular experiment on a particle, regardless of whether I actually perform that experiment. Now how do I connect this to real experiments, which are obviously concerned with possibility b)? I make the crucial assumption, which I expect that you disagree with or think is misleading, that the following two probabilities are always equal:
1. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is actually oriented at angle x.
2. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is NOT actually oriented at angle x, but instead some different angle y.

Now why do I assume that these two probabilities are equal? Because I am assuming that the answers to the following two questions are always the same:
1. What result would you get if you send this photon through a polarizer oriented at angle x, given that the polarizer is actually oriented at angle x?
2. What result would you get if you send the photon through a polarizer oriented at angle x, given that the polarizer is NOT oriented at angle x, but instead some different angle y?

And why do I assume that these questions have the same answer? That seems to me to be a consequence the no-conspiracy condition: the properties (or answers to questions) that are predetermined cannot depend on the specific measurement decisions that are going to be made later, since by assumption those decisions are free and independent.

I hope that clarifies some of my reasoning.

 

[lugita15]

Strangely, I am forced to agree with Bill on that. Step 3 of the 'proof' in #164 is only applicable to case (c), that is hypothetical theoretical result for a single pair which cannot be directly measured.

Yes, but even if we can't directly measure "the result at x" (meaning "the result we would get if we send either of the photons through polarizer oriented angle x") for all angles x, the assumption of counterfactual definiteness still allows us to talk meaningfully about it even if we don't know what it is.

The transition to step 4 of the proof is not made explicit and as such is wide open to the kind of attacks we see here.

The reasoning required in getting from step 3 to step 4 is utterly trivial. It is a basis law of probability that if the statement "If M then N" is true, then the probability of M is less than or equal to the probability of N. So if the statement "If C then A or B" is true, then the probability of C is less than or equal to the probability of A or B.

The missing ingredient is the probability distribution ρ(λ), independent of measurement angles. Having stable probability distribution which does not change from one experiment to another is what holds the whole thing together. It brings the necessary repeatability and allows one to estimate expectation values for different angles in separate experiments.

I don't think any discussion about the distribution of hidden variables is required. All we need to do is use the no-conspiracy condition to link measured probabilities of outcomes to unmeasured (i.e. "hypothetical") probabiities of outcomes.

Frankly, the math of Bell's theorem is so simple, I don't see a point in trying to avoid it. It only leads to unnecessary confusion.

To me, Bell's original math, for instance integration over hidden variables, is unnecessarily complicated. It often obscures salient philosophical points in the reasoning, such as how locality creates the necessity of hidden variables. So I much prefer a proof like Herbert's, even if it carries the vaguaries of natural language, because I am confident that the reasoning can be made precise.

 

[billschnieder]

I hope that clarifies some of my reasoning.

No it doesn't because you did not adress what I said at all:

You said you were referring to (c). However, QM predictions and every possible experiment that can ever be performed falls under case (b). Two outcomes from a single set of particles CAN place constraints on a third outcome from the very same set of particles. However, there is no basis in physics or logic for two outcomes from one set of particles to constrain the outcome of a different set of particles not correlated with the first one!


How does your reasoning survive the above? I do not yet see from your response that you even understand the criticism.

You say:

1. What result would you get if you send this photon through a polarizer oriented at angle x, given that the polarizer is actually oriented at angle x?
2. What result would you get if you send the photon through a polarizer oriented at angle x, given that the polarizer is NOT oriented at angle x, but instead some different angle y?

As explained above, it is well and good so long as we are talking about just the one photon but then you can't compare it to anything QM or experiment will tell you. There is no reason to expect a mathematical relationship between the results at angle "x" and "y" for one photon to be the same as the relationship between the result at "x" for one photon and the result "y" for a different photon not correlated with it. This is pretty much what you are trying to do here and it is wrong.

 

[lugita15]

Ok. When x and x1 are aligned, then you'd always get identical results, and as x is offset from x1 then the percentage of identical results would vary as cos²|x-x1|. Is that correct?

