Quantum mechanics is nonlocal, but what does that mean? (Patreon)

[This is a transcript with links to references.]

According to the headlines, last year’s Nobel Prize in physics was awarded for showing that the universe is not locally real. Or for “spooky quantum behavior” and spooky action at a distance. Or for “exploring quantum weirdness”. What’s that supposed to mean? Is the universe really not locally real, or not really local, or not really real? That’s what we’ll talk about today.

My great plan for this video is that we’ll first look at what locality means, then we’ll talk about quantum mechanics, and then we’ll combine both. Let’s start with locality.

Y’all know what locality means. It means if you want to go to New York City, you don’t just disappear from Heathrow and reappear at JFK. You must get on a plane, fly in a seat through the sky, and then drag yourself into the immigration queue at US customs. There’s no instantaneous going anywhere.

It’s not just travel, it’s generally any interaction. You can only act on things next to you. And it’s not just you. No, really, I swear. Anything can only interact with things right next to it. This is what we mean by “locality”. Non-locality would basically be a portal. Something goes in and instantaneously appears elsewhere.

Now, if you could travel non-locally that *might allow you to go faster than light. But it doesn’t necessarily have to. Alice could go to Tokyo faster than light without taking a portal. Bob could go through a lame portal that brings him to New York even later than Air France, and leaves his bag in Paris, again.

So, non-locality and faster-than-light travel are two separate things. But the two are related, in that Expedia isn’t offering either. Maybe more importantly, though not unrelatedly, neither of them is allowed in Einstein’s theory of space and time. Yes, that guy again

We can make a little drawing to see what this means. This is a space-time diagram. The horizontal axis is one dimension of space, say left or right. The vertical axis is time. If you don’t move but sit still, that is described by a vertical line in this diagram. If you move at a constant velocity, that makes a line at a fixed angle. By convention, a 45-degree angle is the speed of light.

According to Einstein, nothing can go faster than light. This means if an event happens here, then anything that can be influenced by this event must lie within the region that’s bounded by 45-degree lines emerging from the event. This boundary is called the future light cone, and the region that can be influenced from the event is that inside the future light cone. Likewise, there’s a region which contains everything that can influence the event, which is the inside of the past light cone.

If something *could go faster than the speed of light, it could enter your light cone like this. And if non-locality was possible, it wouldn’t have to respect those light cones either, it could just jump into your light cone. For the rest of this video, we’ll assume that faster than light travel can’t happen, and just look at nonlocality. If you’d rather hear me talk about faster than light travel, I have another video on that.

Our experience suggests that reality is local in that these jumps in space-time don’t happen. There’s no action at a distance, no portals. However, our *descriptions of reality are not always local.

Suppose I take two sheets of paper and write the number plus 1 on one, and minus 1 on the other. I put them in two identical envelopes. Then I send one to Alice in Tokyo and one to Bob in New York who is still waiting for Air France to deliver his bag.

Alice and Bob both know what I did but not what’s in their envelope. Either the plus one went to Bob and the minus one to Alice, or the other way round. It’s a 50-50 chance. But the moment Alice opens her envelope and finds plus one, she knows that somewhere in New York there’s an envelope with the number minus one.

You probably didn’t find this particularly remarkable. Well, what did you expect clicking on a video about locality. Only so much I can do to spice it up. Remarkable or not, this simple example will be useful later. Thing is, the numbers in these envelopes are correlated. You don’t know what either is, but if you know one, the other one is not random anymore. And this correlation can stretch over a distance. If you learn what one number in this correlated pair is, you learn something about the other number, elsewhere, instantaneously.

You could call that a kind of non-locality, but it’s not an action at a distance. The only action that happens is an update of knowledge *about something, elsewhere. The action itself is completely local. It happens in Bob’s brain, or Alice’s, unless maybe they’ve become so bored by my number games there isn’t anything left of their brains.

So the lesson of the first part is: There are two types of non-locality. The one is a physical non-locality, an action at a distance, which is like something going through a portal. The other is a non-local correlation that allows us to acquire knowledge about what happens elsewhere. But in this case the only physical changes are local.

