*Wilf Offord*

From The Fly to X-men, teleportation has been a staple of science fiction for decades. The ability to be anywhere in seconds, to communicate across the galaxy – this has been the object of speculation. Nowadays, with words like entanglement and quantum teleportation thrown around in documentaries and fiction alike, you may be wondering: will such a thing ever be possible? In short, no, but don’t let that spoil the fun. ‘Quantum teleportation’ is in fact a real thing – one that has actually been achieved in a lab (plus a satellite in space!). So, if it isn’t the form of teleportation that might immediately come to mind, what is it, and how does it work?

There are a few things we could be talking about when we say ‘teleportation’. One is ‘quantum tunnelling’. This is a phenomenon that happens on a very small scale whereby particles can seemingly be found in places where they shouldn’t intuitively be. Whilst it is interesting, that’s not what we will be talking about here. In this article, we are talking about quantum teleportation, which has a specific technical meaning in the world of quantum computing. If you haven’t already, check out my first article on quantum computing here, but I will do a quick recap of the important bits (no pun intended).

Quantum computers work using what are known as qubits (short for quantum bits). They are kind of like the bits in computers and have two states, ‘0’ and ‘1’, which we can think of as switches being off or on. They are represented like this:

But unlike normal bits, they can also be somewhere in between the two: a certain multiple of ‘0’ and a certain multiple of ‘1’, represented like so:

This means that before we make a measurement, there’s no definite answer to whether the ‘switch’ is on or off – it’s in a superposition of both states. It is the information stored in these qubits that we will be teleporting, but before we get to that, we need to understand what happens when we start adding more qubits.

Suppose we have a total of two qubits – we can think of two ‘switches’. Now we have four possible combinations:

So, a qubit can be in a combination of these four states, written:

The letters a, b, c, and d correspond to the probability that when we measure the qubits, we will see them in the corresponding state.

We are now ready to talk about entanglement – or what Einstein famously called ‘spooky action at a distance’. Fortunately, the idea is not quite as tricky as you are probably imagining, and can be illustrated with an example. Suppose I have two coloured dice, a red one and a blue one (any pair of objects would work). I put each of them into two opaque boxes and shuffle them around behind my back so that there is no way of distinguishing which is in which box. Then, I give one to you, and I take the other, and get on a rocket ship that goes lightyears away. Neither of us are allowed to look inside our mystery boxes and there’s no way either of us can know which die we have.

Now suppose you look inside your box and see you have the red dice. This means you instantly know I have the blue one – common sense, right? The only ‘spookiness’ that comes into play is when now instead of red and blue dice, we consider two tiny, trapped particles that are small enough to allow us to see quantum effects. Quantum theory tells us that before we make a measurement, there’s no definite answer as to who has which particle – we are in a mix of the two states, |*I have particle A*⟩ and |*You have particle A*⟩. But when you look in your box, and ‘measure’ which particle you have, suddenly the answer becomes definite and we say the state has ‘collapsed’ onto either |*I have particle A*⟩ and |*You have particle A*⟩. But then your action of opening the box has affected the quantum state of my box, lightyears away, instantly! This is the ‘spooky action at a distance’ – something on Earth has affected something on the other end of the galaxy faster than it would take light to reach there!

We can realise this in our context of quantum computing by using a pair of qubits. Suppose we initialize our pair to the state:

where the qubit in the first position is the one you keep, and the qubit in the second is the one I take with me on my spaceship. This state has a name, it’s a *Bell pair*^{1}.

Now, when you measure your qubit, if you get a 0, you know the qubit pair has ‘collapsed’ onto the state |00⟩ (because this is the only possibility with a 0 in the first position), so in particular you know I have a 0. And if you measure 1, similarly you know we are in the situation |11⟩ (again this is the only possibility with a 1 in the first position) and I have a 1. Just like with the dice, your measurement has caused my qubit to ‘collapse’ onto a state that you know, from lightyears away. This is entanglement: before we make an observation, we know that information about one qubit will give us information about the other, and we say the qubits are entangled.

So, finally we are ready to talk about quantum teleportation – what is this and how does it work? Well, essentially it’s a way of using a pair of entangled qubits as a method of transporting all the information stored in a third qubit from one place to another. Don’t worry if the maths looks scary, I have just included it for completeness – what is important is what we’re achieving at the end.