It is true that you get always get identical results when x and x1 are aligned. But if they're not aligned, then the probability that you get identical results is messy: it's .5+.5sin^4(x-x1)+.5cos^4(x-x1) (see post #170 for an explanation).

As far as I understand then, your argument does seem valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with the assumption(s) of local determinism.

And that is precisely what I claim the argument achieves. You can come up with (arguably far-fetched) local deterministic explanations of many if not most phenomena in quantum mechanics, but quantum entanglement alone has special preperties that make it different.

But I haven't thoroughly considered bill's or DK's objections yet.

You have a few different options. You can abandon local determinism, you can hide behind an experimental loophole like zonde does, you can escape via superdeterminism like Nobel Laureate Gerard t'Hooft. But I don't think objections of the kind that E.T. Jaynes or billschneider raise carry any water. It seems like these objections generally question whether proponents of Bell are really applying the no-conspiracy condition, i.e. the exclusion of superdeterminism, correctly in the course of the reasoning.

 

[billschnieder]

But I don't think objections of the kind that E.T. Jaynes or billschneider raise carry any water. It seems like these objections generally question whether proponents of Bell are really applying the no-conspiracy condition, i.e. the exclusion of superdeterminism, correctly in the course of the reasoning.

Like I thought, you do not even understand the argument. Assuming that you want to understand it, let me put it in even simpler terms. The triangle inequality gives you a relationship between the three sides of a single triangle AB + BC >= AC. It is valid for a single triangle but it is not valid if you take AB from one triangle and BC from a different triangle and AC from yet another triangle.

The argument which you refused to address above is telling you that for a single triangle, the lengths AB and BC place constraints on the third length AC by virtue of the fact that they belong to the same triangle (this is what the inequality is all about). But the lengths AB and BC from two separate triangles can not possibly place any constraints on the length AC of a third triangle! This is exactly the naive mistake you and others keep making over and over except using probabilities to do it.

 

[lugita15]

No it doesn't because you did not adress what I said at all:

You said you were referring to (c). However, QM predictions and every possible experiment that can ever be performed falls under case (b). Two outcomes from a single set of particles CAN place constraints on a third outcome from the very same set of particles. However, there is no basis in physics or logic for two outcomes from one set of particles to constrain the outcome of a different set of particles not correlated with the first one!


How does your reasoning survive the above?

I thought I did address this, but let me try again. We send numerous photon pairs through the polarizers, orienting the two polarizers at a variety of angle settings. Sometimes we turn the polarizers to the specific angle settings x and y, other times we don't. We observe that we when we do turn the two polarizers to x and y, we get that the probability that the photons do the same thing when sent through these polarizers is a certain number p. From this we conclude, via the reasoning I lay out in my post #182, that even for the photon pairs for which we don't turn the two polarizers to x and y, it is still true that we would have a probability p of the photons doing the same thing if we do turn the polarizers to the angles x and y. We are assuming that the photons we don't subject to a certain experiment would, if we do subject them to that experiment, behave the same as those that we do subject to that experiment, because how can the question of what would a photon do under circumstance x be dependent on whether it is actually under circumstance x? After all, the photons presumably do not "know" yet whether they will face cricumstance x, so don't you believe that the photon's behavior under all possible circumstances is predetermined and thus independent of the circumstances it faces?

 

[lugita15]

Like I thought, you do not even understand the argument. Assuming that you want to understand it, let me put it in even simpler terms. The triangle inequality gives you a relationship between the three sides of a single triangle AB + BC >= AC. It is valid for a single triangle but it is not valid if you take AB from one triangle and BC from a different triangle and AC from yet another triangle.

I am certainly willing to try to understand your point of view, and I apologize if I do not at the moment. I agree that it would be invalid to take sides of arbitrary triangles and claim that they must satisfy the triangle inequality. But I think what I'm doing is akin to arguing that the three triangles must be congruent, and thus we can freely plug in side lengths from any of the three triangles into the inequality.

