 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-60/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

A density operator represents a pure state exactly when it has rank one, equivalently

ρ^{2} = ρ

Tr ρ^{2} = 1

. The spectra just found give

ρ_{1} = ∣ ψ_{1} ⟩ ⟨ ψ_{1} ∣, ∣ ψ_{1} ⟩ = \frac{∣0 ⟩ - ∣1 ⟩}{2}; ρ_{3} = ∣ ψ_{3} ⟩ ⟨ ψ_{3} ∣, ∣ ψ_{3} ⟩ = ∣1 ⟩ .

Their wavefunctions are defined up to a global phase. The other two are mixed states, with purity

(3/4)^{2} + (1/4)^{2} = 5/8

, and have no single-wavefunction representation as rank-one projectors.

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