In 2009, I wrote an explanation of Bell’s Theorem that could be understood by popular audiences. I wanted to repost it, but ended up rewriting it completely.
Although the predictions of quantum theory are well-understood, its interpretation is famously difficult. Because quantum mechanics only makes probabilistic predictions, many people have desired a “hidden variable” interpretation, where quantum objects have definite states, despite appearances to the contrary. But hidden variable interpretations are generally not accepted.
It is certainly possible to create a hidden variable interpretation that agrees with the all the predictions of quantum theory, and de Broglie-Bohm theory is an example of such a interpretation. However, de Broglie-Bohm has a number of unsatisfying properties. Indeed there are a few theorems that prove that any hidden variable interpretation must have unsatisfying properties.
The most important of these is Bell’s Theorem, formulated in 1964. What follows is an explanation of the thought experiment, the mathematical proof, and its implications.
The setup
Bell’s Theorem considers a particular thought experiment, in which two electrons are emitted simultaneously from a single source in opposite directions. This source emits electrons that are “entangled”, meaning that their quantum states are correlated with one another. If you perform the same measurement on both electrons, both measurements will always produce the same result.1
Fig. 1: The electrons are entangled such that when you measure vertical spin of both, either both electrons are spin up, or both are spin down.
The particular thing we want to measure is the electron’s spin.2 Electron spin is quantized; if you measure the spin in the vertical direction, you will either measure +1/2 or -1/2 (in the appropriate units), and never anything between. If the result is +1/2, we call it spin-up, and if it’s -1/2, we call it spin down. You can also measure the spin in the horizontal direction, and again you will always measure +1/2 or -1/2.
Now, you may have heard that in quantum mechanics, it is impossible to know both the position and momentum of a particle. This is a common occurrence in quantum theory, when it is impossible to make two measurements at the same time. This is true of horizontal spin and vertical spin. You can measure the vertical spin of an electron, and you can measure the horizontal spin, but you can’t measure both at the same time.3
While we cannot measure both the horizontal and vertical spin of a single electron, we can do the next best thing, and measure two entangled electrons simultaneously. Preferably these measurements are performed sufficiently far apart that no information can be exchanged between electrons without traveling faster than light. And furthermore, we don’t need to restrict ourselves to horizontal and vertical directions, we can make our detectors freely rotate to any angle. In particular, we consider three configurations, with angles 0 degrees, 45 degrees, and 90 degrees from vertical.
Fig. 2: We measure the spin of the two electrons simultaneously. Experiment A: We measure one at 0 degrees, and one at 90 degrees. Experiment B: We measure one at 0 degrees, and one at 45 degrees. Experiment C: We measure one at 45 degrees, and one at 90 degrees. The “coincidence rate” is the probability that both detectors will measure the same spin (in the direction measured, ie both red, or both blue). Quantum theory predicts a coincidence rate of (1+cos(x))/2, where x is the relative angle between measurements.
The probability paradox
Under a hidden variable interpretation, each electron has a definite state even if it’s something you never measure. Every electron is either spin-up or spin-down, and every electron is either spin-left or spin-right. For Bell’s Theorem, we also need to consider the spin along a 45 degree diagonal; every electron is either spin-up-right or spin-down-left. Putting it all together, every electron is in one of the following eight states:
(up, right, up-right)
(up, right, down-left)
(up, left, up-right)
(up, left, down-left)
(down, right, up-right)
(down, right, down-left)
(down, left, up-right)
(down, left, down-left)
These eight possible states can be summarized in the Venn diagram below.
Fig. 3: A Venn diagram illustrating eight possible hidden variable states that an electron may have in the Bell’s Theorem thought experiment.
If you measure both electrons in the same direction, then they produce identical results. This indicates that both electrons are in the same section of the Venn diagram, although we don’t know exactly which section that is. In fact, we don’t even know what the probabilities are for each section. Since we can only measure two of the circles of the Venn diagram at a time,4 we will have to do some sleuthing to find out.
However, if you go through the sleuthing, you will find that the probabilities don’t make any sense! This can be illustrated in a single picture (Fig. 4 below), using the coincidence rates predicted by quantum theory (shown in Fig. 2).
Fig. 4: An illustration of bell’s inequality. We consider the probability that the electrons are in the shaded region of the leftmost Venn diagram. It must be at least as large as the shaded region in the upper right diagram, which we determined was 50% in experiment A (see Fig. 2). Additionally, it must be no more than the sum of the shaded regions in the lower left diagrams, which we determined are 15% each, using experiments B and C.
