Common Channels

This page is a compact lookup table for common quantum channels. It identifies the standard action, typical use, and main caveat for each model. The detailed explanations live in Common Noise Channels, Pauli Channels, Dephasing Channel, Depolarizing Channel, Amplitude-Damping Channel, Erasure and Loss Channels, Gaussian Channels, and Channel Composition and Fixed Points.

Unless stated otherwise, $\Phi$ is a completely positive trace-preserving map and $\rho$ is the input density operator. A Kraus representation

$$\Phi(\rho) = \sum_r K_r\rho K_r^\dagger, \qquad \sum_r K_r^\dagger K_r=I$$

is a representation of the channel, not a unique physical story. To attach meaning to individual Kraus operators, specify the environment basis, detector record, or implementation.

One-Line Selection Guide

If the dominant effect is…	Start with…	Do not forget…
phase coherence decays in one basis	dephasing or phase damping	the basis is part of the model
errors are stochastic Pauli conjugations	Pauli channel	unitality and independence assumptions
all qubit axes are benchmark-randomized equally	depolarizing	this is usually phenomenological
excitation relaxes downward	amplitude damping	the channel is not unital
upward and downward thermal transitions both occur	generalized amplitude damping	detailed balance fixes the stationary state
a loss event is heralded	erasure	the output includes an orthogonal flag
photons or bosonic excitations leak away unheralded	loss or attenuation	number and amplitude statistics change
a harmonic mode damps toward a thermal state	thermal oscillator damping	the bath occupation sets the noise floor
quadrature covariance is attenuated	Gaussian attenuation	vacuum or thermal noise enters
phase-insensitive gain is added	Gaussian amplification	added noise is required by complete positivity

Finite-Dimensional Channels

Dephasing

Dephasing suppresses off-diagonal density-matrix elements in a preferred basis while leaving populations fixed. For a qubit in the $Z$ basis, a common phase-flip form is

$$\Phi_p(\rho) = (1-p)\rho+pZ\rho Z, \qquad 0\le p\le 1.$$

In matrix elements,

$$\rho_{01} \mapsto (1-2p)\rho_{01}, \qquad \rho_{00} \mapsto \rho_{00}, \qquad \rho_{11} \mapsto \rho_{11}.$$

Use for unread which-path information, slow phase noise, nonselective measurement in a basis, or pure dephasing limits. Main caveat: dephasing is always basis-dependent. See Dephasing Channel.

Pauli

A one-qubit Pauli channel applies $I,X,Y,Z$ conjugations with probabilities $p_\mu$:

$$\Phi(\rho) = \sum_{\mu=0}^3p_\mu\sigma_\mu\rho\sigma_\mu, \qquad \sum_{\mu=0}^3p_\mu=1.$$

Use for stochastic Pauli error models, stabilizer simulations, and twirled noise. Main caveat: Pauli channels are unital; they do not model amplitude damping, leakage, or coherent over-rotations without an additional approximation. See Pauli Channels.

Depolarizing

The depolarizing channel shrinks a state toward the maximally mixed state. In the replacement-probability convention for dimension $d$,

$$\mathcal D_q(\rho) = (1-q)\rho + q\frac{I}{d}, \qquad 0\le q\le 1.$$

For a qubit Bloch vector $\mathbf r$,

$$\mathbf r \mapsto (1-q)\mathbf r.$$

Use for symmetric benchmark noise, randomized error models, and compact estimates. Main caveat: real relaxation, leakage, coherent drift, and crosstalk are usually not isotropic. See Depolarizing Channel.

Amplitude Damping

Amplitude damping models zero-temperature relaxation from an excited state to a lower state. For a qubit with $|1\rangle$ excited,

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

Thus

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

Use for spontaneous emission, $T_1$ relaxation, and zero-temperature decay. Main caveat: it is trace preserving but not unital; it drives $I/2$ toward the ground state. See Amplitude-Damping Channel.

Generalized Amplitude Damping

Generalized amplitude damping is the finite-temperature version of amplitude damping. It has both downward and upward transition processes and relaxes to a mixed stationary state. A transparent continuous-time parametrization is

$$\frac{d\rho}{dt} = \Gamma_\downarrow\mathcal D[\sigma_-]\rho + \Gamma_\uparrow\mathcal D[\sigma_+]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

The equilibrium excited population is

$$p_e^\ast = \frac{\Gamma_\uparrow} {\Gamma_\uparrow+\Gamma_\downarrow}.$$

Use when a qubit exchanges energy with a finite-temperature bath. Main caveat: the rates must encode the bath temperature or nonequilibrium reservoir; otherwise the claimed thermal fixed point is only a fit parameter. See Thermal Master Equations and Detailed Balance.

Erasure

An erasure channel replaces the input by an orthogonal flag state with known probability. In its simplest form,

$$\mathcal E_\epsilon(\rho) = (1-\epsilon)\rho \oplus \epsilon\lvert e\rangle\langle e\rvert, \qquad 0\le\epsilon\le1,$$

where $\lvert e\rangle$ lies outside the original input space.

Use for heralded loss, located erasures in error correction, and communication models where the receiver knows that transmission failed. Main caveat: erasure is not the same as unheralded amplitude damping or bosonic attenuation; the flag is part of the output. See Erasure and Loss Channels.

Phase Damping

Phase damping is often used as a synonym for dephasing, especially when the microscopic story is random phase accumulation rather than Pauli phase flips. For a basis $\{\lvert n\rangle\}$, a schematic form is

$$\lvert n\rangle\langle m\rvert \mapsto \lambda_{nm}\lvert n\rangle\langle m\rvert, \qquad \lambda_{nn}=1.$$

Use when populations in the chosen basis are preserved but coherences decay. Main caveat: the coefficients $\lambda_{nm}$ must define a completely positive map; arbitrary damping factors need not be physical.