 

[San K]

Assuming they end up polarized parallel, they will go through the polarizers and reach the second pair of polarizers a and b. Then the probability for each of the photons to go through its respective polarizer is the cosine squared of the difference between the polarization angle of the photon and the setting of the polarizer it now encounters. But now the photons are no longer entangled, so these probabilities are independent of each other: even if a and b have the same orientation, it need not be true that the two photons must do the same things. Instead, each photon has an independent probability cos²(x-x1) of going through, where x is the angle setting of polarizers a and b, and x1 is the angle of polarizers a1 and b1.

just to make sure we got your post above correctly...

Case 1: let's assume a1 and b1 are aligned to each other and polarizers a and b are aligned to each other (but not to a1 and b1)

i.e. a1 and b1 are at, say -- 0 degrees and a1 and b1 are at, say -- 30 degrees.

in that case the photons will show exactly same behavior at a1 and b1...however when they reach a and b...they might not show same behavior?
i.e. photon a might pass through polarizer a, however its not necessary that photon b will pass through polarizer b?

Case 2: all four are aligned i.e. a1 is aligned to b1 and a and b

i.e. a1, b1, a, b all are at, say -- 0 degrees

if the photons pass through a1 and b1, what can we say about their behavior at a and b?

 

[lugita15]

just to make sure we got your post above correctly...

Case 1: let's assume a1 and b1 are aligned to each other and polarizers a and b are aligned to each other (but not to a1 and b1)

i.e. a1 and b1 are at, say -- 0 degrees and a1 and b1 are at, say -- 30 degrees.

in that case the photons will show exactly same behavior at a1 and b1...however when they reach a and b...they might not show same behavior?
i.e. photon a might pass through polarizer a, however its not necessary that photon b will pass through polarizer b?

Yes, exactly. One photon may go through and the other may not (see post #170 for all the probabilities).

Case 2: all four are aligned i.e. a1 is aligned to b1 and a and b

i.e. a1, b1, a, b all are at, say -- 0 degrees

if the photons pass through a1 and b1, what can we say about their behavior at a and b?

They will both be guaranteed to go through. Think about it, once they go through a1 and b1, they lose their entanglement and become polarized in the 0 degree direction. Thus they have a 100% probability of going through another 0 degree polarizer.

That's a general rule: once a photon has come out of one polarizer, regardless of whether it's entangled or not, it will now be polarized in the direction of the polarizer and thus will go through any number of subsequent polarizers oriented in the same direction.

 

[Delta Kilo]

I agree that it would be invalid to take sides of arbitrary triangles and claim that they must satisfy the triangle inequality. But I think what I'm doing is akin to arguing that the three triangles must be congruent, and thus we can freely plug in side lengths from any of the three triangles into the inequality.

Well, no, I'm afraid this is not going to work. The individual triangles are not congruent and it would be wrong to plug side lengths from different triangles into the inequality. This is exactly the kind of things Bill is talking about.

The correct analogy would be a source continuously spewing out lots and lots of triangles of all sorts of sizes and shapes. The inequality obviously holds for the average values of side lengths of all triangles. Now the key feature of the theorem is that given large enough run, we can correctly estimate the averages by measuring only some small percentage of all triangles and only one side at a time, provided the source is not influenced by our choice of which side of which triangle to measure.

 

[harrylin]

Well, no, I'm afraid this is not going to work. The individual triangles are not congruent and it would be wrong to plug side lengths from different triangles into the inequality. This is exactly the kind of things Bill is talking about.

The correct analogy would be a source continuously spewing out lots and lots of triangles of all sorts of sizes and shapes. The inequality obviously holds for the average values of side lengths of all triangles. Now the key feature of the theorem is that given large enough run, we can correctly estimate the averages by measuring only some small percentage of all triangles and only one side at a time, provided the source is not influenced by our choice of which side of which triangle to measure.

Finally a good example - thanks! (And of course, the sample must not be biased in some way).

 

[lugita15]

Well, no, I'm afraid this is not going to work. The individual triangles are not congruent and it would be wrong to plug side lengths from different triangles into the inequality. This is exactly the kind of things Bill is talking about.

