Hidden phase in polarization tests of Bell's inequality?

[Nugatory]

One thing that is still not clear to me: why do you group up the different measurements?

You might try https://static.scientificamerican.com/sciam/assets/media/pdf/197911_0158.pdf which is @PeroK’s explanation presented in a more visual way. But it’s not as complicated as it looks at first glance:

When we have one detector set at angle A and the second detector set at angle B, and we’re assuming your hypothesis about a hypothetical phase angle as the hidden variable…. What is the probability that we will record + at the first detector and + at the second detector? That can only happen if the phase angle is set to produce + at the first detector at angle A, + at the second detector at angle B, and either + or - if we had instead set either detector to angle C. That’s two mutually exclusive possibilities so we add the probabilities, which is what @PeroK is doing when he writes $p(a+,b+)=p_3+p_4$.

 

[PeroK]

It's possible Im missing some probability calculus rule here, it's not my forte.

The point you may be missing is that a particle would have to have hidden variables for all possible measurement angles. If you have one angle, then a single (relevant) variable will do. But, if you have $3$ angles you must have effectively $3$ relevant hidden variables that form a sample space of $8$ possibilities. I.e. every permutation of up/down for each angle. These possibilities are mutually exclusive.

An understanding of probability theory is a prerequisite to studying QM. I can't teach you that here: you have to study that for yourself.

Post #34 in my view is elementary and a misunderstanding of it is hard to comprehend, I'm sorry to say.

PS I suspect you may also not understand the concept of correlation. This seems to be a stumbling block for many in understanding Bell's inequality. It's not only the correlation between the measurements on the two particles, but the $\cos^2$ correlation between measurement angles on the same particle, that create the classically impossible overall correlation between measurement results.

 

[Leureka]

The point you may be missing is that a particle would have to have hidden variables for all possible measurement angles. If you have one angle, then a single (relevant) variable will do. But, if you have $3$ angles you must have effectively $3$ relevant hidden variables that form a sample space of $8$ possibilities. I.e. every permutation of up/down for each angle. These possibilities are mutually exclusive.
An understanding of probability theory is a prerequisite to studying QM. I can't teach you that here: you have to study that for yourself.

Post #34 in my view is elementary and a misunderstanding of it is hard to comprehend, I'm sorry to say.

PS I suspect you may also not understand the concept of correlation. This seems to be a stumbling block for many in understanding Bell's inequality. It's not only the correlation between the measurements on the two particles, but the $\cos^2$ correlation between measurement angles on the same particle, that create the classically impossible overall correlation between measurement results.

Bear with me here. I understood your math, as you said it’s very simple. What I did not understand are your premises. I don’t see why you need a single variable for all three angles, when the most you can ever do is sample two angles. In this model the hidden variable is an area overlap which you can determine through multiple measurements of the photon, but only with always the same angle settings. Once you change your setting on one of the angles, the area overlap you are determining will be different. Since entanglement is limited to only two photons, there is no way to do this operation without losing information of the overlap on one of the angles.

To visualise what I mean, here are some other drawings I made. This time we’re going to deal with spheres, since we have an extra degree of freedom in a third angle. Note that since we’re dealing with a perfect sphere the overlaps will vary linearly with angle and so won’t reproduce a cosine distribution, but again this is a matter of choosing the correct 3d shape (like in the ellipse case).

Here, the figure is actually 3 spheres overlapped, each representing one detector setting. The coloured volume represents a “blip” when the detector interacts with the photon, and each sphere is only half coloured to represent each detector has an intrinsic 50% chance of detection. In this example, A and B are at 90 degrees, while B and C are at 45 degrees. The actual values are not important for my argument.

In this second figure, A and B are still at 90 degrees while B and C are still at 45 degrees, but the relationship between A and C changed. This is possible because for each measurement, I can only really define two sets of spheres (one for each photon-detector couple) with a third degree of freedom (left unmeasured) that has no fundamental effect on the outcome of the measurement; in fact, I can freely rotate each sphere independently on the axis defined by its partner without changing the volume overlap. If I wanted to also keep A+C at 45 degrees I would have changed the relationship between A and B, but this would not affect my measurement of A + C.

