# Introduction to Quantum Teleportation

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.

The discussion below is part of a talk I gave at the International Summer School for Young Physicists held at the Perimeter Institute in Waterloo, Canada. The talk was intended for some very smart high-school students, but most of the discussion below should be accessible to a wide audience.

Only the non-technical sections of the discussion (i.e., those that don't use math notation) are reproduced below. However the full set of notes is available as a PDF. A recorded version of the talk is also available via PIRSA.

## Classical teleportation

Let's start by assuming that the world is perfectly classical, that is, let's not worry about the effects of quantum mechanics. Can we do teleportation?

As stated above the problem is trivial and the solution is called a truck. We load the cargo of box A onto a truck, we drive the truck over to lab B, and unload the cargo into box B. Presto exchange-o, we have teleportation!

But that is not the solution we really wanted, so let's build a wall between labs A and B. Now no trucks can get through.

Unfortunately, if this wall is perfect and separates Labs A and B into two different universes, then there is nothing that can be done to move things between the two universes and our poor telenaut Jim will be forever stuck in Lab A.

To make the problem both possible and interesting let's allow a single telephone line between universes A and B. Can we teleport Jim from A to B now?

What we are trying to build now is essentially a fax machine. A giant 3-D fax machine, but a fax machine nonetheless. Into the fax machine at A goes Jim and out of the fax machine at B we get a copy of Jim.

The first objection that you could raise is that we now have two copies of Jim, which may not be ideal. But this is an easily fixed problem. We buy a shredder and attach it to the fax machine at A so that it destroys the originals after they pass through the fax.

So we run Jim through the shredder at A and now there is only on copy at B. Will this be painful for Jim? Maybe (hence the title "brave" telenaut). But remember that the surviving copy at B was made before the "original" at A was put into the shredder. From the point of view of the copy at B, he entered the box at A and exited at B and no pain was ever felt.

A second objection is that we are only getting an approximate copy of Jim at B. Certainly a standard fax machine has a fairly poor resolution, however there is no reason why we can't build very very accurate fax machines.

Now it is true that the copy at B will never be perfect. But that shouldn't be a problem. Even if we used a truck to transport an object from A to B, the object that arrives at B would be slightly different from the one that left A. Along the way it will be shaken a bit or it might get hit by some cosmic rays which will change the state of a few atoms. Our goal should be that the errors that appear when we teleport Jim via the fax machine should be comparable to the changes that would have occurred when moving Jim in a truck. That is, a few very very small errors should be acceptable.

An important thing to notice is that our giant fax machine is not intended to transfer matter and energy, just like a regular fax machine would not be used to transmit blank papers. We always assume that we have the appropriate matter and energy available in Lab B and our goal is simply to assemble it into the pattern of the object placed in Box A.

So can we build a classical teleportation device as described? The answer appears to be yes. That doesn't mean that it is easy. It would be an incredible engineering feat to build a giant 3-D super-accurate fax machine. But it really is just a difficult engineering problem. From the point of view of a physicist there is no reason why this shouldn't be possible.

## Quantum teleportation

But now we remember that the world is quantum mechanical, and realize that there is a problem...

What is the fax machine supposed to do?

1. Fully measures the state of the input
2. Transmits the results via the phone
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.

It appears that even before running Jim through the shredder, the measurement process will likely destroy the only good copy without obtaining the required information to rebuilt Jim anew.

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.

## The mathematics of quantum teleportation

To view the full discussion of the mathematics of teleportation you will need to continue reading the PDF version of these notes. Alternatively, you can skip ahead to the discussion below covering a few important issues involving teleportation.

## Can quantum teleportation be used for superluminal communication?

If we tried to define a colloquial notion of teleportation it would probably have two main properties: That objects move from A to B without "passing" through the space in between and that it be done instantaneously, or at least very very fast.

Roughly speaking, our teleportation schemes satisfy the first property. However, thus far we haven't discussed the speed at which teleportation should occur.

Teleportation as defined here requires sending a message from Lab A to Lab B using a regular phone. The message will travel at the speed of light from A to B. Therefore, our version of teleportation cannot be instantaneous and does not allow for travel faster than the speed of light.

In fact, teleportation might be significantly slower than light travel if the measurement and reconstruction procedures are slow. However, if we are teleporting a person (or some other system that is not static) then the measurement and reconstruction procedures need to be performed nearly instantaneously. After all, if you get to see as your feet are slowly measured and disassembled, the process would likely not be pain-free.

