Philosophia

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Is quantum teleportation possible?

by Katie Howard 

What is teleportation? Roughly speaking, there is a Lab A and a Lab B, and each lab has a box. The goal of teleportation is to take any object that is placed in Box A and move it to Box B.

Of special interest to science fiction fans (among others) is human teleportation, where a brave telenaut (whom we shall call Jim) enters Box A and uses the teleportation machine to travel to Lab B.

It turns out that human teleportation appears possible in principle, though is probably impossible in practice. Nevertheless, teleportation of much smaller objects like individual spins is not only possible, but has been accomplished in the laboratory. Our goal here is to explain both how teleportation is done and why it is interesting.

Quantum information has a number of uses, such as to create quantum computers or to perform quantum cryptographic protocols. One of the most basic tasks you can imagine performing for quantum information is moving it around. Of course, you could just encode the information in a single photon and send it through an optical fiber, but it might get lost along the way. This is a serious problem, since quantum information cannot be copied without losing its essential quantumness. One possible solution is to use a quantum error-correcting code to protect the qubits being sent. This will work, but is still technically quite difficult. A rather more straightforward way is to use a protocol known as quantum teleportation.

Quantum teleportation, in some sense, can be viewed as splitting up the “quantum” and “information” parts of quantum information. For

Alice to teleport a quantum state to Bob, they must first share a particular type of entangled quantum state, an EPR pair. That is their “quantum” resource. Then to send her state to Bob,

Alice first measures the state and her EPR pair, getting two classical bits. That gives her the “information” part, which she can then send to Bob. In the process, the original quantum state is destroyed, but Bob reconstructs it exactly. The quantum state has been “teleported” to Bob, even though only classical information passed to him from

Alice.

In the above circuit, time moves from left to right. The top two lines represent qubits held by

Alice, and the bottom line represents a qubit held by Bob. As you can see, Alice must make a particular type of measurement called a “

Bell measurement” on her two qubits, and send to Bob the two classical bits that come out of it. Bob performs the operation P, which depends on the value of the two bits he receives from

Alice.

Note that quantum teleportation has very little to do with moving about large objects: it is simply a way of substituting classical information for quantum information when we want to move the latter about. A large object like a chair needs an enormous amount of classical information to describe it completely, but the exact quantum state of a chair is unlikely to be of any interest. Therefore quantum teleportation is not going to be useful if you want to move a chair from Los Angeles to

New York. On the other hand, if you want to move a single atom, then the quantum state might be crucial, and quantum teleportation can save you a good deal of trouble.

What is the teleportation machine supposed to do?

1. Fully measures the state of the input
2. Transmits the results via the line
3. Reconstructs the original from the received description.

Step 1 is already impossible in a quantum world because of the Heisenberg uncertainty principle. We could measure the position of all the particles forming Jim but then we wouldn’t get a chance to measure the momentum of those particles. Alternatively, we could measure the momentum but then not the position. One can also envision a mixed strategy where we measure some positions and some momenta, however the uncertainty principle basically guarantees that we will never obtain enough information to rebuild even a modestly good copy of Jim.

The surprising result of quantum teleportation is that even though the “measure and reconstruct” procedure does not work, there is an alternative procedure that effectively realizes teleportation in the quantum world.

In fact, it was not until the publication of a 1993 paper by Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters that we realized quantum teleportation was possible. That is some 70 years after the formulation of the theory of quantum mechanics!

Effectively we realized that quantum teleportation, which we thought to be impossible, is only very very hard. What is the difference between the two notions? Traveling faster than the speed of light is impossible, traveling at say 99% of the speed of light is possible but very hard to do.

The upgrade in status from impossible to very very hard may not be very significant to those who would like to actually build such a device. But to a physicist it makes a world of difference, and is a very exciting discovery.

So let me begin by describing the setup for quantum teleportation, which is almost identical to the setup for classical teleportation described above. Again, we will have Labs A and B, each with a box, and we will try to move the contents of box A to box B. The two labs will be separated by a wall and only connected by a phone.

We have to be careful in specifying what kind of phone. If this phone allows sending quantum information back and forth, then the problem of quantum teleportation becomes relatively trivial. It is similar to the classical case when we allowed trucks to move objects between A and B.

The interesting case is when the phone allows only the passage of classical information. You can think of the phone as measuring all signals as they pass through the phone. All standard phones are classical phones.

In effect, what we are asking here is can we use our standard classical communication tools to transmit the state of a quantum system.

Thus far our setup for quantum teleportation is equal to the one for classical teleportation. But there is one important difference. In the quantum case, Labs A and B must begin with something called an entangled quantum state, which will be destroyed by the teleportation procedure.

Roughly speaking an entangled state is a pair of objects that are correlated in a quantum way. Below we will describe a specific example known as the “singlet state” of two spins. However, let us first explore the consequences of this extra requirement for quantum teleportation.

To prepare an entangled state of two particles, one essentially has to start with both particles in the same laboratory, let’s say Lab A. Now we have the problem of sending one of the particles to Lab B. In principle, we could use quantum teleportation to send this particle to B, but this process would destroy one entangled state to create another entangled state, a net gain of zero. In any case, we have to worry about how the first entangled state is created.

The only solution is that sometime in the past the wall that separates Lab A and Lab B must not have been there. At that time the scientists from the two labs met, created a large number of entangled states, and carried them to their respective laboratories.

Think of two friends who lived nearby, but now one is moving away. They can create some entangled states that the friend who is moving can carry with him when he leaves, and then they can use those to teleport things back and forth. However, if they had never met in person and have no friends in common (who could have met with both of them) then quantum teleportation becomes impossible.

So returning to our brave telenaut Jim, he will be able to teleport to the labs of his friends. But also he could use two teleportations to travel to the labs of people whom he has never met personally, but who are friends of his friends. Similarly, he can teleport to the labs of the friends of his friends of his friends, and so on. However, teleporting to say a distant planet or to some other place we have never had contact with is impossible.

The entanglement requirement poses a second problem, since as we mentioned above it is destroyed when used. Entanglement is effectively a resource that is slowly depleted as teleportations occur. It can be renewed by meeting in person and then carrying entanglement back from Lab A to Lab B, but it has to be transported without the use of teleportation. In principle this is difficult, otherwise we wouldn’t have bothered using teleportation from A to B in the first place. However, the idea is that one difficult journey from A to B can allow in the future many quick transfers from A to B.

I should mention one last important detail of quantum teleportation. In the classical case we decided to run Jim through the shredder in Lab A after “faxing” him to lab B. But it seems like this step was optional, and we could have chosen to end up with two copies of Jim. In the quantum case this is not possible, because quantum information cannot be copied. The only way to teleport an object to Lab B is to destroy the object at Lab A.

Philosophically, one can say that if there can only ever be one copy of Jim at any time, and the copy of B survives the teleportation process in a pain free manner, then whatever is destroyed at in Lab A could not have been a copy of Jim.

However, we shall leave moral questions of this sort to the philosophers, and instead turn our attention now to the mathematics of quantum teleportation.