 

[PeroK]

I don't understand what you are doing there. The numbers are important. I think you should try to calculate to what extend your classical probability distribution can violate the Bell inequality. You need to calculate a number for the Aspect experiment.

Otherwise, all you have are ambiguous diagrams that may not achieve the miracle you claim.

 

[PeroK]

PS the Aspect experiment does sample three different angles. So, your claim that this cannot be done is false. This also suggests, I'm sorry to say, that you haven't understood the Bell test at all!

 

[Leureka]

I don't understand what you are doing there. The numbers are probably important. I think you should try to calculate to what extend your classical probability distribution can violate the Bell inequality. You need to calculate a number for the Aspect experiment.

Otherwise, all you have are ambiguous diagrams that may not achieve the miracle you claim.

I'm not trying to achieve any miracle. As I said, I'm trying to understand why some things are the way they are.

No the numbers are not important. It's a visual way to say that (in my mind) if your measurement can only output two values, there's no reason to introduce a third that can't be determined by the previous two values. If you measure at angles A and B, whatever your angle C is your measurements will not be affected. It feels wrong to assume that what you get from measuring A and C will be affected by what you had before, because each time you're sampling completely different photons.

Just take two overlapping spheres instead of three, A and B at 90 degrees. You can rotate by 360° the sphere A around B (or B around A) keeping the colored emispheres orthogonal. Any rotation by any amount always corresponds to the exact same probability of both detectors bleeping, which in turn corresponds to the overlapping volumes of the colored hemispheres (a quarter of the volume of the whole sphere). In other words, if the photon "vector" (a unit vector that starts from the centre of the sphere) lands inside this volume, both detectors will blink. But the number of photon vectors that are capable of landing in that volume is not limited to a fixed part of the sphere: any vector in the sphere can land there if you rotate sphere A and sphere B the right amount. That is why determining the correlations for A and B won't tell me anything about those for A and C or B and C: there's infinitely many different distributions of vectors also landing in C, one for each infinitesimal rotation of either A or B.

I can't provide any math because I don't know which 3D shape can reproduce a cos^2 non linear overlap dependent on angle, if that's even possible (because I'm not claiming it is, but i don't have evidence it isn't). What I already showed is that non linear overlaps ARE possible, with anything but a perfect sphere.

 

[Nugatory]

I don’t see why you need a single variable for all three angles, when the most you can ever do is sample two angles.

That is a requirement for a realistic theory, which is to say one that asserts that unmeasured quantities still have a definite value. If your hypothetical phase vector does not allow us to predict the results of the measurement on all three axes, you haven't proposed a local realistic theory as a counterexample to Bell's theorem - at best you have a more complicated model of quantum mechanics.

To see why all three angles matter for a realistic theory even when we're only measuring two, consider the experiment (which has been done!) where the measurement angles are not set until after the pair has been created and the particles are in flight. We can choose to measure on any of the three angles and the phase vector is already set (if it's not, then your theory is not local) so the phase vector must be sufficient to determine the result on any of the three axes even though we won't measure one of them.

 

[Leureka]

That is a requirement for a realistic theory, which is to say one that asserts that unmeasured quantities still have a definite value. If your hypothetical phase vector does not allow us to predict the results of the measurement on all three axes, you haven't proposed a local realistic theory as a counterexample to Bell's theorem - at best you have a more complicated model of quantum mechanics.

It might be that my model is indeed nonlocal (I just haven't got a clear answer one way or another); but the point I'm trying to get across is that if that's true, then the source of non-locality is not mysterious at all. It really boils down to how we compare measurements of a phenomenon in which both detector and photon play a role. All that is happening at the local level is asking whether the detector is able to bleep, and that will depend by both the photon vector and the local phase of the instrument. The correlations will be a natural result of the difference in phase of the two detectors with respect to the photon vector.

