Where is the flaw with predetermined entanglement state?

[LsT]

Not to repeat what is already said, my thoughts are more or less the 4 points mentioned here:

http://physics.stackexchange.com/qu...ement-an-illusion-based-on-a-wrong-assumption

The upvoted answer to what gpgemini's said is that this assumption is essentially what local hidden variables theory propose, and by that its ruled out because experiments demonstrate Bell's inequality violation.

But to my understanding, a predetermined state is a different thing than local hidden variable theory:

Within a local hidden variable theory, entangled particles somehow supposed to have infinite (?) embedded information with them about what will the outcome be in any possible measurement, such as the outcome will emulate what is interpreted as puzzling FTL instantaneous action in other QM interpretations. Referring to this, Bell showed that there are limits to the probability distribution of all possible outcomes if the information the entangled particles carry, is predetermined the moment the entangled pair is created and not changed until measurement. Experiments repeatedly violate local hidden variables probability limits, ruling them out (assuming no loopholes)

But with a predetermined entanglement, particles do not carry ANY information, just exists a definite "opposing" state between them (such as a specific polarization angle difference or opposing spin direction in x axis). Bells inequality clearly does not apply here, because you cannot calculate probability distribution limits of NO information. And to my understanding (wrong?) any entanglement experiment using polarization direction of photons, because of Malus law CAN violate the inequality, without ruling out, this predetermined state hypothesis.
...
I am for sure missing something fundamental, but what?

 

[DrChinese]

But with a predetermined entanglement, particles do not carry ANY information, just exists a definite "opposing" state between them (such as a specific polarization angle difference or opposing spin direction in x axis). Bells inequality clearly does not apply here, because you cannot calculate probability distribution limits of NO information. And to my understanding (wrong?) any entanglement experiment using polarization direction of photons, because of Malus law CAN violate the inequality, without ruling out, this predetermined state hypothesis.
...
I am for sure missing something fundamental, but what?

You are saying there is predetermined entanglement and yet it is local. How does that make sense? If there is predetermined entanglement between 2 particles, and it is an actual state (not just a correlation), then it cannot be local. Do you see why?

And if it IS local and predetermined, and is a correlation, then it cannot agree with the predictions of QM (a la Bell's Theorem).

Please keep in mind that measurement of 2 entangled particles always show 100% predictable results when measured at the same angles. If it is local, this implies predetermination. And the particles MUST be carrying that information, since they cannot be in contact with each other.

 

[LsT]

Hello DrChinese and thank you for your answer.

(Here I think that I should make clear that I am trying to do nothing more than to understand some subjects using my own thought experiments, I am completely aware of my ignorance, and grateful to anyone helps me to reduce it :) )

You are saying there is predetermined entanglement and yet it is local. How does that make sense? If there is predetermined entanglement between 2 particles, and it is an actual state (not just a correlation), then it cannot be local. Do you see why?

Hmm..I guess I have used wrong terminology here and I apologise for this. What I am talking about is an actual state and local of course. Not non-local. Not hidden information. I am hypothesizing an entangled particle pair eg photons that each has a definite very specific polarization angle determined at their inception/creation. The polarization angle of each photon can be "random" but their difference is allways constant. Obviously that is not properly called predetermined (maybe just determined is better?) but that is what I am talking about.

And if it IS local and predetermined, and is a correlation, then it cannot agree with the predictions of QM (a la Bell's Theorem).

If by correlation we exclude that it is an actual state, no I am not talking about this. I understand that Bell's Theorem applies here. But as I stated I think that Bell's inequality does not apply to an actual state because of the lack of (hidden) information. If I am wrong here please enlighten me.

Please keep in mind that measurement of 2 entangled particles always show 100% predictable results when measured at the same angles. If it is local, this implies predetermination. And the particles MUST be carrying that information, since they cannot be in contact with each other.

Please elaborate this. I fail to see why a (local) definite state as I hypothetized above will fail to produce the same results along the lines of QM. (please keep this talk about photons/polarization angle if you get into details).

 

[DrChinese]

Please elaborate this. I fail to see why a (local) definite state as I hypothetized above will fail to produce the same results along the lines of QM. (please keep this talk about photons/polarization angle if you get into details).

By the way, welcome to PhysicsForums, LsT!

This is the conclusion of Bell's Theorem, which shows that QM and a theory as you describe cannot be reconciled. You are saying the state is local and predetermined. Therefore the choice of measurement for Bob's photon cannot influence the outcome of a measurement by Alice on her photon, correct? Your premise is certainly reasonable - Einstein essentially believed as much. But he didn't have the benefit of Bell and Aspect et al (the team who first tested Bell Inequalities).

Please check out one of my Bell Theorem web pages. The easiest for math is:

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

Or read the original Bell:

http://www.drchinese.com/David/EPR_Bell_Aspect.htm

 

[zonde]

I understand that Bell's Theorem applies here. But as I stated I think that Bell's inequality does not apply to an actual state because of the lack of (hidden) information. If I am wrong here please enlighten me.

I like this informal proof of special case for entangled photons https://www.physicsforums.com/showthread.php?p=2817138#post2817138
If you think this proof does not apply to your model it should be easy to point out the place where is the difference.

 

[LsT]

@DrChinese
I like your website! And your explanation of the Bell's Theorem is really nice. So as I started reading your article I stumbled upon this phrase:

EPR provided a proof that says in essence: either there are Hidden Variables, OR particle attributes (such as position, velocity, energy, polarization, etc.) are not real and defined until they are observed

AFAIK hidden variables as an idea came after the EPR paper. If possible please elaborate how this proof is fomulated (just a simple description, give me some points). And I will repeat again that my point of view is that local hidden variables are not the same as a determined state in the way I am describing it in this thread.

@zonde

I like this informal proof of special case for entangled photons https://www.physicsforums.com/showthread.php?p=2817138#post2817138
If you think this proof does not apply to your model it should be easy to point out the place where is the difference.

That is really easy to follow even for me! I'll try to point out where I see the difference(s):

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.

Ok

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.

