Can particles really be in two places at once?

[This is a transcript of the video embedded below. Some of the explanations may not make sense without the animations in the video.]


Today I want to talk to you about what happened when I wrote an opinion piece for the Guardian about quantum computing, had to explain what a qubit is, and decided against using the phrase that it “can be in two states at the same time”. What happened next and what did I learn from it? That’s what we’ll talk about today.

Three years ago, just before Google’s first demonstration of quantum supremacy I wrote a brief essay for the Guardian about how that isn’t going to change the world. Quantum supremacy has since been renamed to quantum advantage, but as you can see it indeed hasn’t changed the world. That I wrote this so briefly before the publication of the Google paper was of course totally entirely coincidental.

Now, you can’t write a popular science piece about quantum computing without explaining how a quantum computer works, and that normally involves fumbling together a paragraph that no one understands who doesn’t already know how a quantum computer works, but that has to give the reader the impression they understood it. This means you can’t use words like “superposition”, “Hilbert space”, “complex number” or “Bloch sphere”. Wait, don’t leave. I’ll explain what the Bloch sphere is in a minute.

So what I wrote was

“while a standard computer handles digital bits of 0s and 1s, quantum computers use quantum bits or qubits which can take any value between 0 and 1” and “when qubits are connected by quantum entanglement... such machines can rattle out computations that would take billions of years on a traditional computer”.

Though I am pretty sure the phrase “rattle out” came from the editor because I’m not usually that eloquent.

By writing this I wanted to get across two points. First, the phrase that “a qubit can be in two states at the same time” which you have probably read or heard somewhere makes no sense and would in my opinion better be avoided. Second, it’s the entanglement that makes the difference between a conventional computer and a quantum computer.

Why do I say that a qubit can take any value between 0 and 1? Well, a qubit is the simplest example of a wave-function. Here’s the mathematical expression. Remember what I told you in my earlier video that these mysterious looking brackets really just mean these things are vectors. So the zero and the one are two basis vectors. And then a qubit is a sum of those two basis vectors with coefficients in front of them. That sum is what’s called a superposition.

You might think this is like having vectors in a two dimensional flat space but this isn’t quite right. That’s because the wave-function describes probabilities. This means if you square the coefficients in front of the basis vectors, they have to add up to one. And also, these coefficients can be complex numbers. This is why, if you want to draw all possible qubit states, those do not lie in a flat grid, they lie on the surface of a sphere. This is the Bloch sphere.

The Bloch sphere is commonly drawn so that the state 0 points to the north and the state 1 points to the south pole. So what’s an arbitrary qubit state? Well, all places on the surface of Earth lie between the north and south pole, and all qubit states lie on the Bloch sphere between 0 and 1. That’s why I wrote what I wrote in my article.

Before we look at what happened in the comments, a big thank you our supporters on Patreon, and especially those in tier four. If you want to see more of our videos, you can help by joining us on Patreon or right here on YouTube by clicking on the join button below. Let’s then see what happened the comments. Pretty much as soon as the piece was published, someone wrote: “That’s an analogue computer, not a quantum computer! A qubit can have a superposition of the values 0 and 1.”

Yes, but you can’t just write “superposition” in a popular science article without explaining what that is. And the superpositions in quantum computers are indeed similar to analogue computers, just that the values “in between” can be complex numbers. This doesn’t mean that quantum computers are just conventional analogue computers. The relevant property that sets quantum computers apart from conventional computers is that you can entangle those qubits which you can’t do with a conventional computer, regardless of whether it’s digital or analogue. As I’d written in my article.

Next time I looked at the comments someone had replied: “Never understood why they give this kind of story to an arts grad.” Next person: “Actually she’s a physicist, a string theorist and a good one at that.” Ah, arts grad, string theorist, same thing really. Next. “I’m an “arts grad” with over 30 years experience in IT. I expect better writing and research about this subject from an “arts grad”.” Yes. Let that be a lesson to all the string theory arts grads writing about quantum computing. I couldn’t think of anything polite to respond, so I instead replied to some other comments. And luckily, next time I looked, two people had shown up to explain the matter. The first wrote:

“the author meant x |0> + y|1> as lying “between” the pure states |0> and |1> ; “between” in state space, not on the real number line.”

