Entanglement with the simplest maths possible

This is part of a project to get people involved with quantum error correction. See here for more info.

You have probably heard that quantum mechanics is weird. You may also know that it has some strange correlations, known as entanglement, which are a big contribution to that weirdness.

In this post we will try to show that it's not all that weird, and that you can understand it without complicated maths. We'll also show how you can use entanglement to do something that no boring normal correlations can do: Teleportation!

This is a sequel to my earlier, and surprisingly successful, post on the maths of qubits (also known as 'Quantum mechanics with the simplest maths possible'). It also provides some of the maths behind my recent posts on quantum error correction. Nevertheless, I'll try to keep it pretty self-contained.

The story so far...

We are dealing with the simplest kind of quantum system. Some call them qubits. Some call them spin-1/2s. It doesn't matter what names we use though, so we'll just choose the ones that make our life easy.

A qubit can be in one of two states: up or down. You can ask it whether it's in state up or down: we call this a measurement. When it is up, it is not down. When it is down, it is not up. If it were neither, it would not be a qubit. So far, so normal.

But a qubit is quantum. It can be undecided about whether to be up or down. This is described by two numbers, which we call the upness and downness. When it is definitely up, its upness is 1 and its downness is 0. When its down, its upness is 0 and its downness is 1. When its undecided, both are somewhere in between. These are called quantum superpositions.

If we measure whether it is up or down, it will be forced to choose. The probability of choosing up is the upness squared (i.e. upness × upness = upness²). The probability of down is the downness squared. Since it must be up or down, adding up the probabilites for the two must give us 1. So

upness² + downness² = 1 .

In some sense, these probabilities are the true measures of how biased the qubit is towards up and down. So the actual upness and downness can be a bit crazy as long as they still make sense as probabilities when squared. The consequence of this is that the upness and downness can actually be negative numbers if they want, because negative numbers still square to positive numbers.

For example, imagine a qubit with an upness of 1, and one with an upness of -1. What's the difference? Well, the one with an upness of 1 is just the state up. For the other, if we ask whether it's up or down it'll answer 'up' with probability (-1)² = 1, which makes it a certainty. And if it is certain that its own state is up, with no regard to the strange minus sign, who are we to argue?

To do maths we need some way of writing the superpositions down. If we have some state that we want to call S, we write it down like this.

S = (upness of S) x up + (downness of S) x down

Here we have written quantum states in an equation as if they were numbers, which might be a bit confusing. We have to remember what is a number and what isn't. So, from now on, all quantum states will be underlined.

S = (upness of S) x up + (downness of S) x down

That's better! This includes all the relevant information, and looks nice and mathsy.

This equation has a x and a + in there, which look like multiplication and addition. However, unlike normal multiplication, the one here doesn't multiply numbers together. Instead it multiplies a number with a quantum state, whatever that means. Nevertheless, when we do algebra type stuff with them, they follow the same rules as multiplication and addition. So let's just pretend they make sense.

Two of our favourite states are called left and right. The state called right is a superposition of up and down that has the same upness as downness.

right = √½ × up + √½ × down,

Left is the same, but it has a negative downness.

left = √½ x up + (-√½) x down

It turns out that these are as different from each other as up and down. This is because of the minus sign, which does interesting things with the maths when part of a superposition as we found out last time. And because these states are completely different, we are also allowed to measure whether our qubit is left or right.

What if we have a qubit that is in the up state and we measure whether it is left and right? Well, just as left is a superposition of up and down, up can be thought of as a superposition of left and right.

up = √½ × left + √½ × right

So if we do the left/right measurement, the qubit will have to decide one way or the other. 

It's not just the up state can be thought of as a superposition of left and right. Anything can

S = (leftness of S) × left + (rightness of S) × right

= (upness of S) x up + (downness of S) x down

Entanglement

In the original draft of this post, there was a bunch of stuff about how to describe pairs of qubits,  and some of the details about some of the more interesting measurements for them work. But that made everything far too long. All that stuff was moved to its own post: Fun measurements for pairs of qubits. So if you want the full experience, check that out before you continue.

