 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-323/4/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

{∣ j ⟩}_{j = 1}^{d}

of the input space and define operators from the input to the

k

-dimensional outcome register by

K_{a, j} = ∣ a ⟩ ⟨ j ∣ E_{a} .

\sum_{a, j} K_{a, j} X K_{a, j}^{†} = \sum_{a} Tr (E_{a} X) ∣ a ⟩ ⟨ a ∣ = Φ (X) .

The POVM completeness relation gives

\sum_{a, j} K_{a, j}^{†} K_{a, j} = \sum_{a} E_{a} (\sum_{j} ∣ j ⟩ ⟨ j ∣) E_{a} = \sum_{a} E_{a} = I .

Therefore

Φ

is a completely positive map and is trace preserving: it is a quantum channel representing a deterministic measurement channel.

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