Well, I think I'm interpreting the triangle analogy a bit differently from you. For me, an individual triangle corresponds to the set of all photon pairs for which the polarizers are set to some particular angle settings x and y. And the side lengths of a triangle correspond to probabilities. So we have three triangles we're interested in, corresponding to orienting the polarizers to -30° and 0°, 0° and 30°, and -30° and 30°. For each of the triangles, we only know one of the side lengths. For instance, for the angle settings -30° and 0° we only know the answer to the question "what is the probability that, for a randomly chosen particle pair, the result at -30° differs from the result at 0°?"

Now billschneider is raising the objection that I am taking one side length out of each of these triangles, since that's all I know, and I am plugging them into the triangle inequality. I plead guilty to that charge, but I have good reason, namely that I am arguing that the three triangles are congruent. In other words, I am arguing that following two questions have the same answer, even though the first one is found experimentally and the second is not:
1. Out of the particle pairs for which you actually orient the polarizers at -30° and 0°, what is the percentage of particle pairs for which the two photons would give different results at -30° and 0°?
2. Out of the particle pairs for which you actually orient the polarizers at (say) 0° and 30°, what is the percentage of particle pairs for which the two photons would give different results at -30° and 0°?

And my basis for asserting this, for equating the measured probability and the unmeasured (and presumably unmeasurable) probability, is an application of the no-conspiracy condition. So this seems to be the dividing line between proponents of Bell and people like billschneider: it is a disagreement about whether the no-conspiracy condition is being applied property.

 

[ThomasT]

My only objection is to replacing math with handwaving. It makes it all too easy to overlook some small detail and leave the proof open to attacks. Sorry if I didn't make myself clear.

My misunderstandings are nobody's fault but my own. What I'm trying to do is precede/accompany/interpret the mathematical representation(s) with some sort of conceptual understanding.

I currently have no doubt that Bell's (and Herbert's and lugita15's) local hidden variable (local realistic, local determistic) formulation(s) is (are), definitively, incompatible with QM (and experiment) wrt entanglement setups. The only question in my mind has to do with interpreting the meaning (ie., what might be reasonably inferred about the underlying reality) of Bell's (and equivalent) theorem(s). That is, do experimental violations of Bell-type inequalities imply the existence of nonlocal transmissions in nature, or is there some other explanation for their violation that would preclude/supercede that inference? It's wrt that question that I find Bill's (and others) objections to Bell's (and Bell-type) formulation(s) at least worthy of consideration (insofar as I think that I don't yet really understand them, and insofar as I have this intuitive sense that there really is a way of understanding BI violations that does preclude/supercede the inference of nonlocality in nature).

 

[ThomasT]

As far as I understand then, your argument does seem valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with the assumption(s) of local determinism.

And that is precisely what I claim the argument achieves.

Ok, we're basically agreed on that then. And thanks for taking the time for us to hash out your argument via email.

Let me qualify my above statement somewhat. To wit, in my understanding, your argument is valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with a particular formulation involving the assumption(s) of local determinism.

... quantum entanglement alone has special preperties that make it different.

I'm not so sure about that. That is, I think it just might boil down to entangled entities having motional properties that are related in classically understandable ways. And, of course, maybe not. The problem is that the extant experimental evidence doesn't definitively answer that question. Just not enough data yet.

 

[lugita15]

Ok, we're basically agreed on that then. And thanks for taking the time for us to hash out your argument via email.

Let me qualify my above statement somewhat. To wit, in my understanding, your argument is valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with a particular formulation involving the assumption(s) of local determinism.

What do you mean a particular formulation? Do you believe that there is any step in my proof that any local determinist could possibly disagree with, even in principle? I thought you agreed with me that step 1 is an experimental prediction of QM, step 2 is a logical consequence of local determinism, step 3 follows from steps 1 and 2, and steps 4-6 are just application of the laws of probability. So where am I "formulating" or restricting things in any way?