I'll assume you're skipping it, so let's take a few paragraphs for a very brief overview of the world of two qubits. For this we need to know what the possible states of two qubits are. Each qubit has two possible states: up and down. So two qubits have four possible states: both up, both down, the first one up and the second one down and vice-versa. Let's call these uu, dd, ud and du. The kind of superpositions that you can get for two qubits then look like this

uuness x uu + udness x ud + duness x du + ddness x dd .

If we simply have something like uu, the two qubits are in their own state and don't really care what the other is doing. This is also true for the state

½ uu + ½ ud + ½ du + ½ dd

Despite having lots of fancy superpositions going on, it turns out that this is just another way of writing that both qubits are in the state left. Again, they are both doing their own thing, without any relation to the other. There are lots of states like this. They are called separable states.

Separable states are not so intesting. More interesting would be something like

 √½ x uu + √½ x dd

Here the qubits are in a superposition of being up and down, but they will always be the same. So if one gets measured and decides to be up, the other will somehow instantly know about it. Then if it gets measured, it can give the same result. States that are correlated like this are more interesting than separable states. They are called entangled. In the rest of this post we'll look at how the kind of measurements that can create entanglement, and see how it can be used to teleport.

This will by no means tell you everything you need to know about entanglement. As a quantum information theorist, I find it best to try and understand entanglement by looking at what it can do. Teleportation is part of that, but we'll also look at other things like non-locality in the future.

Bell Measurements

What kind of measurements can we do for two qubits? Well we could just measure both to see if they are up or down. But there's another possibility, that's a bit less disruptive. This looks at the up and down states, but only tells you whether these states for the two qubits are the same or different. So if they are in the state uu or dd, it gives you the result same. Crucially, it would do this also for any superposition of uu and dd, without messing up the superposition. Similarly, ud, du or any superposition thereof would give the result 'different'.

This measurement is called ZZ. It has a friend, XX, which does the same thing but with the states left and right. These measurements are also ones that play nicely together. Suppose you have a state for which you know what the outcome of the ZZ measurement would be, but instead measure XX. This might force a decision out of the qubits and change the state. But whatever changes happen, the outcome for the ZZ measurement will stay the same. This sounds quite normal, given that this is exactly how measurements work in our world of big things. But in quantum mechanics, measurements working well together like this is a rarity.

After doing a ZZ measurement, we know exactly how the up/down states of our two qubits are correlated. After doing an XX we know the same for the left/right states. Because they play nicely together, after doing both of these measurements we know about both types of correlation at the same time. This gives us a state of two qubits that are correlated state in an interesting way. The combination of an ZZ and XX measurement must therefore be quite an interesting measurement. For that reason, it gets its own name: a Bell measurement.

Suppose the results to both XX and ZZ came out 'same'. Because they play nicely together, we know that they'd definitely give the same results if we did the same measurements again. There is only one state that give these definite answers. It's one with a lot of quantum entanglement, and is called Φ

Φ = √½ x uu + √½ x dd = √½ x ll + √½ x rr
 

For the other three possible combinations of answers we would get three similar states, called Φ', Ψ or Ψ'.

Bell measurements are pretty useful. Suppose we take some boring state like uu, which is not entangled. If we then we do a Bell measurement, they will be forced to be one of these four entangled states. They are boring no longer.

Bell measurements are named after J. S. Bell, who was the first to really show that entanglement could do things beyond the abilities of normal correations. The four states Φ, Φ', Ψ or Ψ' are call the Bell states, or Bell pairs.

Quantum Teleportation

Now we have Bell measurements we can do fun things. One of those fun things is teleportation! 

Here's the situation. There are these two guys: Alice and Bob. They met ages ago, at a conference or something. They did a Bell measurement on a couple of qubits and got the state Φ. Then they went home to their own labs, each taking one of the qubits of the Bell pair.

Much later, Alice gets a spare qubit. She doesn't know what the state is. Maybe its even entangled to some other stuff. She doesn't know. She doesn't care. She doesn't want it.

