Amplitude-Damping Channel

The amplitude-damping channel is the standard one-qubit channel for irreversible energy relaxation from an excited state $\lvert1\rangle$ to a lower state $\lvert0\rangle$. It is the finite-time channel version of zero-temperature spontaneous emission or qubit $T_1$ decay.

With decay probability $\gamma$, the channel is

$$\mathcal A_\gamma(\rho) = K_0\rho K_0^\dagger + K_1\rho K_1^\dagger, \qquad 0\le\gamma\le1,$$

where

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

The basis is part of the model. The state $\lvert1\rangle$ is the state that can decay, and $\lvert0\rangle$ is the state that receives the population.

What the Channel Means

Amplitude damping is population relaxation with energy flowing into unobserved degrees of freedom. It is not pure dephasing and not isotropic randomization.

For a density matrix

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

the channel gives

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

Thus

$$\rho_{11}\longmapsto(1-\gamma)\rho_{11}, \qquad \rho_{00}\longmapsto\rho_{00}+\gamma\rho_{11},$$

and

$$\rho_{01}\longmapsto\sqrt{1-\gamma}\,\rho_{01}.$$

The population decay and coherence damping are tied together by complete positivity. The same parameter that transfers population from $\lvert1\rangle$ to $\lvert0\rangle$ also reduces the coherence involving $\lvert1\rangle$.

For the broader model-selection context, see Common Noise Channels.

For repeated damping steps, the survival probabilities multiply; see Channel Composition and Fixed Points for the general composition rule and fixed-state language.

Trace Preservation

The channel is trace preserving because

$$K_0^\dagger K_0 = \begin{pmatrix} 1&0\\ 0&1-\gamma \end{pmatrix}, \qquad K_1^\dagger K_1 = \begin{pmatrix} 0&0\\ 0&\gamma \end{pmatrix},$$

so

$$K_0^\dagger K_0+K_1^\dagger K_1=I.$$

It is completely positive because it is written in Kraus form. See Kraus Representation for the general structure.

The branch $K_1$ is trace nonincreasing. It represents the decay event in a quantum-jump interpretation. The branch $K_0$ represents the no-jump operation. Their sum is the deterministic channel obtained when the environment record is not kept.

Stinespring Picture

The channel can be realized by an isometry from the system to system plus environment:

$$\begin{aligned} \lvert0\rangle\lvert0\rangle_E &\longmapsto \lvert0\rangle\lvert0\rangle_E,\\ \lvert1\rangle\lvert0\rangle_E &\longmapsto \sqrt{1-\gamma}\, \lvert1\rangle\lvert0\rangle_E + \sqrt{\gamma}\, \lvert0\rangle\lvert1\rangle_E. \end{aligned}$$

The environment state $\lvert1\rangle_E$ records that a decay quantum was emitted. Tracing out the environment gives the amplitude-damping channel.

For an input superposition

$$\lvert\psi\rangle = \alpha\lvert0\rangle+\beta\lvert1\rangle,$$

the joint output is

$$\alpha\lvert0\rangle\lvert0\rangle_E + \beta\sqrt{1-\gamma}\, \lvert1\rangle\lvert0\rangle_E + \beta\sqrt{\gamma}\, \lvert0\rangle\lvert1\rangle_E.$$

The two environment alternatives are orthogonal, so the reduced system state loses coherence with the emitted-decay record. This is a concrete example of reduced dynamics and of a Stinespring dilation.

Selective Jump and No-Jump Updates

If the environment is monitored, the two Kraus operators define two possible conditional updates.

The jump probability is

$$p_1 = \operatorname{Tr}(K_1\rho K_1^\dagger) = \gamma\rho_{11}.$$

When the jump is observed and $p_1>0$, the normalized state is

$$\rho_1 = \frac{K_1\rho K_1^\dagger}{p_1} = \lvert0\rangle\langle0\rvert.$$

The no-jump probability is

$$p_0 = \operatorname{Tr}(K_0\rho K_0^\dagger) = 1-\gamma\rho_{11}.$$

The corresponding normalized state is

$$\rho_0 = \frac{K_0\rho K_0^\dagger}{p_0}.$$

Even the absence of a detected jump changes the state: it reduces the relative amplitude of $\lvert1\rangle$. This is a measurement-backaction statement, not just a dissipative one. Without access to the record, the observer uses $\mathcal A_\gamma(\rho)$.

