What is the explanation behind the binary result in the EPR thought experiment?

[Tomer]

Hello dear readers,

I have a problem. I'm giving a lecture tomorrow on the EPR experiment and Bell's theorem and so on.
I had the feeling I really understand the essence of the EPR thought experiment.
However, a few hours ago I just had a disturbing thought (while practicing the lecture in front of a wall!) that made me stutter. I'll try to shortly explain my problem:

The formulation I'll have in my lecture consist of correlated photons and aligned polarization filters. I hope it's familiar to you / obvious why it's analog to the famous spin one.
The idea, as I understand it, is that by measuring the first photon's polarization we immediately know precisely what the polarization of the other photon is, although it might be light years away. Assuming locality is held, we conclude that the faraway photon always had this specific polarization - therefore QM isn't a complete theory, for it failed to assert that in the first place. (simply put, of course I won't portray it as simplistic)

My problem: QM predicts that the wave function collapses to either "horizontal" or "vertical" eigenstates (for example |x>1|x>2), assuming the filters are for example aligned with the x-axis. So in the EPR thought experiment, if we assume the first photon is transmitted, we infer that the second one has a polarization in the x-axis direction.
QM always end up saying "second photon is polarized horizontally" or "second photons is polarized vertically". But as I understand EPR's logic, the polarization could be in any direction whatsoever? The claim that by measuring the first photon's polarization we discover the real value of the second one's polarization - but not create it. How do they therefore explain this binary result? How come the second photon never turns out to be polarized in an angle of 45 degrees (for example).

I hope my problem is clear. It doesn't contradict their claim or anything, it just looks like another indication for a "collapse" of the wave function, which they opposed, and I wonder what their view on the subject was, and I can't seem to find a reference to it anywhere.
For me, they either have to treat this problem, or it could be that I'm misunderstanding something.

Thanks a lot,

Tomer.

 

[DrChinese]

First, EPR was essentially incorrect on the point. But let's forget that for a moment.

Second, the H and V bases are arbitrary. You can set H=45 degrees and the results are the same. Knowledge of Alice's outcome at any angle allows you to predict Bob's at that angle plus/minus any multiple of 90 degrees.

The presumed reason for this "coincidence" is more or less as follows: each photon has a preset group of parameters (the hidden variables) that precisely determine the outcome of any operation by either Alice or Bob. Imagine they each had identical decks of cards, and testing the polarization at any angle X was akin to pulling the Xth card from the deck and looking to see if it is red or black.

That is the EPR concept. Which as I said cannot be correct; due to Bell's Theorem.

 

[spoirier]

QM predicts that the wave function collapses to either "horizontal" or "vertical" eigenstates

According to John Baez's page of http://math.ucr.edu/home/baez/open.questions.html,

...Does "wavefunction collapse" actually happen as a physical process? If so, how, and under what conditions? If not, what happens instead?

Many physicists think these issues are settled, at least for most practical purposes. However, some still think the last word has not been heard. Asking about this topic in a roomful of physicists is the best way to start an argument, unless they all say "Oh no, not that again!". There are many books to read on this subject, but most of them disagree.

I would describe things like this : QM does not predict anything like a preference for [either "horizontal" or "vertical"] polarization over diagonal polarization, because there is not such a thing as a possible measurement whose probability result may be influenced by this alternative.
The alternative between diagonal and [horizontal or vertical] polarization for one photon, is only the choice of the observer (or measurement apparatus); the only way the state of the photon may have an influence on the result, and thus what may be seen as its "real" property (though there are anyway troubles whether anything may be called "real"), is its preference (effect on probability, thus not anything all clear) for vertical over horizontal IF it is measured according to this alternative.
When you have more time you can see how I introduce the structure of predictions in quantum physics in a clear and simple form.

 

[Tomer]

I wrote a reply, but it didn't feel clear, so I deleted it.
I'm giving the lecture in about 2 hours, so I'll probably just have to skip the matter, cause I haven't yet understood the resolution :-)

Thanks a lot for replying, I'll write later!