But Bob does. He really wants it. I don't know why. Alice could just send it to Bob, but sending it so it doesn't get messed up by noise is really expensive. He doesn't want it that much! Is there a cheaper way from Alice to send it to Bob?

Perhaps Alice could measure it, work out what it is and just tell Bob. Unfortunately, it cannot be so simple. What measurement would she do? If she did an up/down measurement, it would have to choose to be up or down. If it was actually left or right, we wouldn't know. If it was entangled to something else, that entanglement would be destroyed. So she won't have given Bob what he wanted. She would have messed it up. But Alice knows her quantum mechanics, so she doesn't do this.

What could she do instead? Maybe they could use that Bell pair they've been sitting on.

Before we look at how to do this, lets just recap a little. Alice and Bob have three qubits between them, one is a some state S and the other two are a Bell pair. Alice has two (the spare one and half of a Bell pair) and Bob has one (half of a Bell pair).

Alice wonders what would happen if she forced her to qubits to become a Bell pair, by doing a Bell measurement. Then there would still have a Bell pair, but now Alice would have it all. So what would Bob have? Alice thinks that, to make everything nice and symmetric, he must end up with the state of the spare qubit.

Bob is skeptical. He thinks that the Bell measurement might end up extracting some information about the spare qubit. As we saw before, that would be bad and mess it up.

Alice can easily counter this. Her half of the Bell pair is randomly either up or down, and no-one knows which. This randomness will make the outcome of a ZZ measurement random too, whatever the state of the spare qubit is. The same is true for an XX measurement. In neither case do we find out anything about the state of the spare qubit.

Bob is still not convinced. He thinks that it might cause the state of the spare qubit to just be destroyed. This perhaps seems reasonable. His half of the Bell pair has never even been near the spare qubit, so how could it magically acquire the spare qubit's state? And information gets lost all the time in our everday human world. Why wouldn't qubits suffer the same.

Alice nevertheless sticks to her guns, and demands they do the maths to see who is right. For that we'll to work out how to describe three qubits. But that's not much different to two qubits, which isn't too much different to one qubit, so we'll dive straight in.

For the unknown state S of Alice's spare qubit, we'll need to write it in the most general way possible.

S = (upness of S) x up + (downness of S) x down

The Bell pair is in the state

Φ = √½ x uu + √½ x dd

Now lets write the state of all three together, and do so as some superpoistion of the three qubit states uuu, uud, udu, etc. For this we'll think of the first qubit as Alice's spare one, the next one as Alice's half of the Bell pair and the last as Bob's half of the Bell pair. So the state uud would mean that both of Alice's are up, and Bob's qubit is down.

To write down the correct three qubit state, we'll need to calculate the uuuness, the uudness and all the rest. We can get the uuunes by multiplying the upness of the spare qubit and the uuness of Φ

uuuness = (upness of S) x √½ ,

And so on for all the others. Because

uddness = (upness of S) x √½ ,

duuness = (downness of S) x √½ ,

dddness = (downness of S) x √½ ,

All the rest are zero because they would need the two qubits of the Bell pair to be different, and Φ does not allow that.

All in all, we get the three qubit state

(upness of S) x √½ x uuu + (upness of S) x √½ x udd

+ (downness of S) x √½ x duu + (downness of S) x √½ x ddd .

If Alice makes a ZZ measurement on the first two qubits, there's some parts of this state that will give the result 'same' (namely uuu and ddd), and some parts that'll give 'different' (udd and duu). This means the qubits have to decide one way or the other when she makes the measurement. Let's suppose they decide to be the same. The state after measurement will then be

(upness of S) x uuu + (downness of S) x ddd .

Here we've also removed a the √½ for each state. This is just something we need to do after measurements, to make sure the numbers that should add up to 1, still do.

The state we get here is quite interesting. We now have all three qubits entangled together, and the information about the state of the spare qubit is now spread over all three. It's as if all three qubits are now working together to be the spare qubit. If you wanted to measure whether it was up or down, you could do it by measuring any of the three.

