Measurement and the Born rule

So far we’ve shown you a qubit’s state vector and Bloch-sphere point. But there’s one thing we haven’t actually done yet: we haven’t looked. In quantum mechanics, looking is a verb that does something. It’s called measurement.

Here’s the punchline upfront: when you measure a qubit in superposition, you always see exactly one of the basis states — either $|0\rangle$ or $|1\rangle$, nothing in between. The superposition is gone. The qubit is now a bit.

Which one you see is random. How random is governed by the amplitudes.

The Born rule

If a qubit is in state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, then measuring it gives:

• $|0\rangle$ with probability $|\alpha|^2$
• $|1\rangle$ with probability $|\beta|^2$

This is called the Born rule. It’s the bridge between the quantum world (complex amplitudes) and the classical world (yes/no outcomes).

Notice the absolute value and the square. You take the amplitude, find its magnitude (the length of the complex number’s arrow), and square it. That gives a real number between $0$ and $1$, which is your probability.

The normalization condition from Lesson 2.1 now makes sense: $|\alpha|^2 + |\beta|^2 = 1$ is just “the probabilities of the two outcomes add to $1$.” Something has to happen.

See it happen

The widget below starts with a qubit in the $|+\rangle$ state — an equal superposition. Measure it. Then measure it again. Then mash the ×10 and ×100 buttons and watch the observed frequencies (bottom bar) catch up with the theoretical probabilities (top bar).

Before you read on, play with it. Measure once, watch the state collapse. Reset. Measure a hundred times. Notice how the observed frequencies wander at first and then lock in.

Two things to notice

First: after a single measurement, the qubit is no longer in $|+\rangle$. It’s in whatever basis state you just measured. If you measure it again right away, you’ll get the same answer — the superposition is gone. This is called wavefunction collapse.

Second: the ×100 observed histogram is about $50/50$, not exactly $50/50$. Run it again and you’ll get a slightly different result. The randomness is fundamental. It is not a statement about our ignorance of the qubit’s “real” state — there is no hidden “real” state. The qubit was genuinely in superposition until you looked.

Why this is different from the coin

Remember the coin from Lesson 1? Under the cup, the coin is already heads or tails. We just don’t know which. When we lift the cup, we’re learning about a pre-existing fact.

A qubit in $|+\rangle$ is not that. It is not “really” $|0\rangle$ or “really” $|1\rangle$ with us just being ignorant. It is in a genuinely new kind of state — one that has no classical counterpart. Measurement doesn’t reveal a pre-existing answer; it creates one.

How do we know? Because superposition states can interfere. Two paths leading to the same $|0\rangle$ can cancel each other out if their phases are opposite — something that classical probabilities can never do. (Negative probabilities make no sense. Negative amplitudes make total sense.) This interference is what gives quantum computers their edge, and you’ll see it for the first time in the next lesson.

What’s next

You’ve seen a qubit, you’ve moved it around the Bloch sphere, and now you know what happens when you look at it. Time to do the thing every beginner wants to do after they learn what a qubit is: make one go into superposition on purpose.

In the next lesson, you’ll meet your first quantum gate — the Hadamard — and put a qubit into $|+\rangle$ with a single click.