 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-65/2/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

ρ = \frac{1}{2} (1 + s_{z} s_{x} + i s_{y} s_{x} - i s_{y} 1 - s_{z}) .

Λ (ρ) = (ρ_{00} + p ρ_{11} 1 - p ρ_{10} 1 - p ρ_{01} (1 - p) ρ_{11}) .

(s_{x}, s_{y}, s_{z}) ⟼ (1 - p s_{x}, 1 - p s_{y}, p + (1 - p) s_{z}) .

Here

0 \leq p \leq 1

. The shift in

s_{z}

shows that this quantum channel is not unital unless

p = 0

; its fixed ground state lies at the north pole, rather than the center of the Bloch sphere.

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