I’m in a pop-sciency kind of mood this morning, so I’m going to dash off a quick post about “quantum teleportation”. It will have embarassingly few equations, but it might be entertaining. If you don’t follow me below the fold, I want you to remember this: quantum teleportation isn’t quite teleportation, not in the Star Trek sense of the word anyways.

So what is quantum so-called “teleportation”? It’s a laboratory technique, and so it’s best to begin by explaining what problem it solves.

Let’s suppose that you and I each have a baseball in hand, and that — for some reason — you’ve decided that you want your baseball to be in the exact same rotational state as mine. How are we going to accomplish this? Well, one approach would be to call me on the phone and ask “Hey, A.J., what state is your baseball in?” I would then look at the baseball, note that it was spinning around some axis with a certain angular velocity, and tell you what axis and what velocity. Then you’d align your baseball, spin it, and we’d be done.

This game is easy because baseballs are well described by classical mechanics. They’re very large objects, relative to atoms, and when I measure my baseball’s rotational state, I only have to disturb it very very slightly. I have to interact with it slightly (otherwise I won’t know anything about it), and this will change its state, but only in a very small way. Obviously, I could disturb it more, say by dropping it on the floor and seeing which way it rolled, but I don’t have to.

So what if you and I each have an electron instead of a baseball? Electrons do have a kind of rotational degree of freedom, called “spin”, which is like baseball rotation in that when you measure it, you should get an axis and a number. Things are a little weirder here in that the number can only take the values +1 and -1 (in appropriate units), but the basic idea is similar.

But our little game is vastly more difficult with electrons, for two closely related reasons. First, electrons don’t have to exist in definite spin states. My electron’s spin state could be +1, or it could be -1, or it could be 35% + 1 and 65% -1. This is the fundamental weirdness of quantum mechanics. Very large, very “classical” objects are (to a good approximation) always in definite rotational states, but very small, very “quantum” systems don’t have to be. Of course, if we couple a quantum object to a classical object, the combined system is basically a classical object.

And this is pretty much what is meant by “measurement” in quantum mechanics. We take a very quantum object and bludgeon it with a classical object. The delicate superpositions don’t actually go away, but for all practical purposes they are absorbed into the state of the classical measuring apparatus. (In particle physics, for example, one frequently measures a neutron’s momentum by allowing it to collide with a wall of steel.) The result: our quantum object is forced into a classical state, and we can tell what state that is by looking at our classical measuring device.

This is the second problem: You want to know what state my electron was in *before* I hammered it into a definite spin state. You want to know what the delicate superpositions were. So we’re going to need some better laboratory technique.

At this point, you can almost guess what correct technique is. I can’t hit the poor electron with a classical measuring device without misplacing a lot of information. But if I allow the electron to gently interact with another quantum object, I can perform a different kind of measurement. This time I won’t actually find out what state the electron is in; I won’t get a number from this experiment. Instead both the electron and the measurement apparatus will change state, but they’ll do so in a much more controlled way. The new state of the measurement apparatus will reflect the old state of the electron.

At this point, I could look at the measurement apparatus, get a number and ruin all of the superpositions. Or I can put the measurement apparatus in a box and mail it to you. Then, if we’ve chosen the right kind of measurement apparatus, you can allow the apparatus to interact with your system, effectively reversing the measurement I made with the apparatus. (The actual experimental technique is a bit more complicated, but not different in principle. We create an entangled pair of “quantum measuring devices” and each take one. Then I allow my half of the entangled pair to interact with my electron, make some classical measurements and pass them to you over the phone. Having this information, you can then let your electron and your half of the entangled pair interact in a way which puts your electron in my electron’s old state.)

The net effect, in any case, is that your electron and my electron have changed state. Your electron is now in the same state as my electron was, and my electron is in who knows what state.

This is “quantum teleportation”. It doesn’t let you transfer information faster than light; we still had to transfer the information by earthly means. (Frankly, talk of FTL is a dead giveaway that you’re dealing with lousy journalism.) It doesn’t let you “clone” electron states; we still had to disturb my electron.

But it is useful if you want to build a quantum computer, because it allows you to move information around between different logic gates without destroying any of the delicate superpositions that make quantum computing work. Some sort of quantum copying is almost certainly going to be necessary to combine lots of “quantum transistors” into a functioning quantum computer.

I actually just saw an interesting generalization of this a couple weeks ago. In essence, teleportation works because we have an adjunction — in particular a duality.

I never really put it together like this before, but I remember drawing the teleportation diagram and seeing that it was exactly a Temperley-Lieb diagram, and the Temperley-Lieb category is exactly the free monoidal category with duals on a single self-dual object.

Whenever you have a monoidal category with duals where an object can be interpreted as a “state space”, you’ll have a teleportation. Use one duality morphism to create a “particle/antiparticle” pair, move the particle to your target and the antiparticle to your source. Then pair off the antiparticle with the source and get information out with the other duality morphism. Transport that “classical” information to the remaining particle and use the unit isomorphism to put it into the proper state. Bingo!

In my view, this posting has too much gee-whiz in it and not enough math, here is a different explanation of what quantum teleportation is.

Quantum teleportation is a means of transmitting a qubit. A qubit is a certain (small) container of quantum information which should be defined rigorously. So first, what is a container of classical information? A piece of classical information is modeled by a probability distribution in a measure space. Usually the measure space is a finite set, but it doesn’t have to be. The probability distribution is the distribution over messages. For example, a letter of the alphabet is a small lump of information, whose distribution could perhaps be the letter frequencies in English. The container is the measure space itself, in this example the set of 26 letters.

