= Solution
For <Kraus operators> $A_j$, the <Kraus formula for entanglement fidelity> is
$$
F_e(\rho,\Lambda)=\sum_j|\operatorname{Tr}(\rho A_j)|^2.
$$
Indeed each Kraus contribution to the purified-state overlap is $|\langle\Psi|(I\otimes A_j)|\Psi\rangle|^2$, and the expectation in a purification is $\operatorname{Tr}(\rho A_j)$. In this case,
$$
\operatorname{Tr}(\rho A_1)=\frac{1+s_z+\sqrt{1-p}(1-s_z)}2,\qquad
\operatorname{Tr}(\rho A_2)=\frac{\sqrt p}{2}(s_x+is_y).
$$
The <entanglement fidelity of amplitude damping> is thus
$$
\boxed{F_e(\rho,\Lambda)=\frac14\left[1+s_z+\sqrt{1-p}(1-s_z)\right]^2
+\frac p4(s_x^2+s_y^2).}
$$
For $p=0$, this is one. For a ground-state input it is one for every $p$, whereas for an excited-state input it is $1-p$. These checks agree with the physical relaxation mechanism.
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