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.

Written by

Wrangler of qubits. Drinker of tea. Father.

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