In my "model/thought experiment", step Three is INDENTICAL to step Two, because no single particle carries any information regarding the the outcome of a single detector, CONTRARY to what hidden variables theory postulates. The "information" eg the existing state is defined only between them, and so only in relation to the angle DIFFERENCE of the detectors. So what we have done until now is to repeat two times a 30 degree difference. And, irrespective of my assumptions, I think there is an error in this statement: the mutual misalignment (lets say that mutual misalignment = AnlgleA - AngleB) is not -30 degrees if we turn the B detector in the opposite direction. The mutual misalignment is 30 degrees, exactly the same with step 2.

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?

Nothing can be expected. We tried only a 30 degree difference, which gives 25% errors. The 60 degree difference results are unknown and depended (in the case of photons/pollarization angle) only in the real/defined pollarization angle between the photons, the angle between detectors and Malus' law.

We assume, following John Bell's lead, that REALITY IS LOCAL.

Maybe local hidden variables are not including all cases of local realism?

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.

As I have pointed, the above is correct only for a local hidden variable theory. In my "model" the outcome is defined/makes sense only in relation to the angle difference of the detectors. So yes, detector A output does not depend on the output of detector B, because there is no defined output of A at its own and vice versa. (because the output of A or B at its own is random (1/2), as is the output of a single polarization filter against a single photon). But the Output = OutputA - OutputB is of course related to both.

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.

...

 

[atyy]

I am for sure missing something fundamental, but what?

In the Copenhagen, the quantum state is not real. But there is also nothing wrong with treating the quantum state as if it is real for all practical purposes (reality is just a tool to calculate the probability of measurement outcomes). Here I will therefore treat the state as real.

In quantum mechanics, one can see that the EPR and Bell tests clearly violate locality because after Alice measures her particle, the wave function will collapse. The wave function collapse is nonlocal, as it occurs on a slice of simultaneity. So the quantum mechanical explanation of the result is clearly nonlocal, for all practical purposes.

The question arises whether there is another formalism without collapse, without quantum mechanics that can provide a local explanation of the correlations predicted by quantum mechanics. Bell's theorem tells us the answer is no - there is no local hidden variables explanation of the correlations predicted by quantum mechanics - with some loopholes such as retrocausation, superdeterminism, or allowing an experiment to produce more than one outcome.

 

[LsT]

atyy, thank you for you answer

Aside from the more general information you provided, and is more than welcomed by me, what I understand from your answer is that my "hypothesis" falls clearly under the local hidden variables category (or not?). Maybe you can be more specific, as to WHY this is the case? I am not asking for some simplified easy understandable answer btw (it would be better for me of course), just a specific answer.

 

[atyy]

atyy, thank you for you answer

Aside from the more general information you provided, and is more than welcomed by me, what I understand from your answer is that my "hypothesis" falls clearly under the local hidden variables category (or not?). Maybe you can be more specific, as to WHY this is the case? I am not asking for some simplified easy understandable answer btw (it would be better for me of course), just a specific answer.

If I understood you correctly, my thinking was that your example is a nonlocal hidden variable hypothesis. Here the hidden variable is the entangled state that is initially created. The simplest way to see that this is nonlocal is to have the Alice measure before Bob in the Bell test. Then Alice will collapse the wave function, which is manifestly nonlocal.

 

[zonde]

As I have pointed, the above is correct only for a local hidden variable theory. In my "model" the outcome is defined/makes sense only in relation to the angle difference of the detectors.

The above is correct for any local model. With variables we sort of describe the potential certainty of some interaction.
Your model is clearly classified as non-local. Because angle difference of distant detectors is non-local variable.

 

[LsT]

@zonde

Because angle difference of distant detectors is non-local variable.

I don't see how the angle difference is non-local. Irrespective of their distance, let's assume that the 2 detectors are physically connected (part of a single stiff object) such as when I change the objects angle, B and A detectors change angle (their difference remains constant). With the right construction, B can easily "follow" A a lot faster than if B was receiving information from A with the speed of light.

And if the angle difference of the detectors is indeed non-local (in a way I don't understand), how can Bell's Theorem also use it to in/validate locality?

 

[stevendaryl]

@zonde

I don't see how the angle difference is non-local. Irrespective of their distance, let's assume that the 2 detectors are physically connected (part of a single stiff object) such as when I change the objects angle, B and A detectors change angle (their difference remains constant). With the right construction, B can easily "follow" A a lot faster than if B was receiving information from A with the speed of light.

And if the angle difference of the detectors is indeed non-local (in a way I don't understand), how can Bell's Theorem also use it to in/validate locality?

The angle difference is nonlocal because the two experimenters (I always call them Alice and Bob) can choose their angles independently at a moment's notice.

As to the second question, that's the point of Bell's argument. It's easy to explain the result using nonlocal, instantaneous effects. But it is not possible to explain the result using local effects.

 

[LsT]

The angle difference is nonlocal because the two experimenters (I always call them Alice and Bob) can choose their angles independently at a moment's notice.

So, in a Bell experiment with the two detectors connected in a way that makes their angle difference local, my hypothesis is to be considered?

 

[stevendaryl]

So, in a Bell experiment with the two detectors connected in a way that makes their angle difference local, my hypothesis is to be considered?

Well, it would be weird if there were one mechanism for the case where the angle difference is fixed ahead of time, and a different mechanism for the case where they are not fixed ahead of time.

 

[ZapperZ]

Not to repeat what is already said, my thoughts are more or less the 4 points mentioned here:

http://physics.stackexchange.com/qu...ement-an-illusion-based-on-a-wrong-assumption

The upvoted answer to what gpgemini's said is that this assumption is essentially what local hidden variables theory propose, and by that its ruled out because experiments demonstrate Bell's inequality violation.

But to my understanding, a predetermined state is a different thing than local hidden variable theory:

Within a local hidden variable theory, entangled particles somehow supposed to have infinite (?) embedded information with them about what will the outcome be in any possible measurement, such as the outcome will emulate what is interpreted as puzzling FTL instantaneous action in other QM interpretations. Referring to this, Bell showed that there are limits to the probability distribution of all possible outcomes if the information the entangled particles carry, is predetermined the moment the entangled pair is created and not changed until measurement. Experiments repeatedly violate local hidden variables probability limits, ruling them out (assuming no loopholes)

But with a predetermined entanglement, particles do not carry ANY information, just exists a definite "opposing" state between them (such as a specific polarization angle difference or opposing spin direction in x axis). Bells inequality clearly does not apply here, because you cannot calculate probability distribution limits of NO information. And to my understanding (wrong?) any entanglement experiment using polarization direction of photons, because of Malus law CAN violate the inequality, without ruling out, this predetermined state hypothesis.
...
I am for sure missing something fundamental, but what?

