Need help with some aspects of Bell’s theorem

[miosim]

PeterDonis

Let me think about your response.

 

[miosim]

So what are these key properties of HV (local) theories?

... they are that the probability distribution of measurement results at detector A cannot depend on the settings of detector B, and vice versa. That was Bell's key assumption. It is usually referred to as "locality", and is why the conclusion of the theorem applies only to "local" hidden variable theories.

Indeed the probability distribution of measurement results at detector A should not depend on the settings of detector B, and vice versa. Therefore if we rotate the first detector the probability distribution at second detectors shouldn't change. Does it contradict with the QM prediction? Does it contradict with experiment?

 

[PeterDonis]

Therefore if we rotate the first detector the probability distribution at second detectors shouldn't change. Does it contradict with the QM prediction?

Yes. That's what Bell showed, by showing that any probability distributions (of the results at A and B) that meet the conditions will have correlations (between the results at A and B) that satisfy the Bell inequalities. Specifically, he showed that the correlation function between A and B, if A's and B's distributions meet the conditions, must factorize into a product of a function depending only on A's settings, and a function depending only on B's settings. Then he showed that any distribution that factorizes in this way must satisfy the Bell inequalities.

The QM prediction violates the Bell inequalities, as can be shown either by just looking at the correlation function it gives and its values at various angles, or by showing that it doesn't factorize in the way described above.

Does it contradict with experiment?

Yes, since experiments match the QM prediction.

 

[miosim]

Therefore if we rotate the first detector the probability distribution at second detectors shouldn't change. Does it contradict with the QM prediction? Does it contradict with experiment?

Yes. That's what Bell showed, by showing that any probability distributions (of the results at A and B) that meet the conditions will have correlations (between the results at A and B) that satisfy the Bell inequalities

Bell's experiment is about CORRELATION between A and B. But what about measurements at one detector only. Does it depend on the position for other detector? Would the actual number of detections at detector A is changing from max to zero while we rotate detector B?

 

[Nugatory]

Indeed the probability distribution of measurement results at detector A should not depend on the settings of detector B, and vice versa. Therefore if we rotate the first detector the probability distribution at second detectors shouldn't change. Does it contradict with the QM prediction? Does it contradict with experiment?

The probability distribution at the second detector does not change - every incident particle has a 50% chance of passing and a 50% chance of not passing; the experimental results at each individual detector are as random as if they had been generated by flipping coins. This is both the quantum mechanical prediction and the experimentally observed result (and If it were not, we could use entanglement to send faster-than-light messages).

However, the probability distribution of the coincidences, which only becomes apparent when Alice and Bob get together after the fact and compare notes, does change. Bell's theorem states that the quantum mechanical prediction for the coincidences can only be produced by theories in which the probability of a detection at one detector is affected by the position of and result at the other detector.

You may be finding yourself confused by the claim that the probability distribution of the detections at either detector is completely random, yet the correlations between them may not be. Here's an easy example of how that could happen: You are watching someone tossing an honest coin, and you see a random sequence of heads and tails. I am also watching someone tossing an honest coin, and I see a random sequence of heads and tails. But when we get together afterwards and compare notes, we might find that the correlation is not random; for example if we happened to be watching the same guy flipping the same coin but we were looking at different sides of the coin, the every time that I saw a heads you would see a tails and vice versa. Yet the coin flips would still be as random as random can be.

 

[miosim]

The probability distribution at the second detector does not change - every incident particle has a 50% chance of passing and a 50% chance of not passing;

It is also my understanding.

You may be finding yourself confused by the claim that the probability distribution of the detections at either detector is completely random, yet the correlations between them may not be

I have no problem with this concept

Bell's theorem states that the quantum mechanical prediction for the coincidences can only be produced by theories in which the probability of a detection at one detector is affected by the position of and result at the other detector.

Another word the correlation predicted by QM is higher that any local HV theory can offer. Is it correct?

 

[Nugatory]

The correlation predicted by QM is higher that any local HV theory can offer. Is it correct?

