Teleportation of Toddler Toys

Today is the official release date for the paperback edition of How to Teach Physics to Your Dog, so I wanted to write up something cool about quantum physics to mark the occasion. I looked around the house for inspiration, and most of what we have lying around the house is SteelyKid’s toys. Thus, I will now explain the physics of quantum teleportation using SteelyKid’s toys:

“Wait, wait, wait… You’re not seriously planning to explain something quantum without me, are you?”

“I could hardly expect to get away with that, could I. No, I’m happy to have your contributions– the book is about talking physics with you, after all. Just break in if you have something to add, and I’ll put it into the blog post.”

“When I have something to add, you mean. Because I’m going to have stuff to add, you know.”

“Oh, I know it…”

The phenomenon known as “quantum teleportation” (a slightly unfortunate name) involves using quantum entanglement– the strange phenomenon Einstein called “spukhafte fernwirkung” or “spooky action at a distance”– to transmit the state of a quantum system from one place to another without disturbing it. Explaining teleportation thus involves both a sender and a receiver of the quantum state, who are traditionally referred to as “Alice” and “Bob.” Since SteelyKid doesn’t have many anthropomorphic toys, though, we’ll use these two:

Appa and Bertha the Big Bear (It’s a little disconcerting, by the way, to watch SteelyKid playing with these two, and remember that not all that long ago, she was significantly smaller than either of them…).

So, the scenario is this: Appa has a quantum object, whose exact state he doesn’t know, so it is in some superposition of all the available states at the same time. This could be anything– an atom with two possible states, an electron spin, a polarized photon. We’ll represent Appa’s unknown state by this little brown-and-white dog:

And he wants to transmit this state to Bertha, who is a long distance away:

So, how can he send this state to Bertha and make sure she gets exactly the state that he starts with?

“What do you mean, how can he send it to her? He just makes another one just like the one he’s got, and sends her the copy. End of experiment.”

“That does seem like the obvious course of action, but in fact, it’s not possible. There’s a mathematical rule about quantum physics, helpfully called the ‘no-cloning theorem,’ that says it’s impossible to make a perfect copy of a single quantum state unless you already know something about the state.”

“Wait, how can you not make a copy of the state? You just measure the state, and make another one that’s identical to the result of your measurement.”

“That would work for a classical system, but remember, in quantum mechanics, you can only determine probabilities, not absolute outcomes. So, when you make your measurement, you get one or the other of the two possible outcomes. But that doesn’t tell you anything useful about the probability of that outcome.”

“So? Just measure it lots of times, and determine the probability that way.”

“Well, once you’ve measured the state, you can’t un-measure it. If you make repeated measurements of the state of the original object, you’ll just get the same result as the first measurement, over and over again. To determine probability through repeated measurements, you need to have lots of identical states. But that’s exactly the thing that you’re trying to obtain by measuring the state in the first place. So it’s impossible to make a copy of a single unknown quantum state.”

“Well, that stinks. Anyway, what’s Appa doing with a single quantum object whose state he doesn’t know, anyway? I mean, he’s got it– surely he knows what state he put it in.”

“Not necessarily. It could be, say, the result of a calculation on a quantum computer, that he needs to send to Bertha for her to use in her own quantum computations.”

“OK, I guess that makes sense. Carry on.”

Appa wants to send the state of his system to Bertha without disturbing it by measuring it, and he can’t just make a copy to send along. Having thought of just this sort of problem, though, Appa and Bertha have previously shared two objects in an “entangled state,” which is a state of two quantum systems whose states are correlated in such a way that measuring one absolutely and immediately determines the state of the other. So, for example, we could use these two dogs:

If Appa measures his entangled particle and determines that he has the Golden Retriever, then he knows immediately and with absolute certainty that Bertha has the chocolate Lab. If he finds that he has the chocolate Lab, then he nows immediately and with absolute certainty that Bertha has the Golden Retriever.

The entangled states aren’t restricted to just these two options, though. They could do different measurements, say, asking whether the entangled particles were elephants rather than dogs:

Or even whether they were hedgehogs:

In every case, the result of Appa’s measurement absolutely and instantaneously determines the outcome of Bertha’s measurement. They will always find the states of their entangled particles to be correlated in this way. This entanglement correlation will also hold no matter how far apart they are when they make the measurements.

Since the state is indeterminate until measured, we’ll represent the pre-measurement entangled state using these two stuffed animals of indeterminate species:

So, Appa and Bertha share a pair of systems in an entangled state, each taking one:

“Wait, wait, wait. If Appa can send half of an entangled state to Bertha, why doesn’t he just send the original system to her? I mean, if I had an unknown quantum state and wanted to send it to another dog– which, by the way, I do not. Other dogs don’t deserve my unknown quantum states. If I did want to send one, though, I would just send it. None of this faffing about with entangled pairs.”

“Well, there are lots of reasons why you might not be able to just send the system. The usual demonstrations of this use polarized photons, which are actually pretty robust, and easy to send from one place to another, but you can teleport the states of atoms or electrons, or even ensembles of atoms. Those don’t travel nearly as well as photons.”

“Oh, ok.”

“And even with photons, it’s not hard to imagine ways in which you could end up with a situation where Appa and Bertha can’t simply send photon polarization states to one another:”

“Quantum entanglement will work even if there’s a barrier of some sort between them.”

“OK, this isn’t as stupid as I initially thought. Carry on.”

“Thanks for the vote of confidence.”