To see why all three angles matter for a realistic theory even when we're only measuring two, consider the experiment (which has been done!) where the measurement angles are not set until after the pair has been created and the particles are in flight. We can choose to measure on any of the three angles and the phase vector is already set (if it's not, then your theory is not local) so the phase vector must be sufficient to determine the result on any of the three axes even though we won;t measure one of them.

If the measurement is a result of both photon vector AND instrument phase, whether the choice happens after or before pair creation makes no difference. The measurement will always be either a yes or no at a single detector, but the correlation will depend on both detector choices as I explained above. The photon by itself can't know whether it will pass a barrier that might or might not be there.

 

[DrChinese]

... No the numbers are not important. It's a visual way to say that (in my mind) if your measurement can only output two values, there's no reason to introduce a third that can't be determined by the previous two values. If you measure at angles A and B, whatever your angle C is your measurements will not be affected. It feels wrong to assume that what you get from measuring A and C will be affected by what you had before, because each time you're sampling completely different photons.

...I can't provide any math because I don't know which 3D shape can reproduce a cos^2 non linear overlap dependent on angle, if that's even possible (because I'm not claiming it is, but i don't have evidence it isn't). What I already showed is that non linear overlaps ARE possible, with anything but a perfect sphere.

You haven't shown anything that relates to Physics. Your model does not pass the Bell test, and not because you are having a difficult time trying to come up with overlapping shapes. It is sad that you continue to ignore Bell's Theorem, as you have been advised.

I will try to explain what others have said about Bell in terms of what you have said. According to you: "If you measure at angles A and B, whatever your angle C is your measurements will not be affected." And yet you model produces a prediction at every possible angle. A, B, C, D and so on, according to a hard and fast rule you have given us. Oh, and yes, you may plan to tweak it. But...

It's that hard and fast rule that's the problem. Because there are NO consistent models of the type you describe. (And by consistent, I mean consistent with the quantum predictions.) That's because the type of model you propose is of type "objective realistic" - meaning the experimenter's choice of angle settings in no way affects the outcomes. Which is just what you think is the case.

But that's what Bell showed is not possible. That's because quantum mechanics is what is called "subjective realistic". It is the specific 2 angles, taken as a pair, that determines the statistical results. In your model, I should be able to select as many angles as I want and get a statistical prediction for any pair. But the resulting statistics won't match QM. Yes, it looks that way to you - because you are only looking at the outcomes in front of you. Hey, Einstein thought the same exact thing! But he didn't live to read Bell.

You freely acknowledge that the match % for 0 & 22.5 degrees is 85%. You freely acknowledge that the match % for 22.5 & 45 degrees is also 85%. So how do you explain that the match % for 0 and 45 degrees is 50%? According to standard probability theory: the absolute minimum match % would actually be 85% * 85% which is about 73%. That doesn't match your model's prediction of 50%. Your model's assertion that there is a hard and fast rule - which says that the outcome is objectively realistic - leads to inconsistency.

QM says that the outcomes are only dependent on the relative difference between 2 settings. There is no meaning to the possible outcomes at other settings. This is closely related to the Uncertainty Principle, which says that spin (or polarization) has mutually non-commuting bases. Regardless, you will quickly find that the statistics don't work out.

If you like, check out a simple proof on one of my web pages. My example uses 120 degrees (match=25%) for photons instead of your 22.5 degrees (match=85%). This angle makes the problem of a model of your type more clear, and you shouldn't care what the angle is. In fact, for the angle settings of 120 degrees, you don't even need a model. You can't even hand pick them and match the quantum predictions with an objective realistic model. The closest you can get with hand picked outcomes is at least 33%.

Bell's Theorem with Easy Math

 

[Nugatory]

I should add that the impossibility of making a local realistic model work across three settings is not just a quibble - It goes directly to some of the weirdest aspects of quantum mechanics.

