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Applying the amplitude damping channel to the Bloch vector form of a qubit density operator gives an affine map. Transverse components contract by $\sqrt{1-p}$; the longitudinal component contracts by $1-p$ and shifts toward the ground-state north pole. Reading the entries of the output density operator proves the map. The displacement distinguishes this channel from centered depolarization.

The Kraus formula for entanglement fidelity applies to any input, including mixed qubit states. For the amplitude damping channel, the diagonal Kraus trace is $[1+s_z+\sqrt{1-p}(1-s_z)]/2$ and the jump Kraus trace has squared modulus $p(s_x^2+s_y^2)/4$. Summing them gives the formula. Ground-state inputs have unit entanglement fidelity; excited-state inputs have fidelity $1-p$.

In the computational basis, writeThe Kraus representation of the amplitude damping channel givesReading off its Bloch vector gives the Bloch-vector map of amplitude damping:Here $0\le p\le 1$. The shift in $s_z$ shows that this quantum channel is not unital unless $p=0$; its fixed ground state lies at the north pole, rather than the center of the Bloch sphere.

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.