Figure 4 shows a contradiction. The shaded region of the leftmost Venn diagram must be at least 50%, but also no more than 30%. It cannot be both!
Reconsidering our assumptions
Our paradox arose from a number of different assumptions, so we can conclude that at least one of those assumptions was incorrect. Let’s review our assumptions.
First, we assumed that the quantum theory predictions are correct. Of course, it may be the case that when you reproduce Bell’s thought experiment with a real experiment, quantum theory turns out to be wrong. However, a number of such experiments have been performed, always confirming quantum theory.
Second, we assumed that the electrons cannot affect each other, since they’re far apart and the measurements are simultaneous. This assumption is called “locality”.
Finally, we assumed that there is a hidden variable theory. More specifically, we assumed that each electron has a state that determines the outcome of every possible measurement (even though we can only pick one of those measurements).
So the conclusion is that any hidden variable theory must be nonlocal.5 You can have hidden variables, but if you do, then particles must exchange this hidden information at speeds faster than light.
This is unsatisfying, because under relativity theory, nonlocality implies travel backwards through time. It brings to mind experiments where you send a message back in time that kills your grandfather. Of course, you can’t actually do that, because the only thing that travels back in time is hidden information that can never be uncovered by any experiment. Even so, what exactly is carrying this information, and what rules govern it? As de Broglie-Bohm theory shows, nonlocal hidden variable interpretations are possible, but understandably not very popular.
[Next up: the Kochen-Specker Theorem]
1. Strictly speaking, the entangled electrons could also be anti-correlated with each other, always producing opposite results. However for simplicity I’m considering positive correlation. (return)
2. A note on experimentally measuring electron spin: it’s very difficult. There’s a spin detector in our lab, and its operation is the subject of a former student’s dissertation. In practical realizations of the Bell’s Theorem thought experiment, they use light instead of electrons, and measure polarization instead of spin. However, I’m sticking to electron spin for simplicity of the explanation. (return)
3. What you can do is measure horizontal spin and vertical spin in succession, but that’s different, since the first measurement changed the quantum state of the electron. If you measure a single electron’s vertical spin twice without measuring anything between, then you will always get the same result both times. However, if you insert in the middle a measurement of the electron’s horizontal spin, then the results will be completely uncorrelated. (return)
4. One thing that long puzzled me about the Bell’s Theorem is that it seems like you could prepare not just two, but three entangled electrons. You could then perform all three measurements simultaneously, determining exactly where the electrons appear in the Venn diagram. However, it turns out that this is not possible. While you can indeed prepare three entangled electrons perfectly correlated in the vertical direction, you will find that they are no longer perfectly correlated in the horizontal direction. The proof is easy for anyone with a physics background, but the rest of you will have to trust me. (return)
5. Of course, theoretical physicists have been prodding Bell’s Theorem for decades, so some of them have come up with other assumptions of the theorem which could be rejected in place of locality. I include this footnote as a shout out to the pedants looking for an opportunity. (return)
For those with some physics and math background, Bell’s original paper is well worth reading. Short and sweet.
It’s also worth noting that Bell himself favoured de Broglie-Bohm.
But nothing is traveling faster than light according to de Broglie-Bohm! You merely need to appeal to nonlocality to explain the observed correlations, not claim that anything (like information) is propagating through spacetime from A to B and/or B to A. It doesn’t work by traveling or propagating like that, and de Broglie-Bohm makes no claim that it does. No-signaling theorems still apply, so special relativity isn’t violated in that sense, just like it isn’t violated by orthodox quantum mechanics which must agree with de Broglie-Bohm on this.
To put it another way, the wavefunction (aka the pilot wave) just is nonlocal, which should’ve been obvious from the beginning since it is mathematically defined on an abstract 3N-dimensional configuration space. That doesn’t imply any of the actual physical things which are configured move through our actual 3+1 spacetime faster than light. The fact that de Broglie-Bohm put things in certain definite places (because it actually has real particles unlike other theories) shouldn’t persuade you that special relativity won’t like that, because Einstein’s theory also has things in definite positions which have definite trajectories… that’s the only coherent way you could’ve understood all of the talk about lightcones and such in the first place. If you get rid of that, it’s not at all clear how you could be coming into closer agreement with relativity (and you still wouldn’t be getting rid of nonlocality, which is just a fact that needs to be explained by any sort of quantum theory).