If you have read that Scientific American article I linked earlier in the thread you will have seen that the proof of Bell’s inequality uses the same logic that I use when I assert that the number of male redheads in a room must be at least at great as not greater than the number of red-headed cigarette smokers plus the number of non-smoking men. I can verify this while measuring only two of the three properties at a time by asking all the non-smoking men to raise their hands, counting, then asking all the red-headed smokers to raise their hands, counting, then finally asking for all the red-headed men. This is basically what an experiment testing for violations of Bell’s inequality is doing, and the profoundly weird thing is that the inequality is not respected by quantum mechanics.

 

[PeroK]

No the numbers are not important.

The numbers are everything! As @DrChinese points out:

You freely acknowledge that the match % for 0 & 22.5 degrees is 85%. You freely acknowledge that the match % for 22.5 & 45 degrees is also 85%. So how do you explain that the match % for 0 and 45 degrees is 50%? According to standard probability theory: the absolute minimum match % would actually be 85% * 85% which is about 73%. That doesn't match your model's prediction of 50%.

 

[Leureka]

If you have read that Scientific American article I linked earlier in the thread you will have seen that the proof of Bell’s inequality uses the same logic that I use when I assert that the number of male redheads in a room must be at least at great as the number of red-headed cigarette smokers plus the number of non-smoking men.

3 properties:
- is male
- is redheaded
- is smoker

Male + redhead <= (redhead + smoker) + (male + nonsmoker).

Is this right? I think you meant "is less or equal to", but I might misrepresenting your example.But what happens if everytime you ask a question, the people in the room change? You might still be getting the exact same number of redheaded males, but the number of smokers is not constant. The three properties are completely independent from one another.

@DrChinese @PeroK for the same reason, why do you think AC(22.5) and BC(22.5) have anything to do with AB(45)? They are completely different measurements. This is obvious from the fact that the system is rotationally invariant, calling AC and BC with different names doesn't make them distinct.

 

[PeroK]

@DrChinese @PeroK for the same reason, why do you think AC(22.5) and BC(22.5) have anything to do with AB(45)?

In a word, because QM predicts that the results are statistically correlated. This is confirmed by experiment. That's why random hidden variables were proposed (and, are being proposed by you!!) as an alternative to QM. I suspect you don't understand the concept of correlated measurement results in the context of light polarization.

There are many experiments that prove these correlations between different measurement angles. So, when you say that "one measurement has nothing to do with another", you are right that they are different measurements, but wrong that they are uncorrelated.

It's difficult to know what you understand and what you are fundamentally misunderstanding. I suspect you are missing a lot of the fundamentals of QM, so we are just chasing shadows here.

1) Classical physics would normally predict a single (deterministic) result for measurements on a sequence (or ensemble) of identically prepared systems. There would be no statistics at all. Like, say, a cricket or baseball machine that can fire cricket balls or baseballs with the same velocity and spin every time. It's fundamental that you cannot do that in QM.

2) QM is fundamentally probabilistic. I.e. measurements on identically prepared systems produce a distribution of results. For example, if you build a machine that produces photons with a given polarization in one direction, then polarization measurements in other directions will be probabilistic.

3) If you want a classical model of polarization, then you must introduce hidden random variables to mimic the probabilistic behaviour. There are severe constraints on your variables, as described above, as they must reproduce the measurements results and correlations of QM. Not just the measurement results, as you may believe, but all the correlations as well.

And, of course, this is proved (quite simply) to be mathematically impossible.

Bell's inequality is all about correlations, correlations, correlations.

 

[berkeman]

Thread closed temporarily for Moderation...

 

[Nugatory]

The question in the original post has been answered - no, a hidden variable of the type that OP is considering cannot explain violations of Bell’s inequality - and the subsequent argument is based on OP’s misunderstanding of Bell's inequalities.

This thread will remain closed.
As with all thread closings, you can PM any mentor to ask that it be reopened to allow new and relevant content to be added.