One way to model a measure space is by its L^infty algebra of bounded random variables, which is (in usual cases) a commutative von Neumann algebra. (Or an ell^infty algebra if the measure space is a finite set.) A probability distribution can for brevity be called a “state”. In quantum mechanics, a lump of information is a state in a von Neumann algebra that doesn’t have to be commutative. The von Neumann algebra describes the vessel of the information. In particular, the algebra of random variables of a qubit is the 2 x 2 matrix algebra M_2.

Of course, both classical and quantum information can be correlated. “Entanglement” is a name used for a special case of quantum correlation that violates certain classical inequalities. In particular, classically, if a joint state has no entropy, then its two halves cannot be correlated. Quantumly a zero-entropy joint state can still be correlated. If it is, it is called entangled. However, it is still wise to think of entanglement as just a kind of correlation.

Classically, you can see when one container of information is bigger than another. You can encode a bit in a trit but not vice-versa. Quantumly life gets more interesting. Given a classical digit A with a states and a quantum digit B with b states, you can encode A in B when a 1. No classical telephone line has enough bandwidth to carry even a single qubit, even though a qubit cannot hold more than one classical bit.

Quantum teleportation is a loophole that shows that two classical bits can almost hold a qubit. Suppose that Alice has a qubit that she would like to send to Bob, but can’t directly. Suppose further that an enabler, Ned, sends Alice and Bob a maximally entangled pair of qubits. (A maximally entangled pair of qubits is something like a maximally correlated pair of bits, but more interesting.) Then Alice can perform a certain four-valued measurement on her qubit together with the qubit that she got from Ned. She can then send the outcome of that measurement, which is two classical bits to Bob. Bob can then manipulate his qubit from Eve, in a way that depends on these two bits, to make it equivalent to the qubit that Alice had wanted to send to Bob.

The enabler Ned does not need any direct access to anything that Alice wants to send to anyone. All that he has to do is create and distribute entangled pairs. His role is like a classical crypotological enabler who distributes randomly chosen pairs of one-time pads. In fact, Ned’s entangled pairs can also be used to make invulnerable one-time pads — random bit streams that cannot be intercepted by undetected eavesdroppers — a la quantum key distribution.

Unlike quantum key distribution, quantum teleportation is not directly useful for any Star-Trek-like or non-Star-Trek-like application. It is related to quantum error correction and quantum key distribution, which are both useful in principle. Its direct role is to demonstrate a weak equivalence of different types of information. For instance, why would one qubit be weakly equivalent in this sense to two bits rather than one? The reason is that the algebra of random variables of a qubit, M_2, is four-dimensional, just like the ell^infty algebra of two classical bits.

his qubit from Evehis qubit from Ned, sorry. (Too bad there is no edit function for comments in WordPress.)

Given a classical digit A with a states and a quantum digit B with b states, you can encode A in B when a 1.Argh, another error coming from a collision between math notation and HTML. Given a classical digit A with a states and a quantum digit B with b states, you can encode A in B when a ≤ b. You cannot encode B in A if b > 1, no matter how big a is.

In the absence of an edit function, it would be really nice to have a preview facility.

Hi Greg,

Thanks for adding the math. I hope I didn’t offend your sensibilities too badly. (The post

wasmeant to be pure pop-sci.)We’ll do something about the previews one of these days. I know Peter Woit has it working…

I hope I didn’t offend your sensibilities too badly.I completely applaud your choice of topic and your courtesy to me too, and I do not mean to denigrate your real knowledge of quantum information theory. But I have to say that the style of this post could strengthen a number of misconceptions. For instance you say, “our quantum object is forced into a classical state.” Actually, there is no such thing as forcing a quantum object into a classical state. A classical probabilistic system is a commutative algebra of a random variables, while a quantum probabilistic system is a non-commutative algebra of random variables. But states are dual vectors on these algebras, not elements of them. And even if you do express states as elements (which is slightly unnatural but excusable), then every element is contained in

somecommutative subalgebra, namely the subalgebra that it generates.A more careful statement would be: When you measure a random variable x, the state is forced into one which is classical relative to x. But even here the word “forced” is wrong, because it is nothing other than passing to a conditional state. If you roll a die and you measure that it is a 3, you do not say that your measurement “forces” it to be 3. It is a little different quantumly (more precisely if X is not a central element) because then to measure is to alter. But still, “forced” is melodramatic.

That’s a fair criticism.

I should say, in case anyone is wondering what’s going on, that I was describing the measurement process using language inspired by decoherence theory. As a matter of principle, this is misleading, as you rightly point out: decoherence doesn’t actually explain the collapse of the wavefunctions. But for practical matters, I don’t think it’s a bad way of thinking about measurements. It suffices for me, at any rate, until we’ve actually sorted out the matters of principle and interpretation. (Of course, I generally prefer to avoid arguing about the interpretation of quantum mechanics, so it’ll probably be someone else who actually does the sorting out.)

The way that I would say it is that decoherence is a process in quantum probability, not a theory; and that decoherence should not be used to define conditional states. Although in fact if Alice performs a hidden measurement on Bob’s state (she measures this state of his quantum system without telling him the answer), the effect on his state is equivalent to decoherence.