How to Teach Physics to Your Dog (22 page)

*
Or even longer, depending on what the dog is doing when called.

*
If you prefer the Copenhagen view, this projection involves a real collapse of the wavefunction into a single state. If you prefer many-worlds, the apparent projection onto a single state comes because we perceive only a single branch of the wavefunction. In either case, the resulting correlation is the same, and the effect is instantaneous.

*
We get the maximum value of 100% if the system has a 50% chance of being in state 1 and a 50% chance of being in state 8. We get the minimum value of 33% by never letting the system be in state 1 or state 8, and making the other six states equally likely. If you look at states 2 through 7, you’ll see that no matter what two different angles you choose, there are always two states that give you the same answer for both detectors.

*
As a result the predictions of Bell’s theorem are often called “Bell inequalities.”

Emmy trots into my office, looking pleased with herself. This is never a good sign.

“I have a plan!” she announces.

“Really. What sort of plan is this?”

“A plan to get those pesky squirrels.” They keep escaping up the trees in the backyard, and she’s getting frustrated.

“Is this a better plan than the one where you were going to learn to fly by eating the spilled seed from the bird feeder?”

“That was going to work,” she says, indignantly. “And for your information, yes, it’s a much better plan than that.”

“Well, then, I’m all ears. What’s this brilliant plan?”

“Teleportation.” She looks smug, and wags her tail vigorously.

“Teleportation?”

“Yep.”

“Okay, you’re going to have to unpack that a little.”

“Well, I figured, the problem is, they can see me coming from the house, and they get to the trees before I do. If I could get between them and the trees, though, I could get them before they get away.”

“Okay, I’m with you so far.”

“So, I just need to teleport out into the backyard, instead of going through the door.” Her whole back end is wagging now.

“Uh-huh. And how, exactly, did you plan to accomplish this feat?”

“Well . . .” The tail slows down, and she does her very best cute-and-pathetic look. “I was hoping you would help me.”

“Me?”

“Yeah. I read where some physicists have done quantum teleportation, and you’re a physicist, and you’re really smart, and you know about quantum, so I was hoping you would help me build a teleporter.” She puts her head in my lap. “Pleeeeease? I’m a good dog.”

I scratch behind her ears. “You are a good dog, but I really can’t help. For one thing, I don’t do teleportation experiments in my lab. But even if I did, I wouldn’t be able to help you use teleportation to catch squirrels.”

“Why not?”

“Well, the existing teleportation experiments all deal with single particles, usually photons. You’re made up of probably 10
²⁶
atoms—a hundred trillion trillion—which is way more than anybody has ever teleported.”

“Yeah, but you’re really smart. You can just . . . make it bigger.”

“I appreciate your confidence, but no. The bigger problem is that the quantum teleportation people do in the real world isn’t like the teleportation you see with the transporters on
Star Trek.
”

“How so?”

“Well, all that quantum teleportation does is transmit the
state
of a particle from one place to another. If I have an atom here, for example, I can ‘teleport’ it to the backyard, and end up with an atom there that’s in the exact same quantum state as the atom I started with here. At the end of the process, though, I still have the original atom here where it started—it doesn’t move from one place to another.”

“That’s pretty lame. What’s the point of that?”

“Well, quantum mechanics won’t let you make an exact copy of a state without changing the original state, and quantum states of things like atoms are pretty fragile. If you really needed to get a particular quantum state from one place to another, your best bet might be to teleport it.” She looks a little dubious. “You could use it to make a quantum version of the Internet, if you had a couple of quantum computers that you needed to connect together.”

“Well, okay. So just teleport my state into the backyard, and I’ll use it to catch squirrels.”

“Even if I knew how to entangle your state with a whole bunch of photons—which I don’t—I would need to have raw material out in the backyard. There would need to be another dog out there, one that looked just like you.”

Her tail stops dead. “We don’t like those dogs,” she says. “Dogs that look just like me. In my yard. We don’t like those dogs at all.” She looks distressed.

“No, we don’t. One of you is all the dog we need.” She perks up a bit. “So, you see, teleportation isn’t a good plan, after all.”

“No, I guess not.” She’s quiet for a moment, and looks thoughtful. “Well,” she says, “I guess it’s back to plan A.”

“Plan A?”

“Can I have some birdseed?”

“Quantum teleportation” is probably the best-known application of the nonlocal correlations discussed in the previous chapter. The name certainly fires the imagination, conjuring up images of
Star Trek
and other fictional settings in which people, either through fictional science or just plain magic, can instantaneously transport objects from one place to another. The object starts at point A, disappears with a soft
*bamf*,
and reappears at point B, some distance away.

The high expectations created by science fiction make the reality of quantum teleportation seem disappointing. Real
quantum teleportation involves only the transfer of a quantum state from one location to another, and not the movement of complete objects. The transfer is also slower than the speed of light, because information needs to be sent from one place to another. This is a great disappointment to dogs hoping to beam themselves out into places where unsuspecting critters are waiting.

Nevertheless, it’s a marvelously clever use of quantum theory, tying together several of the topics that we’ve already talked about. In this chapter, we’ll see how indeterminacy and quantum measurement make it difficult to transmit information about quantum states from one place to another. We’ll see how the “quantum teleportation” scheme makes ingenious use of nonlocality and entangled states to avoid these problems, and why you might want to.

