Amplitude-Damping Channel

One-Sentence Description

The amplitude-damping channel is the standard one-qubit channel for irreversible relaxation from $\lvert 1\rangle$ to $\lvert 0\rangle$ with probability $\gamma$.

For the fuller derivation, physical interpretation, Choi matrix, and master-equation limit, see Amplitude-Damping Channel.

Physical Setup

The model describes zero-temperature energy relaxation in a chosen computational basis, often used as a simplified model of a qubit’s $T_1$ process. It is basis dependent: the states $\lvert 0\rangle$ and $\lvert 1\rangle$ must be identified with lower and upper energy states.

For Markovian exponential relaxation, one often writes

$$\gamma(t) = 1-e^{-t/T_1}.$$

Channel Definition

A standard Kraus representation is

$$K_0 = \begin{pmatrix} 1&0\\ 0&\sqrt{1-\gamma} \end{pmatrix}, \qquad K_1 = \begin{pmatrix} 0&\sqrt{\gamma}\\ 0&0 \end{pmatrix}, \qquad 0\le\gamma\le 1.$$

The channel is

$$\mathcal E_\gamma(\rho) = K_0\rho K_0^\dagger + K_1\rho K_1^\dagger.$$

For

$$\rho = \begin{pmatrix} \rho_{00}&\rho_{01}\\ \rho_{10}&\rho_{11} \end{pmatrix},$$

the output is

$$\mathcal E_\gamma(\rho) = \begin{pmatrix} \rho_{00}+\gamma\rho_{11} & \sqrt{1-\gamma}\,\rho_{01}\\ \sqrt{1-\gamma}\,\rho_{10} & (1-\gamma)\rho_{11} \end{pmatrix}.$$

Bloch-Vector Form

Using the convention

$$\rho = \frac12 \left( I+x\sigma_x+y\sigma_y+z\sigma_z \right),$$

where $\lvert 0\rangle$ has $z=+1$, the channel maps

$$(x,y,z) \longmapsto \left( \sqrt{1-\gamma}\,x, \sqrt{1-\gamma}\,y, (1-\gamma)z+\gamma \right).$$

The affine shift in $z$ shows that the channel is not unital.

What It Teaches

Amplitude damping separates energy relaxation from pure dephasing and isotropic Pauli noise. It is the simplest channel where coherence decay and population flow are tied together by complete positivity.

Canonical Links

• Kraus Map
• Lindblad Equation
• Bloch Vector
• Trace Distance
• Open Systems Crosswalk

Common Mistakes

• Treating amplitude damping as a Pauli channel.
• Forgetting that it depends on the energy basis.
• Confusing the probability $\gamma$ with the amplitude factor $\sqrt{1-\gamma}$.
• Using the zero-temperature channel when thermal excitation is relevant; that requires generalized amplitude damping.

Quick Check

Why is the amplitude-damping channel nonunital?

Solution

Applying the channel to $I/2$ gives a state with diagonal entries $(1+\gamma)/2$ and $(1-\gamma)/2$. This is not $I/2$ unless $\gamma=0$, so the channel does not preserve the maximally mixed state.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• J. Preskill, Lecture Notes on Quantum Computation, California Institute of Technology.