Bell's Theorem and other things Bell

[mugaliens]

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics." - Physicist John S. Bell

In the Lorentz Contraction 2 thread, grav (I think it was grav) proposed Bell's Spaceship Paradox, which generated much discussion along Lorentzian/contractual lines.

Bell was famous for many things other than exploring the reality of length contractions as v --> c. One of his principle focuses involves "hidden variables" involved in quantum mechanics. He challenged several "impossibility proofs" against hidden variables, and throughout his life he "remained interested in objective 'observer-free' quantum mechanics," where emphasis should be placed on what is, rather than what's observed.

Although an experiement in 1972 appeared to violate Bell's Inequality, many believe that future, more precise experiements will reveal that one of the known loopholes had been biasing the interpretations.

Question: How does the EPR paradox figure in with Bell's Inequality?

[hhEb09'1]

Question: How does the EPR paradox figure in with Bell's Inequality?

As is mentioned at that wiki page, Bell's inequality concerns local variables: "if one takes quantum mechanics and adds some seemingly reasonable (but actually wrong, or questionable as a whole) conditions (referred to as locality, realism, counter factual definiteness, and completeness; see Bell inequality and Bell test experiments), then one obtains a contradiction"

The EPR set up has been tested. Einstein was right again

[Ken G]

Actually, I think the normal interpretation is that Einstein was wrong, but Bell figured out exactly in what way Einstein was wrong, thereby salvaging the parts of Einstein's position that were not wrong. A subtle state of affairs, I admit.

Experiments validate quantum mechanics, so we are pretty much past wondering if it's right or not and focusing instead on what is the proper way to interpret it. The part you cannot escape is that quantum mechanics encodes holistic elements of any experimental setup that transcend relativistic causality when interpreted purely locally, and you can either interpret that holism as a kind of "spooky action at a distance", or you can just say the universe is not completely atomistic in regard to information, it includes holistic forms of information "storage" (if one insists on preserving the storage concept of realism). Then you face the issue of how you will interpret your quantum mechanics to be consistent with the answer you like to that issue.

As I understand it, there are three main valid ways to go with quantum mechanics interpretations, depending on personal taste (which basically comes down to how seriously you take your physics, and how willing are you to tolerate metaphysical baggage to support that seriousness). They are:

1)Bohr's approach: quantum mechanics is a prescription for making predictions. The wave function is its fundamental tool. It will never be more than that, never be anything real to be taken literally, because of the fundamental gap we must cross to get a classical system (us) to understand a quantum system (it). The act of crossing that gap is what gives us "wavefunction collapse", which is outside the theory itself and always will be, owing to said insurmountable obstacle. For example, no one will ever use the "wave function of a cat", so it is silly to even suppose there is meaning to such a thing.

2)Everett's approach: quantum mechanics is to be taken completely literally, and the wave function is perfectly real, and the universe, being a closed system, has a wave function that evolves in such a way that no "collapse" can occur to the global system. The collapses we think of are simply what happens when you restrict to a particular "realization" of an experiment, which our minds are trapped into doing (echoing Bohr's "gap" problem). The "truth" is, when you get away from the limitations of our minds, all results in some sense occur, generating "many worlds" that are in superposition.

3) Einstein's approach: quantum mechanics is correct, but incomplete because it does not allow realism-- it rests on a flimsy ontological foundation that should not be tolerated. Realism is a way to treat quantum mechanics that does not refer to the observer or the act of observation in any way. It says that the universe is doing what it does, with or without being observed, and all observation does is convey that information to our minds. It does not need to change that information, or interfere with it in any fundamental way, but there is more information there than we can ever extract with experiment, so that's why we conclude there must be fundamentally random aspects. But Einstein also wanted that information to be "stored" locally in the objects, via "hidden variables", and the EPR paradox was intended to show that if you adopt that approach, quantum mechanics makes impossible predictions. But he was wrong about that part, so now people following that path have to look for nonlocal types of hidden variables, that can restore realism at the expense of locality.

As far as I know, all three of these approaches are perfectly consistent with all experimental data. Personally, I simply ask that we serve with our salad of physics knowledge all metaphysical dressing entirely "on the side".

[mugaliens]

A subtle state of affairs, I admit.

I wouldn't say so. Preventing people from throwing out the baby with the bathwater is no small feat.

The part you cannot escape is that quantum mechanics encodes holistic elements of any experimental setup that transcend relativistic causality when interpreted purely locally, and you can either interpret that holism as a kind of "spooky action at a distance", or you can just say the universe is not completely atomistic in regard to information, it includes holistic forms of information "storage" (if one insists on preserving the storage concept of realism).

I'm not so sure I buy some of the tenets of quantum entanglement. Let us suppose that instead of dealing with quantum particles, we were dealing with a penny that could be split thinwise into two parts such that one showed heads, and the other showed tails. We would not be able to determine which was which until we measured them, thereby destroying them, but even split up, one will be heads, and one will be tails. 100% of the time, when you measure one, then the other, you find that the other is opposite of the first.

Spooky action at a distance?

No! Just the fact that one is heads and it's complement is tails. It's said that "hidden variables" are determined when the pair is created, but I propose that the very creation of the particle pair itself will always result in one of the pair having one attribute and the other particle it's complement.

As for how I interpret it... I like portions of all three. Bohr's approach relies on statistical predictions, which is in line with data on large numbers of particles. Everett's function is borne out by expriements with quantum dots, wires, and wells, and I believe that collapses results in the form of photons emitted in discrete frequencies for any given substance bears witness to Bohr's gap issue, and it's solution by realizing that Bohr's particle model didn't account for angular velocity, whereas Heisenberg's use of Dirac's reduction of Planck's constant accounted for the 2pi factor which facilitated a wave approach. While I believe the universe has a wave function, it's inconsequential due to it's value.