Let’s then talk about quantum mechanics. In quantum mechanics everything is described by a wave-function, usually denoted with the Greek letter Psi. The wave-function is a device that we use to calculate the probability of getting a particular measurement outcome. But it doesn’t predict the outcome with certainty. We say that quantum mechanics is not deterministic.

Imagine for example you have a single quantum of light, a photon, and you send it through a semi-transparent plate. If you still have the space-time diagram in your head, please take it out and put it aside for now. This illustration just shows two dimensions of space, think table top. The photon has a 50 percent probability of going through the plate and a 50 percent probability of being reflected.

So long as we don’t observe the photon, its wave-function changes locally, from one place to the next by spreading into these two directions. But once we make a measurement, we know where the photon went. Then, the probability jumps to one hundred percent on the side where you have measured it, and drops to zero percent on the other side. This is like when Alice opens her envelope and sees the number plus one. Now she knows that the probability that Bob had plus one is zero.

And like Alice adjusts her probability for Bob when she opens the envelope, once we measure the photon on one side, we have to update the wave-function to reflect this change in probability on the other side. Why? Because otherwise it doesn’t agree with our observations. This update is also sometimes referred to as the collapse or reduction of the wave-function and it’s non-local. You make a measurement in one place, and the wave-function changes elsewhere. So, quantum mechanics is non-local in some sense. The question is, in which sense.

As we saw previously, just because you gain knowledge about something that happened elsewhere doesn’t mean there was a non-local action. Maybe the wave-function just describes knowledge rather than being real itself. Maybe the photon really went one way or the other, you just didn’t know it, and the measurement revealed its real state.

Being a wave-function isn’t easy. Not only do people reduce and collapse you, they also then argue that it doesn’t matter because you aren’t real. Tsk.

The quantum mechanical version of the envelope example goes like this. Imagine you have a particle with spin zero that decays into two particles that can each either have spin value plus or minus one. One flies to the left, the other one to the right. The total spin is conserved in the decay, so you know that either the particle going left has spin value plus one and the one going right has minus one, or the other way round. But you don’t know which has which spin. These two particles are “entangled”. Entanglement is a correlation, like the correlation between the numbers in the envelopes. And like the correlation between the envelopes, it can stretch over a distance.

The difference between the two cases is that if you think that quantum mechanics describes all there is, then before you make a measurement the particles really are in both states at once. And then the update is really physically non-local.

The idea that this update is physically real is what Einstein called a “spooky action at a distance”. He thought that can’t be right. Einstein believed that the real process must be local, and that the update just reveals something you didn’t previously know. That something you reveal in the measurement is usually called the “hidden variable”. It could be for example the value of the spin that the particle “really” had before you did the measurement and found out what it is.

Bohr held against that. He said there are no hidden variables, and no underlying information that you can reveal. The wave-function is all there is, and the collapse is really non-local. Bohr seems to have liked quantum mechanics because he thought that its indeterminism makes place for free will, though that’s rather questionable. We just talked about that a few weeks ago.

Einstein did not use the phrase “spooky action at a distance” to refer to entanglement. To begin with, the example in which he coined the phrase didn’t use entangled particles. I went through the history in a previous video. But more importantly, entanglement has no “action”. It’s just a correlation that stretches over a distance, like the correlation between the envelopes.

When I read the headlines about last years’ Nobel Prize, I got the impression that some science writers believe entangled particles are somehow mysteriously linked to each other. They seem to think if you do something to one particle in an entangled pair, then that will immediately affect the other one. But this isn’t so. It’s only when you *measure one particle, then you have to update the wavefunction of both. So long as you don’t measure them, doing something to one particle won’t affect the other one.

You could for example flip the spin of one of the particles in the entangled pair, so that one becomes minus one and the other way round. You can do that without measuring the spin. Like, you can do this in reality, in the laboratory, not just with mathematics. If you flip the spin of one particle, without measuring it, this will not do anything to the particle it’s entangled with. It’s not as weird as they made you believe, is it?
Ok, so entanglement is just a correlation over long distances. But does it at least allow us to send a signal faster than light? Unfortunately not. It doesn’t work because you can’t force the outcome of your measurement on one of the entangled particles to be a particular value, it’ll just be randomly distributed according to the probability given by the wavefunction.

