 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-33/3/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Φ (I) = I

. In equal input and output dimensions, it therefore preserves the maximally mixed state. For amplitude damping channel,

A_{p} (I) = K_{0} K_{0}^{†} + K_{1} K_{1}^{†} = (1 + p 0 0 1 - p) = I + pZ .

Hence the channel is not unital for $p > 0$ ; it is unital only in the identity case $p = 0$ . Trace preservation involves the opposite products

K_{i}^{†} K_{i}

and does not imply unitality.

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