Take the bit-as-a-vector idea and allow the entries to be complex numbers. That single change — squaring amplitudes instead of reading off probabilities — is the whole of quantum superposition.
A probabilistic bit stored two odds that add to one. A qubit stores two amplitudes — and it's their squared sizes that add to one. Amplitudes can be negative or complex, which lets them cancel and reinforce. That's the door probabilities can't open.
Dial a qubit below. Mix tilts it between 0 and 1; phase rotates the second amplitude in the complex plane.
Noise-cancelling headphones play a second sound wave to silence the first. Each wave alone is just as loud — yet line them up out of step and they cancel to quiet, or in step and they roar. The loudness is like a probability (always positive); the timing is like the phase of an amplitude. That a thing with a fixed “size” can still cancel against another is the one trick plain probabilities can never do — and the whole reason amplitudes exist.
Physicists write a state vector inside an angle bracket, |ψ⟩ — read "ket psi." The two basis states are the same axes as before, just renamed:
A general qubit is a weighted sum — a linear combination — of the two, with complex weights α and β:
"Linear combination," "basis," "vector" — yes, this is literally linear algebra over the complex numbers. Quantum mechanics is, to a first approximation, that subject taken seriously.
You never observe α and β. What you observe is a 0 or a 1, with probability equal to the amplitude's squared magnitude — the Born rule:
Since a measurement always returns something, those probabilities sum to one. In vector language that's the statement that |ψ⟩ has length one. We measure length with the inner product ⟨ψ|ψ⟩ — flip the ket into a bra ⟨ψ| (a row, with each entry conjugated) and multiply:
Multiplying the whole state by a unit complex number e^{iγ} changes nothing measurable — every probability is unchanged. This global phase is physically invisible, so we're free to make α real and positive, as the experiment above does.
What does matter is the relative phase φ between α and β — the angle on that little dial. It's invisible to a measurement right now, but Chapter 5 shows how a gate can rotate it into plain view, where it drives interference.
Six states show up constantly. The poles |0⟩ and |1⟩ are certain; the other four are perfect 50/50 superpositions that differ only by phase. Tap one to load it into the qubit above.
The inner product also measures how much one state overlaps another. Project your qubit onto |0⟩ and you recover α; onto |1⟩, you recover β. The Born rule is just: probability = squared overlap.
Two states with zero overlap are orthogonal — perfectly distinguishable, like |0⟩ and |1⟩. That single idea, orthogonality = distinguishability, quietly runs every measurement and every gate ahead.
“The amplitude is the probability.” No — you must square it. An amplitude of 0.8 means a 64% chance, not 80%, because 0.8² = 0.64.
“A qubit is secretly already 0 or 1, we just don’t know which.” That is plain ignorance — a coin under a cup. A superposition is different: its amplitudes carry signs that can cancel, and Chapter 3 shows the answer can even depend on which question you ask. There is no hidden value waiting.
“Superposition means the qubit is in two places / doing two things at once.” Gentler and truer: it is in one definite state that happens to be a blend of |0⟩ and |1⟩. You only ever read a single 0 or 1; the blend lives in the amplitudes, not in the result.