This means on the other end they won’t know whether you have made a measurement any more than Bob knows whether Alice has opened her envelope. If quantum mechanics is correct, then no information can be sent faster than light. One can mathematically prove that this is the case, not just for this experiment, but for any possible experiment. It’s called the no-signalling theorem.

But because of this randomness of quantum mechanics, it’s also rather unclear in which sense the collapse is even non-local. If making the measurement on one of a pair of entangled particles doesn’t change the measurement outcome on the other one, then what is non-local in the first place?

Enter John Bell. Bell tried to find a way to pin down what it means that quantum mechanics is non-local.

Please put the space-time diagram back into your head because we need it again. We’ve seen earlier that, without faster than light travel, everything that an event might be influenced by has to be in the past light cone. But if the theory is also local, then you don’t need the *entire inside of the light cone.

This is because if you know what happens at one moment in time, you can calculate what happens later. If the theory is deterministic, you can calculate it exactly. If it’s not deterministic, probabilistically. This means all you really need is the information at one particular moment in time, for anyone’s notion of time. In one dimension, that moment is line going through the inside of the past light cone. In three dimensions it’s what’s called a space-like hypersurface, just so you don’t go away disappointed about the lack of incomprehensible terminology.

The point is that the information at one moment in the past light cone of the event is all you need to calculate what happens at the event. *If the theory is local. If the theory isn’t local, then that doesn’t work, because information could jump into your light cone later.

Bell’s insight was now that one can use this to decide whether a theory is local, and it works even if the theory has a random element. Suppose you have all available information on this slice here. And you use that to make your best prediction for what happened here. Then the theory is local if information from over here does not tell you anything new about what happened at A. Bell called this property local causality.

If on the other hand, the theory is not locally causal, then information can jump into the light cone after your slice. And that can tell you something new about what happens at that event.

Quantum mechanics is not locally causal. Think back of the earlier example with the entangled particles. Let’s say they are created here. One goes left one goes right. In the space-time diagram, this means they go diagonally, no less than a 45-degree angle. You make two measurements here and here, let’s call them A and B. We want to know what happens at measurement A.

All the information you could have about it is down here in the wave-function. This wave-function, as we’ve seen earlier, does not allow you to predict the measurement outcome at A, just the probability of a measurement outcome. It’s either spin minus one or plus one with 50 percent probability each.

Here’s the thing. If you make a measurement over here at B, and you find it’s plus one, then you know the spin at A is minus one. So what happens at B *does add information to the inside of the past light cone and quantum mechanics is not locally causal.

The important thing is now that local causality is a property of a model. In this case, a property of quantum mechanics. The relevant question is whether it’s also a property of reality. If quantum mechanics was all there is, then that would be the case. The wave-function would be how the world really works. And it would have a spooky action at a distance.

But if quantum mechanics was not all there is, then the wave-function down here was not the entire information. There’d be a hidden variable there from which the outcome followed. And if you had had the entire information, then adding the measurement outcome from B would not provide any further information. Quantum mechanics would still not be locally causal, but reality would be. So which one’s right? That’s what John Bell was trying to figure out.

Wow you’re still with me, that’s lovely. I really appreciate the company. Let’s look again at that semi-transparent plate because it can teach us something very important.

We will now try to build a simple hidden variables model that explains what happens at that semi-transparent plate, also called a beam-splitter. We want to do this with hidden variables, and we want to do it locally. So, no action at a distance.

The hidden variables are usually called \lambda. So we’ll say if \lambda is 1 then the photon goes that way, if \lambda is three, then it goes the other way. Why three and not 2? Because I know what’s coming next and you don’t.

Now we put two mirrors here and here so that the photons don’t run off our screen, and another two semi-transparent plates there. Okay, so now we have four different paths, each with 25 percent probability. We’ll give those the numbers one two three four.

