Bell’s Theorem - when local variables are NOT hidden

[robinpike]

From Wikipedia: Bell's Theorem draws an important distinction between quantum mechanics (QM) and the world as described by classical mechanics. In its simplest form, Bell's theorem states:

“No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.”

 

When trying to understand why the above is true, I find it difficult to follow the explanations without a classical particle to use as an example. I would like to step through Bell’s Theorem using a classical particle, albeit a hypothetical classical particle, but nonetheless a classical particle.

 

The purpose of this post is to understand how Bell’s Theorem proves that all classical particles must, at some point, give a different experimental result to a quantum particle. The experiment that I would like to focus on is the spin measurement of a particle.

 

The hypothetical particle that I have in mind has internal movement that is at a fixed, single speed. The particle is structured like a Slinky toy that is bent around into a circle. If you take a Slinky toy and stretch it out, it is just a long strand that has been coiled up. It is the same for this particle; the particle is basically a long strand that moves at a fixed speed, with the strand coiled up and then the whole thing bent around into a circle to form a torus.

 

The particle as a whole has spin because of the movement around the torus (or ring). If the plane of the ring is horizontal, then the particle’s axis of spin is vertical.

 

The particle’s axis of spin can be at any angle to a chosen x, y and z axis – these angles being the ‘hidden local variables’ mentioned in Bell’s Theorem. However, for this particle we can SEE these angles at all times.

 

Unlike light, which moves at the same fixed speed to all frames of reference, the circular movement of the particle has to be against just one frame of reference, a universal frame of reference. (Otherwise which frame of reference is the circular movement, circular to? Overall the particle is stationary – stationary to whom?)

 

This highlights a problem, for the fixed speed to a single frame of reference makes it difficult for the particle as a whole to move in any direction at all. To get around this, a slightly more complex structure is required, one that consists of multiple strands rather than a single strand (rather like joining together many short Slinkys into one long Slinky). And in addition, the fixed speed only applies to the head and tail of each strand, allowing the body of each strand to stretch and compress.

 

This set up allows the particle to bunch up on one side and stretch out on the other side of the ring (say by an electric field, without going into detail), allowing the particle as a whole to move in a direction that is in the plane of its ring.

 

Note that it is difficult for the particle to move in a direction that is not in the plane of its ring. For example, if the plane of the ring is horizontal, then to keep this orientation and move upwards would be difficult: the opposite sides of each coiled strand would need a different distortion and this distortion would need to be applied in the same manner to each and every coiled strand in the ring. It is easier for the particle to swivel the plane of its ring into the vertical position, and then move upwards.

 

This behaviour means that in a beam of these particles, the axis of spin of every particle is at right angles to the direction of movement. In the picture below, for a beam moving in the x-axis direction, the particles can have any axis of spin in the y-z plane, but not an axis of spin outside of that y-z plane, i.e. the axis of spin is at right angles to the x-axis.

Bell’s theorem applies to any classical particle, so it must apply to this hypothetical classical particle as well. I would like to use this particle to step through Bell’s Theorem so that I can understand why any classical particle must disagree with what is found experimentally.

 

 

[Mordred]

You might want to watch This excellent video. Posted in a thread by Swansort on the subject.

 

http://www.scienceforums.net/topic/87347-why-hidden-variables-dont-work/

The video would be a good one for being in a pinned thread

[robinpike]

Thanks Mordred for adding that link. When Swansont posted that video I still wasn't able to follow every part of how Bell's Theorem was being applied, so maybe this post will help others as well as myself.

 

Part of the explanation requires an understanding of how particle spin detectors work, so maybe someone could describe how they measure the spin of a particle. The ones I need to understand are the particle spin detectors, rather than photon spin detectors.

 

I have seen descriptions of the Stern-Gerlach set up, but is that actually used in these kind of experiments? I understand that it is used to measure the spin of an unpaired electron in a silver atom and when a beam of silver atoms is passed through the device, the outline of a circle is produced on the detector screen. In principle, the hypothetical particle that I have chosen to step through Bell's Theorem, would it seems, also produce a circular outline if a beam of the particles were to pass through such a device.

 

A lot of the discussions that I have seen on Bell's Theorem use 'paired particles' and three detectors, angled equally between themselves at 0 degrees, 120 degrees and 240 degrees.

[swansont]

Thanks Mordred for adding that link. When Swansont posted that video I still wasn't able to follow every part of how Bell's Theorem was being applied, so maybe this post will help others as well as myself.

 

Part of the explanation requires an understanding of how particle spin detectors work, so maybe someone could describe how they measure the spin of a particle. The ones I need to understand are the particle spin detectors, rather than photon spin detectors.

The effect doesn't rely on how you detect the spin, just that you can detect it. This would seem to be an unnecessary complication. But to answer the question, another way is fluorescence detection — the spin states are at different energies (as long as the nucleus is not spin 0), and will be resonant with different frequencies of light. You can see which source "lights up" the atom

 

 

I have seen descriptions of the Stern-Gerlach set up, but is that actually used in these kind of experiments? I understand that it is used to measure the spin of an unpaired electron in a silver atom and when a beam of silver atoms is passed through the device, the outline of a circle is produced on the detector screen. In principle, the hypothetical particle that I have chosen to step through Bell's Theorem, would it seems, also produce a circular outline if a beam of the particles were to pass through such a device.

