When one hears the words "quantum teleportation," one cannot help but imagine a futuristic science fiction world in which people can just "beam" themselves from place to place like in Star Trek. Sadly, this has very little to do with what quantum teleportation is actually about.

### Why do we need this process?

To appreciate what "quantum teleportation" is, you first have to appreciate the fact that you take for granted your ability to copy bits and send them to your friends. In the classical computing world, there's no problem with having as many copies of a piece of information as you can make --- this is what lets us have such nice systems as the Internet where thousands and potentially millions of people can have access to the same piece of information.

Sadly, in the Quantum world you cannot do this. The reason why you can't is given formally by the No Cloning Theorem, but in short: if you could do this, you'd essentially be making a measurement without changing the state, and that's not allowed. To draw a simplified picture, suppose that someone gives you a coin to toss so that you and he/she can settle a dispute. You would like to know whether the coin is "fair" or not -- that is, whether it has equal probabilities of landing on heads or tails. The only way to determine this (short of doing a very complicated physical analysis of the coin) is to flip the coin many, many times, and see whether you get significantly more heads than tails, or vice versa.

However, suppose that the coin you've been given has the odd property that once you've flipped it, it "sticks" so that you can't flip it again. Now you have no way of finding out whether it was fair or not! This turns out to be the way that Quantum information works: a qubit starts (very roughly speaking) as a probability distribution, with certain odds of being "1" or "0", but once you look at it you force it to one state or the other and so you have no way of learning what the original probabilities were.

Now, there is a way that you could conceivably get around this: if you could copy the qubit a thousand times, then you could look at 999 of them and use that data to get a very good estimate of the probability distribution of the one you haven't looked at. This would be "cheating", in a manner of speaking, because it would let you effectively perform a measurement on a state without destroying it, which for some reason our natural universe does not allow. The only way to close this loophole is for copying itself to be illegal -- and, by the No-Cloning Theorem, it turns out that this is the case!

As a result of this, merely sending information from one place to another becomes a whole lot more difficult! In order to prevent the bit from being copied, you have to somehow get it to not only appear at the new location, but also to disappear from the old location! Thus, sending information from one place to another becomes the odd process that we call “quantum teleportation”.

### So how do you do it?

There is a classical analogue to this process which I’ll describe first just so you can get a handle of what’s going on. There is a classical gate called the XOR which takes two classical bits and outputs a “0” if they have the same value and a “1” if they have different values. It turns out that if

A xor B = C

then

C xor B = A

that is, the process is completely reversible -- if you know C and B, then you know what A was.

So suppose you and a friend have access to the same string of random bits -- but you cannot actually look at these bits, you can only XOR them with other bits. So now, to send a bit you own to your friend, one procedure you could use is to XOR your bit with the next random bit in the string, send the result of this, to him/her, and then he/she will XOR that with the same bit on his/her string. This cancels out your XOR, and so the friend now has a copy of your original bit. (This procedure is also known as encoding via a one-time pad.)

The “quantum teleportation” procedure is very similar to this. We’ll call the qubit that you want to send “A”. The first thing that you have to do is entangle two qubits, and then put one of these qubits at the old location and one at the new; we’ll call these respectively “B” and “C”. This is analgous to you and a friend sharing the random string of bits in the classical procedure.

Next, at the old location you need to perform a controlled-not operation between A and B, and then a Hadamard on A. These operations are the analogue of XORing your bit with the random bit -- except that unlike a classical XOR which preserves the original bit, this process actually scrambles both bits together in such a way that you no longer have access to the originals. Thus, you can now measure both your of these new bits without learning anything about the old ones.

Finally, you send your measurement results to the new location via some classical means, where they are used to perform operations on C, the other half of the entangled qubit. Specifically, if A is 1 then you apply a bit-flip operator to C (kinda like a logical NOT gate), and if B is 1 then you apply a phase-flip operator to C (there's no classical gate analogue here). Both of these operations are independent -- you could perform neither, both, just a bit-flip, or just a phase-flip. When the smoke clears, it turns out that you’ve now turned C into your original qubit!

You can see that this was a very long and elaborate process to get a qubit from one location to another, but it does the job! Now the original qubit exists at the new location, and it has been destroyed at the old location; furthermore, the fact that you “scrambled” the qubit with a second random bit before taking a measurement of it means that you cannot have learned anything about it.

So in short, “quantum teleportation” is not a process which has anything to do with transporting physical objects made of matter from one place to another, but just an elaborate “dance” that you have to perform to send a piece of quantum information to a different location since you are not allowed to make any copies.

### Can we do this?

Keeping in mind that a “qubit” is an abstract object and that there are many possible ways to physically create one, the answer is: yes, for certain physical manifestations of qubits, we can perform quantum teleportation. See the above writeup for examples.

### But what does this process mean?

This is a deep question. A qubit could be in any of an infinite number of states -- that is, the probability of being “1” could take any number from 0% to 100% -- so in a manner of speaking you’ve just sent an infinite amount of information from one place to another, using only two classical bits! This is really strange. The thing that makes this process work is that the two locations shared an entangled qubit. It’s as if somehow, the information got from one half of this qubit to the other, even though the two are completely separated! Mysteriously, each half seems to “know” something about what has happened to the other. (This is, in fact, another manifestation of the Einstein-Podolsky-Rosen paradox.) Orson Scott Card postulated the existence of a device called the “ansible” which supposedly tapped into this property to allow people to instantly communicate with each other across the galaxy, but it turns out you cannot do this. Just because the two qubits share some kind of “quantum information” does not mean that you can use them to send “classical information”; indeed, even in quantum teleportation the information had not truly arrived at the new location until the two classical bits had arrived. Furthermore, most of the “infinite information” disappears as soon as you look at the bit, so in some sense it really doesn’t even “exist” in the same way that classical information does.

We see here a common theme in Quantum Mechanics: it often seems seems as if you could use these funky properties to send classical information faster than the speed of light, but it always turns out that there’s a catch that screws things up!

### Conclusion

“Quantum teleportation” is a very misleading term; probably something like “qubit transportation” would have been a more apt description. Regardless, even though this process has nothing to do with the transportation of physical objects, it is nonetheless incredibly useful because without it we’d have no way of sending quantum information from one location to another.
Source: David Bacon's Quantum Computing Lecture Notes, http://www.cs.washington.edu/education/courses/cse599d/06wi/