Quantum teleportation is a complex and subtle subject, probably the most difficult topic discussed in this book. It’s also the best example we have of the strangeness and power of quantum physics.

We can’t teleport in the way envisioned in science fiction and fantasy, but the essence of teleportation is just duplication at a distance—you take an object at one place, and replace it with an exact copy at some other location. By that definition, we do have an approximation of teleportation using classical physics: a fax machine.

If you have a document that you want to send instantly from one place to another—for example, if Truman has just gotten a really nice bone, and wants to taunt RD by sending him a picture of it—you can do this with a fax machine. The machine works by scanning the document, converting it to electronic instructions for creating an identical document, and sending
that information over telephone lines to another fax machine at a distant location, which prints a copy. What’s transmitted is not the document itself, but rather information about how to make that document.

The operation of a fax machine is different from the fictional idea of teleportation, but the differences are not all that significant. When you fax a document from one place to another, you end up with two copies in different locations, but if you regard this as a problem, you could always attach a shredder to the sender’s fax machine to destroy the original. The copy produced by a fax machine isn’t perfect, but that’s just a matter of the resolution of the scanner and printer, and you can always imagine getting a better scanner and printer. The transmission is limited by the time it takes to transmit the information from one place to another, so it’s not perfectly instantaneous, but that’s not a major problem for most transactions involving a fax machine.

If you wanted to approximate the fictional ideal of teleportation in a classical world, the best you could do would be to upgrade the concept of the fax machine. Truman would take a bone, and place it in a machine, which would scan the bone to determine the arrangement of atoms and molecules making up the bone. Then he would send this information to RD’s “teleportation” machine, which would assemble an identical bone out of materials at hand and present it to him to chew.

When we turn to quantum teleportation, we’re talking about “teleporting” a quantum object. This means not just getting the right physical arrangement of the atoms and molecules making up the object, but also getting all those particles in the right quantum states, including superposition states. Truman could use an upgraded fax machine to send RD a cat in a box, but he
would need a quantum teleportation device to send a cat in a box that was 30% alive, 30% dead, and 40% bloody furious. This turns out to be vastly more difficult than the classical analogue, due to the active nature of quantum measurement.

While in theory it is possible to do quantum teleportation with any object, in practice, all of the experiments done to date have used photons, so we’ll imagine that Truman is trying to send a single photon of a particular polarization to RD.
*
As we saw back in
chapter 3
(page 65), a polarized photon can be thought of as a superposition of horizontal and vertical polarizations, with some probability of finding either of those two allowed states.

When we describe a photon with a polarization between vertical and horizontal, we write a wavefunction for that photon that is a superposition state: it’s
a
parts vertical, and
b
parts horizontal:

The numbers
a
and
b
tell us the probability of finding vertical or horizontal polarization.
^†
In fact, any object in a superposition state will be described by a wavefunction exactly like this one. If we can find a way to teleport a photon polarization from Truman to RD, we can use the same technique to teleport the state of a cat in a box—it’s just a matter of increasing the number of particles involved.

So, Truman has a photon that he wants to send to RD. The classical recipe tells him to simply measure the polarization
of the photon, then call RD on the phone, and tell him how to prepare an identical state. But the only way Truman can measure the polarization is if he already knows something about the state, and can set his polarization detector appropriately. For example, if he knows that the photon is either vertical or horizontal, he can send it at a vertically oriented polarizer. If it passes through, he knows that the polarization was vertical, and if it gets absorbed, he knows it was horizontal. He can then send that information to RD, who can prepare a photon in the appropriate state.

Unfortunately, if the polarization is at some intermediate angle—
a
parts vertical and
b
parts horizontal—it’s impossible for Truman to make the necessary measurement. The numbers
a
and
b
tell us the probability of the photon passing through a vertical or horizontal polarizer, but there’s no way of measuring both
a
and
b
for a
single
photon—either it passes through a polarizer or it doesn’t. Even if the photon passes through, the superposition is destroyed and it’s left in one of the allowed states.

You can only determine both probabilities by repeating the measurement many times using identically prepared photons. That doesn’t help us to transmit the polarization of a
single
photon, though, which is our goal.

This polarization measurement problem is a specific example of the no-cloning theorem. William Wootters and Wojchiech Zurek proved in 1982 that it is impossible to make a perfect copy of an unknown quantum state. Unless you already have some idea what the state is, you change the state when you try to measure it, and can never be sure that your copy is faithful. If Truman really needs to send RD a perfect copy of a single photon, without knowing its polarization in advance, he’ll need to find a more clever way of doing it.

“Why not just send the photon?”

“Pardon?”

“I mean, it’s a photon. They travel places at the speed of light—that’s what they
do
. If I had a photon and I wanted to send it to some other dog—which I don’t, by the way. Other dogs don’t
deserve
my photons. If I did, though, I would just point the photon at the other dog, and let it go.”

“Oh. Well, there are a lot of things that can happen to a photon on the way from one place to another that would change the polarization. If you want to be sure that the dog on the other end gets exactly the polarization you started with, teleportation is a sure way of doing that.”