Simple proof of Bell's theorem

[stevendaryl]

Your question is actually backwards. Your questions should be: what is the essential difference between the premises/assumptions of LHV theories and those of QM, that lead to the former predicting a straight-line rather than a S-curve coincidence/angle relation?

QM predicts the "S-curve" relationship due to specific theoretical considerations (which I will not go into). There is no specific LHV theory which predicts a the straight line relation because it has been known for over 200 years that is incorrect as compared to observation (Malus, ca. 1809).

I think that's a little bit misleading. Malus' equation is about sequential operations on a single beam of light---send it through a polarizing filter at this orientation, then send it through a filter at that orientation. But Bell's remarks about linear relationships is about correlations between distant measurements. It happens to be true that for the twin-photon EPR experiment, the correlations between measurements performed on correlated pairs of photons is described by Malus' equation, as well, but that prediction was certainly not made 200 years ago. They didn't know how to produce entangled photon pairs 200 years ago, did they?

 

[jeremyfiennes]

I found an en.wikipedia quote that sums up nicely my doubt:
"All Bell inequalities describe experiments in which the predicted result assuming entanglement differs from that following from local realism."
What exactly does "assuming entanglement" here involve, in everyday terms?

 

[Nugatory]

I found an en.wikipedia quote that sums up nicely my doubt:
"All Bell inequalities describe experiments in which the predicted result assuming entanglement differs from that following from local realism."
What exactly does "assuming entanglement" here involve, in everyday terms?

To be sure, you'd have to ask the author of that quote (although it appears elsewhere on the internet, so there is some possibility that whoever added it to wikipedia was copying and pasting without complete understanding).

However, it seems likely that they're trying to say that the situations in which the quantum mechanical predictions will be different from the predictions of a theory that agrees with equation #1 in Bell's paper (which is to say, any LHV theory) will be the situations that involve entanglement. Thus, any experiment that will go one way if QM is right and another way if there is a valid LHV theory will involve entanglement.

 

[Simon Phoenix]

What exactly does "assuming entanglement" here involve, in everyday terms?

I would adopt Nugatory's interpretation here with the proviso that, strictly speaking, if one is only interested in violating the inequality then entanglement is not actually necessary.

That seems like it runs counter to accepted wisdom, but I believe it's important to understand because it highlights the essential features of QM from which the possibility of violation emerges.

If we look at the maths of Bell's proof there's a very critical step which is the locality assumption. In the maths it's the bit where $P(A| \alpha , \beta , \lambda )$ gets written as $P(A| \alpha , \lambda )$. Here $\alpha$ and $\beta$ are the settings of the detectors, $A$ is the result at Alice's detector and $\lambda$ stands for the hidden variables. So we're making the assumption that the probability of getting a certain result at Alice, conditioned upon the device settings and the hidden variables, does not depend upon the setting of the remote device.

There's no requirement that the devices of Alice and Bob are spacelike separated - it's irrelevant for the proof of the inequality. The ansatz that probabilities of results 'here' are not affected by settings 'there', the locality assumption, is assumed to hold whether or not the devices are spacelike separated.

Now it's possible that there is some unknown, and strange, mechanism that allows the device 'here' to know about the settings 'there' - some unknown field that carries the information about remote settings whatever experiment we set up and for whatever measurement device. In this case we couldn't make our ansatz because the existence of something like this field would affect the probabilities.

The importance of the spacelike separation step is to force any information about remote settings to have to be transmitted FTL. Now it becomes a very big deal. Before this step we could, conceivably, have some hitherto unknown weird and wonderful physics going on that allows the probabilities to be affected. With this spacelike separation step this hypothesized new physics would have to violate the principles of relativity.

So what about entanglement? Well if we ditch the requirement for spacelike separated measurements then it's possible to observe a Bell inequality violation with single, non-entangled, particles. The violation occurs in this instance between the preparation statistics of Alice and the measurement statistics of Bob. I won't go into the details but suffice it to say that it's possible. What this is telling us is that the violation of the mathematical inequality is not dependent on the devices being spacelike separated (which we already knew from the maths anyway). Furthermore, it's telling us that in this case we can obtain violations even without entangled particles. So something about QM allows this violation even without considerations of entanglement.

The spacelike separation - a very critical step if you want to rule out local hidden variable theories - is the icing on the cake - but it's not the essential reason why we see a violation of the math inequality. Nor is entanglement, per se.

If you want to see violation for spacelike separated measurements, then you need entanglement.

 

[DrChinese]

I think that's a little bit misleading. Malus' equation is about sequential operations on a single beam of light---send it through a polarizing filter at this orientation, then send it through a filter at that orientation. But Bell's remarks about linear relationships is about correlations between distant measurements. It happens to be true that for the twin-photon EPR experiment, the correlations between measurements performed on correlated pairs of photons is described by Malus' equation, as well, but that prediction was certainly not made 200 years ago. They didn't know how to produce entangled photon pairs 200 years ago, did they?

Yes, all true. But what I said was not misleading, as there was never a point in time (certainly after 1809) in which the polarization we are talking about was considered "straight-line". The starting point for entanglement (I think electron entanglement was first) was always a cos function of some type. So probably since the 1940's, perhaps. was that specifically considered?

 

[jeremyfiennes]

Thanks all. Thinking-cap time needed.