Exact amplitude-damping distance

Let $a=\sqrt{1-\gamma }$, $q=1-a$, and $\Delta ={\mathcal{A}}_{\gamma }-\mathrm{id}$. In input-reference/output order $00,01,10,11$,

For $\gamma >0$, set

The matrix $Z+K_{\Delta }$ is diagonal and nonnegative. The nonzero off-diagonal block of $Z-K_{\Delta }$ is

which is positive semidefinite because its determinant is zero and its diagonal is nonnegative. Thus $Z⪰\pm K_{\Delta }$.

Since $q=\gamma /(1+\sqrt{1-\gamma })\le \gamma$,

Lemma lem:dual gives error at most $\gamma$. Input $|1⟩⟨1|$ has output distance exactly $\gamma$ from its ideal output. Therefore

At $\gamma =0$, the channel is identity.