Amplitude-Damping Channel

The amplitude-damping channel transfers population from an excited state $\lvert1\rangle$ to a lower state $\lvert0\rangle$. It models zero-temperature relaxation over a specified interval. A time-independent decay rate gives the familiar exponential $T_1$ law; the channel definition itself does not require that time dependence.

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.

Helpful background. Kraus operators distinguish the operation for each monitored outcome from the channel obtained by ignoring the outcome.

Population transfer and coherence

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

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

For a map with this population transfer and $\rho_{01}\mapsto c\rho_{01}$, complete positivity requires $|c|^2\le1-\gamma$. The amplitude-damping channel saturates this bound, with real $c=\sqrt{1-\gamma}$. Additional pure dephasing can reduce $|c|$ further: complete positivity does not require the saturation.

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

For ideal monitoring that distinguishes the two displayed environment states, the Kraus operators define jump and no-jump updates. Other environment measurements give different instruments with the same unobserved channel.

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

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

When $p_0>0$, the corresponding normalized state is

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

Even the absence of a jump changes a superposition by reducing the relative amplitude of $\lvert1\rangle$. For $|+\rangle$ and $\gamma=0.36$, the jump probability is $0.18$; otherwise the state is

$$|\psi_0\rangle=\frac{|0\rangle+0.8|1\rangle}{\sqrt{1.64}}.$$

Both conditional states are pure. Ignoring the record gives

$$\rho'=\begin{pmatrix}0.68&0.40\\0.40&0.32\end{pmatrix}, \qquad \operatorname{Tr}(\rho'^2)=0.8848.$$

A detector of efficiency $\eta_{\mathrm d}<1$ can miss emissions. Its no-click operation is then $K_0\rho K_0^\dagger+(1-\eta_{\mathrm d})K_1\rho K_1^\dagger$ in the simplest independent-detection model. No click no longer implies no jump; conditioning on no click generally gives a mixture.

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 a time-homogeneous zero-temperature relaxation model with constant $\Gamma\ge0$, 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

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

For $\Gamma>0$, this convention gives $T_1=1/\Gamma$. 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 explains how measured $T_1$, $T_2$, and equilibrium populations determine a finite-duration idle model.

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.

In a weak-coupling Markov model with constant rates $\Gamma_\uparrow,\Gamma_\downarrow\ge0$ and positive sum, populations 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 positive spacing $\hbar\omega$, a thermal bath in the detailed-balance regime, and $\Gamma_\downarrow>0$.

Writing $\Gamma_\Sigma=\Gamma_\uparrow+\Gamma_\downarrow$, the solution is

$$\rho_{11}(t)=\rho_{11}^{\mathrm{ss}}+ [\rho_{11}(0)-\rho_{11}^{\mathrm{ss}}]e^{-\Gamma_\Sigma t}.$$

Without additional pure dephasing, coherences decay as $e^{-\Gamma_\Sigma t/2}$ in the rotating frame. This gives generalized amplitude damping with transfer parameter $1-e^{-\Gamma_\Sigma t}$ and fixed Bloch coordinate $z_{\mathrm{ss}}=(\Gamma_\downarrow-\Gamma_\uparrow)/\Gamma_\Sigma$.

For example, $\Gamma_\downarrow=3\,\mathrm{s}^{-1}$ and $\Gamma_\uparrow=1\,\mathrm{s}^{-1}$ give $\rho_{11}^{\mathrm{ss}}=1/4$. Starting excited, after $t=(\ln2)/4$ seconds the population is $1/4+(3/4)/2=5/8$. Omitting upward transitions would give the wrong stationary state and decay law.

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

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018). Author’s text.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). DOI.