At first glance, though, there seems to be a way to use the teleportation procedure for superluminal communication. That is, by measuring the spins in Lab A, we are somehow instantaneously modifying the spin in Lab B. Whether or not this is a good description of what is going on depends which interpretation of quantum mechanics is used to describe the system (there are actually many interpretations of quantum mechanics which describe the above process in very different ways). However, all interpretations of quantum mechanics agree on one fact: that such tricks cannot be used for superluminal communication.

The basic idea of such a proof is to check that, when averaged over all the outcomes obtained in Lab A, any measurement done in Lab B will always yield 50-50 outcomes, no matter what state is being teleported. Therefore the measurements in Lab B cannot convey any useful information, at least until such a time when the correction operators have been applied.

Unfortunately all modern theories of physics predict that both faster than light travel and faster than light communication are impossible.

## Real experiments that do teleportation

A number of groups conducted experimental realizations of the quantum teleportation procedure described above in the years 1997 and 1998, using a variety of different systems such as the spin (or polarization) of photons and the spin of atoms. In many cases Labs A and B were the left and right side of a table, and the spins were teleported roughly 50 cm.

The reason distance becomes relevant has to do with the distribution of entanglement which becomes harder as the separation between the two "labs" increases. A second related problem is the storing of entanglement which can only be done for very short periods, so in practice most early experiments distribute the entanglement only moments before it is to be used for teleportation. However, these experiments were sufficient to convince most physicists that teleportation of spins is possible.

Since 1997 there have also been many improved versions of the teleportation experiment. For instance, the distance has been increased in one experiment to 600 m, and the accuracy of the teleported state has also been slowly improving.

However, at the time this document was written, most experiments have only teleported a single spin. In principle, if you can teleport one spin, then you can teleport many spins simply by repeating the experiment in series many times. But this roughly only works on disjoint spins. To teleport a single object comprised of many spins is still out of reach of present day experiments.

In the future, though, we should see experiments that teleport large numbers of spins. Certainly, if a practical quantum computer is ever built then the same technology would likely allow us to teleport a few thousand spins. It is likely that this will happen within the next 30--50 years, if not sooner.

## But will we ever be able to teleport people?

There are some 10^29 matter particles comprising a human person, each of which has position and momentum degrees of freedom in addition to spin. In principle, we might also need to teleport the photons, gluons and other energy particles comprising a person. Teleporting all that is going to be significantly harder than a few thousand spins. It is probably a good guess that teleportation of humans will never be possible.

Are we at least sure that it is possible to teleport humans in principle?

While most scientists expect that ten, hundreds and maybe even thousands of spins will be teleported in practice some day, the teleportation of a human being, even in principle, is actually still a controversial subject.

I would roughly divide people into three schools of thought.

The first group of physicists would argue that there is a soul, consciousness or spirit that permeates the human body that cannot be described by science. Unfortunately, in this view by definition we are prevented from using science to determine if teleportation is feasible.

A second group of physicists would disagree with human teleportation because of something known as the measurement problem. Roughly speaking, any object that is capable of performing quantum measurements cannot itself be a quantum object, and therefore cannot be teleported using quantum teleportation. In this view, small numbers of particles are quantum but at some point when you combine enough particles you end up with a classical or "observer" object, which cannot be described by the laws of quantum mechanics.

In principle, such a belief will have experimental consequences, as we should be able to determine at what point do objects stop being quantum mechanical. At the moment there is neither any experimental evidence for such observer objects nor even a consistent theory that could describe them. On the other hand, it is also true that presently it is very hard to experimentally study large quantum systems, and so it is quite possible that something interesting will happen when a large enough system is examined.

The third school of though (which I am partial to) would say that all objects big and small are quantum mechanical, and therefore in principle can be teleported. What happened with the measurement problem? I would argue that measurements never actually occur. What happens is that the observer becomes entangled with the system he is measuring, and this appears to the observer as if a measurement was performed. The mathematics for this process works out quite nicely, but it does leave the nagging question of why does it feel like we are constantly measuring the world?

Of course, the final answer to whether teleportation of people is possible even in principle must wait for the formulation of a complete theory of physics, one which unifies relativity with quantum mechanics.

In the meantime, one can ask if there any applications for teleporting thousands of spins?

The answer is probably yes. In the future it is likely that quantum computers (i.e., computers capable of processing quantum information) will be built and may even be as ubiquitous as classical computers are today. These computers will need to exchange quantum information. One way these exchanges of information can occur is via a quantum phone, that is, a device capable of sending and received quantum messages. But when such phones are not available, the alternative is to do teleportation using a regular phone. So don't be surprised if some day in the next 100 years you see a quantum teleportation device for sale in your local computer store.