Tomer.

 

[Tomer]

Had my lecture, was great, but I avoided this topic, for I have not yet come to an resolution of my problem. :-)

Thanks again for your replies.
DrChinese - I realize of course that EPR were wrong (that's were Bell comes in). I also realize we can choose to align our filters arbitrarily. But this is exactly the essence of my problem.
Spoirier - I'm afraid after reading your post a few times I'm still not quite sure what you wanted to convey or whether it exactly answers my question.

Let me try and explain my problem again. It is important that you guys realize that I'm not trying to understand some underlying physical reality. All I'm trying to understand is Einstein's (or EPR's) motivation.

Imagine the two photons travelling, as imagined by EPR, very distant from one another.

scenario I:
We have two aligned filters, let's say they're both H-H (horizontal).
Now assume we measure the first photon and it's transmitted. QM says: the second photon's polarisation state has now collapsed to the eigenstate H (or, if you'd like, |H>). This means experimentally that there's a 100% probability that when measuring the second photon's polarisation with a horizontal filter, it'll be transmitted.
Correct me if I'm wrong, but didn't Einstein interpret that and imagined that as: "the photon has the polarisation H"? Now, whether talking about "real" properties without measuring them makes sense or not doesn't play a role here because I'm just trying understand Einstein's motivation.
Einstein then said: "well, the second photon has the polarisation H. But since we didn't effect it (locality), it always had this polarisation, H (well, since its creation) - and we merely found that out"

scenario II:
Now imagine we had performed exactly the same scenario, with the same two photons (we traveled back in time, if you will). But this time, the two filters are aligned with the angle of 30 (compared to the previous system).
Again, let's assume the first photon is transmitted. QM says: the second photon's wavefunction collapsed to the eigenstate |30>.
And by the same logic, Einstein says: "the second photon has the definite polarisation of 30, correct. But we couldn't have influenced him, and therefore he always had the polarisation 30, and we merely found that out".

So Einstein's "elements of reality" seem to depend on the orientation of our filters! If the photon had one specific polarisation, as Einstein believed it had (please correct me if I'm wrong by saying this again and again) - how could he explain the fact that the orientation of our filters would always lead to a binary choice as to the definite polarisation of the second photon?

I hope this is clear.

Thanks!

Tomer.

 

[DrChinese]

OK, here is what Einstein would say (of course these are my words in his mouth):

"Photons have definite polarizations at all angles at all times. If you choose to think of there being 360 possible angles, there are 360 'answers' embedded (hidden) in the photon. If you choose to think of it a 3600 possible angles, there are 3600 possible answers in that photon. Of course, no one can say precisely how many or how few there are. But clearly there must be quite a few. I take it for granted that all must exist simultaneously, for any other viewpoint is unreasonable*."

Now, Bell would say: "There is only 1 polarization maximum per photon at a time, consistent with the HUP."

*EPR actually state this assumption explicitly, which I paraphrased, and this is the specific point Bell attacked on. It turns out there is no such set of 'answers' which can match experiment. Ergo the assumption is wrong, or there exist superluminal processes which are otherwise unknown.

 

[Nugatory]

So Einstein's "elements of reality" seem to depend on the orientation of our filters! If the photon had one specific polarisation, as Einstein believed it had (please correct me if I'm wrong by saying this again and again) - how could he explain the fact that the orientation of our filters would always lead to a binary choice as to the definite polarisation of the second photon?

We can save the Einstein view of the world by conjecturing that each photon has some physical properties (we'd call them "hidden variables" in the QM lingo) that determine how it will react at any given angle - now all that's necessary is that the two members of a pair also have opposite values for these properties.

As long as the detectors are always aligned, there's no way of distinguishing this hidden variable theory from the classical quantum mechanical explanation in which measuring one particle somehow causes the other particle to magically assume the opposite value (because as you say above, we can't run the experiment repeatedly on the same pair).

