Amplitudes are private. The only way information leaves a qubit is a measurement — which returns a single bit and destroys the blend. Here is how the Born rule plays out in practice, and the matrices that describe it.
Measure a qubit and you get back a single 0 or 1; the state collapses onto that answer and forgets the rest. One shot reveals almost nothing. But prepare the same state again and again, and the tally of outcomes converges on |α|² and |β|².
Set the qubit's bias, then run shots and watch the bars climb onto the dashed odds.
A coin spinning on a tabletop is, for that moment, neither heads nor tails. Slap your hand down and it commits — instantly, and at random — to one face, and there it stays. Measurement does the same to a qubit: it forces a single answer and leaves it stuck. The one twist a coin lacks: with a qubit, which way you slap (which basis you measure) changes the odds you see.
Here is the part that trips everyone up: a measurement doesn't just read the qubit, it changes it. The first look lands on a 0 or 1 at random — and from then on the qubit is stuck there. Ask again and you get the same answer, forever. The blend is gone.
Prepare a 70/30 qubit, then keep measuring the same copy — not a fresh one each time.
Compare with §1: there, every shot re-prepared a fresh qubit, so the tally spread across both outcomes. Here a single qubit is measured repeatedly — and it never wavers after the first look.
A measurement in the 0/1 basis is described by two projector matrices, P₀ = |0⟩⟨0| and P₁ = |1⟩⟨1|. Each picks out one slot:
The probability of outcome 0 is the "sandwich" ⟨ψ|P₀|ψ⟩ = |α|² — exactly the Born rule, now as matrix arithmetic.
Label outcome 0 as +1 and outcome 1 as −1 — that's the Z observable, the matrix diag(+1, −1). Its average over many shots is
It runs from +1 (a sure 0) through 0 (an even superposition) to −1 (a sure 1) — and it's exactly the height of the Bloch arrow you'll meet next chapter.
So far we always asked the same question: “are you 0 or 1?” But that is a choice of measurement — the 0/1 axis, called the Z basis. You can just as well ask along the X basis, whose two answers are |+⟩ and |−⟩.
The punchline: a state that is perfectly certain in one basis is a fair coin in the other. Switch the state and the question and watch.
The reason is overlap (Chapter 2): |0⟩ = (|+⟩ + |−⟩)/√2, so a definite |0⟩ is an even split along X — and vice versa. The two bases are maximally incompatible; sharpness in one is total blur in the other. That trade-off is the seed of the uncertainty principle.
“The qubit secretly was 0 the whole time — measuring just revealed it.” Tempting, but false. There was no hidden 0 or 1 waiting to be found: a state on the equator is genuinely both, and §5 shows the answer you get even depends on which question you ask. Bell's theorem in Chapter 6 closes the door on every “it was decided all along” story — experimentally.