Common Noise Channels

Noise channels are effective maps for how a quantum state changes when uncontrolled degrees of freedom are ignored, randomized, lost, or only partially observed. They are not all the same physical mechanism.

The most useful first classification is:

dephasing       loss of phase coherence without energy exchange
damping         population relaxation, usually with energy exchange
depolarizing    symmetric benchmark randomization
Pauli noise     stochastic Pauli conjugations
erasure/loss    known loss event or attenuation into inaccessible modes
Gaussian noise  continuous-variable noise preserving Gaussian states

This page is a map of standard models. It emphasizes what each model means, which assumptions are built in, and which mistakes to avoid.

Common Noise Models is the complementary device-facing reference. It owns simulator parameter conversions, matched $T_1$–$T_2$ composition, leakage and seepage population models, coherent overrotation cards, and correlation checks. This open-systems page remains the canonical home for the channel constructions and physical distinctions.

Channel Language

A deterministic noise model is a completely positive, trace-preserving map

$$\Phi(\rho) = \sum_\alpha K_\alpha\rho K_\alpha^\dagger, \qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$

The Kraus representation is one way to write the channel. The Stinespring representation writes it as a reversible system-environment interaction followed by discarding the environment. The physical noise model is the map $\Phi$ together with an interpretation of the system, basis, timescale, and environment.

A selected measurement branch is instead trace nonincreasing before normalization. That belongs to an instrument, not to a deterministic channel by itself.

For reproducible finite-dimensional checks of these maps, use Simulating Quantum Channels.

For serial noise models, repeated channel powers, and invariant states, see Channel Composition and Fixed Points.

Quick Comparison

Channel	Basic action	Physical reading	Main caution
dephasing	suppresses off-diagonal terms in a preferred basis	phase randomization, unread which-path information, low-frequency noise	basis dependent
Pauli	applies stochastic $I,X,Y,Z$ conjugations	error models, stabilizer simulations, twirled noise	unital and often approximate
depolarizing	shrinks the state toward $I/d$	symmetric benchmark noise	rarely microscopic
amplitude damping	transfers excited population to lower states	zero-temperature relaxation, spontaneous emission, $T_1$ process	not a Pauli channel
thermal damping	relaxes toward a Gibbs state	finite-temperature bath	needs bath temperature and detailed balance
erasure	replaces the state by a flagged loss state	known loss event	output Hilbert space changes
bosonic loss	attenuates field amplitude	photon loss, imperfect transmission	not the same as qubit erasure
Gaussian additive noise	broadens quadrature uncertainty	continuous-variable classical noise or thermal background	convention dependent

No table can determine the correct model by itself. One must know which degrees of freedom are ignored, which basis is physically selected, and whether the noise is Markovian, correlated, or conditioned on a record.

Dephasing Channel

Dephasing suppresses coherence in a preferred basis while leaving populations fixed. For a qubit in the $Z$ basis, a common phase-flip form is

$$\Phi_p(\rho) = (1-p)\rho+pZ\rho Z, \qquad 0\le p\le1.$$

For

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

the output is

$$\Phi_p(\rho) = \begin{pmatrix} \rho_{00}&(1-2p)\rho_{01}\\ (1-2p)\rho_{10}&\rho_{11} \end{pmatrix}.$$

Often one writes the coherence factor as $\lambda$:

$$\rho_{01}\longmapsto \lambda\rho_{01}, \qquad |\lambda|\le1.$$

For Markovian pure dephasing, $\lambda=e^{-\Gamma_\phi t}$ in an appropriate convention. See Dephasing Channel for the canonical channel formulas.

Physical sources include fluctuating energy splittings, which-path information leaking into an environment, nonselective measurement in the dephasing basis, and random phases accumulated over an ensemble.

The main warning: dephasing is always dephasing in some basis. If the Hamiltonian basis changes or the noise couples through another operator, the channel may not be diagonal in the computational basis.

Depolarizing Channel

The depolarizing channel is an isotropic benchmark model. In the replacement-probability convention for a $d$-dimensional system,

$$\mathcal D_q(\rho) = (1-q)\rho+q\frac{I}{d}, \qquad 0\le q\le1.$$

For a qubit Bloch vector,

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right),$$

this gives

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

The channel treats all Bloch-sphere directions equally. This makes it convenient for benchmarking, randomized models, and simple estimates.