It isn’t very popular, true. But it’s not understandable and it’s very odd to fault de Broglie-Bohm for making the right experimental predictions and agreeing with orthodox quantum mechanics about them. It’s just explicit and transparent in the guiding equation that there is nonlocality (when you have entangled states), whereas I guess some people would prefer to sweep it under the rug somehow or pretend it’s something they don’t need to worry about. Like Bell and others have also said, I think it’s a credit to this theory that it isn’t making this fact any more obscure than it needs to be, and there’s no appeal to mystical principles like complementarity which are supposed to seem profound, because it’s just a theory of physics which tells you in the most straightforward and sober way what there is and how it behaves. It’s also not making fishy terms like “measurement” or “observable” fundamental terms in the basic formulation of the theory, since those are derivable from the actual stuff in the world, which a physical theory describes as behaving in a certain consistent way under all circumstances.
cr @2:
You can quibble about “traveling”, but BM does have instantaneous effects by construction. It’s right there in the guiding equation; the velocity of a particle depends explicitly on the positions of all other particles at the same time. Lorentz invariance is broken, whether you can observe it or not.
More generally, as Bell himself wrote in the paper I linked to;
@Consciousness Razor,
I didn’t say that. I was talking about hidden variable theories more generally. One can imagine that there is a hidden variable theory (not de Broglie-Bohm) which is local and does not make the same experimental predictions. So there are experimental tests of Bell’s theorem, but they just don’t say one thing or another about de Broglie-Bohm.
Right. (Well, I think the talk of “all other particles” could be somewhat misleading, because their causal dynamical interactions with each other still have to be local or in accordance with SR, and only entangled particles exhibit such nonlocal behavior. But if that’s understood, then never mind.)
Maybe you think this is quibbling (and I’m not saying this for your information, since I think you understand, but for others here). It violates SR in a very particular way, not just any old way you might dream up such as the superluminal motion of a physical object, and not in a way that disagrees with SR or orthodox QM about any observable experimental outcomes. If you thought, like Einstein did, that you could account for literally everything at a certain point by looking at its past light cone, then you were wrong about that. That’s because in certain specific cases where there are entangled (or “non-separable”) quantum states, you also have to account for certain events which are spacelike separated from that point which in some sense “influence” what happens there. Einstein in the EPR paper thought you could prevent A from being disturbed by separating it from B, but like it or not, in fact it gets disturbed according to any theory which correctly and fully describes what’s going on in reality.
I would disagree with this from Bell: “Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.” It may not be Lorentz invariant, but we’re not forced to conclude that a “signal” must “propagate instantaneously” … This is kind of a sloppy way to talk about it, and Bell had to know de Broglie-Bohm is a counter-example which does account for an “influence” of some sort, one which is not literally a “propagating signal” in the sense of some matter or light or what-have-you that is moving from one place to another.
But the fact is that nothing is moving from Alice to Bob or vice versa, except the particles (or strings, whatever) within their own separated regions of space. Not in the real world, and not according to de Broglie-Bohm — so that’s not what the problem is. The problem is whether or not any quantum theory (or anything which gets these predictions right) can be totally compatible with relativity in various other respects, since as you point out the lack of signaling isn’t a guarantee that it’s relativistic in all the ways that we might think are important. That’s a problem because we all agree relativity gets a whole lot of stuff right, and it isn’t at all obvious what needs to go in order to bring them together into a single consistent framework.
cr @5:
No, they’re not local. That’s the point! Bell’s theorem gives you two options*: ditch locality and keep realism (BM), or ditch realism and keep locality. Ditching locality means ditching SR, even if you don’t specify a mechanism for “stuff” moving from one point to another. BM really is non-local, and the guiding equation makes that explicit. It is the mechanism, if you like. There is no “stuff” moving from one point to another; just a relationship, like Coulomb’s law, which implies instantaneous response. The way Bohmians keep SR on firm empirical ground is to introduce an unobservable preferred foliation of spacetime which defines hypersurfaces of equal time.
Yes you are, really. BM-like theories require that instrument settings** in an EPR scenario affect readings at the other location/time, even at space-like separations. It’s an instantaneous effect, whether you use the word “propagate” or not.
*Maybe there are more, but these are the options I’m aware of.
**”instrument setting” implies observers, so is misleading. But “preferred basis due to the physical conditions at each spacetime event” seems rather awkward.
FYI the next scheduled post is about the Kochen-Specker theorem, which has more cool things to say about de Broglie-Bohm.
I don’t know what you’re even talking about. If you’re not entangled with, say, a galaxy billions of light years away from you, the particles in that galaxy simply do not affect what’s going on with the particles where you are. Only if you have, say, a pair of photons leaving a calcium atom (or numerous other circumstances obviously) do you get the result that those entangled photons have those statistics which cannot be explained by a local theory because they violate Bell’s inequality. As a general statement, the guiding equation says the position of one depends on the rest, but in actual practice you have to essentially factor out that distant galaxy and the rest, just not in the case of entangled states.