So the EPR thought experiment was advanced to suggest that quantum mechanics was incomplete in the sense that there was a deeper realistic theory underneath it; we had more to learn about what was "really" going on with these spin/polarization measurements.

That's a very intuitively compelling argument: "I reach into the bag and pull out a left-handed glove; I know the one still in the bag is right-handed because there was a left-handed and a right-handed glove in there and I just happened to grab the left-handed one first".
It's even more intuitive when you consider how the quantum mechanical explanation works: "The two gloves are in a 'superposition' (whatever that is!) of left-handedness and right-handedness, and when I pull one out and look it, it decides to settle down and be a proper glove - and it miraculously and instantaneously also tells its mate in the bag what handedness it should have too". This is a parody of the QM position, but it fairly conveys the essential weirdness.

So we can forgive Einstein his preference for the former explanation. It was only after his death that John Bell discovered Bell's Theorem, which pointed out a way of experimentally comparing the prediction of the two models. The experiments were done, and they actually support the QM explanation. The world really is that weird. Although we will never know, I think that Einstein would have been convinced at that point.

 

[Tomer]

Thanks for the replies.
I'll continue until I get my answer, or until you get tired of me.
DrChinese - did Einstein really believe that? That a photon has all possible polarisation all of the time?
Was I wrong by saying that Einstein would have interpreted an eigenstate in QM to be a well defined, "real" state? Wouldn't he say, in my first scenario, "the second photon has the polarisation H"?

Nugatory - I understand the EPR thought experiment and the Bell theorem that follows pretty well. I still lack a resolution to the problem I have shown.
How can it be* that the second photon could both have such a polarisation that would with a 100% probability be transmitted through a horizontal filter (first scenario), and hence always had a horizontal polarisation, but at the same time would be with a 100% chance be transmitted through a 30° filter (second scenario), and hence always had a 30° polarisation?

* It can be, of course, under quantum mechanical assumptions. But it seems to contradict Einstein's realist view.

 

[yuiop]

* It can be, of course, under quantum mechanical assumptions. But it seems to contradict Einstein's realist view.

Contradicting Einstein's realist view is probably a good thing, because Einstein's realist view has been proven wrong by experiment.

How can it be* that the second photon could both have such a polarisation that would with a 100% probability be transmitted through a horizontal filter (first scenario), and hence always had a horizontal polarisation, but at the same time would be with a 100% chance be transmitted through a 30° filter (second scenario), and hence always had a 30° polarisation?

If there is a 100% probablity that the photon is transmitted through the second filter if the first first photon is transmitted and both polarisers have the same orientation, then there is only a 75% probablity (not 100% as you say) that both photons will pass through their respective polarisers, if they have a 30 degree difference in orientation to each other. Einstein's incorrect local realist expectation is that their is only a 66% (or less) probablity of both photons passing through both polarisers with a 30 degree difference in orientation.

Note. I think in actual experiments, there is 0% correlation (100% anti-correlation) between the passing photons if both polarisers have the same orientation and 100% correlation if the polarisers have a 90 degree difference in orientation.

 

[Tomer]

Contradicting Einstein's realist view is probably a good thing, because Einstein's realist view has been proven wrong by experiment.

As I've said, I'm trying to understand Einstein's view, or more accurately, I'm trying to understand how the EPR experiment didn't already imply to Einstein that there's a contradiction, or a logical leap, portrayed by the example I described in my previous posts.

If there is a 100% probablity that the photon is transmitted through the second filter if the first first photon is transmitted and both polarisers have the same orientation, then there is only a 75% probablity (not 100% as you say) that both photons will pass through their respective polarisers, if they have a 30 degree difference in orientation to each other. Einstein's incorrect local realist expectation is that their is only a 66% (or less) probablity of both photons passing through both polarisers with a 30 degree difference in orientation.

Please read carefully what I wrote. In no stage I discussed filters that have an angle difference of 30 - I just discussed filters aligned to one another, but that have an orientation of 30 relative to the x-axis.