It is not a generic model of laboratory noise. Energy relaxation, leakage, coherent calibration errors, and correlated crosstalk are not automatically depolarizing. Also check conventions: the Pauli-error probability convention differs from the replacement-probability convention. See Depolarizing Channel for the canonical discussion and the reference model card for a compact formula.

Amplitude Damping

Amplitude damping models irreversible relaxation from an excited state to a lower state. For a qubit with $|1\rangle$ excited and $|0\rangle$ ground, a standard zero-temperature Kraus representation is

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

The channel is

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

It maps

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

Amplitude damping is trace preserving but not unital. It moves the maximally mixed state toward the ground state. For exponential relaxation one often writes

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

Use this model when zero-temperature relaxation is the intended approximation. If upward transitions are relevant, use a thermal or generalized amplitude-damping model instead. See Amplitude-Damping Channel for the canonical discussion and the reference model card for a compact formula.

Thermal Damping

Thermal damping describes relaxation toward a finite-temperature equilibrium state

$$\rho_\beta = \frac{e^{-\beta H}}{Z}, \qquad Z=\operatorname{Tr}e^{-\beta H}.$$

For a qubit, this means both downward and upward transitions occur, with rates constrained by detailed balance:

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

In a Markovian approximation, thermal damping is usually described by a thermal master equation rather than a single memorized finite-time formula. The finite-time channel depends on the Hamiltonian, temperature, coupling operator, and approximations used.

The warning is simple: do not use zero-temperature amplitude damping when the bath can thermally excite the system.

Erasure and Flagged Loss

The erasure channel represents a known loss event. The output space contains the original Hilbert space plus an orthogonal flag state $|e\rangle$:

$$\mathcal H_{\mathrm{out}} = \mathcal H_{\mathrm{in}}\oplus\mathbb C|e\rangle.$$

For erasure probability $p$,

$$\mathcal E_p(\rho) = (1-p)\rho + p\,\operatorname{Tr}(\rho)|e\rangle\langle e|.$$

The receiver knows whether erasure occurred because $|e\rangle$ is orthogonal to all valid input states.

This is different from unknown depolarizing noise. In erasure noise, the damage location is flagged; the quantum information may be gone, but the event is identifiable.

See Erasure and Loss Channels for the full distinction between flagged erasure, postselected survival, leakage, detector inefficiency, and bosonic attenuation.

Bosonic Loss

For an optical or oscillator mode, loss is often modeled by coupling the mode to an environment mode through a beam splitter of transmissivity $\eta$. In the Heisenberg picture,

$$a_{\mathrm{out}} = \sqrt{\eta}\,a_{\mathrm{in}} + \sqrt{1-\eta}\,e_{\mathrm{in}},$$

where $e_{\mathrm{in}}$ is an environment annihilation operator.

If the environment is vacuum, this is the pure-loss channel. If the environment is thermal, the channel adds thermal noise as well as attenuation.

Bosonic loss is not the same as a qubit erasure channel. The photon number changes probabilistically, and the loss event is not necessarily flagged.

The channel-level loss conventions are developed in Erasure and Loss Channels.

Gaussian Channels

Continuous-variable systems are often described by quadratures

$$R=(q_1,p_1,\ldots,q_n,p_n).$$

A Gaussian channel maps Gaussian states to Gaussian states. At the level of first moments $d$ and covariance matrices $V$, it has the form

$$d\longmapsto Xd+d_0, \qquad V\longmapsto XVX^{\mathsf T}+Y.$$

The matrices $X$ and $Y$ must satisfy a complete-positivity inequality determined by the symplectic form and the convention for $\hbar$. Physically important examples include pure loss, amplification, additive classical noise, and thermal noise.

Gaussian channels are powerful because many optical and oscillator noise models stay inside the Gaussian sector. They are also limited: photon counting, strong nonlinearities, and non-Gaussian state preparation require more than covariance-matrix dynamics.

See Gaussian Channels for the complete-positivity condition, attenuation and amplification formulas, and convention warnings.

