2.5

# Measurement

We just described a single qubit and its important characteristics, its ability to support superposition and phase. But it’s important to note that we can’t directly see either of those characteristics. Instead, we must measure the qubit, which has some behaviors that are relatively intuitive and some that aren’t.

As we noted, the waves we have been describing are called probability amplitudes in quantum mechanics. These amplitudes determine the probability of finding a value when we measure the state.

## Measuring a Qubit

When we measure a qubit, we always find one of two states (usually either zero or one, but we’ll relax that a bit later). For a 50/50 state like our “ket plus” state

$\sqrt{1/2}|0\rangle + \sqrt{1/2}|1\rangle$,

there is a fifty percent probability of finding zero, and a fifty percent probability of finding one. We can calculate this by taking the absolute value of the square of the amplitude. For both $|0\rangle$ and $|1\rangle$, that is $|\sqrt{1/2}|^2 = 1/2 = 50\%$.

For the state

$1/2|0\rangle + \sqrt{3/4}|1\rangle$

there is a $|1/2|^2 = 1/4 = 25\%$ probability of finding zero and a 75% probability of finding one. We don’t get 0.75 when we measure it; instead, our quantum probability amplitudes determine what the probability is of finding one of the states.

## Collapse of the Wave Function

Not only does measuring the qubit give us a value based on the probability amplitudes, we also say that it collapses the wave function. What does this mean?

It means that after we find, for example, a zero, any amplitude for one has disappeared. We are left with 100% zero. There is no way to work backwards from the measurement result to determine anything about the original probability amplitudes, other than obviously the amplitude for the value that we measured was non-zero.

If you want to know more, you’ll just have to rerun your experiment a bunch of times, starting from the preparation of your qubit, to collect some statistics. For example, if you run your experiment 100 times and you find zero 49 times and one 51 times, you can infer that your experiment is creating a state that is about 50% zero and 50% one.

## Three Ways of Measuring One Qubit

The Bloch sphere is useful for thinking about measurements. (It’s one of the main reasons we introduced the concept.) The qubit’s vector can point to an arbitrary position on the sphere. If the vector is in the “northern” hemisphere, we are more likely to find a $|0\rangle$ state when we measure (recalling that $|0\rangle$ is at the north pole), whereas if the vector is in the “southern” hemisphere, we are more likely to find a $|1\rangle$ (since the $|1\rangle$ state is at the south pole). The state is then projected onto the state we measured, so that we only have $|0\rangle$ or $|1\rangle$ left. (Projected means that we move from our arbitrary vector onto one of those states.)

The $|0\rangle$ / $|1\rangle$ axis of the Bloch sphere is known as the $Z$ axis. In our discussion so far, we have assumed that measuring a qubit means looking at it in a way that projects it into one of those two states. Measuring a qubit is actually more general than that: we can pick any line through the center of Bloch sphere and measure the qubit, which will project the qubit onto one of the two places where the line meets the sphere.

To keep things relatively simple, we will assume that our measurements are along the $X$, $Y$, or $Z$ axis of the Bloch sphere. When we measure on the $X$ axis, we will project our qubit to one end of the $X$ axis, which we called our $|+\rangle$ and $|-\rangle$ states.

Measuring along the $Y$ axis is less common, so unlike the $Z$ and $X$ axes, the two ends don’t have such simple nicknames. The states at the two ends are
$\sqrt{1/2}|0\rangle + (\pi/2)\sqrt{1/2}|1\rangle$ or $\sqrt{1/2}|0\rangle + (3\pi/2)\sqrt{1/2}|1\rangle.$

## Relationship to Algorithms

Just a quick peek ahead: in fact, the wave function collapse can affect not just the particular qubit we are looking at, but the state of all of the qubits in our system, under some circumstances. We’ll see a little more of that when we talk about entanglement shortly, and a lot more when we talk about algorithms.

We will see that the entire goal of an algorithm is to use interference, which we have already talked about, to create states where measuring the outcome has a high probability of being the solution to our problem.