The <Kraus representation> is
$$
\boxed{T(\rho)=\sum_jK_j\rho K_j^\dagger},
\qquad
\boxed{\sum_jK_j^\dagger K_j=I}.
$$
The second identity is precisely trace preservation. Different Kraus families related by an isometry represent the same channel.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 1 / a / ii / Solution

A <quantum channel> is a linear completely positive trace-preserving map. Its <Kraus representation> is
$$
\mathcal E(\rho)=\sum_\alpha K_\alpha\rho K_\alpha^\dagger,
\qquad
\boxed{\sum_\alpha K_\alpha^\dagger K_\alpha=I}.
$$
Kraus operators are not unique: two representations of the same channel are related by an isometry on the Kraus index, and by a unitary when both lists are minimal and equally long. A differentiable Markovian family of infinitesimal channels gives the <Lindblad equation>
$$
\dot\rho=-i[H,\rho]
+\sum_\alpha\left(
L_\alpha\rho L_\alpha^\dagger
-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}
\right).
$$

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 1 / a / Solution

{c}
{wiki}

Every finite-dimensional completely positive map has $T(\rho)=\sum_jK_j\rho K_j^\dagger$. It is trace preserving exactly when $\sum_jK_j^\dagger K_j=I$.


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