I'll try to approach this in the simplest, possible manner. What you are describing here is the case for what is commonly known as "realism". This means that you are saying that in the EPR-type experiment, for example, the two entangled pair actually already have a definite state before they are measured. So when one is finally measured, then it is automatic what the other one will be in. Is this correct?

The problem here is that such a concept of classical realism is not compatible with what we have observed in experiments. An example is this one. Tony Leggett has also formulated a case where it is almost implausible to maintain the idea of classical realism.

So in this case, the experiments just simply do not match the idea of an already predetermined state.

Zz.

 

[LsT]

@stevendaryl
Yes, I understand what you mean, but no one can argue for the case that the angle difference can be local, which is not the case for non-local. For example is there a proof saying that you can know the angle difference of the detectors at all times without violating non-locality? To my hypothesis, if you have to know the angle difference at all times, is the equivalent of a making a local connection between the detectors, however fast you move them. Maybe I can say that by using a clock as reference, you make the system equivalent of a local.

@ZapperZ
Quoting from S. Groeblacher et al.

According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality. 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.

But that is saying nothing about local realism except it has been disproved by Bell's experiments. I here make a hypothesis about local realism and the possibility that it is out of the scope of Bell's theorem.

So in this case, the experiments just simply do not match the idea of an already predetermined state.

In fact, that is my bet also. But I am not aware of such an experiment.

 

[ZapperZ]

@ZapperZ
Quoting from S. Groeblacher et al.

But that is saying nothing about local realism except it has been disproved by Bell's experiments. I here make a hypothesis about local realism and the possibility that it is out of the scope of Bell's theorem.In fact, that is my bet also. But I am not aware of such an experiment.

Then you are contradicting the concept of quantum superposition, which has been shown to be valid in numerous observation, especially in Chemistry. You will have to start from there.

Zz.

 

[atyy]

For example is there a proof saying that you can know the angle difference of the detectors at all times without violating non-locality?

There is no such proof, and in fact this is one of the well-known loopholes to Bell's theorem - superdeterminism. The other famous loophole is retrocausation. Bell's theorem does not forbid a superdeterministic, or a retrocasual theory from being local and also explaining the correlations predicted by quantum mechanics.

http://arxiv.org/abs/quant-ph/0301059
"Now “free choice” is a notion belonging to philosophy and I would prefer not to argue about physics by invoking a physicist’s apparently free choice. It is a fact that one can create in a laboratory something which looks very like randomness. One can run totally automated Bell-type experiments in which measurement settings are determined by results of a chain of separate physical systems (quantum optics, mechanical coin tossing, computer pseudo-random number generators). The point is that if we could carry out a perfect and successful Bell-type experiment, then if local realism is true an exquisite coordination persists throughout this complex of physical systems delivering precisely the right measurement settings at the two locations to violate Bell’s inequalities, while hidden from us in all other ways."

http://arxiv.org/abs/1303.2849
"A second problem is that we may never be sure that the choices of measurements are really “random” or “free”. For instance, in the experiments (Scheidl et al., 2010; Weihs et al., 1998) the measurement choices are decided by processes that are genuinely random according to standard quantum theory. But this need not be the case according to some deeper theory. Some people have argued that a better experiment for closing the locality loophole would be to arrange the choice of measurement setting to be determined directly by humans or by photons arriving from distant galaxies from opposite directions, in which case any local explanation would involve a conspiracy on the intergalactic scale (Vaidman, 2001).

The point of this discussion is that an experiment “closing” the locality loophole should be designed in such a way that any theory salvaging locality by exploiting weaknesses of the above type should be sufficiently conspiratorial and contrived that it reasonably not worth considering it."

 

[LsT]

There is no such proof, and in fact this is one of the well-known loopholes to Bell's theorem - superdeterminism.

Maybe I am not getting somethin right (again), I mean, when conducting a Bell's experiment you have to know the angle difference at all times right? So you can corelate it with coincidences. Why is this not the equivalent of locality, if all you care about is this angle difference? What difference does it make if each in relation to you appears to rotate completely random, with superfast speeds. You make the equivalent of a physical connection just with software and clocks.

Then you are contradicting the concept of quantum superposition, which has been shown to be valid in numerous observation, especially in Chemistry. You will have to start from there.

Yes, maybe that's what I am doing :)

Please point to me why you came to that conclusion.

Can you please also give me a good place to start reading about that (actual) experiments?

 

[atyy]

Maybe I am not getting somethin right (again), I mean, when conducting a Bell's experiment you have to know the angle difference at all times right? So you can corelate it with coincidences. Why is this not the equivalent of locality, if all you care about is this angle difference? What difference does it make if each in relation to you appears to rotate completely random, with superfast speeds. You make the equivalent of a physical connection just with software and clocks.

Bell's theorem assumes that the measurement settings on each side are independent of each other, independent of the hidden variable, and independent of the measurement outcomes. In order to satisfy this assumption, we make the measurement settings as random as we can. So from our point of view, a good effective description of the measurement settings is that they are random. If in fact we failed in our attempt to make the measurement settings random, then the assumptions underlying Bell's theorem do not hold, and the theorem does not apply.

 

[LsT]

Please make this clear to me:

Is not the output of a Bell's experiment in essense coincidence-count per detector-angle-difference and this repeated at different (all or specific) angles?

 

[Nugatory]

Please make this clear to me:

Is not the output of a Bell's experiment in essence coincidence-count per detector-angle-difference and this repeated at different (all or specific) angles?

In an ideal experiment (and the Weihs experiment here comes pretty close to this ideal) at each measurement station the observers choose an axis at random, measure, and record the time, the result and the axis. They repeat this process until they each have an interestingly large number of measurements... and only then do they get together and compare notes. When they both have an observation at the same time, they know they're looking at the two members of an entangled pair, and they can consider the relative angles and the measurement results.

 

[bahamagreen]

Just a quick question... What exactly is the justification for this assumption?:

"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%."