Exactly, they lie between 0 and 1 in state space, which can be illustrated by the Bloch sphere. All states on the Bloch sphere are pure states. Then another comment:

“you fail to take into account the near impossibility of explaining the concept of a superposition in PopSci language, which does not allow for concepts like “complex number”... Try by those rules yourself and see if you can produce anything that does not amount to “sort of like an average”, which would in this context be equivalent to “any number between 0 and 1”.”

This indeed captures the difficulty well. Finally, someone points out that I’m not a string theorist, and they lived happily ever after, the end.

So why am I telling you this? Well for one I want to belatedly thank those commenters for taking the time to sort this out. But also, I’ve been thinking about this episode quite a bit and wondered what went wrong there.

I believe the problem is that when we write about quantum mechanics we’re faced with the task of converting mathematical expressions into language. And regardless of which language we use, English, German, Chinese, or whatever, our language didn’t evolve to describe quantum behavior. So all the words that we can come up with will be wrong and will be misleading. There’s no way to get it right.

What’s a superposition? A superposition is a sum of vectors in a Hilbert space. Alright. But if one of the vectors is a particle going left and the other a particle going right, what does this superposition mean? I don’t know. Could you say it’s a particle going into both directions? I guess you could say that. I mean, you just said it, so arguably you can. But is that what it actually is? I don’t think so.

For one it’d be more accurate to say that the wave-function “describes” a particle instead of saying that it “is” a particle. But maybe more importantly, I don’t think such a superposition is anything in the space we inhabit. It’s a vector in this mathematical structure we call the Hilbert space. And what does that mean? I don’t know. I don’t think there are any words in our language to explain what it “means”.

I still think that the explanation that I gave for a quantum bit was more truthful to the mathematics than the more commonly used phrase that it can be in “two states at once”. But I also think we have to accept that regardless of what language we use to describe quantum mechanics, it will never be correct. Because our language isn’t fit to describe something we cannot experience.

Should this worry us? Does this mean there’s something wrong with quantum mechanics as a scientific theory? I don’t think so. I think it’d be surprising if it was otherwise. Quantum mechanics describes the behavior of matter in circumstances we don’t observe in daily life. We’ve never needed the language to explain quantum behavior so we don’t have it.

To give you a second opinion I've asked Arvin Ash to tell us what he thinks. Arvin is an expert in science communication in general and quantum mechanics in particular. He told me the following.

Hi Sabine, as I tried to illustrate in a recent video, the root of the problem and cause of so much confusion in quantum mechanics is the fact that when we measure things, that is, whenever we have the opportunity to actually observe a quantum object, it seems to lose its quantum behavior. The superposition is lost and what we see is something that looks like it's behaving classically. This is the case with the double slit experiment when individual photons or electrons always show up as dots on the screen rather than some kind of wave and we only see the wave interference pattern when we shoot many photons through the slits or when we observe quantum particles and cloud chambers, they leave trails as if they were little cannonballs.

This is also the case for quantum computers. While the math describes superposed states of quantum bits as, pardon the language, taking on any value between zero and one when the computer actually makes a measurement the bits are either zero or one, it's never in between. So while the calculation is quantum, the result is binary. We are surrounded by a quantum world we can't directly observe, and when we sample this world by taking measurements the quantum phenomena convert to classical results.

It's hard to describe something you can't ever experience. We humans like to make connections with familiar things. If we could see it, we could describe it. The math of quantum mechanics has no obvious classical analog. We could however certainly be more precise in our language. But I do think that in the future quantum mechanics will become more intuitive to more people through watching videos like this thanks for having me.

I think that’s a good point. Arvin has his own YouTube channel and if you find my channel interesting, I’m sure you’ll like his too, so go check it out.

When it comes to quantum mechanics, I think what will happen in the long run is that the mathematical expressions will just become better known and we will use them more widely. Like we’ve become used to talking about electromagnetic radiation. That was once a highly abstract mathematical concept, waves that travel through empty space, rather than traveling in some medium. But we now use electromagnetic radiation so frequently that it’s become part of our everyday language. I think that it’ll go the same way with qubits and superpositions.