Bloch-Vector Form

Write

$$\rho = \frac12 \left( I+xX+yY+zZ \right),$$

with $\lvert0\rangle$ at $z=+1$ and $\lvert1\rangle$ at $z=-1$. Then amplitude damping acts as

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

This is an affine map, not a simple contraction about the origin. The Bloch ball is pulled toward the ground-state pole.

In particular,

$$\mathcal A_\gamma\!\left(\frac{I}{2}\right) = \frac12 \begin{pmatrix} 1+\gamma&0\\ 0&1-\gamma \end{pmatrix}.$$

Therefore $\mathcal A_\gamma$ is not unital for $\gamma>0$. Since every mixture of unitary conjugations is unital, amplitude damping is not a random-unitary channel unless $\gamma=0$.

Choi Matrix

Using the output-input convention of Choi Matrix, and the ordered basis

$$\lvert00\rangle,\, \lvert01\rangle,\, \lvert10\rangle,\, \lvert11\rangle,$$

where the first slot is the output system and the second slot is the input reference, the Choi matrix is

$$J_{\mathcal A_\gamma} = \begin{pmatrix} 1&0&0&\sqrt{1-\gamma}\\ 0&\gamma&0&0\\ 0&0&0&0\\ \sqrt{1-\gamma}&0&0&1-\gamma \end{pmatrix}.$$

Its nonzero eigenvalues are

$$2-\gamma, \qquad \gamma.$$

Thus $J_{\mathcal A_\gamma}\ge0$ for $0\le\gamma\le1$, and the channel has Kraus rank $2$ for $0\lt\gamma\le1$. At $\gamma=0$ it reduces to the identity channel and has Kraus rank $1$.

The trace-preservation condition is

$$\operatorname{Tr}_{\mathrm{out}}J_{\mathcal A_\gamma}=I_{\mathrm{in}}.$$

The unitality condition fails for $\gamma>0$:

$$\operatorname{Tr}_{\mathrm{in}}J_{\mathcal A_\gamma} = \begin{pmatrix} 1+\gamma&0\\ 0&1-\gamma \end{pmatrix} \ne I.$$

Markovian Limit and T1 Decay

For Markovian zero-temperature relaxation with decay rate $\Gamma$, the finite-time channel has

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

The excited-state population obeys

$$\rho_{11}(t) = e^{-\Gamma t}\rho_{11}(0),$$

and the coherence obeys

$$\rho_{01}(t) = e^{-\Gamma t/2}\rho_{01}(0)$$

apart from any Hamiltonian phase rotation or additional pure dephasing.

The corresponding Lindblad equation is

$$\frac{d\rho}{dt} = \Gamma \left( \sigma_-\rho\sigma_+ - \frac12 \{\sigma_+\sigma_-,\rho\} \right),$$

where

$$\sigma_-=\lvert0\rangle\langle1\rvert, \qquad \sigma_+=\lvert1\rangle\langle0\rvert.$$

In this convention $T_1=1/\Gamma$. Additional pure dephasing changes the transverse coherence time but not the population decay. See Lindblad–GKSL Equation and the Formula Sheet for the general master-equation conventions.

For the generator-level treatment, including jump interpretation, finite-temperature rates, and oscillator damping, see Amplitude Damping Master Equation.

Relation to Dephasing and Depolarizing Noise

Amplitude damping differs from dephasing because it changes populations in the energy basis. Pure dephasing leaves $\rho_{00}$ and $\rho_{11}$ fixed.

Dephasing and Amplitude Damping owns the QI-facing conversion from protocol-qualified T1–T2 and equilibrium-population records to one physical, composable idle channel with held-out checks; this page retains the zero-temperature channel definition, Kraus and Choi forms, dilation, fixed point, and formal finite-temperature extension.

It differs from depolarizing noise because it is not isotropic and not unital. Depolarizing noise pulls every state toward $I/2$; amplitude damping pulls states toward $\lvert0\rangle\langle0\rvert$.

It differs from a Pauli channel because Pauli channels are unital mixtures of Pauli conjugations. Amplitude damping has an affine shift in the Bloch ball, so it cannot be written as a Pauli channel for $\gamma>0$.

Finite Temperature and Generalized Amplitude Damping

The channel above is a zero-temperature model: it allows downward transitions but no upward thermal excitation. If the bath can excite the system, the correct qubit model includes both downward and upward processes.

