How to Teach Physics to Your Dog (23 page)

“That’s a silly thing to want to do, anyway.”

“Not really, but you’ll have to wait until the end of the chapter to find out why.”

It’s hard to find a classical analogue for quantum teleportation, because the issues involved are inherently nonclassical. But we can get a little of the flavor of what’s involved by thinking of the photon teleportation process in graphical terms. We can also get a hint of what quantum teleportation will really require.

As we saw in
chapter 3
(page 66), we can represent a photon polarization by an arrow indicating the direction of polarization. We can think of the horizontal and vertical components in terms of the number of steps we take in the different directions: you take
a
steps in the vertical direction, and
b
steps in the horizontal direction.

In this graphical picture, teleportation is a problem of aligning arrows. Truman has an arrow pointing in some direction, and both dogs will get steak if RD can make his arrow point in the same direction. How do they manage this?

The only way the two dogs can get their arrows aligned is if
they have some shared reference. If they each have a compass, Truman can compare his arrow to the direction of the compass needle, and tell RD to point his arrow, say, 17° east of due north. The compass provides a reference that they both share, and any scheme for photon teleportation will need a similar reference.

Representing light polarization as a sum of horizontal and vertical components. The larger arrows represent two different photon states, while the smaller arrows are the vertical and horizontal components.

The problem of teleporting a photon is much harder than simply aligning arrows, though, because of the no-cloning theorem. Truman can’t measure the direction of his arrow without disturbing it. Somehow, he needs to communicate the direction of his arrow to RD without measuring it. What he needs is a
nonlocal
reference, a kind of magic compass that can communicate a direction to RD’s compass without making a measurement. Quantum teleportation is possible because the quantum entanglement that we discussed in
chapter 7
provides this kind of nonlocal reference.

Quantum teleportation was developed in 1993 by a team of physicists working at IBM (including William Wootters of the no-cloning theorem). It uses a four-step process to transfer an unknown state from one place to another:

1. Share a pair of entangled particles with your partner.

2. Make an “entangling measurement” between one of the entangled particles and the particle whose state you want to teleport.

3. Send the result of your measurement to your partner by classical means.

4. Tell your partner how to adjust the state of his particle according to the measurement result.

This recipe for teleportation exploits quantum entanglement to generate a copy of an arbitrary state at a distant location through one measurement and a phone call. It uses the active nature of quantum measurement to align one of the two entangled photons with the state to be “teleported.” In the process, the second entangled photon is instantly converted to a polarization that depends on the original state. The no-cloning theorem still applies, so the state of the original particle is altered by the measurement, but at the end of the process, the second entangled photon is in the same state as the original photon before “teleportation.”

Here’s how it works: let’s imagine that Truman has a single photon in a particular polarization state, and he wants to get exactly that state to his old friend RD (but he can’t just send it straight there). Anticipating that this situation might come up, Truman and RD have previously shared a pair of photons
in an entangled state, each taking one. The polarizations of these photons are indeterminate until measured, but they are guaranteed to be opposite each other. So, the two dogs have a total of three photons: Photon 1 is the state that Truman wants to convey to RD (at some randomly chosen angle, described by
a
|V> +
b
|H>), Photon 2 is Truman’s photon from the entangled pair, and Photon 3 is RD’s photon from the entangled pair. The teleportation procedure outlined above will allow RD to turn his Photon 3 into an exact copy of Photon 1.

Teleportation works because quantum physics is nonlocal. We saw in
chapter 7
that any measurement Truman makes on
Photon 2 will instantaneously determine the polarization of RD’s Photon 3. Of course, it’s not as simple as measuring the individual polarizations of Photon 1 and Photon 2—we already saw that that won’t work. Instead, what Truman does is to make a
joint
measurement of the two photons together. He measures whether the two polarizations are the same or different—not what they are, just whether they’re the same.

A cartoon version of quantum teleportation. At the beginning of the process, Truman has two photons, Photon 1 in a definite (though unknown) state that he wants to send to RD, and Photon 2 in an indeterminate state that is entangled with RD’s Photon 3. After the teleportation procedure is completed, Truman has two photons in an indeterminate state entangled with each other, and RD has a photon whose polarization is identical to the original polarization of Photon 1.