 

[ThomasT]

What do you mean a particular formulation? Do you believe that there is any step in my proof that any local determinist could possibly disagree with, even in principle? I thought you agreed with me that step 1 is an experimental prediction of QM, step 2 is a logical consequence of local determinism, step 3 follows from steps 1 and 2, and steps 4-6 are just application of the laws of probability. So where am I "formulating" or restricting things in any way?

The particular form of steps 1 and 2 doesn't matter, but their content has to be some sort of expression of the assumptions of locality and determinism.

It then follows that:

3. If C, then A or B
4. P(C)≤P(A or B) (P denotes probability)
5. P(A or B)≤P(A)+P(B)
6. P(C)≤P(A)+P(B)

Where:
C = different results at (-30_30)
A = different results at (-30_0)
B = different results at (30_0)
-------------------------------------------
Your conclusion, based on the way you've formulated your line of reasoning, is that the assumptions of locality and determinism necessarily lead to the prediction in your step 6. (a prediction which is incompatible with QM and experiment).

But I, being a local determinist, might have a somewhat different way of looking at the experimental situation, which might be compatible (or at least not incompatible) with the assumptions of locality and determinism.

The salient point will be that you have not taken into consideration the known behavior of light wrt crossed polarizers in situations where locality and determinism are not (at least not normally or historically) in question.

 

[harrylin]

The particular form of steps 1 and 2 doesn't matter, but their content has to be some sort of expression of the assumptions of locality and determinism.

It then follows that:

3. If C, then A or B
4. P(C)≤P(A or B) (P denotes probability)
5. P(A or B)≤P(A)+P(B)
6. P(C)≤P(A)+P(B)

Where:
C = different results at (-30_30)
A = different results at (-30_0)
B = different results at (30_0)
-------------------------------------------
Your conclusion, based on the way you've formulated your line of reasoning, is that the assumptions of locality and determinism necessarily lead to the prediction in your step 6. (a prediction which is incompatible with QM and experiment). [..]

Hi Tom, that is wrong: as far as we know, no experiment has provided such hypothetical (perhaps even impossible) results. We discussed that in the thread on Herbert's proof, which is still open: https://www.physicsforums.com/showthread.php?t=589134

 

[ThomasT]

Hi Tom, that is wrong ...

Hi Harry, what's wrong?

 

[harrylin]

Hi Harry, what's wrong?

The word "experiment" - as elaborated in the thread on Herbert's proof. Real experimental results are not as Herbert claims.

 

[lugita15]

Your conclusion, based on the way you've formulated your line of reasoning, is that the assumptions of locality and determinism necessarily lead to the prediction in your step 6. (a prediction which is incompatible with QM and experiment).

Yes, but I don't see the point of the qualification "based on the way I formulated my line of reasoning". Who cares how I formulated things? What matters is that I have a valid logical proof that local determinism is incompatible with the experimental predictions of QM.

But I, being a local determinist, might have a somewhat different way of looking at the experimental situation, which might be compatible (or at least not incompatible) with the assumptions of locality and determinism.

It doesn't matter how you look at things, does it? Wouldn't the only way you can dispute my conclusion be to demonstrate that the reasoning in one of my steps is logically invalid?

 

[lugita15]

I currently have no doubt that Bell's (and Herbert's and lugita15's) local hidden variable (local realistic, local determistic) formulation(s) is (are), definitively, incompatible with QM (and experiment) wrt entanglement setups.

I'm not just proving that one particular formulation contradicts the predictions of QM. I'm trying to show that ALL possible local deterministic theories (excluding superdeterminism) must differ from the experimental predictions of QM.

That is, do experimental violations of Bell-type inequalities imply the existence of nonlocal transmissions in nature, or is there some other explanation for their violation that would preclude/supercede that inference? It's wrt that question that I find Bill's (and others) objections to Bell's (and Bell-type) formulation(s) at least worthy of consideration (insofar as I think that I don't yet really understand them, and insofar as I have this intuitive sense that there really is a way of understanding BI violations that does preclude/supercede the inference of nonlocality in nature).

