Unitary dilation of amplitude damping ($U|10⟩=\sqrt{1-p}|10⟩+\sqrt{p}|01⟩$)

With the environment initially in $|0⟩$, take $U$ to fix $|00⟩$ and $|11⟩$, and rotate the span of $|10⟩,|01⟩$ by the displayed action and $U|01⟩=\sqrt{1-p}|01⟩-\sqrt{p}|10⟩$. This is a unitary operator. Its environment matrix elements $⟨0{|}_{E}U|0{⟩}_E$ and $⟨1{|}_{E}U|0{⟩}_E$ 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 $\sqrt{1-p}$.