Pauli Channels

For a qubit, a Pauli channel has the form

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

where $\sigma_0=I$ and $\sigma_1,\sigma_2,\sigma_3$ are $X,Y,Z$.

Pauli channels are unital and diagonal in the Pauli operator basis. They are useful in quantum error correction and randomized noise models, but coherent errors, amplitude damping, leakage, and many correlated errors are not Pauli channels without additional twirling or approximation.

See Pauli Channels for the Bloch-vector eigenvalues, complete-positivity tetrahedron, and multiqubit Pauli-string conventions.

Coherent Errors and Stochastic Errors

A coherent control error may be a unitary channel:

$$\rho\longmapsto U_{\mathrm{err}}\rho U_{\mathrm{err}}^\dagger.$$

This is a valid channel, but it is not stochastic noise. Repeated coherent over-rotations can add systematically rather than diffusively.

Randomizing, averaging, or twirling coherent errors can produce an effective stochastic channel, but that is an additional physical or protocol assumption. Do not replace coherent error by depolarizing noise unless the approximation has been justified.

How to Choose a Model

Noise, Channels, and Error Mitigation owns the operational choice from mechanism and context through diagnostic evidence, intervention, and cost; this page retains the channel constructions, parameter conventions, characteristic actions, and general mathematical comparisons.

Before choosing a noise channel, ask:

• What is the system Hilbert space, and can population leave it?
• Which basis is selected by the Hamiltonian, apparatus, or coupling operator?
• Is energy exchanged with the environment, or only phase information?
• Is the bath effectively zero temperature or finite temperature?
• Is the loss event flagged, unflagged, or only partially observed?
• Are errors independent, correlated in time, or correlated across subsystems?
• Is the channel meant as a microscopic model, an effective fit, or a benchmark?
• Are coherent calibration errors being averaged into a stochastic model?

These questions are often more important than the formula. The same algebraic channel can represent different experiments, and similar experiments can require different channels.

Common Mistakes

• Calling every decay of off-diagonal elements “depolarizing.”
• Using amplitude damping for finite-temperature relaxation without upward transitions.
• Treating depolarizing noise as realistic merely because it is mathematically simple.
• Confusing qubit erasure with bosonic photon loss.
• Ignoring leakage out of the computational subspace.
• Assuming independent single-qubit channels when the environment creates correlated noise.
• Reading a Kraus representation as a unique set of physical histories.
• Fitting a channel outside the domain where the Markovian approximation is valid.

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).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).
• A. S. Holevo, Quantum Systems, Channels, Information, De Gruyter (2012).
• M. M. Wilde, Quantum Information Theory, Cambridge University Press, 2nd ed. (2017).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).

Exercises

1. Dephasing matrix form. Starting from $\Phi_p(\rho)=(1-p)\rho+pZ\rho Z$, derive the matrix form of the qubit dephasing channel in the $Z$ basis.

Solution

Since

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

we get

$$\Phi_p(\rho) = \begin{pmatrix} \rho_{00}&(1-2p)\rho_{01}\\ (1-2p)\rho_{10}&\rho_{11} \end{pmatrix}.$$

The populations are unchanged, and the coherences are multiplied by $1-2p$.

2. Amplitude damping is not unital. Use the amplitude-damping Kraus operators to compute $\mathcal A_\gamma(I/2)$.

Solution

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

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

This equals $I/2$ only when $\gamma=0$. Thus amplitude damping is not unital.

3. Kraus operators for erasure. Let $\{|j\rangle\}_{j=1}^d$ be an input basis and let $V$ embed the input Hilbert space into the non-erased output subspace. Show that

$$K_0=\sqrt{1-p}\,V, \qquad K_j=\sqrt p\,|e\rangle\langle j|$$

defines a trace-preserving erasure channel.

Solution

Compute the completeness relation:

$$K_0^\dagger K_0 = (1-p)V^\dagger V =(1-p)I,$$

and

$$\sum_{j=1}^dK_j^\dagger K_j = p\sum_{j=1}^d|j\rangle\langle j| = pI.$$

Therefore

$$K_0^\dagger K_0 + \sum_{j=1}^dK_j^\dagger K_j =I.$$

The channel is trace preserving. Its output is the original state in the embedded subspace with probability $1-p$ and the flagged erasure state with probability $p$.