The edges of the pattern are unimportant — it's that different spins deflect in different directions

 

I think you'll find that what's important is that the atom will deflect in a predictable direction 100% of the time. However, I'm pretty sure the fact that your spin is not quantized is going to be an issue, since this is a discussion of a classical system imposed on an inherently quantum effect.

[robinpike]

I find descriptions that apply Bell's Theorem difficult to follow.

 

For example, below is an extract from this link http://www.upscale.utoronto.ca/PVB/Harrison/BellsTheorem/BellsTheorem.html

 

and I fall at the first hurdle! For example, where does the 85% and 50% come from in this part of the explanation?

 

But if we try to measure the spin at both 0 degrees and 45 degrees we have a problem.

The figure to the right shows a measurement first at 0 degrees and then at 45 degrees. Of the electrons that emerge from the first filter, 85% will pass the second filter, not 50%. Thus for electrons that are measured to be spin-up for 0 degrees, 15% are spin-down for 45 degrees

And here is the context of the above text...

APPLYING BELL'S INEQUALITY TO ELECTRON SPIN

Consider a beam of electrons from an electron gun. Let us set the following assignments for the three parameters of Bell's inequality:

A: electrons are "spin-up" for an "up" being defined as straight up, which we will call an angle of zero degrees. B: electrons are "spin-up" for an orientation of 45 degrees. C: electrons are "spin-up" for an orientation of 90 degrees.

Then Bell's inequality will read:

Number(spin-up zero degrees, not spin-up 45 degrees) + Number(spin-up 45 degrees, not spin-up 90 degrees) greater than or equal to Number(spin-up zero degrees, not spin-up 90 degrees)

But consider trying to measure, say, Number(A, not B). This is the number of electrons that are spin-up for zero degrees, but are not spin-up for 45 degrees. Being "not spin-up for 45 degrees" is, of course, being spin-down for 45 degrees.

We know that if we measure the electrons from the gun, one-half of them will be spin-up and one-half will be spin-down for an orientation of 0 degrees, and which will be the case for an individual electron is random. Similarly, if measure the electrons with the filter oriented at 45 degrees, one-half will be spin-down and one-half will be spin-up.

But if we try to measure the spin at both 0 degrees and 45 degrees we have a problem.

The figure to the right shows a measurement first at 0 degrees and then at 45 degrees. Of the electrons that emerge from the first filter, 85% will pass the second filter, not 50%. Thus for electrons that are measured to be spin-up for 0 degrees, 15% are spin-down for 45 degrees.

Thus measuring the spin of an electron at an angle of zero degrees irrevocably changes the number of electrons which are spin-down for an orientation of 45 degrees. If we measure at 45 degrees first, we change whether or not it is spin-up for zero degrees. Similarly for the other two terms in this application of the inequality. This is a consequence of the Heisenberg Uncertainty Principle. So this inequality is not experimentally testable.

In our classroom example, the analogy would be that determining the gender of the students would change their height. Pretty weird, but true for measuring electron spin.

However, recall the correlation experiments that we discussed earlier. Imagine that the electron pairs that are emitted by the radioactive substance have a total spin of zero. By this we mean that if the right hand electron is spin-up its companion electron is guaranteed to be spin-down provided the two filters have the same orientation.

Say in the illustrated experiment the left hand filter is oriented at 45 degrees and the right hand one is at zero degrees. If the left hand electron passes through its filter then it is spin-up for an orientation of 45 degrees. Therefore we are guaranteed that if we had measured its companion electron it would have been spin-down for an orientation of 45 degrees. We are simultaneously measuring the right-hand electron to determine if it is spin-up for zero degrees. And since no information can travel faster than the speed of light, the left hand measurement cannot disturb the right hand measurement.

So we have "beaten" the Uncertainty Principle: we have determined whether or not the electron to the right is spin-up zero degrees, not spin-up 45 degrees by measuring its spin at zero degrees and its companion's spin at 45 degrees.

Now we can write the Bell inequality as:

Number(right spin-up zero degrees, left spin-up 45 degrees) + Number(right spin-up 45 degrees, left spin-up 90 degrees) greater than or equal to Number(right spin-up zero degrees, left spin-up 90 degrees)

This completes our proof of Bell's Theorem.

The same theorem can be applied to measurements of the polarisation of light, which is equivalent to measuring the spin of photon pairs.

The experiments have been done. For electrons the left polarizer is set at 45 degrees and the right one at zero degrees. A beam of, say, a billion electrons is measured to determine Number(right spin-up zero degrees, left spin-up 45 degrees). The polarizers are then set at 90 degrees/45 degrees, another billion electrons are measured, then the polarizers are set at 90 degrees/zero degrees for another billion electrons.