I don’t think you can ditch realism and still have a serious physical theory. That is a logical possibility, yes, but only in the sense that solipsism is a logical possibility, which can’t be refuted by any rational or empirical methods or arguments or evidence. I think there is a real world, and physics has a responsibility as an empirical science to attempt to explain that. If you have a solipsist come along and tell me that physics is up to something else other than explaining reality (because there is no such thing, they say), then I can agree with you that there may be a person like that, but I don’t have to take their claims seriously.
There are various loopholes, which is not to say they’re any good. You can say the experiments aren’t good enough to demonstrate nonlocality, because they’re not perfectly efficient (not viable anymore, but decades ago that was still an honest question). Or you could invent some very absurd conspiratorial theories, such that any time you choose to measure spin one way as opposed to another, you had been forced by a past conspiracy to make that choice which (through local interactions) accounts for both you and the spin. Or you could have a many-worlds theory, which may have to be nonlocal in an important sense anyway (but isn’t proven by Bell’s theorem), but I don’t think I have to tell you that many-worlds brings with it a whole other set of problems.
You’re just being silly here. It was on firm empirical ground well before any of this was even discussed, because it’s using the same wavefunction which has to give you the same as Schrodinger with the Born rule probabilities. You just have standard quantum mechanics, except that it lacked particles, so you added them and described their evolution. That’s really it.
A preferred foliation, if one is invoked*, is just a weird insignificant after-thought to all of that, which makes no empirical difference, neither to put the theory on firm ground nor to take it off of such ground. You do have to admit that the theory is constructed in a way which implies some such thing**, but that doesn’t mean it’s doing anything useful for you (or against you for that matter) or that the theory’s empirical adequacy actually depends on it. (I mean ask yourself, why would it? It’s unobservable, so how would observable stuff depend it?)
* Some are trying versions which don’t involve anything like that, since it isn’t central to the theory and isn’t in fact useful for doing any empirical work. That said, I’m not too hopeful they’ll be successful, since it’s apparently an extremely difficult challenge to excise a piece of math like that and keep the rest intact. I just don’t think you should lose any sleep over it, because there’s a piece of math which doesn’t have any real-world consequences…. I mean, come on, an unobservable foliation has done exactly nothing to bother you and will never do anything to bother you, so as unnecessary or as unpleasant as it may seem, why should you let it bother you?
** Sort of like inflation theories implying there’s a multiverse. Any other universes are unobservable (in most respects at least), and they’re not a part of how the theory goes about giving you empirical predictions about our universe, doesn’t make them better/worse/different. They don’t really do anything at all. It’s just an implication of the theory, according to some, that there are those other universes somewhere. But who really cares? If that’s the best cosmological theory we’ve got, given the competitors, then so what, that your theory implies such things? If we’re just plain stuck with that, even though many would be happy to drop it like it’s hot if they could, then I think we’ll have to make do and it’ll be okay.
@Consciousness Razor,
In the context of quantum interpretations, “realism” is a technical term that most definitely does not mean what you said. These days the term “counterfactual definiteness” is preferred as a more precise term. But anyway, there’s a reason I didn’t use either term in a basic explanation of Bell’s theorem.
Siggy, I think you misunderstand. If the claim is “physics is up to something else other than explaining reality,” I’m not obliged to accept that. A solipsist is just one example of the sort of person who might say something along those lines, one I chose because it’s clearly a ridiculous example of a logical possibility that some people might grasp for just because they’re informed it’s a possibility that they could entertain.
But anti-realists of whatever stripe are claiming that physics shouldn’t be construed as being about reality, because they have some other conception of what it’s about it and why it’s done. They often seem impervious to reason and evidence, much like our ridiculous friend the solipsist. So while I don’t know if there’s any point in trying to persuade them, it’s certainly not like I have to take it as a serious position which deserves thought on my end or that it gives them anything worth arguing about.
I mean, if you want to insist your theory which isn’t about reality is not “nonlocal,” because you say it’s “local” and not about reality, then okay … whatever floats your boat, I guess. You put those words together in that form, but what the hell is it even supposed to mean? I know there are all sorts of sophisticated-sounding formulations of anti-realism, particularly in foundations of quantum mechanics. But when it boils down to you talking about something other than the world and which words you feel like using to describe that non-world, then I personally don’t have a reason to care. And at a very impersonal level, physics has a job to do — explaining our world, if you ask everybody and their brother, except people in deep the grips of a dilemma they don’t like and don’t want to think about clearly — so that kind of woolly-headed crap is counterproductive at best.
cr @8:
I’m not talking about particles in distant galaxies, but all the particles in the system you’re considering; you know, the N particles in the 3N-dimensional configuration space you mentioned earlier.