 

[Nugatory]

Just a quick question... What exactly is the justification for this assumption?:

"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%."

There are two ways that A and B can end up with a given bit being different in their copies of the message: Either A didn't flip that bit and B did, or B didn't flip that bit and A did. If neither of them flipped the bit, then they'll both have the same value and it will match the original value; if both of them flipped it then they'll both have the same value and it won't match the original value.

Thus, the maximum number of mismatches happens when A flips 25% of the bits and B flips a different 25%; now 50% of the bits will have been flipped by A or B but not both.

 

[bahamagreen]

There are two ways that A and B can end up with a given bit being different in their copies of the message: Either A didn't flip that bit and B did, or B didn't flip that bit and A did. If neither of them flipped the bit, then they'll both have the same value and it will match the original value; if both of them flipped it then they'll both have the same value and it won't match the original value.

Thus, the maximum number of mismatches happens when A flips 25% of the bits and B flips a different 25%; now 50% of the bits will have been flipped by A or B but not both.

That's not quite what I am asking about... I'm concerned with the angles.
The walk through background for the set up at Nick Herbert's site and many others begin by presenting that facts of polarization and the polariscope - Herbert's site uses the Single Photon Orientation Tester (SPOT) to simplify and demonstrate the 100% passage of vertical polarized light when the SPOT is oriented vertically, 0% at 90 degrees, and 50% at 45 degrees.

Although the error math of 25% + 25% = 50% less matched errors seems OK at first glance, I don't feel comfortable assuming it.

For example, if 25% error occurs at 30 degrees when either A or B is offset in opposite rotations, I'm not convinced that these can just be added when both A and B are offset in opposite rotations... one already knows that the 50% is happening at 45 degrees for either A or B.

Also, why use opposite rotations? If opposite rotations of 30 degrees is a net 60 degree rotation, then does one assume that if A at 30 degrees is 25% then one expects A at 60 degrees to be 50%? Again, one already knows that A at 45 degrees is 50%.

One does not even need to know about the cosine squared to ask if it is appropriate to simply add these numbers arithmetically.

What I'm asking is - what is the basis for adding the two 30 degree offset errors of 25% to get 50% (less matches)... what is the basis for assuming that the degrees' errors would add arithmetically?

Assuming measures or magnitudes of 25+25 should equal 50 may not be appropriate here; for example, 250,000km/s + 250,000km/s does not result in 500,000km/s... but it was assumed to be obvious for a very long time.

The locality argument's dependence on this assumption is a stumbling block for me; until I understand it the proof is spoiled because I can't make it to the conclusion.

 

[LsT]

@Nugatory

In an ideal experiment (and the Weihs experiment here comes pretty close to this ideal) at each measurement station the observers choose an axis at random, measure, and record the time, the result and the axis. They repeat this process until they each have an interestingly large number of measurements... and only then do they get together and compare notes. When they both have an observation at the same time, they know they're looking at the two members of an entangled pair, and they can consider the relative angles and the measurement results.

Yes, I can see clearly now that I was wrong about the locality of the detector angle difference. In such an experiment, the angle difference is clearly non-local. But saying this, I cannot accept the fact that this makes my hypothesis non-local, as zonde has said to me, the argument was about:

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.

I had made an argument here that this is not the case for my hypothesis, because it is valid only for for a local hidden variable hypothesis. What I said was wrong. This is indeed a requirement for a local world and it IS the case also for my hypothesis. Here is the crucial difference: Miss A detector rotation does not influence MissB result and vice versa, because the result of miss A in relation to miss B is allways defined for a given relative angle (per malus law), because photons pollarization angle difference is a real state and well defined from their creation. In contrary, because each photons polarization angle, with relation to each detector is random, each detectors output on its own is 1/2 per particle. So can now my hypothesis considered local?

@bahamagreen
Here is what I think:
What are you saying is what Bell has proven. But Bell makes some assumptions BEFORE the formal proof. First of all makes the assumption that in a local world, the result must be predetermined, to agree gith QM prediction. And follows the assumption that in a local world, each particle must carry on with it some information for the outcome of a single detector output. If you consider this, obviously the proof is solid.

What I talk about here is the possibility for a local/realistic hypothesis that lies out of those assumptions. Of course as ZapperZ has said to me, probably I am contradicting some other proven experimental facts even if I am right. But after all as I said, I make this conversation for me, to better understand quantum entaglement through deduction, because of the kind folks here who bother to answer me. I have not enough knowledge on the subject (nor it is my intention) to chalenge anything.

 

[stevendaryl]

What are you saying is what Bell has proven. But Bell makes some assumptions BEFORE the formal proof. First of all makes the assumption that in a local world, the result must be predetermined, to agree gith QM prediction. And follows the assumption that in a local world, each particle must carry on with it some information for the outcome of a single detector output. If you consider this, obviously the proof is solid.

The idea is this: The photons are created initially in some state \lambda. Alice's detector is in some state \alpha. Bob's detector is in some state \beta. The assumption is that there is some probability, P_A(\alpha, \lambda) of Alice detecting a photon, and that it depends on the initial state of the photons and on the state of Alice's detector. There is similarly a probability, P_B(\beta, \lambda) of Bob detecting a photon. Locality is simply the assumption that P_A does NOT depend on \beta, Bob's detector's state, and P_B does not depend on \alpha, Alice's detector's state.

The experimental results show that the only thing that is relevant about the detectors is their orientation, so we might as well assume that \alpha and \beta are angles.

Now, we have one key fact, which is that when \alpha = \beta, the correlation between Bob's result and Alice's result is PERFECT. That is, either both of them detect a photon, or neither detects a photon. This is only possible if P_A(\alpha, \lambda) and P_B(\alpha, \lambda) are both equal to 0, or both equal to 1. To see this, the probability that Alice will detect a photon at orientation \alpha and Bob will not detect a photon at orientation \alpha is:

P_A(\alpha, \lambda) (1 - P_B(\alpha, \lambda))

Experimentally, this quantity is zero. That means that, for any value of \alpha or \lambda, the product
P_A(\alpha, \lambda) (1 - P_B(\alpha, \lambda)) = 0. That means that either P_A(\alpha,\lambda) = 0 or 1 - P_B(\alpha, \lambda) = 0, which means that the probabilities must be either 0 or 1.