And I like Einstein's approach as he, like Bell, was trying to get away from the apparent and get on towards the actual. But his hidden variable approach, I believe, was similar to the penny-pair example, that yes, it's stored in the particle itself, as a fundament nature of the particle, it's just that we can know what that is until we measure the particle.

As far as I know, all three of these approaches are perfectly consistent with all experimental data. Personally, I simply ask that we serve with our salad of physics knowledge all metaphysical dressing entirely "on the side".

Well, there you go. We shouldn't throw the baby out with the bathwater after all, but instead realize that each approach has both contributions as well as limitations, and so long as we're aware of both, out understand of the thing as a whole increases.

[cjameshuff]

I'm not so sure I buy some of the tenets of quantum entanglement. Let us suppose that instead of dealing with quantum particles, we were dealing with a penny that could be split thinwise into two parts such that one showed heads, and the other showed tails. We would not be able to determine which was which until we measured them, thereby destroying them, but even split up, one will be heads, and one will be tails. 100% of the time, when you measure one, then the other, you find that the other is opposite of the first.

My understanding is that you can separate the halves, and then do something that "flips" one coin half from whatever it is to whatever the other is, and even if the other half is measured before a photon created at the "flip" event reaches it, it will turn out to have been flipped as well. It can not transmit information, because you can only flip the coin without knowing what it initially is, and the observer of the other half can not tell you flipped it until you tell them so by other means...but the fact that it was flipped was apparently communicated to the other coin half instantaneously.

[Ken G]

100% of the time, when you measure one, then the other, you find that the other is opposite of the first.

That situation would not violate Bell's inequality, so it's not the issue with quantum entanglement. Experiments involving entanglement do violate that inequality, and that is the problematic fact here.

[mugaliens]

That situation would not violate Bell's inequality, so it's not the issue with quantum entanglement. Experiments involving entanglement do violate that inequality, and that is the problematic fact here.

Please enlighten me! To date, all the experiments about which I've read fall into the split-penny type.

[Ken G]

Please enlighten me! To date, all the experiments about which I've read fall into the split-penny type.

You need to read them with a better understanding. The important experiments are done on entangled quantum systems, which yield results you could simply never get from a penny. You need to start from the basic differences between quantum and classical mechanics, and then look into experimental data involving entanglement. Then you need to google "Bell's inequality" and understand why split pennies don't violate it but entangled systems do. Then you are ready to understand why the experiments (e.g., by Zeilinger on quantum informatics) require a more sophisticated treatment than a classical understanding of split pennies.

[Ken G]

Let me clarify, I'm not saying anyone who doesn't understand entanglement knows no quantum mechanics, because it is really one of the most subtle aspects of that theory. People still argue what it is all about, and I've had my share of disagreements with people who know a lot of quantum mechanics. There are really two levels of weirdness to quantum entanglement-- weirdness at the level of correlation between measurements, and weirdness at the level of causation between them.

As for causation, we can just say correlation is not causation, and dispense with it in the current discussion. So we are focusing on correlation. Now, Bell's inequality specifies a limit to the amount of correlation you can have without there being some causation involved, if you think that all information about any object is encoded in that object. In other words, if you believe in "local realism".

Now, we can do many classical experiments where local realism works fine, like the split penny mentioned above, or we can distribute the cards from a deck to 52 people and have 52 correlated observations of what those cards are. None of that is entanglement, because none of it violates Bell's inequality of how much correlation you can have without causation, and still have all the information about the cards stored right in the cards.

But entangled quantum systems cannot have all the information about the system stored in its individual components, because Bell's inequality is violated for those systems in actual experiments. So you either have spooky action at a distance, meaning a causation that not only violates the speed of light but can even have an effect precede its cause, or you have a sense of holism to the system, where some types of information (such as can be extracted via looking at correlations) is encoded in the system as a whole, not purely as logical statements about its individual parts (like if she has four aces, he can't have any, because the locally real property of "aceness" is already exhausted by her four aces). The statements about locally real properties encoding all the information is not the way quantum mechanics works, because a multiple-particle system has a wave function that cannot be associated with its individual parts independently, it can only be written as a whole entity with all kinds of interconnections that can seem quite bizarre when you project onto a single part of the system (such as wavefunction "collapse").

This is actually quite a common theme in quantum mechanics, I'm not sure why it causes such consternation in the context of entanglement. Another example that has no direct connection to entanglement is the way quantum mechanics treats identical particles. If you have fermions, like electrons, that are identical, like electrons, then the wave function of a system of them has to be antisymmetric to exchanging the particles. That means the wave function has to obtain a negative sign when you swap any two electrons, and that constrains allowable wave functions for such a system to be holistic, not independently writable for each particle. Atoms even exhibit an "exchange energy", which is a measureable shift in the allowed energies of the atom based on this requirement on its wave function. That would certainly seem to violate local realism to me, so entanglement is just another example of a very common theme in wave functions.

Getting back to entanglement, one may still wonder what is the source of the difference between entanglement of quantum systems and correlations between classical systems like the cards in a deck (if I'm playing poker, and someone shows a deuce, I know I have less chance of drawing one-- that's a classical correlation but does not do violence with local realism). The key difference is that wave functions have something that cards don't-- they have a phase. That means that correlations have to be tracked with a different algebra, an algebra like that of complex numbers, in which you can have phase cancellation.

Phase cancellation means that something that can occur in two ways might end up with no chance of happening, whereas it would have had a chance had there only been one way. That is the behavior that split pennies never have, and it is the source of all quantum weirdness, like the way entanglement can violate Bell's inequality without requiring any spooky action at a distance. But you do have to dump local realism-- which you have to dump in quantum mechanics anyway.