And now, as usual, you end up talking about what you personally like and dislike. You dislike the idea that an electron, or neutrino, or whatever, is not in a definite state (spin or flavor or position, etc) until it interacts with the environment. Fine.
You seem to have misunderstood what I wrote. The formulation of BM begins nonrelativistically, and you have to choose a preferred way of slicing up spacetime to reproduce the results of special relativity. And of course, you go on to sweep this under your rug. As you did with the question of quantum field theory in a previous thread. You attach a lot of significance to BM agreeing with nonrelativistic QM, but this agreement is trivial; it agrees by construction.
Wow. “Nothing to see here, folks. Move along.”. Actually, I don’t lose any sleep over it. Just shake my head at the desperate need of some people to reject what quantum mechanics seems to be saying about how the world works.
@CR
Yeah but that isn’t what “realism” means in this context so what you wrote is irrelevant.
@Rob Grigjanis,
Correct me if I’m wrong here: My understanding of Bohmian mechanics, is that the state of a system is described by a pilot wave (which is identical to the wavefunction in the many worlds theory) and a set of positions, momenta, and spins for each particle. The laws that govern the evolution of position, momentum, and spin depend on your choice of space-time coordinates (what you called the “foliation of space-time”). Thus, two theorists could choose the same initial conditions (including all the hidden variables) but different space-time coordinates. These two theorists would make different predictions about the outcomes of certain measurements.
Please tell me what you think realism means in this context and explain how it is irrelevant.
On the foliation issue, a possibly helpful paper on arxiv: Can Bohmian mechanics be made relativistic? https://arxiv.org/abs/1307.1714
@CR,
At the risk of being flippant, you can find definitions of “local realism” and “counterfactual definiteness” in the context of quantum theory in any basic resource, e.g. Wikipedia. I am not sure how precisely defined it is, so it may be that you will find multiple disagreeing definitions. Personally I don’t think the terms are very helpful. But “physics is up to something else other than explaining reality” is not a remotely accurate characterization, and “I think there is a real world” really doesn’t address the issue.
Rob:
The guiding equation isn’t talking about the same thing. You have to get an “conditional wavefunction” for that, which isn’t the universal one. When you analyze how “measurements” and (effective) “collapses” take place in particular situations, that’s what you would be using, if you’re going to be using that theory instead of your own.
Does the environment, including your measuring apparatus, not have a definite state? It’s all made of the same stuff, after all, if it’s made of anything. So when does it obtain one … when you measure your measuring apparatus? When does the thing you use to measure you measuring apparatus obtain its definite state? Is there a reason why this should ever end?
QFT is not an interpretation of quantum mechanics. I don’t know whether you even have one and still don’t get whatever your question is supposed to be about.
What does it say about that, according to you? I don’t know what I may or may not reject.
Then say what is. I’ve read wikipedia’s entry, for what that’s worth, and it says nothing helpful about what you think I’ve misunderstood or what I’ve said which is inaccurate.
I’m thinking of this as I think about scientific realism in general, since QM is science and realism/antirealism about it (or parts of it) has the same essential features as other forms of scientific realism/antirealism. Certain issues are of course specific to the field, but it’s not like this is a new or unique problem only to be understood in the context of QM itself.
Please read the whole sentence you’re quoting:
The part in bold is where the emphasis should be.
Siggy @12: Here’s the claim made by Dürr et al (page 12).
Make of it what you will. Reads like handwaving to me.
@CR,
If you don’t understand Wikipedia’s definition on local realism, it is doubtful that it would be worth my time to replicate the definition here. I’m not paid enough to debug your views regarding quantum mechanics.
@Rob,
That sounds like a confirmation of my understanding. Bohmian theory will reproduce the same probabilities regardless of preferred foliation, but the evolution of the hidden variables does depend on preferred foliation.
The arxiv paper linked by CR apparently does not deny that you need a preferred foliation. Rather, it agrees that a preferred foliation is necessary, and then constructs one.
EPR=ER
General relativity has its own version of non-locality; an Einstein-Rosen bridge. This does not violate the no signaling limits of Special Relativity. It is possible that entanglement is a wormhole like phenomena.