So, for a given value of \lambda, Alice's result is determined--she either definitely detects a photon, if P_A = 1, or she definitely doesn't, if P_A = 0.

So we can see that the fact that Alice's result is predetermined by the state \lambda is not an assumption---it is a conclusion drawn from the fact of perfect correlations when Alice and Bob choose the same detector orientations.

 

[Nugatory]

That's not quite what I am asking about... I'm concerned with the angles.

There's no addition of angles involved in the inequality that you're asking about. We're just counting the number of mismatches between four long strings of measurement results: what A would record at some angle; what B would record at the same angle; what A would record if we rotated A by some angle; what B would record if we rotated B by some angle. The number of mismatches between A-rotated and B-rotated has to be no greater than the number of mismatches between A-rotated and A plus the number of mismatches between B and B-rotated because we can't have a mismatch between A-rotated and B-rotated without also having a mismatch between one or both of A/ A-rotated and B/B-rotated. That's just counting.

The addition of angles only comes onto play when we say that turning A to the left by 30 degrees and B to the right by 30 degrees has to yield the same number of mismatches between A and B as turning B to the right by 60 degrees and leaving A alone, or turning A to the left by 5 degrees and B to right by 55 degrees, or turning A to the left by 5 degrees and B to the left by 65 degrees, or whatever.

 

[LsT]

@stevedaryl
I quote from the beggining of the original Bell's paper (bold is mine):

II. Formulation

With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins $\vec{\sigma_1}$ and $\vec{\sigma_2}$. If measurement of the component $\vec{\sigma_1} \vec{\alpha}$ where $\vec{\alpha}$ is some unit vector, yields the value +1 then, according to quantum mechanics, measurement of $\vec{\sigma_2} \vec{\alpha}$ must yield the value -1 and vice versa. Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not infulence the result obtained with the other. Since we can predict in advance the result of measuring any choosen component of $\vec{\sigma_2}$ by previously measuring the same component of $\vec{\sigma_1}$, it follows that the result of any such experiment must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.(...)

So, Bell does make such an assumption.

The idea is this: The photons are created initially in some state λ. Alice's detector is in some state α. Bob's detector is in some state β. The assumption is that there is some probability, PA(α,λ) of Alice detecting a photon, and that it depends on the initial state of the photons and on the state of Alice's detector. There is similarly a probability, PB(β,λ) of Bob detecting a photon. Locality is simply the assumption that PA does NOT depend on β, Bob's detector's state, and PB does not depend on α, Alice's detector's state.

Ok that is what I am saying also. Probability $P_A$ does not depent on $\beta$ and vice versa, because $P_A = P_B = \frac{1}{2}$. So we don't violate locality.

Now, we have one key fact, which is that when α=β, the correlation between Bob's result and Alice's result is PERFECT. That is, either both of them detect a photon, or neither detects a photon.

Ok

This is only possible if PA(α,λ) and PB(α,λ) are both equal to 0, or both equal to 1. To see this, the probability that Alice will detect a photon at orientation α and Bob will not detect a photon at orientation α is:

PA(α,λ)(1−PB(α,λ))

Experimentally, this quantity is zero. That means that, for any value of α or λ, the product
PA(α,λ)(1−PB(α,λ))=0. That means that either PA(α,λ)=0 or 1−PB(α,λ)=0, which means that the probabilities must be either 0 or 1.

Maybe not. And I can easily demonstrate it visually:

(A = detector A output, B = detector B, C = coincidense, PC = coincidence probabillity)

PA = PB = 0, PC = 1
A -> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B -> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
C -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PA = PB = 1, PC = 1
A -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
C -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PA = PB = 1/2 (random), PC = 1
A -> 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0
B -> 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0
C -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

 

[bahamagreen]

There's no addition of angles involved in the inequality that you're asking about. We're just counting the number of mismatches between four long strings of measurement results: what A would record at some angle; what B would record at the same angle; what A would record if we rotated A by some angle; what B would record if we rotated B by some angle. The number of mismatches between A-rotated and B-rotated has to be no greater than the number of mismatches between A-rotated and A plus the number of mismatches between B and B-rotated because we can't have a mismatch between A-rotated and B-rotated without also having a mismatch between one or both of A/ A-rotated and B/B-rotated. That's just counting.

The addition of angles only comes onto play when we say that turning A to the left by 30 degrees and B to the right by 30 degrees has to yield the same number of mismatches between A and B as turning B to the right by 60 degrees and leaving A alone, or turning A to the left by 5 degrees and B to right by 55 degrees, or turning A to the left by 5 degrees and B to the left by 65 degrees, or whatever.

This is the problem assumption right here - "The number of mismatches between A-rotated and B-rotated has to be no greater than the number of mismatches between A-rotated and A plus the number of mismatches between B and B-rotated because we can't have a mismatch between A-rotated and B-rotated without also having a mismatch between one or both of A/ A-rotated and B/B-rotated. That's just counting."

These are my problems with it -
1] To say it "has to be" is based on the same line of thought that says "it has to be" that velocity additions may exceed c.
2] To say "That's just counting" is similar to insisting that velocity addition is an arithmetic sum and not bounded by c.
3] Ever since I first learned that velocity addition is not arithmetic I have kept a suspecting watch for similar assumptions.
4] To say "it has to be" does not agree with the experimental facts of the cosine squared curve; how can an argument presented for logical evaluation and hypothesis testing be justified when it is known up front from experiment to be incorrect?
5] All the above suggests to me that the logical math accounting for the number of mismatches is flawed, or incomplete, or missing something critical.

How can I be comfortable with the "That's just counting" methodology when I already know it does not match the experimental results, and I already know of other principles in physics where the simple addition of values does not result in their arithmetic sum?

Please understand, I'm not challenging or denying Bell, I'm just describing the road block I hit in understanding the argument, the point where I cannot continue to follow the argument because of a lack of conviction that the argument is tight. This may be an artifact of trying to follow popularizations of the Bell logic because every time I read a Bell explanation I hit the same uncomfortable feeling about this assumption... until I understand how this assumption is justified I can't follow through with it to its conclusion.

