 Solution

ID: past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2021/3/c/iii/solution

Use the unnormalized Choi matrix convention

J_{r} = \sum_{j, k = 0}^{1} ∣ j ⟩ ⟨ k ∣ \otimes E_{r} (∣ j ⟩ ⟨ k ∣)

, which has trace two. In the ordered basis

∣00 ⟩, ∣01 ⟩, ∣10 ⟩, ∣11 ⟩

J_{r} = \frac{1}{2} 1 + r 00 2 r 0 1 - r 00 00 1 - r 0 2 r 00 1 + r .

Equivalently

J_{r} = 2 r ∣Ω ⟩ ⟨ Ω∣ + (1 - r) I_{4} /2

. The convention obtained by applying the channel to one half of the normalized Bell state gives

J_{r} /2

; its positivity criterion is the same.

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