And now. We’ll combine these two paths by directing both towards the same beam splitter. If our hidden variable explanation was correct, then we should get half of the photons here in detector A, and half here in detector B. But this isn’t what happens.

Indeed, you have probably noticed that this setup is what’s called a Mach-Zehnder interferometer. If the paths are the same length, then the photon will always go to detector B. You can measure it and all. It’s definitely real. The interference in these detectors is exquisitely sensitive to the length of the paths, which is why one can use this type of interferometer to measure tiny deformations. This is how a gravitational wave interferometer works.

But *how does it work? Well, in quantum mechanics the photon is described by a wave-function. And waves can interfere with themselves, either constructively or destructively. In quantum mechanics, the photon goes over both paths, and on this output the wave interferes destructively, with itself, so no signal, and all photons go the other way. But the price you have to pay for this is that if you measure which path the photon went, you need the non-local collapse.

How can we use hidden variable to get this done locally? Well, no one ever said that we can’t have waves that go two paths in a local hidden variable model. All we need to do is to say, if we measure which path the photon goes, then we use these variables 1, 2, 3, 4, and the photon behaves like a particle. If we measure the interference between the paths then, that’s hidden variable number 5, the photon behaves like a wave and goes only into detector 1. Problem solved. It’s local because the photon goes on one continuous path into the detector where it’s measured.

For this to work, the value of the hidden variable that determines what the photon does must depend on what you measure. It’s called a violation of measurement independence.

So the lesson is: If we want a local hidden variable model that reproduces quantum mechanics correctly, then it needs to violate measurement independence.

This finally brings us to Bell’s theorem. I’m not going to explain Bell’s theorem because you’ve already watched 20 videos about it, and it didn’t help.I’ll just tell you what it says.

Bell’s theorem says if you want a local hidden variables model that reproduces quantum mechanics correctly, then it needs to violate measurement independence.

Isn’t that what we just concluded from this little example with the interferometer? Yes, Bell’s theorem says the same thing.  

You see, what happened is that Bell originally just assumed that measurement independence is fulfilled. He then proved that any local hidden variables theory which fulfils this assumption will obey an inequality for the correlations of certain measurement outcomes. It’s now called Bell’s inequality.

Quantum mechanics can violate this inequality. What the Nobel Prize was awarded for is for experimentally demonstrating that these inequalities are indeed violated by observed measurement outcomes. This means that either reality respects measurement independence but is not locally causal, or it is locally causal and violates measurement independence. Just logically, it could be a combination of both, but I don’t know why you’d want to do that.

Since Bell had forgotten about this assumption of measurement independence, he mistakenly thought that demonstrating an experimental violation of his inequality would rule out all local hidden variables models, and thereby show that spooky action at a distance is real. Indeed, most physicists still seem to believe this.

At this point you might think Sabine is talking complete rubbish because if that’s right why doesn’t anyone ever mention this? Well, first of all, you don’t have to take my word for it, it’s all in the published literature and easy to check. In fact, one of the people who pointed out that Bell had forgotten this assumption was John Clauser who was one of the recipients of the 2022 Nobel Prize. Why physicists refuse to mention it is a very interesting question to which I don’t have an answer. My best guess is that they want reality to be weird because they like the mystery.

There are several hidden variables models that are local and that reproduce the predictions of quantum mechanics by violating measurement independence. I’ll leave you links in the info below. And yes, a violation of measurement independence has also been called “superdeterminism”, though this term is extremely misleading. I talked about this in an earlier video.

In summary. Quantum mechanics is not local. What makes it non-local is the collapse of the wave-function. The question is whether this non-locality is physically real. If the collapse of the wave-function was a physical process, it would be what Einstein called a spooky action at a distance, and reality would be non-local. It has not been shown that this is indeed the case, no matter how many headlines say otherwise. Entanglement is a non-local correlation that can stretch over long distances. But it’s not what Einstein meant by action at a distance because there’s no action in it.

If you are a fan of the many worlds interpretation of quantum mechanics, you probably took issue with some things I said. But this video is already too long, so I’ll talk about this some other time. If you want to hear me complain about many worlds, don’t forget to subscribe.