Nothing in this chapter is new. Every piece has already been built, one chapter at a time: vectors and norms (Ch. 1–2), unitary evolution (Ch. 4), tensor products (Ch. 5), projectors (Ch. 2) and eigenvalues (Ch. 1) feeding into Hermitian observables (Ch. 4 and Ch. 6). This chapter's only job is to state, plainly, the four rules that say how those pieces actually govern a physical system — the postulates of quantum mechanics.
Postulate 1 — State
The state of an isolated physical system is a unit vector |ψ⟩ in a complex vector space. "Unit" means ⟨ψ|ψ⟩ = 1 (Ch. 2) — exactly the |α|²+|β|²=1 constraint you first met in Ch. 3, now stated as the general rule it always was.
Postulate 2 — Evolution
A closed system's state evolves over time according to |ψ′⟩ = U|ψ⟩ for some unitary U (Ch. 4). Every gate in Ch. 3 and every circuit in the Composer was already an example of this postulate in action — you were applying it before you had a name for it.
Postulate 3 — Measurement (the Born rule)
An observable is represented by a Hermitian operator M (Ch. 4). Measuring it can only ever produce one of M's eigenvalues (Ch. 1) — that's the actual mechanism behind "measurement outcomes are real numbers," since Hermitian operators guarantee real eigenvalues. If |φᵢ⟩ is the eigenvector for eigenvalue λᵢ, the probability of getting that outcome is:
P(λᵢ) = |⟨φᵢ|ψ⟩|²
— the squared magnitude of an inner product, exactly the projector machinery from Ch. 2. After the measurement, the state collapses to |φᵢ⟩: the outcome you got is now, with certainty, what the system is in.
Postulate 4 — Composite systems
The state space of several systems together is the tensor product of their individual state spaces (Ch. 5). This is the entire reason a 2-qubit system needs 4 amplitudes rather than 4 — it's where every two-qubit state in the Composer and Ch. 3's circuit widgets actually came from.
Watching the Born rule converge
P(λᵢ) = |⟨φᵢ|ψ⟩|² is a single-shot probability — it tells you the odds for one measurement. It doesn't say what one measurement will give you (that's genuinely random), but it makes a sharp prediction about what happens over many measurements: the fraction of each outcome should converge to that exact probability as the count grows. That convergence is something you can actually watch happen.
Why collapse matters for what you can know next
The collapse part of Postulate 3 is easy to skim past, but it's the reason measurement order matters at all. Set θ = 90°, φ = 0° (state |+⟩) and run some shots in the Z basis — you'll see roughly 50/50, as expected. But once a single measurement actually happens and collapses the state to, say, |0⟩, a second Z-basis measurement immediately after will give |0⟩ again with certainty — the randomness was only ever about the first measurement of a given observable on a given state, not a property of the universe re-rolling dice forever.
- TextbookNielsen & Chuang, §2.2The formal statement of all four postulates this chapter assembles from earlier ones.
- ArticleWikipedia — Born ruleHistory and generalizations of the probability rule in Fig. 7.1.
- ToolNumPy docs — random.choiceReference for the shot-sampling used in Fig. 7.2.
End of Part I
Everything from here on — density matrices, entanglement, error correction, algorithms — is built entirely out of these four postulates and the linear algebra underneath them. There's no new foundational machinery coming; just new things to build with what you already have.