If Truman measured the two photons individually, asking whether they’re horizontal or vertical, there are four possible outcomes. Both photons can be vertical (we write this as V
₁
V
₂
, where the first letter indicates the polarization of Photon 1 and the second that of Photon 2), both can be horizontal (H
₁
H
₂
), Photon 1 can be vertical and Photon 2 horizontal (V
₁
H
₂
), or Photon 1 can be horizontal and Photon 2 vertical (H
₁
V
₂
). These four outcomes will occur with different probabilities, depending on what the polarization of the original state was.

For teleportation, Truman doesn’t measure the individual polarizations, but instead asks whether they’re the same. This still gives four possible outcomes, two with the same polarization, and two with opposite polarizations. These “Bell states” are the allowed states for a pair of entangled photons, and when Truman makes his measurement, he’ll find Photons 1 and 2 in one of these four states:

These states are superpositions of the four possible outcomes from the independent measurements, just like Schrödinger’s
famous cat is in a superposition of “alive” and “dead.”
*
Each of the polarizations is still indeterminate—if you go on to measure the individual polarization of Photon 1, you are equally likely to get horizontal or vertical. When you do measure Photon 1, you determine the state of Photon 2 to be either the same or opposite, depending on which of the four states you’re in.

“Wait a minute—why are there four outcomes? Shouldn’t there just be two? What’s with the pluses and minuses? Either they’re the same, or they’re not.”

“That’s true, but in quantum mechanics, there are two different states where they have the same polarization, State I and State II, and two where they have opposite polarizations, State III and State IV. That gives four states.”

“But what’s the difference between State I and State II?”

“They’re different states, in the same way that |V> + |H> and |V> – |H> are different states for a single photon.”

“Wait—they are?”

“Sure. You can see it by thinking of how they add together to give a single polarization at a different angle. You can imagine the |H> as being one step either left or right, and the |V> being one step either up or down. |V> + |H> is then one step up, and one to the right, while |V> – |H> is one step up, and one to the left.”

“So, |V> + |H> is 45° to the right of vertical, and |V> – |H> is 45° to the left of vertical?”

“Exactly. They both give a fifty-fifty chance of being measured as horizontal or vertical, but they’re different states. If you rotated your polarizer 45° clockwise, the |V> + |H> photons would all make it through, while the |V> – |H> photons would all be blocked.”

“So, State I is up and to the right, while State II is up and to the left?”

“Well, it’d be more complicated than that. There are two particles, so you’d need to do it in four dimensions, or something, but that’s the basic idea.”

“Okay, I guess I buy that. Wait—you said the original two entangled photons need to have opposite polarizations. Shouldn’t they be in State III or IV, then?”

“You’re absolutely right. In the usual teleportation procedure, Photons 2 and 3 need to be in State IV. I didn’t mention that earlier, because I thought it would complicate things needlessly. Good catch.”

“I’m a very smart dog. You can’t get anything past me.”

When Truman makes his measurement asking whether Photons 1 and 2 have the same polarization, Photons 1 and 2 are projected into one of these four states. At that instant, the entanglement between Photons 2 and 3 means that RD’s Photon 3 is put into a definite polarization state that depends on which state Truman measured. There are four possible results for the polarization of RD’s Photon 3, whose horizontal and vertical components are related to the horizontal and vertical components of Truman’s original Photon 1.

Each result is a simple rotation of the original polarization state—the arrows point in a different direction, but still involve
a
steps in one direction (up, down, left, or right), and
b
steps in another. Given the outcome of Truman’s measurement, RD knows how to recover the original state of Truman’s photon, even though he doesn’t know what that state was.

All Truman has to do, then, is call RD and tell him the result of the measurement. At that point, RD knows exactly what he needs to do to get Photon 3 into the right state. Based on the result of Truman’s measurement, RD can rotate the polarization of Photon 3, and know that he’s got exactly the state that Truman started with.

The state of RD’s Photon 3 after “teleportation,” for each of the four possible outcomes of Truman’s entangling measurement. Each state is a simple rotation of the initial polarization of Photon 1 (dotted arrow).