For certain arrangements of the detectors, yes. For other arrangements (such as the ones where QM predicts and experiments confirm perfect correlation or perfect anti-correlation) a local hidden variable theory could in principle produce the same results.

 

[miosim]

For other arrangements (such as the ones where QM predicts and experiments confirm perfect correlation or perfect anti-correlation) a local hidden variable theory could in principle produce the same results.

Understand.I have a question:
When photon pass the detector, is the polarization of photon that passed detector and orientation of detectors matches? If they match,
does the wave function collapse first produces a photon with some polarization, and then this original polarization is rotated by detector and then pass/or_not_pass result depends on match between orientation of the detector and the original polarization of photon?

 

[jtbell]

Here's an easy example of how that could happen: You are watching someone tossing an honest coin, and you see a random sequence of heads and tails. I am also watching someone tossing an honest coin, and I see a random sequence of heads and tails. But when we get together afterwards and compare notes, we might find that the correlation is not random; for example if we happened to be watching the same guy flipping the same coin but we were looking at different sides of the coin, the every time that I saw a heads you would see a tails and vice versa. Yet the coin flips would still be as random as random can be.

Or to make things a bit more interesting, imagine the two of you watching someone rolling a six-sided die, but from different directions, say from the top and from one side.

 

[miosim]

Or to make things a bit more interesting, imagine the two of you watching someone rolling a six-sided die, but from different directions, say from the top and from one side.

I understand these examples. I just need a better understanding of physical side of the story.

 

[Nugatory]

does the wave function collapse first produces a photon with some polarization, and then this original polarization is rotated by detector and then pass/or_not_pass result depends on match between orientation of the detector and the original polarization of photon?

The best description in terms of wave function collapse says that neither particle has a definite polarization (we still have the superimposition) until one of them interacts with its detector. When this interaction happens, the particle acquires a definite polarization, either parallel to or perpendicular to the polarizer, and that's the moment of wave function collapse. The entire wave function collapses, so when the first photon acquires its definite polarization the second one immediate collapses into the corresponding state and the proceeds on to interact with its polarizer. In this model, if the first polarizer is set to angle $\alpha$ then the second photon will always interact with its polarizer as if its polarization is $\alpha$ or $\alpha\pm\pi/2$, and this is the spooky action at a distance that you hear so much about.

But do remember that this is an explanation in terms of wave function collapse and collapse is not a fundamental part of quantum mechanics; it's just one way of interpreting the statistical predictions that the theory makes. You'll hear this interpretation a lot because it makes a sort of intuitive sense (as long as you're willing to swallow the spooky bit) and because it is a very helpful way of thinking about many single-particle problems. However, it also has some very serious conceptual problems. The most serious might be that it only makes sense if we can say that one interaction unambiguously happened before the other, and as we discussed earlier in this thread, we cannot.

 

[miosim]

But do remember that this is an explanation in terms of wave function collapse and collapse is not a fundamental part of quantum mechanics; it's just one way of interpreting the statistical predictions that the theory makes. You'll hear this interpretation a lot because it makes a sort of intuitive sense (as long as you're willing to swallow the spooky bit) and because it is a very helpful way of thinking about many single-particle problems. However, it also has some very serious conceptual problems. The most serious might be that it only makes sense if we can say that one interaction unambiguously happened before the other, and as we discussed earlier in this thread, we cannot.

I understang what are you saying and will try to use the language compatible with interpretation.

In this model, if the first polarizer is set to angle α \alpha then the second photon will always interact with its polarizer as if its polarization is α \alpha or α±π/2 \alpha\pm\pi/2, and this is the spooky action at a distance that you hear so much about.

After entangled photon pass their corresponding detector/polarizers what their mutual polarization are? Do they still have polarization differences ±π/2 as entangled pair or this symmetry is distorted due to interaction with detectors?

Or may be I should ask the more general question: What is polarization of a photon that pass polarizer in reference to orientation of this polarizer?

 

[Nugatory]

Or may be I should ask the more general question: What is polarization of a photon that pass polarizer in reference to orientation of this polarizer?