At the master-equation level, populations then satisfy

$$\dot\rho_{11} = -\Gamma_\downarrow\rho_{11} + \Gamma_\uparrow\rho_{00},$$

with steady excited population

$$\rho_{11}^{\mathrm{ss}} = \frac{\Gamma_\uparrow} {\Gamma_\uparrow+\Gamma_\downarrow}.$$

Thermal detailed balance gives

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta\hbar\omega}$$

for a two-level spacing $\hbar\omega$ coupled to a thermal bath. Using zero-temperature amplitude damping when $\Gamma_\uparrow$ is appreciable is a physical error, not just a convention choice.

Common Mistakes

Treating the computational basis as arbitrary

Amplitude damping is tied to an ordered energy basis. Swapping the labels $\lvert0\rangle$ and $\lvert1\rangle$ changes the channel.

Confusing probability and amplitude

The population survival factor is $1-\gamma$, while the coherence factor is $\sqrt{1-\gamma}$. Forgetting the square root gives the wrong $T_1$ contribution to coherence decay.

Calling amplitude damping a Pauli channel

Amplitude damping is nonunital for $\gamma>0$. A Pauli channel is unital. They are not the same model.

Using zero-temperature damping at finite temperature

If the environment can drive upward transitions, use a thermal master equation or generalized amplitude-damping model rather than the two-Kraus zero-temperature channel.

Treating Kraus operators as unique histories

The two-Kraus representation matches a natural jump/no-jump monitoring picture, but Kraus representations are not unique. The physical trajectory interpretation depends on the measurement record and unraveling.

Confusing damping with leakage

Amplitude damping keeps the system inside the two-dimensional Hilbert space. Leakage out of the computational subspace requires a different output space or a larger system model; see Erasure and Loss Channels.

Exercises

Verify trace preservation

Show that the Kraus operators $K_0$ and $K_1$ satisfy the trace-preserving completeness relation.

Solution

Compute

$$K_0^\dagger K_0 = \begin{pmatrix} 1&0\\ 0&1-\gamma \end{pmatrix},$$

and

$$K_1^\dagger K_1 = \begin{pmatrix} 0&0\\ 0&\gamma \end{pmatrix}.$$

Adding gives

$$K_0^\dagger K_0+K_1^\dagger K_1 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} =I.$$

Matrix action

Starting from the Kraus representation, derive the matrix action of $\mathcal A_\gamma$ on

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

Solution

The no-jump term is

$$K_0\rho K_0^\dagger = \begin{pmatrix} \rho_{00}&\sqrt{1-\gamma}\,\rho_{01}\\ \sqrt{1-\gamma}\,\rho_{10}&(1-\gamma)\rho_{11} \end{pmatrix}.$$

The jump term is

$$K_1\rho K_1^\dagger = \begin{pmatrix} \gamma\rho_{11}&0\\ 0&0 \end{pmatrix}.$$

Adding them gives

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

Nonunitality

Compute $\mathcal A_\gamma(I/2)$ and show that the channel is not unital for $\gamma>0$.

Solution

Set $\rho_{00}=\rho_{11}=1/2$ and $\rho_{01}=0$ in the matrix formula:

$$\mathcal A_\gamma(I/2) = \begin{pmatrix} (1+\gamma)/2&0\\ 0&(1-\gamma)/2 \end{pmatrix}.$$

This equals $I/2$ only when $\gamma=0$. Therefore the channel is not unital for nonzero damping.

Coherence decay from T1 relaxation

For Markovian amplitude damping, use $\gamma(t)=1-e^{-t/T_1}$ to show that the coherence factor is $e^{-t/(2T_1)}$.

Solution

The finite-time channel multiplies the coherence by

$$\sqrt{1-\gamma(t)}.$$

With

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

we have

$$\sqrt{1-\gamma(t)} = \sqrt{e^{-t/T_1}} = e^{-t/(2T_1)}.$$

Thus population relaxation alone contributes a transverse coherence decay rate $1/(2T_1)$.

Choi rank

Use the Choi matrix to determine the Kraus rank of the amplitude-damping channel for $0\lt\gamma\le1$.

Solution

The Choi matrix has nonzero eigenvalues

$$2-\gamma, \qquad \gamma.$$

For $0\lt\gamma\le1$, both are positive. Therefore the Choi rank is $2$, which equals the minimal number of Kraus operators. At $\gamma=0$, the eigenvalue $\gamma$ vanishes and the channel reduces to the identity, whose Choi rank is $1$.

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• 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).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).