Measurement, the Born rule and POVMs

Start from the observable, not the outcome. A Hermitian A on Cⁿ has real eigenvalues aₖ and orthogonal eigenspaces, so the spectral theorem writes A = Σ aₖ Pₖ with Pₖ the projector onto the k-th eigenspace, Pⱼ Pₖ = δⱼₖ Pₖ and Σ Pₖ = I. The measurement is the set {Pₖ}; the numbers aₖ are labels. For σ_z the projectors are P₀ = (I + σ_z)/2 = |0⟩⟨0| and P₁ = (I − σ_z)/2 = |1⟩⟨1|, and for σₙ = n⋅σ along any unit axis n they are P_± = (I ± n⋅σ)/2, which the spin-measurement topic UPV-10.07 builds and turns into the probabilities (1 ± r⋅n)/2 for a qubit with Bloch vector r. Degeneracy makes a projector rank higher than one: σ_z ⊗ I on two qubits has eigenvalue +1 on the whole plane spanned by |00⟩ and |01⟩, so P₊ = |0⟩⟨0| ⊗ I has rank 2 and asks nothing about the second qubit. Name the operator, the basis it is diagonal in, and the rank of each projector before touching a state; that is where most errors are decided.

The Born rule is pₖ = Tr(ρ Pₖ). For a pure state ρ = |ψ⟩⟨ψ| the trace collapses to ⟨ψ|Pₖ|ψ⟩, which is |⟨k|ψ⟩|² when Pₖ has rank one; for a mixture it weights each member's probability by its ensemble weight, which is exactly what the linearity of the trace does for you. That linearity has a sharp consequence: pₖ depends on ρ and on nothing else about how the state was made. An equal mix of |0⟩ and |1⟩ and an equal mix of |+⟩ and |−⟩ are both I/2, so no measurement, projective or POVM, can ever tell the two recipes apart; the density operator is not shorthand for an ensemble, it is everything any ensemble behind it can show. The maximally mixed state I/d answers each question in proportion to the size of the subspace asked about, pₖ = Tr Pₖ / d = rank(Pₖ)/d: on two qubits, P = |0⟩⟨0| ⊗ I fires with probability 2/4 = 0.500 and P = |00⟩⟨00| with 1/4 = 0.250. The same trace gives a POVM's statistics unchanged, pₖ = Tr(ρ Eₖ), so everything in this section holds for non-projective measurements too.

After outcome k the state is Pₖ ρ Pₖ / pₖ, or Pₖ|ψ⟩/√pₖ for a pure state. Two consequences matter. First, a rank-one projector produces a pure state whatever went in: measure σ_z on any qubit, read 1, and you hold |1⟩ exactly, the earlier amplitudes gone rather than hidden. Second, a higher-rank projector preserves structure inside its eigenspace. Measuring σ_z on qubit A of (|00⟩ + |01⟩ + |10⟩)/√3 and reading 0 leaves (|00⟩ + |01⟩)/√2: the relative amplitudes of the surviving branch are untouched, which is Lüders' refinement of von Neumann's rule and the reason later measurements on B still make sense. If the outcome is taken but not read, the state is the average Σ pₖ (Pₖ ρ Pₖ / pₖ) = Σ Pₖ ρ Pₖ, which keeps the diagonal of ρ in the measured basis and deletes the off-diagonal terms. For |+⟩ under an unread σ_z measurement, purity Tr ρ² falls from 1 to 1/2. No unitary does that: unitaries preserve Tr ρ² and are deterministic, so the update is a different kind of map.

Suppose a source sends |ψ₁⟩ or |ψ₂⟩ with equal probability, their overlap s = |⟨ψ₁|ψ₂⟩| strictly between 0 and 1, and you must never guess wrong, though you may answer 'don't know'. A projective measurement on C² has two outcomes, (|v⟩⟨v|, I − |v⟩⟨v|). An outcome that names ψ₂ must annihilate ψ₁, which fixes |v⟩ = |ψ₁⊥⟩, the state orthogonal to ψ₁; its partner |ψ₁⟩⟨ψ₁| then fires on ψ₂ with probability s², so it can only mean 'don't know'. The best projective scheme therefore names one state and never the other, an average success of (1 − s²)/2. Naming either state needs three outcomes in two dimensions, and only a POVM supplies them. Take E₁ = a|ψ₁⊥⟩⟨ψ₁⊥|, blind to ψ₁ and so a certain sign of ψ₂; E₂ = a|ψ₂⊥⟩⟨ψ₂⊥|, a certain sign of ψ₁; and E_? = I − E₁ − E₂. The sum rule holds for every a, so positivity is the binding constraint. The two rank-one operators overlap by |⟨ψ₁⊥|ψ₂⊥⟩| = s, their sum has eigenvalues 1 ± s, and E_? ≥ 0 only while a ≤ 1/(1 + s). At that value each state is identified with probability a(1 − s²) = 1 − s, the Ivanovic–Dieks–Peres bound, which beats (1 − s²)/2 for every s between 0 and 1. The figure draws E₁ and E₂ as spokes pointing away from the states they must ignore, because orthogonal states sit at opposite ends of the Bloch sphere.

A POVM element Eₖ ≥ 0 factorises as Eₖ = Mₖ† Mₖ, and the natural update is ρ → Mₖ ρ Mₖ† / pₖ. The factorisation is not unique: Mₖ = Uₖ √Eₖ works for any unitary Uₖ, and every such choice gives identical statistics, so a POVM fixes what you will see and not what you will hold afterwards. Naimark's theorem removes any suspicion that POVMs are a second postulate. Attach an ancilla in |0⟩ₐ, apply a unitary U on system plus ancilla, and measure the ancilla projectively; the effective operator on the system is Eₖ = V†(I ⊗ Pₖ)V with V|ψ⟩ = U(|ψ⟩ ⊗ |0⟩ₐ) an isometry. A qubit with three outcomes is the standard example: the trine Eₖ = (2/3)|φₖ⟩⟨φₖ|, with the three Bloch vectors nₖ 120° apart on the equator, sums to I because the three vectors sum to zero: Σ Eₖ = (1/3)(3I + Σ nₖ⋅σ) = I. In NumPy this is three 2 × 2 matrices and one assertion, and the Naimark isometry is the 6 × 2 stack of the √Eₖ, completed to a 6 × 6 unitary with scipy.linalg.qr.

Gleason's theorem tells you the form of the probabilities is forced once you accept that a probability is assigned to each projector and adds over orthogonal sets, in any dimension three or more. It does not tell you why one outcome happens. Unitary evolution maps |+⟩ to a single definite vector; the postulate maps it to |0⟩ half the time and |1⟩ half the time, and no linear operator does that, so the update is added by hand. Decoherence, treated in the last topic of this unit, explains why the off-diagonal terms vanish in a pointer basis when the environment is traced out; it produces the mixture Σ Pₖ ρ Pₖ and stops there, one outcome short. Two things the rule does guarantee: repeated projective measurements agree, and measuring one half of an entangled pair leaves the other half's reduced state exactly as it was, Σₖ Tr_A[(Pₖ ⊗ I) ρ (Pₖ ⊗ I)] = Tr_A ρ = ρ_B, so no choice of measurement can send a signal. The postulate is a statistical contract, precise to every decimal place ever tested, and silent about mechanism by construction.