Past exam of the computer science course of the University of Oxford / 2023 / Quantum information / 1 / a

d \times d

ρ = ρ^{†}, v^{†} ρ v \geq 0 for every v \in C^{d}, Tr ρ = 1.

Equivalently, its eigenvalues are nonnegative and sum to one, with an orthonormal basis of eigenvectors.^[ref]

ii

{∣ b_{j} ⟩}_{j = 0}^{d - 1}

ρ

gives the outcome probabilities

p (j) = ⟨ b_{j} ∣ ρ ∣ b_{j} ⟩ = Tr (ρ ∣ b_{j} ⟩ ⟨ b_{j} ∣) .

These probabilities are nonnegative by positive semidefiniteness and sum to one by the resolution of the identity and unit trace. In the computational basis, the answers are simply the diagonal entries

ρ_{jj}

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