Just to be clear, I could suggest that because of pure logic and math ("That's just counting") velocities must add arithmetically without an upper limit and I could claim that this addition "has to be" the underlying basis of experimental measurements. When I discover experimentally this simple addition is not correct for velocities, among other things I will be very careful about other arguments based on just counting, simple addition, and in general any logic/math that self suggests it has to be that way. This Bell assumption above is one of those.

To me, it is similar also the the pure logic/math of assuming that two polarizers set at relative 90 degrees must pass no light no matter what is placed between them. That seems like an unassailable conclusion until a third filter at 45 degrees is inserted between them... the simple logic/math is found to be inadequate to account for the subsequent passing of light.

 

[carllooper]

Bell doesn't make an assumption as such. He is introducing the hypothesis in question: the EPR hypothesis.

He will go on to test that hypothesis, rather than simply assume it.

Assumptions are like an hypothesis, but assumptions ask to be treated as if they were not in question.

C

 

[LsT]

@carlhooper
As I understand it, Bell clearly makes the assumption (not hypothesis) that in a local world the result must be predetermined to agree with QM. Upon this he continues, and in the end undoubtly "confines" this predetermined-result locality (aka local hidden variables) within testable limits. And local hidden variables is proved experimentally wrong by Bell's theorem, I have no doubts about this. But a locality without predetermined results is addressed in Bell's paper only through this assumption. (maybe just I am unaware of a proof of this assumption?)

 

[atyy]

As I understand it, Bell clearly makes the assumption (not hypothesis) that in a local world the result must be predetermined to agree with QM. Upon this he continues, and in the end undoubtly "confines" this predetermined-result locality (aka local hidden variables) within testable limits. And local hidden variables is proved experimentally wrong by Bell's theorem, I have no doubts about this. But a locality without predetermined results is addressed in Bell's paper only through this assumption. (maybe just I am unaware of a proof of this assumption?)

No, Bell's does not assume that a local hidden variables theory must agree with QM. Bell defines what he means by a local hidden variables theory, and shows that this entire class of theories cannot match the predictions of QM.

 

[Nugatory]

These are my problems with it -
1] To say it "has to be" is based on the same line of thought that says "it has to be" that velocity additions may exceed c.
...

Suppose I write down a long random sequence of the letters U and D (standing for "up" and "down"). I make two copies of this sequence on two separate sheets of paper, give one to you and keep one for myself. You change three randomly chosen letters on your sheet of paper and I change three randomly letters on mine. We then get back together, compare our lists.

You're not seriously suggesting that we might find more than six disagreements between our two lists, I hope? All we're doing is counting the number of places where your list doesn't match mine, and noting that everywhere that happens then one or both of our lists must also not match the original list.

 

[stevendaryl]

So, Bell does make such an assumption.

It's not an assumption, since it follows from the fact of perfect correlations. If something is provable from other assumptions, then it isn't an additional assumption.

 

[stevendaryl]

Ok that is what I am saying also. Probability $P_A$ does not depent on $\beta$ and vice versa, because $P_A = P_B = \frac{1}{2}$.

No, you don't get to assume that P_A(\alpha, \lambda) = P_B(\alpha, \lambda) = \frac{1}{2}. We don't measure the particle state \lambda. So the probabilities that we measure are averaged over all possible values of \lambda:

\int P(\lambda) P_A(\alpha, \lambda) d\lambda = \int P(\lambda) P_B(\beta, \lambda) d\lambda = \frac{1}{2}

If it's always the case that P_A = P_B = \frac{1}{2}, then you would always have that the probability of both Alice and Bob measuring spin-up would be:

P_A \cdot P_B = \frac{1}{4}

That isn't what we observe. Instead, we observe:
P_A \cdot P_B = \frac{1}{2} cos^2(\theta)

where \theta is the angle between Alice's setting and Bob's setting. (In the spin-1/2 EPR experiment, the prediction is \frac{\theta}{2}

 

[stevendaryl]

@carlhooper
As I understand it, Bell clearly makes the assumption (not hypothesis) that in a local world the result must be predetermined to agree with QM. Upon this he continues, and in the end undoubtly "confines" this predetermined-result locality (aka local hidden variables) within testable limits. And local hidden variables is proved experimentally wrong by Bell's theorem, I have no doubts about this. But a locality without predetermined results is addressed in Bell's paper only through this assumption. (maybe just I am unaware of a proof of this assumption?)

I sketched the proof. If the result is not predetermined, but is instead probabilistic, then you could not have perfect correlation in the case where Alice and Bob choose the same setting.

 

[stevendaryl]

These are my problems with it -
1] To say it "has to be" is based on the same line of thought that says "it has to be" that velocity additions may exceed c.
2] To say "That's just counting" is similar to insisting that velocity addition is an arithmetic sum and not bounded by c.
3] Ever since I first learned that velocity addition is not arithmetic I have kept a suspecting watch for similar assumptions.
4] To say "it has to be" does not agree with the experimental facts of the cosine squared curve; how can an argument presented for logical evaluation and hypothesis testing be justified when it is known up front from experiment to be incorrect?
5] All the above suggests to me that the logical math accounting for the number of mismatches is flawed, or incomplete, or missing something critical.

It seems to me that you are arguing at cross-purposes to Bell. He's not saying that there is no way to reproduce the EPR correlations. There certainly is: QM. What he's saying is that there is no way to reproduce the correlations using a classical notion of local state information propagating no faster than the speed of light.

The comparison with SR would be: There is no way for the speed of light to be the same in every reference frame if we are to use the classical notion of velocity addition. That's true. And so is what Bell claimed. He's saying that there is no classical, local, hidden-variables explanation. He's not saying that there is no explanation.

 

[LsT]

No, Bell's does not assume that a local hidden variables theory must agree with QM.

We agree, because I have never said that.

It's not an assumption, since it follows from the fact of perfect correlations. If something is provable from other assumptions, then it isn't an additional assumption.

