 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/2/b/solution

Each term is a Hermitian rank-one projection and

p_{j}

is real, so

ρ^{†} = ρ

. Using the rank-one trace,

Tr ρ = \sum_{j} p_{j} ⟨ ψ_{j} ∣ ψ_{j} ⟩ = \sum_{j} p_{j} = 1 .

The remaining admissibility condition follows from

⟨ v ∣ ρ ∣ v ⟩ = \sum_{j} p_{j} ∣ ⟨ ψ_{j} ∣ v ⟩ ∣^{2} \geq 0

: $ρ$ is positive semidefinite.

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