Suppose Alice and Bob (this pair seems to always be doing thought experiments together in physics!) meet up and generate an entangled Bell pair, before moving to opposite ends of the earth (or galaxy). Then, say Alice has a third, unknown qubit, which she wants to send that we will call

This setup means that we can write the initial state of all three qubits like this:

It’s sort of a ‘product’^{2} of Alice’s unknown qubit (on the left) and the shared entangled pair (on the right). This is essentially shorthand for the three-qubit state we get by ‘expanding the brackets’ (just like in high school algebra):

Where the first two qubits in each triple are the ones on Alice’s end, and the third is Bob’s. By applying a certain sequence of physical processes, known as quantum logic gates^{3} to her two qubits in her quantum computer, Alice can entangle her two qubits as well, in such a way that now all three are entangled and the state transforms into:

Now what is most useful for us is to group these qubits into those belonging to Alice and those belonging to Bob – remember, the first two are Alice’s and the last is Bob’s. Grouping them up using the ‘product’ notation mentioned earlier, we get:

But now by measuring her two qubits, Alice can know which state Bob’s qubit is in – for instance if she measures |00⟩, she knows Bob’s qubit is in the state

by looking at the first term in the brackets. But hold on a second – this has the same numbers 𝑎 and 𝑏 as our original state |𝜓⟩! In fact it is an identical copy of |𝜓⟩, so if Alice measures this we are done – |𝜓⟩ has been successfully teleported to Bob.

Suppose now she measures |01⟩, so she knows Bob’s qubit has state

Whilst this isn’t exactly the state we are looking for (note the 1 and 0 are swapped), if Alice measures |01⟩ she can then send a classical message^{4} to Bob telling him to apply his own quantum logic gate, transforming it in a certain way to ‘fix’ it back to the state |𝜓⟩. In fact, for each of the four possible outcomes Alice can observe, she can send Bob a message telling him how to fix his state into |𝜓⟩. Alice sends one classical message and in return has ‘teleported’ the exact state of an (unknown) qubit!

So, what have we achieved? We have ‘spent’ one pair of entangled qubits plus one classical message to transport the exact state of a qubit. On Alice’s end, the information in the unknown state |𝜓⟩ has been destroyed (when she measures her qubits), and it is reassembled on Bob’s end. Note that no information has actually travelled faster than light – Alice still had to send a classical message to Bob that will travel at some sub-light-speed. But, nevertheless this is impressive! A qubit is a real, physical object (e.g. a particle) stored somewhere in a quantum computer, and we have created an exact copy of it in a completely different location. Normally, this would take infinite classical data to do, if it were even possible, as we would have to measure all the properties of the particle to infinite precision^{5}, but here using entanglement we have done exactly this – the classical message we have to send only involves the relatively tiny amount of information about which of four cases we are in.

This form of quantum teleportation is exactly what was achieved in 2017 by the University of Science and Technology in Shanghai – they managed to quantum teleport a qubit in the form of a photon from the ground to a satellite in orbit – the farthest quantum teleportation that has been achieved to date!

So, whilst quantum teleportation may not let you go on holiday in an instant, or let us establish faster-than-light communication, it is an engineering reality and has many applications in the exciting new field of quantum information. This seeks to understand what is possible in the realm of sending information from one place to another, in the relatively recent framework of quantum mechanics. We’ve seen that entangled pairs are an important resource that we can use to accomplish things like quantum teleportation, and new ways are being developed of measuring and quantifying entanglement. Many countries are trying to construct networks that can share quantum information. Qubits and quantum information really seem like the next generation of technology – rather than teleporting away, it looks like they might be here to stay!

1. Named after John Stewart Bell, nothing to do with real bells!

2. For people who have learned about a kind of maths called linear algebra, this is more specifically the tensor product

3. Again, for mathematicians: quantum logic gates correspond to a certain class of linear transformations or matrices acting on the qubit state

4. e.g. an email – but importantly she only has to send normal bits, not qubits!

5. actually, it’s a general fact in quantum mechanics that such a measurement is in fact impossible – so we wouldn’t have hope to begin with