# Quantum Computation with the simplest maths possible (part 2)

Do you want to know what makes quantum things so awesome? Then you need to do a little maths. But only a little. Promise!

If you missed part 1, check it out here. Otherwise, let’s press on.

**Left and Right**

Now we’ve made up a nice bunch of maths, let’s do something with it. Let’s consider a specific superposition state, that has the same upness as downness. We can give it a name that also lies between up and down. Let’s call it **right**.

We already decided that **right** has upness =downness. And we know that sum of their squares must be 1. With this we can get a specific number for the upness and downness: They will both be √½. Now we can write down right in our mathsy way

**right** = √½ **×** **up** + √½ **×** **down**,

Since we have called this **right**, it would be nice to know if there is a **left** to go with it. Is there as state that is as different from right as **up** is from **down**? If there is, we must also be able to write it down in our mathsy way

**left** = (upness of **left**) **×** **up** + (downness of **left**) **×** **down**

The overlap of **left** and **right** will then be, using the equation from before for **S** and** T**,

overlap of **left** and **right** = (upness of **left**) × (upness of **right**)

+ (downness of **left**) × (downness of **right**)

= (upness of **left**) × √½ + (downness of **left**) × √½

If **left** and **right** are to be as different as **up** and **down**, their overlap needs to be zero. This will happen only if -

(downness of **left**) = - (upness of **left**)

because then we get the following, which comes out as zero.

overlap of **left** and **right** = (upness of **left**) × √½ - (upness of **left**) × √½

So one possible choice for left is

**left** = √½ × **up** + (-√½) × **down**

This has an upness of √½, just as right does, but it has a downness of -√½. This is a negative downness, just like the equation earlier hinted might be possible. What does this mean? Does it make any sense? Well it is rather strange. But the equation earlier also told us to only expect the squares of numbers to be completely sensible, and (-√½)² does give us a nice sensible ½. So let’s not worry too much yet.

So far we’ve been thinking about everything as a superposition of **up** and **down**. But now we have **left** and **right**, which are two states just as different as **up** and **down**. What does a superposition of these look like? Something like

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

As an example, let’s start with a state that has the same leftness as rightness. So

**S** = √½ **×** **left** **+ **√½ **×** **right**

The overlap of this state and **left** is obviously √½, and the overlap for **right** is √½ too. What is the overlap for our trusty old friends **up** and **down**? Using the same sort of method as before, and calculating the overlaps of both left and right with up, we find

overlap of **S** and **up** = √½ **×** (overlap of **left** and **up**) **+ **√½ **×** (overlap of **right** and **up**)

= √½ × √½ + √½ × √½ = ½ + ½ = 1

So the upness of this state is 1! Let’s check the downness.

overlap of **S** and **down** = √½ **×** (overlap of **left** and **down**) **+ **√½ **×** (overlap of **right** and **down**)

= √½ × √½ + √½ × (-√½) = ½–½ = 0

It has a downness of 0! We already know a state with an upness of 1 and a downess of 0. It’s **up**! This superposition of **left** and **right** turns out to be our good old friend **up**.

**up** = √½ **×** **left** **+ **√½ **×** **right**

We can similarly find a superposition for **down**, but let’s do this differently (for fun!). Let’s trust that the weird **×** and **+** that we use on states really do have properties like multiplication and addition. Now let’s look at the superposition

**T** = (-√½) **×** **left** **+ **√½ **×** **right**

Since we have equations for **left** and **right** as superpositions of **up** and **down**, let’s stick these in.

**T** = (-√½) **×** (√½ **×** **up** + (-√½) **×** **down**) **+ **√½ **×** (√½ **×** **up** + √½ **×** **down**)

Doing a bunch of the maths from algebra lessons at school (see, it did come in useful), we can make this into

**T** = (-√½) × (√½) **×** **up** + (-√½) × (-√½) **×** **down** **+ **√½ × √½ **×** **up** + √½ × √½ **×** **down**

= -½ **×** **up** + ½ **×** **down** **+ **½ **×** **up** + ½ **×** **down**

= (½–½) **×** **up** + (½ + ½) **×** **down**

= **down**

The up part disappears and the down part gets strengthened. This is exactly the sort of thing that we call an *interference effect*, and which causes the strange quantumness in the double slit experiment.

**All states are created equal**

With **left** and **right** we have another pair of states that are completely different from each other, just as different as **up** and **down**. We saw that **left** and **right** are both superpositions of **up** and **down**, but **up** and **down** can also be thought of superpositions of **left** and **right**.

This shows us an important fact about the states of quantum things, like qubits. There are no states that can claim to be the ‘proper’ states, with everything else just being their superpositions. Any state is as important as any other. Any state can be thought of as a superposition of other states.

This fact often gets obscured when we are talking about quantum computation. We like to think of this as a quantum version of a normal computer, so the stories we tell ourselves have to be full of 0’s and 1’s. But the qubit states we choose to call 0 and 1 are not really any more special than any other.

This fact also gets obscured by the intuition we get from the real world. For a cat in a box, as in Schrödingers famous thought experiment, the states ‘alive’ and ‘dead’ do seem more sensible than all the possible superposition states. This is just because big and complicated objects, like cats, are constantly interacting with other big and complicated things. These interactions, like getting looked at or and having light and the air bashing into you all the time, tend to be easier to describe using some states rather than others. When and how things bash into into you, for example, depends on where you are. So the interaction is much easier to describe using states for which you have a definite position than ones for which you are a superposition of being in different places. But for tiny things like electrons, which haven’t had all the quantumness bashed out of them, superpositions of different positions are perfectly fine. And that’s what we see in the double slit experiment.

Since **left** and **right** are just as important as **up** and **down**, it means we can also use them for measurements. When ask a qubit what state its in, we can ask it whether it is **up** or **down**, and a qubit in state **left** will have to randomly decide one or the other. But we could also ask it whether it is **left** or **right**, and an **up** qubit would need to randomly decide. In fact we can do an infinite number of possible measurements, where we ask if it is some state **S** or another state **S’**. The only restriction is that **S** and **S’** must always be completely different from each other: they must have an overlap of 0.

Now you know more about quantum mechanics than most. Have fun laughing at rubbish popular science articles.

The original version of this article is available at the link below.