Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 1 / b

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- Solution b

Solution

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

ρ^{2} = ρ

or

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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