Amplitude damping channel ($A_0=|0⟩⟨0|+\sqrt{1-p}|1⟩⟨1|,A_1=\sqrt{p}|0⟩⟨1|$)

A zero-temperature energy-relaxation channel transfers the excited population of a qubit to its ground state. Its Kraus operators are $A_0=\mathrm{diag}(1,\sqrt{1-p})$ and $A_1=\sqrt{p}|0⟩⟨1|$ for $0\le p\le 1$. They satisfy ${\sum }_j{A}_j^{†}{A}_j=I$. The excited population is multiplied by $1-p$ and coherences by $\sqrt{1-p}$. For nonzero $p$, it is not a unital quantum channel.

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 $1-p$.

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.

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.