Note. I think in actual experiments, there is 0% correlation (100% anti-correlation) between the passing photons if both polarisers have the same orientation and 100% correlation if the polarisers have a 90 degree difference in orientation.

Nope - that's the case with electrons. Photons are bosons and have a fully symmetrical entangled wave function, that looks like: |x>|x> + |y>|y> (with a normalization coefficient of course)

 

[Delta Kilo]

In EPR model the photon is assumed to posses a set of hidden variables which, when combined with polarizer angle, fully determine the outcome of a measurement.

Example: let's imagine that each photon has a hidden polarization angle. During the measurement it deterministically 'snaps' to the angle either parallel or perpendicular to the polarizer axis, whichever is closer. The choice of axis determines the outcome of a measurement. New polarization angle after the measurement is then chosen randomly in +/-45 degrees range around the selected axis. In EPR experiment, we assume the initial value for this hidden variable to be random but the same for both photons.

This model qualitatively reproduces observed effects, including single photon passing through a series of polarizers as well as EPR-type experiment with 2 photons. For EPR experiment, it produces perfect correlation with polarizers at the same angle, no correlation at 45 degrees and perfect anticorrelation at 90 degrees. Unfortunately, the numbers for the angles in between agree with Bell's inequalities but disagree with the actual experiment.

Also, in this model, you can't really say "I know the polarization of the second photon is now 30 degrees". All you can say is "the state of the second photon is the same as if it just went through the 30 degrees polarizer", which is almost, but not quite the same thing.

 

[Tomer]

Delta - we're getting there :-)
So does it make any sense at all (did it make any sense to EPR) to imagine a photon that would contain "hidden variables" that could simultaneously be responsible for him to pass with a 100% probability a filter oriented with 0° (first scenario) as well as a filter oriented with 30° (second scenario, traveling back in time...)?

If so, maybe it was a over-simplification to imagine Einstein saying "Photon 2 had an H polarization all along" (like I have read many times!). Maybe the correct way, as you suggest, is to imagine him saying "the photon's hidden variables are responsible for him passing the H filter with a 100% probability" (which might not be equivalent to "the photon has the polarization H").

Did I understand you correctly?

 

[Delta Kilo]

So does it make any sense at all (did it make any sense to EPR) to imagine a photon that would contain "hidden variables" that could simultaneously be responsible for him to pass with a 100% probability a filter oriented with 0° (first scenario) as well as a filter oriented with 30° (second scenario, traveling back in time...)?

Yes. With hidden variables the photon is supposed to have an answer prepared in advance for any polarizer angle. If the answer for a given angle is 1 then the photon will pass with 100% probability, otherwise it will be absorbed with 100% probability. So yes, i is possible in EPR model to have an outcome of 1 "programmed" into the photon for both 0 and 30 degrees.

If so, maybe it was a over-simplification to imagine Einstein saying "Photon 2 had an H polarization all along" (like I have read many times!). Maybe the correct way, as you suggest, is to imagine him saying "the photon's hidden variables are responsible for him passing the H filter with a 100% probability" (which might not be equivalent to "the photon has the polarization H").

Did I understand you correctly?

Yes, but these are just words, I wouldn't look too much into them. I mean if you say "the photon has a definite polarization of that many degrees", you cannot possibly refer to the actual value of its hidden variable because it is, uhm.., hidden. You can only refer to the observable effect which is "the photon will pass the polarizer at that many degrees with 100% probability". As it turns out, there is at most one angle at a time which we can be 100% sure about, so it sort of makes sense to refer to it as the photon's definite polarization angle.

 

[Tomer]

Well, it's subtle, but I think that settles it. I thought the mere possibility of a photon having such a polarization (or state, or whatever) that allows him with a 100% probability to be transmitted in two distinct angles would somehow violate some EPR realist principle, but I guess it doesn't necessarily do so.

Thanks a lot for the replies you all, and especially Delta, you really helped me.