So, Appa and Bertha share a pair of particles in an entangled state. To initiate the teleportation process, Appa makes a joint measurement of the state of the two particles he has: the state he’s trying to send, and his half of the entangled pair:

The measurement he makes is not a measurement of the individual states of the particles, because that wouldn’t do him any good. Rather, he makes a joint measurement of the two together, basically asking “Are you in the same state, or in two different states?”

The effect of this measurement is to entangle the state of Appa’s particle with the state of the entangled particle that he has. And, since the state of his entangled particle is entangled with the state of Bertha’s particle, this has the effect of putting Bertha’s half of the entangled pair into a state that depends on the initial state Appa was trying to send her in the first place:

“So now Bertha has Appa’s state. Hooray, quantum is magic, let’s go for a walk.”

“Not so fast. Bertha’s particle is put into a state that depends on the state Appa is trying to send, not the exact state he’s sending. There are four possible outcomes, corresponding to the four possible outcomes of Appa’s measurement:”

“Wait, so how does this do anybody any good?”

“Well, the state depends on Appa’s result, so once Bertha knows that, she knows what she has to do to her state in order to turn it into Appa’s initial state. And the operations required don’t require her to know the exact state of her particle. She can just follow the procedure indicated by Appas’s result, and know that she’ll end up with the right final state.”

“How does she know the result of Appa’s measurement, though?”

“Oh, Appa sends the result to her via a classical communications channel. Basically, they call each other on the phone:”

“I thought you were only going to use toddler toys?”

“These count. They’re old phones that we gave SteelyKid to play with. She pretends to talk to her grandparents and great-grandparents on them all the time.”

“I think that’s a little weaselly, but whatever.”

So, the end result of the teleportation protocol is to put Bertha’s half of the entangled pair into one of four definite states, which depend on the outcome of Appa’s measurement:

Once she gets the result of Appa’s measurement via a classical communication, she just does the appropriate operation needed to obtain the initial state. If she’s dealing with polarized photons, that just means doing a simple rotation of the polarization state using standard optical techniques. If the initial system was something like an atom or an electron, she uses some more sophisticated operations to map the state from the entangled pair onto the system of interest. At the end of the operation, she has exactly the state that Appa started with, without ever having measured it.

This might seem like it’s really making a copy of Appa’s state at Bertha’s location, because Bertha ends up with exactly the state that Appa started with, while Appa still has his original system. It doesn’t violate the no-cloning theorem, though, because Appa’s state was subjected to an entangling measurement as part of the teleportation process. That means that even though he still has the physical object whose state he was sending, it is no longer in the state it started out in:

So, at the end of the process there is still one and only one object in the exact state of interest. The state hasn’t been cloned, it’s just been shifted from one place to another.

And that’s how you teleport quantum states, using toddler toys.

“OK, before you go off to doing something else, I still have a few questions.”

“Fire away.”

“First of all, if Appa doesn’t know the state of the thing he was sending, and Bertha doesn’t know the state of the thing she’s receiving, how do we know this worked?”

“Well, obviously, you test it with known states first, so you can confirm that the teleportation was successful. There have been a whole bunch of quantum teleportation experiments over the years, starting with Anton Zeilinger’s group in Innsbruck and then Vienna. They’ve teleported photon polarization states across the lab, across the Danube river, and even from one mountain to another in the Canary islands. Chris Monroe’s group at Maryland has even teleported the state of ytterbium ions between two separate ion traps. Everything works out exactly as expected from the theoretical prediction.”

“OK, I guess I’ll buy that. Next question: This really isn’t much like teleportation at all, is it? I mean, it’s nothing like Star trek.”

“Despite what’s claimed by the occasional idiot, no. Nothing physical moves from one place to another. Appa and Bertha each need to have an identical object of the type that is being teleported at the start of the experiment, and they each have the same object at the end of the process. The only thing that moves from one place to another is the state of the object.”

“And it’s not really instantaneous, is it?”

“No. The entangling of Bertha’s particle with Appa’s state happens instantaneously, but the teleportation isn’t complete until after the result of Appa’s measurement is transmitted to Bertha. That transmission has to be at the speed of light or below, which means that there’s no faster-than-light communication taking place here. Which is good, because if you could communicate faster than the speed of light, you could violate causality, and end up with effects happening before the things that caused them.”

“How does that work?”

“That’s relativity, and something we’ll talk about in our next book.”

“Oh, right. OK, last question: This isn’t going to let me teleport into the back yard and catch the squirrels before they get up the trees, will it?”

“Sadly, no. You’re not going to see anybody teleporting macroscopic objects any time soon. It’s probably not really necessary, anyway– if you want to send a physical object from one place to another, all you really need to do is get the right sorts of atoms in the right positions relative to one another, and not worry so much about the exact quantum state of everything. You might possibly need quantum teleportation to move things like brain states, if people like Roger Penrose who think there’s something quantum about brain function are right, but that’s still wildly impractical, given the sheer number of atoms whose states would need to be transmitted.”

“Bummer. I wanted to catch some squirrels.”

“You wouldn’t like it anyway. Teleporting your state from inside the house into the back yard would require us to have another dog identical to you out in the back yard waiting to receive your state. And given the way you freak out when perfectly harmless dogs walk by on leashes on the other side of the street, I doubt very much you’d want identical-to-you dogs running around the back yard.”

“No. We don’t like those dogs. Not one bit. also, they’d scare the squirrels off.”

“Also a good point. So, any final questions?”

“Just one: Can we go for a walk, now?”

“Sure, we can go for a walk.”

“Yippee!”