Loss and Bosonic Channels

Loss

Loss means degrees of freedom leave the retained Hilbert space or mode. In finite-dimensional qubit language, loss might mean leakage to a state outside the computational subspace. In bosonic language, it usually means attenuation into an inaccessible environment.

For a single bosonic mode, the pure-loss channel with transmissivity $\eta$ has the Heisenberg input–output relation

$$a_{\text{out}} = \sqrt{\eta}\,a_{\text{in}} + \sqrt{1-\eta}\,e_{\text{in}}, \qquad 0\le\eta\le1,$$

with the environment mode $e_{\text{in}}$ usually taken to be vacuum for quantum-limited loss.

Use for photon loss, imperfect transmission, detector inefficiency, and cavity leakage. Main caveat: unheralded loss is not an erasure channel unless a reliable flag is produced. See Erasure and Loss Channels.

Thermal Oscillator Damping

Thermal oscillator damping is the finite-temperature damping channel for a harmonic mode. The Markovian master-equation form is

$$\frac{d\rho}{dt} = \kappa(\bar n+1)\mathcal D[a]\rho + \kappa\bar n\,\mathcal D[a^\dagger]\rho,$$

where $\bar n$ is the bath occupation at the oscillator frequency.

Use for damped cavities, vibrational modes, microwave resonators, and harmonic approximations coupled to thermal reservoirs. Main caveat: the model assumes a stable bath occupation and weak enough coupling for a Markovian description. For structured reservoirs or strong coupling, the finite-time map may not have this simple form.

Gaussian Attenuation

Gaussian attenuation is the continuous-variable channel that reduces displacement and adds the noise required by coupling to an environment mode. For quadrature displacement vector $d$ and covariance matrix $V$,

$$d \mapsto \sqrt{\eta}\,d, \qquad V \mapsto \eta V+(1-\eta)V_E, \qquad 0\le\eta\le1.$$

For quantum-limited attenuation, $V_E=I/2$ in the common $\hbar=1$ convention. For thermal attenuation, $V_E$ is the covariance of the thermal environment.

Use for optical fiber attenuation, beam-splitter loss, inefficient detection, and noisy Gaussian transmission. Main caveat: covariance conventions differ; always state the quadrature normalization and whether the environment is vacuum or thermal. See Gaussian Channels for the full covariance-matrix convention.

Gaussian Amplification

Gaussian amplification is phase-insensitive gain for a bosonic mode. For gain $G\ge1$,

$$d \mapsto \sqrt G\,d, \qquad V \mapsto GV+(G-1)V_E'.$$

Quantum mechanics requires added noise. In the quantum-limited phase-insensitive amplifier, $V_E'=I/2$ in the same $\hbar=1$ covariance convention.

Use for idealized linear amplifiers, microwave readout chains, and communication models with gain. Main caveat: noiseless deterministic phase-insensitive amplification is not a physical channel; complete positivity enforces the added-noise term. See Gaussian Channels for the complete-positivity inequality.

Parameter Translations

The same physical model appears with different parameters in different communities:

Model	Common parameter	Continuous-time relation
pure dephasing	coherence factor $\lambda$	$\lambda=e^{-\Gamma_\phi t}$ in a Markovian convention
phase-flip dephasing	flip probability $p$	$1-2p=e^{-\Gamma_\phi t}$ for real positive decay
amplitude damping	decay probability $\gamma$	$\gamma=1-e^{-t/T_1}$
depolarizing	replacement probability $q$	often $q=1-e^{-\Gamma t}$, convention dependent
bosonic attenuation	transmissivity $\eta$	$\eta=e^{-\kappa t}$
Gaussian amplification	gain $G$	$G=e^{gt}$ for an idealized amplifier rate $g$

These translations are only bookkeeping. They do not prove that the model is microscopically correct.

Choosing a Channel Responsibly

Before using a channel in a derivation, simulation, or fit, record:

• the system Hilbert space and whether leakage states are included,
• the basis in which dephasing or phase damping acts,
• whether loss is heralded, unheralded, or postselected away,
• whether the channel is a one-step phenomenological map or the finite-time map of a master equation,
• whether the environment is vacuum, thermal, squeezed, structured, or merely unspecified,
• whether the parameters are probabilities, rates, coherence factors, transmissivities, or gains.

For computational checks, see Simulating Quantum Channels. For assumptions behind master-equation limits, see the Approximation Checklist.

Self-Checks

1. A qubit loses phase coherence in the energy basis, but its measured energy populations do not change. Which channel family is the natural first model?

Solution

Use dephasing or phase damping in the energy basis. The unchanged populations rule out ordinary amplitude damping as the leading effect.

2. A photon is lost in transmission, but the receiver is not told whether loss occurred. Is this automatically an erasure channel?

Solution

No. Erasure requires a known orthogonal flag. Unheralded photon loss is better modeled as bosonic attenuation or as a loss channel appropriate to the retained Hilbert space.

3. A proposed phase-insensitive amplifier maps $V\mapsto GV$ with no added noise for $G>1$. What is wrong?

Solution

It is not a physical deterministic quantum channel. Complete positivity requires added noise; for a quantum-limited amplifier the covariance has an additional term proportional to $G-1$.

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. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• A. S. Holevo, Quantum Systems, Channels, Information, De Gruyter, 2012.
• C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Reviews of Modern Physics 84, 621, 2012.
• A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press, 2017.
• M. M. Wilde, Quantum Information Theory, Cambridge University Press, 2017.