Now we are part way there. Bob has a share in the spare qubit state, and could even do an up/down measurement if he wanted. But he might want to do other, more interesting things. For that he needs complete control of the spare qubit state, so Alice needs to somehow give up her part.

By doing an XX measurement she will not get any information about the spare qubit state, as we discussed earlier. She will also force her two qubits to be a Bell pair. This is because she will know definitely the outcomes of both and XX and ZZ measurement, and only Bell pairs can do that. So she'll have no information about the spare qubit, either in her measurement results or her qubits. So Bob must have it all.

Let's make sure by doing the maths. For that we need to know a little more about the state Φ'. Unlike Φ, which gives the result 'same' for both an ZZ and XX measurement, Φ' will always say 'same' for ZZ and 'different' for XX. In the mathsy way, it looks like this

Φ' = √½ x uu + (-√½) x dd = √½ x lr + √½ x rl

Using this, and the mathsy form of Φ from before, we can see that uu and dd can be though of as a superpositon of Φ and Φ'

uu = √½ x Φ + √½ x Φ'

dd = √½ x Φ - √½ x Φ'

So the state that we got after the ZZ measurement can be rewritten

(upness of S) x uuu + (downness of S) x ddd

= (upness of S) x √½ x (Φ u + Φ' u)

+ (downness of S) x √½ x (Φ d - Φ' d)

Here Φ u describes the state where Alice's two qubits are in state Φ and Bob's one is in state u, etc.

When Alice makes the XX measurement of the her qubits, she gets the result 'same' or 'different'. The same result is consistent only with Φ, and the different result is consistent only with Φ'. So if she gets 'same', the state afterwards is

(upness of S) x Φ u + (downness of S) x Φ d

Again we've removed each √½, because that's what you do after measurements. In this state we find that Alice's qubits always have the state Φ. So all the superposition stuff is actually just because Bob's state is

(upness of S) x u + (downness of S) x d

This is exactly the state of the spare qubit, just as he wanted.

It worked! Alice was right! If she makes the ZZ and XX measurements, and gets the result 'same' for both, Bob's qubit magically acquires the state that the spare qubit used to have. Even though his qubit never went anywhere near the spare one!

But what if Alice gets different results? For the other three options for her Bell test, Bob will get the one of the following states

(upness of S) x up + ( - downness of S) x down

(downness of S) x up + (upnness of S) x down

(downness of S) x up + (-upness of S) x down

For the first one the downness has got a minus sign in front of it. But that's okay, Bob just needs to do something called a phase flip to get his qubit into the state it should be. For the second we have the upness and downness the wrong way around. But Bob can use a bit flip to correct that. For the last, Bob just needs to do both

But it is important that Bob knows exactly which flip to do. So he needs Alice to tell him the result of her measurement. She just gets on the phone, tells him and then he does it. Then he always has the state of the spare qubit, just as he wanted.

Without this information, all he knows is that he randomly has one of the four possibilities. That doesn't do him any good. In fact, there's no way to tell the difference between that and his half of the Bell pair, so he can't even tell whether or not Alice has done the measurement yet. That's good, because this would allow Alice and Bob to communicate instantly, which the theory of relativity does not allow. So without knowing anything about the speed of light, the maths we've used to describe quantum states have nevertheless made sure that we can't communicate faster than it.

When Bob does get Alice's result (which is just some random outcome that gives no information about the spare qubit state) and combines it with his qubit (which never met the spare qubit) somehow that combine to form the spare qubit state. That's that magic of entanglement!

I think that one interesting thing here is that quantum mechanics had a choice. It could either allow information to get deleted from the universe, or allow states of qubits to magically be transported far away Star Trek style. It chose Star Trek. Quantum mechanics is weird, but it's good weird.

Footnotes

1. This is the same as asking whether there are an even or odd number of downs, as we do in this post on quantum error correction. If there are even, it means uu or dd, so they are the same. Odd means ud or du, so they are different.