Once the photon has cleared the polarizer, subsequent measurements of its polarization along that axis will yield the same result - the photon passes. (It's also no longer entangled with its partner).

In a collapse interpretation, that's because the wave function collapsed with the first interaction so now both particles are in non-superimposed polarization eigenstates. If you prefer a more ascetic interpretation, then the first measurement is a "preparation procedure" which prepares the system into a state such that a subsequent polarization measurement along that axis will be positive 100% of the time.

 

[miosim]

In a collapse interpretation, that's because the wave function collapsed with the first interaction so now both particles are in non-superimposed polarization eigenstates. If you prefer a more ascetic interpretation, then the first measurement is a "preparation procedure" which prepares the system into a state such that a subsequent polarization measurement along that axis will be positive 100% of the time.

Another word the photon pair after exiting corresponding polarizer lost symmetrical polarization. It is like polarizers align photons, after they lost entanglement, to follow polarizer's orientation. It seams the "ability" of photons to follow orientation of polarizers contributes into correlation making it higher than any local HV theory can be predict.

 

[miosim]

Another word the photon pair after exiting corresponding polarizer lost symmetrical polarization. It is like polarizers align photons, after they lost entanglement, to follow polarizer's orientation. It seams the "ability" of photons to follow orientation of polarizers contributes into correlation making it higher than any local HV theory can be predict.

Actually this language isn't compatible with description of QM system and should be reserved only for classical theories or to local HV theories.
In this case, per classical description, the predicted by QM correlation may be explained in terms of photon polarization sufficiently rotated by detector. Per local HV theory the the predicted by QM correlation may be derived from HV that measures orientation of the corresponding detector and change photon's polarization accordingly (using function similar to Malus' law). In this case there is no need for photon pair to influences each other over the distance to achieve sufficient correlation. Instead they need just "pay attention" to the orientation of local detector only.

Does it make sense?

 

[Nugatory]

Per local HV theory the the predicted by QM correlation may be derived from HV that measures orientation of the corresponding detector and change photon's polarization accordingly (using function similar to Malus' law). In this case there is no need for photon pair to influences each other over the distance to achieve sufficient correlation. Instead they need just "pay attention" to the orientation of local detector only.

I don't understand what you're saying here.
Are you saying that the QM correlation can be derived from a hidden variable theory in which the result at one detector does not depend on the angle and result at the other detector? If so, you might want to try coding up a computer simulation - I expect that you'll find that it doesn't work.

 

[miosim]

I don't understand what you're saying here.
Are you saying that the QM correlation can be derived from a hidden variable theory in which the result at one detector does not depend on the angle and result at the other detector? If so, you might want to try coding up a computer simulation - I expect that you'll find that it doesn't work.

I guess I need to try this, but it would take time for me to do this.
Meanwhile, I have a question.

If the main goal of the Bell theorem is to prove non-locality of QM why does this prove relies on rejecting of local HV theory? The non-locality is a well known feature of QM. So if QM predicts correlation and this prediction is based on non-locality of wave function and the experiment conforms QM prediction - what else do we need? Why do we need to complicate this by mixing with disproving of local HV.
Is proving of non-locality indeed depends of disproving local HV theory? What if we wouldn't have the EPR paper?

 

[Avodyne]

The goal of the Bell theorem is to show that local hidden variable theories cannot mimic QM.

Before Bell, it was not clear that the nonlocality in the wave function could not be mimicked by LHV.

Whether or not you think this is interesting is up to you.

 

[miosim]

The goal of the Bell theorem is to show that local hidden variable theories cannot mimic QM.

Before Bell, it was not clear that the nonlocality in the wave function could not be mimicked by LHV.

Whether or not you think this is interesting is up to you.

Aha...
I am getting closer in understanding your response while trying to mimic the correlation predicted by QM cos (a-b), as suggested by Nugatory. I am looking for the equivalent formula in which A and B are separated (meaning that particles A and B know orientation of the correspondent detector only, but not both). Doesn't look it is possible based on the trigonometry.

cos(A - B) = cos A cos B + sin A sin B

I need more time to think about this.

Thank you for response.