How to Teach Physics to Your Dog

by CHAD ORZEL

  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.

  NO CLONING ALLOWED: QUANTUM LIMITATIONS

  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.”

  “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.”

  A MAGIC COMPASS: CLASSICAL ANALOGUE OF QUANTUM TELEPORTATION

  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.

  BEAM ME A PHOTON: QUANTUM TELEPORTATION

  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:

  FOUR STEPS FOR QUANTUM TELEPORTATION

  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 V1V2, where the first letter indicates the polarization of Photon 1 and the second that of Photon 2), both can be horizontal (H1H2), Photon 1 can be vertical and Photon 2 horizontal (V1H2), or Photon 1 can be horizontal and Photon 2 vertical (H1V2). 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).

  This scheme transfers the polarization state of Photon 1 to Photon 3, transforming it into a perfect copy of the initial state of Photon 1. In the process, though, the entangling measurement made on Photons 1 and 2 has changed the state of Photon 1 so that it is no longer in the same state as when it started—it’s in an indeterminate entangled state with Photon 2. It’s impossible for both dogs to end up with exactly the same state, satisfying the no-cloning theorem.

  We also see that teleportation is not instantaneous. The polarization of Photon 3 is instantaneously determined when Truman makes the measurement on Photons 1 and 2, but there’s one more step, because Photon 3 is not instantaneously put into the correct state. Instead, it goes into one of four possible states, depending on the outcome of Truman’s measurement. The teleportation is not complete until RD makes the final rotation of Photon 3. RD can’t do that until he receiv
es the message containing the outcome of the measurement, and that message has to travel from one dog to the other at a speed less than or equal to the speed of light.

  TELEPORTING ACROSS THE DANUBE: EXPERIMENTAL DEMONSTRATION

  The idea of teleportation was first proposed in 1993, and it was demonstrated in 1997 by a group in Innsbruck headed by Anton Zeilinger.* They produced their entangled photons by sending a photon from an ultraviolet laser into a special crystal that produces two infrared photons, each having half the energy of the original photon. They sent the laser through the crystal twice, to produce a total of four photons. One pair was used as the entangled pair needed for teleportation (Photons 2 and 3), while one of the other two was sent through a polarizer to provide the state to be teleported (Photon 1). The fourth photon was used as a trigger to let the experimenters know when to collect data.

  Photons 1 and 2 were brought together on a beam splitter in a way that performed the entangling measurement. They could only detect one of the four Bell states, but when they did, they knew that Photon 3 was projected into a particular polarization. When they detected Photons 1 and 2 in State IV (25% of the time), they sent a signal to their analyzer to measure the polarization of Photon 3. Because they set the polarization of Photon 1 themselves, they were able to repeat the experiment many times, and confirm that Photon 3 was polarized at exactly the angle predicted by the teleportation protocol.

  Schematic of the Zeilinger group teleportation experiment. An ultra-violet laser passes through a downconversion crystal, where it produces two infrared photons (Photons 2 and 3), which serve as the entangled pair for teleportation. The ultraviolet laser then hits a mirror, and passes through the crystal again, producing another pair (Photons 1 and 4), one of which serves as the photon to be teleported, while the other is a trigger to let the experimenters know that all four photons have been produced. Photons 1 and 2 are brought together for an entangling measurement, and when they are found in the appropriate Bell state, the polarization of Photon 3 is measured to confirm the “teleportation.”