= Trace-preserving and unital Kraus conditions
{title2=$\sum_kA_k^\dagger A_k=I\quad\text{versus}\quad\sum_kA_kA_k^\dagger=I$}
For a <Kraus representation> $\Phi(X)=\sum_kA_kXA_k^\dagger$, trace preservation is equivalent to the first identity, by cyclicity of the trace. The second says $\Phi(I)=I$, namely unitality, and is a distinct constraint. For $0<\gamma<1$, the <amplitude damping channel> has operators $A_0=\operatorname{diag}(1,\sqrt{1-\gamma})$ and $A_1=\sqrt\gamma|0\rangle\langle1|$ and is trace-preserving but not unital. Taking the adjoints as Kraus operators instead gives a unital completely positive map whose output trace on $|0\rangle\langle0|$ is $1+\gamma$.
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