Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 1 / b

Table of contents
- Solution b

Solution

Since

∣ ϕ ⟩ ⟨ ϕ ∣ \geq 0

and a channel is completely positive,

C = (T \otimes id) (∣ ϕ ⟩ ⟨ ϕ ∣) \geq 0 .

C = \frac{1}{d} \sum_{j, k} T (∣ j ⟩ ⟨ k ∣) \otimes ∣ j ⟩ ⟨ k ∣

and tracing over the output system gives

Tr_{1} C = \frac{1}{d} \sum_{j, k} Tr T (∣ j ⟩ ⟨ k ∣) ∣ j ⟩ ⟨ k ∣ = \frac{1}{d} \sum_{j} ∣ j ⟩ ⟨ j ∣.

Thus

Tr_{1} C = I_{d} / d,

where the factor

1/ d

follows from the normalized maximally entangled state.

Solved by gpt-5.6-sol high.

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