Tuesday, March 3, 2009

Bell's Theorem explained

It may help to see my previous post for background information on Bell's Theorem. However, I hope that this is understandable, even if you didn't read the background.

Quantum mechanics is well known for saying that particles can be in many locations at once, smeared out in a probability wave. Why can't we simply say that particles have definite locations, and that only our knowledge is complete? Because as Bell's Theorem shows, we will run into problems.

The set-up is that we have a source which emits pairs of entangled electrons. Each electron will go into a separate detector. These entangled electrons exhibit strange correlations in their behavior. If you measure the vertical spin of each electron, then both electrons are guaranteed to give the same result. However, we need not always measure vertical spin. We can also measure the horizontal spin, or any angle inbetween the horizontal and vertical. Let's tinker around with it.

We're going to use three different settings for the two detectors. Setting A will ask the question, "Is its spin going up?" Setting B will ask, "Is its spin going to the right?" Setting C will ask, "Is its spin going at a 45 degree angle between 'up' and 'right'?" If both detectors are on setting A, then they will both give the same answer for any given entangled pair of electrons. Likewise, they will give the same answer if they are both on setting B, or both on setting C.

Let us assume that each electron has predetermined answers to each of these three questions. We just don't know the answers yet! We can describe each electron as a set of three answers to the questions posed by detectors A, B, and C. For example, we could use (Y,Y,N) to denote an electron which will answer "yes" to A and B, but "no" to C. Similarly, (Y,N,Y) would denote an electron which will answer "no" to B, but "yes" to A and C.

Recall that if both detectors are on the same setting, then both electrons will give the same answer. This means that both electrons in a pair can be described with the same set of three answers. Each electron pair can be placed somewhere in this Venn diagram:

There are eight possible locations on the Venn diagram, but there's no reason to assume that each possibility is equally likely. Perhaps we could determine the probability of each possibility? But it's rather tricky. The problem is that we can only determine two of the answers for any given pair of electrons. Once each electron has been measured, we've changed their original quantum state and can no longer investigate it. Nevertheless, we can be mathematical detectives and figure it all out, right?

So let's say that we set the first detector to setting A and the second one to setting C. How often is the answer to A different from the answer to C? That is, how often do the electron pairs fall into the shaded region shown in figure (a)? The answer is about 15% of the time, or (1-cos(45°))/2. This can be predicted by quantum theory, but just as importantly, it can be directly observed with experiments.

What happens if we set the first detector to B, and the second detector to C, and we ask how often the two detectors give different answers? That is, how often do the electron pairs fall into the shaded region in figure (b)? Again, the answer is 15%.

How often do electrons fall into the shaded region in figure (c)? Figure (c) is the "union" of the two sets (a) and (b). All electrons which fall into the shaded regions in (a) and/or (b) will also fall into the shaded region of (c). This implies that electrons will fall into shaded region (c) no more than 30% of the time.

To summarize:
(a) Detectors A and C give different answers: 15%
(b) Detectors B and C give different answers: 15%
(c) The union of (a) and (b): At most 30%

Now, what happens if we set the first detector to A and the second detector to B? How often will the two detectors give different answers? If they give different answers, then the electron pair must fall somewhere in the shaded region of (c). Therefore, it couldn't happen more than 30%, right? Wrong! Both quantum theory and experiment show that the answers to A and B are different 50% of the time. We've run into a mathematical contradiction!

What we usually do when we find a mathematical contradiction, is we back up. We look around for assumptions. One of our assumptions must have been wrong! At the top, I said, "Let us assume that each electron has predetermined answers to each of these three questions." Under the mainstream interpretations of quantum mechanics, this assumption is wrong. Electrons do not have predetermined answers to every possible measurement. Instead, we must describe their states with probabilities.

I should also mention that there are ways around this conclusion. Perhaps I made other implicit assumptions, and maybe those assumptions are the wrong ones. There are several other ones, which have names like "realism", "locality", etc. But I'm not going to go into that. I'll just leave Bell's theorem as is, a proof that something is going intuitively wrong.


Anonymous said...

How do you know the first measurement doesn't change the second measurement?
(P.S. There is a small typo on the line that says: (c) The union of (a) and (c): At most 30%)

miller said...

Thanks for the correction.

I simply assumed that the first measurement cannot affect the second measurement. This assumption is called locality. The rationale behind locality is that nothing can travel faster than light. If an effect travels faster than light, then according to Relativity theory, there exists a reference frame in which it is traveling backwards in time. Therefore, we think an event at point A cannot instantly affect an event at point B.

The assumption of locality is rejected by certain interpretations of quantum mechanics.

Anonymous said...

Let me explain my understanding, and you tell me if I have it. The first measurement is on one electron of an entangled pair, and then the measurement changes it, so a second meaningful measurement on the same electron cannot be taken. The second measurement is on the other electron of the entangled pair, which can be any distance from the first electron, and another meaningful measurement on the same electron again cannot be taken. The first measurement is of "up", which is a yes or no question, and both electrons are guaranteed to have the same answer. The second measurement is of "45 deg", which is also a yes or no question, and both electrons are guaranteed to have the same answer. Statistically over many pairs, 15% of the pairs are either "up" and NOT "45 deg", or NOT "up" and "45 deg" (giving different answers to the two questions). Similarly, statistically over many other pairs from the same source, 15% of the pairs are either "right" and NOT "45 deg", or NOT "right" and "45 deg". BUT, statistically 50% of the pairs are either "up" and NOT "right", or NOT "up" and "right", which gives the mathematical contradiction which you clearly explained. So the answer, explained somewhat anthropomorphically, is that the electrons don't decide what to be until you take the measurements.
By the way, someone asked me to read this blog and explain it to him if I could.

miller said...

