Classic Edition: Beam Me a Photon, Scotty

Part two of three of an explanation of "quantum teleportation" experiments, from July of 2002. This one goes through the basics how teleportation works. I might be able to do better now, having worked through it in more detail in order to teach about it in my Quantum Optics class, but it's been a busy week, so I'll just repost the old entry for now.

So, the last whopping huge physics post here covered the idea of quantum entanglement-- how do you get from entanglement to "quantum teleportation", which is what the article that kicked the whole thing off was about?

The first step here is to define what's meant by "teleportation" in this context. The idea here is that you've got one person, traditionally called "Alice" who has a quantum particle (we'll say it's a photon, to be concrete). She wants to get the photon to a second person, traditionally called "Bob," who needs it as part of a scheme for world domination, or something. It's critically important that Bob end up with a photon in exactly the same quantum state as the one Alice starts with, and for some reason that's never adequately explained, Alice can't just send him the actual photon she's got.

"Well, the answer is simple," you say, "she just measures the state, and sends Bob that information, and he can generate a photon on his end, and put it in the right state." Unfortunately, quantum mechanics doesn't work that way-- it's impossible to do a complete measurement of the quantum state of a single particle.

That's a little strange, right there, so I'll unpack it a little. To be concrete, we'll deal with horizontal and vertical polarization states again. If Alice's photon is exactly horizontally polarized, or exactly vertically polarized, there's no problem-- she can simply measure the polarization and report it to Bob. But Alice doesn't know what the state is-- it could be anything. Therefore, quantum mechanically speaking, it's some combination of horizontal and vertical polarizations. Again, in the interest of being as concrete as possible, let's say it's a 50:50 mix-- it's simultaneously both horizontally polarized and vertically polarized, and an equal mix of the two-- mathematically, we'd say the state is "H + V".

When Alice tries to measure the polarization state, she's essentially asking "Are you vertical?" of the photon. If the photon were exactly vertically polarized, the answer would always be "yes"; if the photon were exactly horizontally polarized, the answer would always be "no." For a photon in a superposition state, like the "H + V" photon mentioned above, the answer could be either. 50% of the time (completely randomly-- God, having grown tired of dice, flips a coin, and "heads" is "yes"), the answer will be "yes," and the other 50% of the time, the answer is "no." If you did the measurement on a hundred identical photons, you'd get (on average) fifty vertical and fifty horizontal photons; for any single photon, you'll randomly find one or the other.

Here's the tricky bit, though: after a measurement where the photon is found to be vertical, the photon will be vertically polarized, and only vertically polarized. If she tries to do a second measurement asking whether it's horizontal, the answer will always come back "no." Similarly, if her original inquiry about whether the photon is vertically polarized comes back "no," the photon is instantly and absolutely horizontally polarized, and a second measurement will always find it to be horizontal.You can't detect a photon state of "H + V" by first measuring "H" and then measuring "V"-- after the first measurement, the superposition state is destroyed, and the state is definitely and absolutely one or the other.

So, Alice sends a message to Bob saying "vertical", he sets his photon generator to "V", and he scheme for world domination fails, because he's failed to duplicate the original "H + V" state. The same thing happens if Alice detects a horizontal polarization-- after the measurement, her photon is in the "H" state, and Bob's out of luck, because he doesn't end up with "H + V." (It's not as obvious, but the same problem occurs if Alice tries to get clever and asks "Are you H + V?" of a pure vertical photon. Trust me.)

Essentially the situation is the same as with Schroedinger's hapless cat. Before the box is opened, the cat is both alive and dead at the same time, but once you open the box, it's either alive, or it's dead, and there's no going back to the original indeterminate state. What Bob wants is the indeterminate state-- he wants the cat to be both dead and alive when it gets to him-- so it's vitally important that Alice not "open the box" by measuring the photon state.

So how do you get around this problem? The answer is "quantum teleportation," the brainchild of a group of scientists working with Charles Bennett at IBM (Interesting side note: The guy in the middle of the back row in the group photo, Bill Wootters, taught my undergraduate Statistical Mechanics class). The key to the idea is to use the magic of quantum entanglement to transmit the state from Alice to Bob.

Alice, being a clever student of physics, gets herself a system that generates entangled two-photon states-- those "HV + VH" states I talked about last time, colloquially known as "EPR Pairs." She sends one of the two photons to Bob, and keeps the other one for herself. The state of the EPR photons is indeterminate, but she knows that whatever the polarization of her photon is, Bob's photon is the opposite. She's also got a photon of indeterminate polarization that she needs to send to Bob.

The clever trick is this: To "teleport" the unknown photon state to Bob, she makes a joint measurement on both of the two photons she has. She doesn't measure the individual states, but measures some relative property of the two. One way to do it would be to do a measurement to see if the two photons have the same polarization-- not what that polarization is, just whether they're the same or different. (The details of how to do this are a little tricky, but you can set up something that basically does the right thing) Doing this causes the state of the EPR pair to become entangled with the state of the photon she's trying to send to Bob-- if the two are the same, then Bob's photon is the opposite polarization of the one she's trying to send; if they're opposite, then Bob's photon is in exactly the state he's looking for. This happens instantaneously, via the "spooky interaction at a distance" that bothered Einstein so much. All she has to do now is send Bob the result of her measurement-- if the two photons were the same, then Bob rotates his photon's polarization by ninety degrees (changing horizontal to vertical); if they were different, then he does nothing. After that, nothing can stop Bob from taking over the world.

There are a few subtle quirky things about this that are worth mentioning: one is that the state moves from Alice to Bob without either of them ever having the slightest idea what it is. This is a very non-classical sort of operation. Another important quirk is that the process destroys the initial state-- it may not necessarily destroy the initial photon, but the state is necessarily changed in making the entangling measurement. Bob gets a perfect copy of the state that Alice started with, but Alice is left with (at best) a photon in a different state than she started with. "Quantum teleportation" is analogous to having Alice send Bob a fax, where she's not allowed to read what's on the paper, and the fax machine shreds it immediately after sending it. (As opposed to the usual state of affairs, where I write down an order, read it over, then fax it to an electronics vendor, where they shred it, leaving me with a perfectly readable copy, and them with no trace of the order...)

It's also important to note that no useful information is sent faster than light-- Bob's photon changes its state instantaneously, but until he gets the message from Alice telling him what she measured, he doesn't know whether he has the right state, or a state that's ninety degrees off. Alice's message has to go via classical channels, and can't travel any faster than the speed of light.

There have been a number of experiments done to verify that this scheme works. Single-photon states were "teleported" by a research group in Austria. With a little bit of work, you can extend the process to include states with large numbers of photons-- laser beams, for example-- and more information than just polarization states, but the basic idea is the same. The first experiment on laser teleportation was done in 1998 by a group at Caltech (with collaborators from all over the place). The Australian results mentioned in the article which kicked all this off (more comprehensibly explained in this New Scientist article-- they have a general teleportation article as well) are essentially a refinement of the Caltech experiment, using better lasers, and doing a slightly better job of conveying the state across the lab.

So what does this have to do with Star Trek? I'll talk about that in a separate post.

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