ThomasT, you have a few options to evade the conclusion of the argument. You can become a superdeterminist, or you can rely on the various experimental loopholes that plague current Bell tests. Since these options are still open, currently practical Bell tests do not suffice to definitively disprove local determinism.

 

[lugita15]

The word "experiment" - as elaborated in the thread on Herbert's proof. Real experimental results are not as Herbert claims.

OK, but that's just due to various experimental loopholes. But the only way a loophole-free Bell test wouldn't give the expected results is if QM were wrong experimentally, which is certainly possible, but doesn't the fact that each of the loopholes have been closed seperately lead you to the conclusion that loopholes aren't likely to be the cause of observed Bell inequality violations?

 

[ThomasT]

The word "experiment" - as elaborated in the thread on Herbert's proof. Real experimental results are not as Herbert claims.

I see. Even so, I like the consciseness of lugita15's proof, and I agree with him and others who say that experimental loopholes aren't the cause of BI violations.

 

[ThomasT]

Yes, but I don't see the point of the qualification "based on the way I formulated my line of reasoning". Who cares how I formulated things?

The formulation is what leads to the prediction (step 6) that contradicts QM. One could rather, for example, show that similar setups where locality and determinism aren't in question also give a cos²θ correlation between angular difference and rate of detection.

It doesn't matter how you look at things, does it?

I think it can make all the difference.

Wouldn't the only way you can dispute my conclusion be to demonstrate that the reasoning in one of my steps is logically invalid?

What's the reason for stating things in terms of step 3? That is, the first thing I would notice in trying to model the situation where pairs of entangled photons are being analyzed by crossed polarizers is that the correlation between the angular difference and the rate of identical detection can't be a linear one, and that this is understandable in a local deterministic world based on many optics experiments.

I'm not just proving that one particular formulation contradicts the predictions of QM. I'm trying to show that ALL possible local deterministic theories (excluding superdeterminism) must differ from the experimental predictions of QM.

Maybe you've done that. Maybe Bell did that. I don't know. What I do know (I think) is that your, and Bell's, formulations are incompatible with the predictions of QM.

The question, for me, in interpreting your proof is: is it more important that your proof contains the assumptions of locality and determinism, or is it more important how you actually framed the proof?

Anyway, as it stands it does logically arrive at a conclusion that's incompatible with QM. So, forget my ill-formed nitpicking reservations for the time being.

 

[lugita15]

ThomasT, here is something that may make you see things more clearly. We can reduce the number of assumptions of the proof to one: "For any given particle pair, statements A, B, and C have well defined truth values which are independent of what angles the two polarizers happen to be set at, and independent of which of the two particles in the pair we are talking about." I hope you agree that a local determinist, who agrees with identical behavior at identical angle settings, MUST agree with this assumption. And I hope you also agree that this assumption alone is enough to get to "If C, then A or B" and thus all the rest of the proof.

 

[harrylin]

OK, but that's just due to various experimental loopholes. But the only way a loophole-free Bell test wouldn't give the expected results is if QM were wrong experimentally, which is certainly possible, but doesn't the fact that each of the loopholes have been closed seperately lead you to the conclusion that loopholes aren't likely to be the cause of observed Bell inequality violations?

That would be unlikely if the alternative wasn't even more unlikely. There may be a mechanism (or mechanisms) due to which such loopholes favour apparent Bell inequality violations. Also experimenter bias could play a role in the kind of loopholes that remain, as most of these experiments seem designed to "prove" QM.

 

[lugita15]

That would be unlikely if the alternative wasn't even more unlikely. There may be a mechanism (or mechanisms) due to which such loopholes favour apparent Bell inequality violations.

But how is it that the kind of particles for which we have closed the communication loophole just happen to be the kind of particles that exploit the fair sampling loophole, and vice versa?

 

[billschnieder]

Yes, but I don't see the point of the qualification "based on the way I formulated my line of reasoning". Who cares how I formulated things? What matters is that I have a valid logical proof that local determinism is incompatible with the experimental predictions of QM.
It doesn't matter how you look at things, does it? Wouldn't the only way you can dispute my conclusion be to demonstrate that the reasoning in one of my steps is logically invalid?

