= Unitary dilation of amplitude damping
{title2=$U|10\rangle=\sqrt{1-p}|10\rangle+\sqrt p|01\rangle$}
With the environment initially in $|0\rangle$, take $U$ to fix $|00\rangle$ and $|11\rangle$, and rotate the span of $|10\rangle,|01\rangle$ by the displayed action and $U|01\rangle=\sqrt{1-p}|01\rangle-\sqrt p|10\rangle$. This is a <unitary operator>. Its environment matrix elements $\langle0|_EU|0\rangle_E$ and $\langle1|_EU|0\rangle_E$ are the <Kraus operators> of the <amplitude damping channel>. Tracing out the environment multiplies the excited population by $1-p$, transfers the remainder to the ground state, and multiplies off-diagonal entries by $\sqrt{1-p}$.
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