You said that, I am saying otherwise, the question is I am not getting a proof of this (continue reading to see why I insist to this)

No, you don't get to assume that PA(α,λ)=PB(α,λ)=12P_A(\alpha, \lambda) = P_B(\alpha, \lambda) = \frac{1}{2}. We don't measure the particle state λ\lambda. So the probabilities that we measure are averaged over all possible values of λ\lambda:
?P(λ)PA(α,λ)dλ=?P(λ)PB(β,λ)dλ=12\int P(\lambda) P_A(\alpha, \lambda) d\lambda = \int P(\lambda) P_B(\beta, \lambda) d\lambda = \frac{1}{2}

You are probably right, but I am not testing a general model here, I just test a specific hypothesis, so I can definitely say that $P_A = P_B = \frac{1}{2}$ just because it is experimental fact, or this is not the case? If it is, how you claim to disprove it?. (btw, how you quote latex correctly ??)

If it's always the case that PA=PB=12P_A = P_B = \frac{1}{2}, then you would always have that the probability of both Alice and Bob measuring spin-up would be:
PA?PB=14P_A \cdot P_B = \frac{1}{4}

To derive to this result, you have to consider the probabilities independent. Otherwise you cannot do that.
(for example, you cannot do this if Alice and Bob are playing by standing at 180 deg. in a wheel of fortune)

I will show again, in my hypothesis, what happens at zero detectors angle difference:

A -> 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0
B -> 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0
C -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Here probability of Alice's Detection is 1/2 as is Bob's (as is always). And the perfect correlation is a fact. The probabilities here are related of course.

But for the probabilities to be related in a local world, I am saying, it is not necessary to assume predetermined results for each detector.

I don't claim to have a formal proof for this, but I will try demonstrating it by the following thought experiment:

This thought experiment is not identical to particle entanglement of course, but it serves the purpose of demonstrating a perfect correlation between results of two distant, non-local detectors each having a probability outcome of 1/2, without any violation of locality, and without predetermined results per each detector (aka hidden variables). The setup is like this:

Alice and Bob are in separated sites. In each site there is a camera for taking photos. Each camera is looking horizontally. Each camera is mounted in a vertical axis rotating mechanism, such as that by the rotation of the mechanism, the camera stands in the perimeter of a circle and keeps pointing at the center of it, for the purpose of taking photos of around an object, that is placed in the center. Now make the rotation of this mechanism/camera independent in such a way that the two sites are considered non-local. Alice and Bob measure the angle of their mechanism, the time and keep the photo for each object that is going through the center of that circle.

For the purpose of simplicity, a hypothetical object is created (this does not affect the perfectness of the correlations):
This is a coin, with a zero width, such as when you take a photo of it, you have to see one (or the other) side.

In the center of the distance between Alice and Bob, we have another mechanism:

This mechanism, is throwing a pair of coins, each one to the direction of each rotating mechanism/camera center, at the speed of light (the speed of light it is not crucial, just a hypothesis to preserve the non-locality between the detectors and the coin), such as each coin is rotating in the vertical axis with the same speed, in the same direction, and with a very specific, constant angle difference that is preserved between them. Now, the angle that each coin has when it leaves the detector, is random, because let's say the mechanism is problematic. But their relative angle is constant and the speed of rotation and direction is the same

When each coin reaches the center of the photographic rotation mechanism, the camera takes a photo, in a random angle. Alice and Bob record all the results for X coin throws, and they compare their results.

The outcome is obvious: The probability of each is 1/2 and for any given angle difference between the mechanisms, the result cannot be but perfectly correlated.

 

[atyy]

But for the probabilities to be related in a local world, I am saying, it is not necessary to assume predetermined results for each detector.

I don't quite understand your question, but maybe it's related to this? http://arxiv.org/abs/1303.3081 (p10)

Proposition 2.1. A family of probability distributions P_X,Y can be explained with pre- established agreement if and only if it can be explained with deterministic local variables.

In other words, although local probabilistic hidden variables may seem be a more general concept than local deterministic hidden variables, for the purposes of deriving Bell's theorem for local probabilistic hidden variables, it is sufficient to derive it for local deterministic hidden variables.

 

[stevendaryl]

You said that, I am saying otherwise

Okay, but you're wrong.

 

[stevendaryl]

You are probably right, but I am not testing a general model here, I just test a specific hypothesis, so I can definitely say that $P_A = P_B = \frac{1}{2}$

But that's not true. If Alice has already observed a polarized photon at angle 0^0 then the probability of Bob measuring a photon polarized at angle 90^o is ZERO, not \frac{1}{2}. Probability \frac{1}{2} is the average over all possible runs, but each individual measurement doesn't have probability \frac{1}{2}

 

[stevendaryl]

To derive to this result, you have to consider the probabilities independent.

Okay, Bell elaborates in his "Theory of Local Beables" that his notion of locality assumes that if two probabilities for events are not independent, then it implies there is a common causal influence on both of them. If all causal influences propagate at lightspeed or slower, then that common causal influence is in their common backward lightcone. You can certainly reject this notion of causality, but that's really the whole point of Bell's theorem, is to show that the predictions of QM are inconsistent with a certain common-sensical notion of causality.

It doesn't do any good for you to say: Well, there might be some other alternative that he didn't think of. He wasn't trying to be exhaustive. He was laying out a particular type of model, and showing that QM can't possibly be that type of model. Once again, you're arguing at cross-purposes to Bell.

 

[stevendaryl]

(for example, you cannot do this if Alice and Bob are playing by standing at 180 deg. in a wheel of fortune).

That doesn't contradict Bell's assumptions---that's an EXAMPLE of the sort of hidden-variables theory that Bell was talking about. Alice and Bob's results in that case become independent once you take into account the position of the wheel.

 

[bahamagreen]

I was looking at various Bell presentations and in my searching I came across a 2008 paper by Andrei Khrennikov "Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?" reviewing the history of mathematical probability assumptions of Bell type inequalities going all the way back to Boole's original development, showing that mathematicians have taken a somewhat different view of the probability assumptions in these inequalities than subsequent physicists, and suggesting a different interpretation of the experimental results with respect to locality.

From the Conclusion section
"...nonexistence of a single probability space does not imply that the realistic description (a map lamda ->a(lamda) is impossible to construct. Bell’s type inequalities were considered as signs (sufficient conditions) of impossibility to perform simultaneous measurement (of/on) all random variables from a family under consideration." (italics in original).