The result of the experiment is that the inequality is violated. The first published experiment was by Clauser, Horne, Shimony and Holt in 1969 using photon pairs. The experiments have been repeated many times since.

The experiments done so far have been for pairs of electrons, protons, photons and ionised atoms. It turns out that doing the experiments for photon pairs is easier, so most tests use them. Thus, in most of the remainder of this document the word "electron" is generic.

[imatfaal]

I really don't like that explanation. Try Dr Chinese's run through (in three layers of difficulty) - it is also worth understanding the maths of the inequality first

 

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

[swansont]

I find descriptions that apply Bell's Theorem difficult to follow.

 

For example, below is an extract from this link http://www.upscale.utoronto.ca/PVB/Harrison/BellsTheorem/BellsTheorem.html

 

and I fall at the first hurdle! For example, where does the 85% and 50% come from in this part of the explanation?

 

 

If you don't understand that, I suggest that you will not be able to understand the subsequent discussion, because this has everything to do with measuring and superposition with regard to spin. You would be better off learning the QM underlying this first before trying to tackle Bell's inequality.

[robinpike]

The effect doesn't rely on how you detect the spin, just that you can detect it. This would seem to be an unnecessary complication.

 

I'm definitely confused - the 85% and 50% values must be based on something - it is not obvious to me!

 

Also, referring back to the extract, it shows the electron / particle under test travelling in a straight line through detector 1 and 2.

 

Is this really the case? And how is an electron stopped at each filter based on its spin?

 

Thanks

 

The figure to the right shows a measurement first at 0 degrees and then at 45 degrees. Of the electrons that emerge from the first filter, 85% will pass the second filter, not 50%. Thus for electrons that are measured to be spin-up for 0 degrees, 15% are spin-down for 45 degrees.

[swansont]

 

I'm definitely confused - the 85% and 50% values must be based on something - it is not obvious to me!

 

Yes, it's based on quantum mechanics. It's kind of a test — if you understand why it's 85% and 50%, then you know enough to go on to the next part of the discussion. If you don't though, there's no point in moving forward.

[robinpike]

I really don't like that explanation. Try Dr Chinese's run through (in three layers of difficulty) - it is also worth understanding the maths of the inequality first

 

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

 

Thanks, in that explanation it lists a table of combinations of pairs of measurements for three detectors at 120 degree angles.

 

 

Permutations of A, B and C with likelihood of matches (++ or --). All we have done is list all the possibilities, and calculated the averages of random tests of the pairs [AB], [bC] and [AC] for each of them. That is, IF we had some way to actually test 2 settings simultaneously for one photon. Can a way be found to do this?

 

OK, now here is the really hard part (just kidding): it is pretty obvious from the table above that no matter which of the 8 scenarios which actually occur (and we have no control over this), the average likelihood of seeing a match for any pair must be at least .333!

 

I don't understand what the average column is calculating? It appears to be calculating the number of combinations for a 'match' by two detectors - rather than the probability of the combination for a 'match' by two detectors?

 

Or have I misunderstood this?

[imatfaal]

Because each row in that table is one configuration (eg A- B+ C-) of all the 8 possible permutations (A+/- B+/- C+/-) the answer given is not called a probability but an average outcome. Dr C presents all 8 permutations, the response to each measurement, and (because which measurement made is random) the average outcome provided that there is an even mix of measurements. Eg Outcome 6 will always give positive if measured at AC - we will measure at AC one third of the time; therefore on average outcome 6 will have a positive response one third of the time

Each of the 8 permutations gives an outcome with at least one of the AB AC BC measurements matching (ie one out of three is 0.333) - and two give an outcome where all three measurements will match

If you sample a fair spread of AB, AC, and BC you will on average get a greater than 1/3 positive response.

[robinpike]

I don't seem to be able to grasp what is going on here at all. The example table was for the measurement of the polarization of photons, but the polarizers don't seem to have much accuracy, which doesn't help.

 

So suppose electrons are used instead, with filters of much better accuracy, say ones that are able to filter the electrons which have spin within 5% of the angle of the filter?

 

So using the set up of the first filter vertical and the second filter at 45 degrees...

 

When the beam of electrons of different angles of spin pass through the first filter, the electrons that get through have the same angle of spin, or very close to the same angle of spin, which is quantised as either an up spin or a down spin.

 

But when those filtered electrons get to the second filter, I don't see how any of the electrons make it through that filter? It's not a case of 85% or even 50%, why isn't it 0%?

[swansont]

I don't seem to be able to grasp what is going on here at all. The example table was for the measurement of the polarization of photons, but the polarizers don't seem to have much accuracy, which doesn't help.

 

So suppose electrons are used instead, with filters of much better accuracy, say ones that are able to filter the electrons which have spin within 5% of the angle of the filter?

The accuracy of the filter is not the issue — it's assumed to be 100 efficient. I suggest you study up on polarization and/or spin. Ask questions about that if you need to — that what the forums are for — but do that is a separate thread where that's the only subject of discussion.

 

Without that knowledge, you have no chance of understanding Bell inequalities.