Repeated amplitude damping limit (${\mathbf{s}}^{(n)}=((1-p{)}^{n/2}{s}_x,(1-p{)}^{n/2}{s}_y,1-(1-p{)}^n(1-s_z))$)

Under repeated identical amplitude damping channels, all excited population eventually relaxes to the ground state for any fixed $0<p\le 1$. The Bloch vector approaches $(0,0,1)$. For $p=1$ the limit is reached in one action, while $p=0$ is the identity channel and preserves every input. Keeping the zero-damping endpoint separate prevents an incorrect universal relaxation claim.