 

[LsT]

atyy that's interesting, maybe it answers my questions but first I have to read it (lol), thanx

But that's not true. If Alice has already observed a polarized photon at angle 000^0 then the probability of Bob measuring a photon polarized at angle 90o90^o is ZERO, not 12\frac{1}{2}. Probability 12\frac{1}{2} is the average over all possible runs, but each individual measurement doesn't have probability 12

$P_A$ is the probability that "Alice detects a photon"
The probability that "Alice detects a photon if somenthing", is only the same with $P_A$ if you consider that this "something" is independent of Alice's result. I am not doing that if "something" is the result of Bob.

Okay, Bell elaborates in his "Theory of Local Beables" that his notion of locality assumes that if two probabilities for events are not independent, then it implies there is a common causal influence on both of them. If all causal influences propagate at lightspeed or slower, then that common causal influence is in their common backward lightcone.

I am in agreement with this notion. In my hypothesis, each photon caries information from their past, but this information is a real state and only in relation to each other, not in relation to any detector.

It doesn't do any good for you to say: Well, there might be some other alternative that he didn't think of. He wasn't trying to be exhaustive. He was laying out a particular type of model, and showing that QM can't possibly be that type of model. Once again, you're arguing at cross-purposes to Bell.

No, this is my first time, last time was bahamagreen (not the same person) :)

And I am not arguing with anyone. In fact Bell seems very clear to me (emphasis mine):

VI. Conclusion
In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote.(...)

That doesn't contradict Bell's assumptions---that's an EXAMPLE of the sort of hidden-variables theory that Bell was talking about. Alice and Bob's results in that case become independent once you take into account the position of the wheel.

(you mean dependent). Ok. I am not saying that the wheel of fortune is my hypothesis. That was just to demonstrate the previous sentence.

 

[stevendaryl]

$P_A$ is the probability that "Alice detects a photon"
The probability that "Alice detects a photon if somenthing", is only the same with $P_A$ if you consider that this "something" is independent of Alice's result.

That's a very silly distinction. Every probability is a conditional probability. To say that Bob has a probability of \frac{1}{2} is conditional on his having an appropriate photon detector, and there being an appropriate source of photons, and so forth. So you seem to be saying that conditional probability is not a probability? That's absurd.

 

[stevendaryl]

(you mean dependent).

No, I mean INDEPENDENT. That's why I said "independent". Alice's result is independent of Bob's result, once you've taken into account the "hidden variable", which is the position of the wheel.

Let A be Alice's result (either 1 if she detects a photon, or 1 if she doesn't). Let B be Bob's result. Let \alpha be Alice's setting. Let \beta be Bob's setting. Let \lambda be the value of the hidden variable (the position of the wheel, for instance).

So let P(A, B, \alpha, \beta, \lambda) be the probability that Alice will get result A and Bob will get result B, given her setting being \alpha and Bob's setting being \lambda and the hidden variable having value \lambda.

To say that Alice's results and Bob's results are statistically independent means that there are two functions P_A(A, \alpha, \lambda) and P_B(B, \beta,\lambda) such that P(A, B, \alpha, \beta, \lambda) = P_A(A, \alpha, \lambda) \cdot P_B(B, \beta,\lambda). That definitely is the case with the wheel of fortune example.

Now, if you average over \lambda, you will find that the averaged probabilities are NOT independent:

Let P_{av}(A, B, \alpha, \beta) = average over \lambda of P(A, B, \alpha, \beta, \lambda)

In general, the average probability P_{av} will not be factorable into P_{av, A}(A, \alpha) \cdot P_{av, B}(B, \beta). So if you ignore hidden variables, then the probabilities seem dependent, but that's because you threw away information.

For classical systems, it's always the case that probabilities becomes independent once you take into account common causal influences.

 

[stevendaryl]

I guess there is a question about whether Bell's factorizability condition on probabilities is an assumption, or a definition, or is provable from some more basic assumptions.

People are always reminded, whenever statistics show a correlation between A and B that "correlation doesn't prove causality". However, it is usually assumed that correlations are either spurious (that is, they are just artifacts of having insufficient data) or are due to an unknown causal influence affecting both A and B. If there is a correlation between, say, black hair and being lactose-intolerant, then people of course don't assume that black hair causes lactose-intolerance, but they assume that something caused both of them. Maybe having a non-European ancestry makes both black hair and lactose-intolerance more likely.

The usual assumption is that once you've controlled for all the variables that might affect both A and B, either the correlation will go away, or you will have discovered a real causal influence.

I think that's what Bell's factorizability assumption formalizes: If A and B are spacelike separated (so neither can cause the other, if relativity is correct), then either they are statistically independent, or there is a common cause for both of them.

 

[LsT]

I 'll try to say it in another way:

But that's not true. If Alice has already observed a polarized photon at angle 000^0 then the probability of Bob measuring a photon polarized at angle 90o90^o is ZERO, not 12\frac{1}{2}

That is circular, because I allready had made the hypothesis (which is consistent with experiments) that $P_A=\frac{1}{2}$ so even if "the probability that B measures a photon if A measures a photon at 90" (lets name this $P_C$) is indeed zero, that does not make $P_B$ zero. As I have already said, $P_B$ and $P_C$ are different probabilities per definition because you change the conditions of the later. Of course I understand now (with your help), that I cannot consider in this hypothesis $P_A$ as $(\alpha, \lambda)$ and $P_B$ as $(\beta, \lambda)$.

As per your example with the wheel I get your point now and this makes perfect sense to me. If you consider $\lambda$ as the state of the wheel, probabilities obviously become independent. But the wheel was not supposed to be a representative example of my hypothesis. Such an example (in the respect of perfect corelations only) is my thought experiment in the post no. #39 There I demonstrate that you can have perfect correlations without a complete specification of the state $\lambda$ that gives a probability other than 1/2 per detector. In fact this is the crucial point in my hypothesis, the state $\lambda$ has to be defined in such a way that it does NOT influence the probabilities of each detector.

I am thinking something in the lines of:

$P_A(\alpha, \theta)$
$P_B(\beta, \kappa)$

where $\theta$ and $\kappa$ is the polarization angle of each photon such as
$\theta - \kappa = 0$
but
$(\theta+\kappa) \sim U([0,720])$ (aka random).

Because of the condition $\theta - \kappa = 0$ I think this does not violate locality