8. The Postulates

8.1Physical states and state vectors

The first postulate: the state of an isolated physical system is completely described by a state vector — a vector in a complex vector space, normalized to length 1. The word completely is doing heavy lifting: the vector holds everything that can be known about the system. For a qubit the space is two-dimensional and a general state is written α|0⟩ + β|1⟩ with complex α, β. Two engineering habits follow. First, normalization is a constraint your code must enforce or check: np.linalg.norm(psi) must equal 1 (up to floating-point slack). Second, the state vector is not a property you can read out; it is a bookkeeping device from which measurement statistics are derived. Chapters 9–11 unpack exactly that machinery.

8.2Observables and measurement

The second postulate: every measurable quantity — energy, spin, parity — corresponds to an observable, a Hermitian operator A. Hermitian means A = A† (equal to its conjugate transpose), which guarantees real eigenvalues: you never measure a complex number. The possible measurement outcomes are exactly the eigenvalues of A. If the system is in state |ψ⟩, the probability of obtaining eigenvalue aᵢ is |⟨aᵢ|ψ⟩|², where |aᵢ⟩ is the matching eigenvector. For qubits the observables that matter are the Pauli operators X, Y, Z — every hardware readout in this book reduces to measuring one of them. Keep the sequence in mind: choose an observable, project the state onto its eigenbasis, square the amplitudes. That is the whole measurement model.

8.3Unitary evolution and composite systems

The third postulate: while a system evolves unmeasured, its state changes by a unitary operator U, giving |ψ⟩ → U|ψ⟩. Unitary means U†U = I: the operation preserves vector norms, hence total probability, and is invertible — quantum evolution never loses information on its own. This is why quantum gates must be reversible and why you cannot "just copy" a state (no-cloning, 13.1). The fourth postulate covers composite systems: the state space of two systems is the tensor product of the individual spaces. A two-qubit state lives in a 4-dimensional space, n qubits in 2ⁿ dimensions. Critically, the combined space also contains vectors that are not tensor products — entangled states — which is the single most consequential fact in this book. Parts V builds directly on it.

Postulates as a pipeline:

  state  |ψ⟩ ──(unitary U)──▶ U|ψ⟩ ──(measure A)──▶ eigenvalue aᵢ  with probability |⟨aᵢ|ψ⟩|²
                                                     │
                                                     ▼
                                          state collapses to |aᵢ⟩

  composite: state space = space₁ ⊗ space₂  (dimensions multiply, not add)

8.4The Born rule and expectation values

The Born rule is the bridge between amplitudes and data: measuring observable A on state |ψ⟩ yields outcome aᵢ with probability pᵢ = |⟨aᵢ|ψ⟩|². The expectation value ⟨A⟩ = ⟨ψ|A|ψ⟩ = Σᵢ pᵢ·aᵢ is the average over infinitely many repetitions — it is what a finite-shot experiment estimates. Two facts engineers use daily. First, ⟨A⟩² ≤ ⟨A²⟩, with equality only for eigenstates; the gap tells you how noisy your readout is. Second, expectation values are linear in the state but not in the probabilities, which is why interference can move averages in ways classical randomness cannot. Every histogram in this book is an empirical Born rule: sample outcomes, count frequencies, compare against computed probabilities.

8.5Pure states, mixed states, and density matrices

A pure state is one state vector; a mixed state is a classical probabilistic mixture of pure states — "the device prepared |0⟩ half the time and |1⟩ half the time" — and no single vector describes it. The density matrix ρ unifies both: for a pure state ρ = |ψ⟩⟨ψ| (an outer product, a 2×2 matrix for one qubit); for a mixture ρ = Σⱼ pⱼ|ψⱼ⟩⟨ψⱼ|. Measurement statistics are then ⟨A⟩ = Tr(ρA), always. A quick diagnostic you will use constantly: a state is pure exactly when ρ² = ρ, i.e. Tr(ρ²) = 1; mixed states have Tr(ρ²) < 1. Real hardware prepares pure states imperfectly, so after Part XI every "state" in this book quietly becomes a density matrix. Learn ρ now and noise stops being mysterious later.

8.6Quantum channels

A quantum channel is the most general physically allowed evolution of a density matrix: a linear, completely positive, trace-preserving map ρ → E(ρ). Unitary evolution is the special case E(ρ) = UρU†; everything else is noise. The workhorse is the depolarizing channel: with probability p replace the state by the maximally mixed matrix I/2, otherwise apply the intended unitary. Its effect on one-qubit gate fidelity is exactly p = 1 − F. Channels compose like functions: apply a depolarizing channel after every gate and you have a crude but honest circuit-noise model, the kind Part X error correction assumes. Writing your own channel in numpy is ten lines and permanently demystifies phrases like "amplitude damping" and "readout error" in hardware papers.