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

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Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / c / iv / Solution by

Codex  0 2026-10-25

∣Ω ⟩

is an eigenvector of

J_{r}

with eigenvalue

(1 + 3 r) /2

. Its three-dimensional orthogonal complement has eigenvalue

(1 - r) /2

. The Choi criterion for complete positivity therefore gives

- \frac{1}{3} \leq r \leq 1.

For sufficiency one can also write a Kraus representation explicitly:

E_{r} (ρ) = \frac{1 + 3 r}{4} ρ + \frac{1 - r}{4} (XρX + Y ρ Y + ZρZ) .

In this range the four coefficients are nonnegative and sum to one, giving a physical mixture of Pauli conjugations. This is the complete positivity range of the depolarizing channel in the Bloch-scaling parameterization.

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