How can you continue to claim the argument is logical?

When I wrote the following:

b) You measured two different pairs of entangled photons.
c) You are not talking about an actual performable experiment but about a hypothetical theoretical what might have been for a single pair.

You admitted that your "probability" equation was for scenario (c). And you also admit that QM and real experiments are for (b). So you still have to account for that disconnect. Why does it make sense to compare apples (b) with oranges (c) as you do in your arguments? Surely you understand that local causality does not give you that justification since both (b) and (c) mean completely different things even when local causality is true. I do not yet see a genuine attempt by you to justify why you would think your probability expressions should apply to QM predictions or experimental results. Simply saying it should is not enough. Remember for (b) we have 64 degrees of freedom, for (c) you have only 8. You can't just use the probabilities you calculate under scenario (c) and compare with scenario (b). What part of this don't you understand?

When I asked "what physical or logical basis do you have to expect the result of measurement on one set of photons to restrain the results of measurements on a different set of photons?". Unless you can answer this question, you can not link (c) to (b). And you MUST be able to link (c) to (b) in order use the QM predictions and experimental results the way you are doing.

Now, in a recent post in response to the triangle inequality you suggested that the triangle was the whole set of photons and that for some yet specified or justified reason one set of photons was congruent with a different set. This claim is the same as simply stating that (c) and (b) are the same without any justification. This claim is the same as saying measurements performed on one set of photons should be able to restrain the results of measurements performed on a different set of photons. How is that possible? You do not explain or even attempt a justification, yet you continue to think that your argument is logical. To see how ridiculous this argument is consider the following example:

We have a die which we throw on a table with little square depressions which exactly fit the die, point being that when the die settles it will always have one of it's vertical sides facing north, another facing south and the others east and west. The actual numbers on those sides will vary randomly for a fair die. We then throw the die and read of the number facing north (say a N6). For the same die, the outcome N6 (6 is facing North), now restraints the other possibilities for that same die. For example, S6 is impossible. There is no logical or physical reason to expect a different die, *congruent* with the first one, thrown at a different time, from being restrained by the outcome we obtained on the first one.

Your argument and many others like it really just demonstrate a fallacy of logic by misapplication of probability calculated under one scenario to a different incompatible scenario. Interestingly such arguments persist when we discuss measuring 3 properties on 2 photons. If we had 3 entangled photons to measure 3 properties, we would not need to use a different set of photons and we will find that the experiment is the same scenario as the inequalities, and the inequalities are never violated. Also, if we could recover the two photons and measure the third property, we would find no-violation. It is therefore clear that the inability to measure the third property on the same set of photons as the other two is the sole reason for the violation, not non-locality.

This is what Boole discovered more than 100 years ago! In fact inequalities like Bell's were routinely used long before Bell to test the compatibility of different samples in which case violation was interpreted to mean that the samples could not possibly have been obtained from the same population.

 

[harrylin]

But how is it that the kind of particles for which we have closed the communication loophole just happen to be the kind of particles that exploit the fair sampling loophole, and vice versa?

Sorry, I don't understand your question. Experiments (and thus experimenters) exploit loopholes by their design.

To give an illustration from SR: an experimenter wants to prove that distant clock synchronisation is absolute because the speed of light is truly isotropic relative to the laboratory. For that purpose he synchronises two clocks next to each other and slowly transports one of them to a far distance. Next he sends a light pulse in two directions and measures the time intervals with the two clocks, showing that they are equal. Of course there is a loophole which he exploits. Other experiments such as the Michelson interferometer exploit again other loopholes if performed for that purpose.

Thus you could ask, how is it that the kind of particles for which we have closed the length contraction loophole just happen to be the kind of particles that exploit the synchronisation loophole, and vice versa - thus suggesting that SR is wrong. However, that would merely be due to a lack of insight in how SR phenomena work, and it seems reasonable to assume that it's similar with QM. While I think to understand how SR phenomena work, possibly nobody - certainly not me! - thinks to understand how QM phenomena work.