Yes, you understand it correctly. I hardly have anything to add.

Anonymous said...

Hello miller,
Could explain how you got a probability of about 15% or (1-cos(45))/2, without knowing the probability of each event in the diagram?

another thing, it's not crucial but when you say that if two detectors are set to measure the same thing they will have the same result for two entangled particles? what are you basing it on? because if we're talking about spin angular momentum which should be conserved and the initial value was zero than the detectors will always show opposite results.


miller said...

True, the detectors will show opposite results rather than the same results. But I switched it up in order to simplify it slightly.

The 15% is measured through experiment, and predicted by theoretical calculations. The theoretical calculations are difficult to understand without a quantum mechanics background, but they are rather straightforward. We know the initial wavefunction of the entangled electrons, and from this we can predict the probabilities of results of any observation.

Anonymous said...

OK, thanks.

Another question, I know that when you make a measurment for example of the spin on a certain axis you force the particle to choose a spin up or down on that axis, and then due to conservation the other entangled electron will have the opposite spin.
my problem is: when I do the measurement and cause the wave function to collapse, doesn't the act of an outside measurement makes the argument of conservation not applicable anymore because the system is not closed any more bur was interrupted from an outside source, so why should the other electron be in an opposite spin state?

Thanks, by the way if you're an undergraduate what year are you? I mean did you already have a course in quantum mechanics?

miller said...

Spin is not necessarily conserved when the wavefunction collapses. Nor is energy, mass, momentum, etc. However, when the electrons are first emitted, no wavefunction collapse is involved, so they must be in a state of total spin zero. This spin zero state causes the electrons to have opposite spins when measured direction.

However, if we measured one electron at 0 degrees, and the other one at 90 degrees, obviously there's no way that the spin could be conserved. Nonetheless, if we calculate the expected spin (averaged over all possibilities, weighted by probability), then it is zero.

I am going into fourth year. I've taken as much quantum mechanics as is offered to undergraduates here.

Anonymous said...

I've written some art about Bell's Theorem,, you can check it on page: Bell's theorem it is in polish language so try google translate

Anonymous said...

Again, these complete and pure nonsenses.

Quantum guys can not calculate the conditional probability, because they do not understand the concept of probability at all.

This is not a real physical entity that can act as a cause.

The fact that the measurements are independent does not mean that the results must be independent.

BN said...

Hi there. Regarding the chances that the readings from A,B; B,C; and A,C, respectively, will differ, you note that “there's no reason to assume that each possibility is equally likely”. (I like this turn of phrase, so I am going to use it for my own purposes…) So, if the detectors are set at A (0 degrees) and C (45 degrees), there will be a 15% chance that the readings will differ. Likewise if the detectors are set at B (90 degrees) and C. This makes sense- in each of these cases, the detectors are 45 degrees apart.
From what I understand, if the detectors are set at 45 degrees apart or less, they have a relatively higher likelihood of correlating than if they are set between 45 and 90 degrees apart. Not knowing a whole lot about ‘picturing’ electron behavior, this also makes sense: if an electron is detected after passing through at 0 degrees, its ‘orientation’ (and thus the orientation of its partner electron) was ‘tending towards’ going through. Thus it seems that it was also ‘tending towards’ going through 45 degrees MORE SO than it was towards 90 degrees (since 45 is closer in its orientation to 0 than 90 is).
Knowing this ‘tendency’ of electrons, it therefore is not surprising that when the detectors are set at 90 degrees apart, their readings differ by more than 30%. Knowing this behavior of electrons, it seems like a ‘regression in knowledge’ to then “assume that each possibility is equally likely” by applying a statistical method that leaves out this tendency of entangled electrons to tend more towards going through two different angles, respectively, that are relatively close together (0-45 degrees), than when they are relatively far apart (45-90 degrees).
That is, I agree with the logic of Bell’s inequality (what of it I can understand!), but based on the above, what am I missing in thinking that the inequality uses probabilities that don’t readily apply to what goes on in these experiments? We already know, based on QM and experiment, that electrons follow a certain probability of being detected (based on cosine wave). Why then leave this knowledge behind, so to speak, and apply a statistics that (wrongly) treats a 90 degree angle separation as equal to two 45 degree angle separations, as far as electron behavior is concerned?

miller said...

If you think that Bell's inequality basically doesn't apply to the electron spins, you're correct. It does not apply, given our understanding of how electron spins work.

However, if you assume that the electron spins are described by "hidden variables", and exist somewhere on the Venn diagram, then Bell's inequality must apply. This demonstrates that our understanding of how electrons work is incompatible with a basic hidden variables picture.

BN said...

Gotcha. Thanks!