Common Noise Channels

A noise channel predicts how a state changes when uncontrolled degrees of freedom are ignored. Dephasing changes relative phases, amplitude damping transfers population, and erasure produces an identifiable loss flag. Choosing among them requires specifying the system, its basis, and what is observed. This page compares the models; the linked articles derive each channel in detail.

Helpful background. Kraus operators express a channel as a sum of matrix operations. Density operators describe both pure and mixed input states.

For device parameters, combined $T_1$–$T_2$ models, and leakage rates, see Common Noise Models.

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.

Comparing channel actions

Channel	Action	Physical interpretation and limits
dephasing	preserves populations in a chosen basis	No energy change only when that basis is compatible with the Hamiltonian.
Pauli	randomly conjugates by Pauli operators	Unital; multiqubit errors may still be correlated.
depolarizing	contracts all directions equally	A symmetric benchmark, seldom a complete microscopic model.
amplitude damping	transfers excited population downward	Approximates relaxation with negligible upward transitions.
thermal damping	includes upward and downward transitions	Gibbs equilibration requires appropriate bath and weak-coupling assumptions.
erasure	replaces the input by an orthogonal flag	The output space is larger, and the loss event can be identified.
bosonic loss	attenuates a mode through environmental coupling	Photon loss is not automatically flagged.
Gaussian additive noise	adds random quadrature displacements	Distinct from attenuation; covariance conventions matter.

One input is not enough to identify a channel. For an input $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$, phase flips with $p=0.1$ and depolarization with $q=0.2$ both give

$$\rho'=\begin{pmatrix}0.5&0.4\\0.4&0.5\end{pmatrix}.$$

Try $|0\rangle$ instead: phase flips leave it unchanged, while depolarization gives $\operatorname{diag}(0.9,0.1)$. Both channels preserve $I/2$, whereas amplitude damping with $\gamma=0.2$ sends it to $\operatorname{diag}(0.6,0.4)$. Testing several inputs distinguishes mechanisms that a single decay curve cannot.

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 a time-homogeneous pure-dephasing semigroup, in a frame removing the Hamiltonian phase, $\lambda=e^{-\Gamma_\phi t}$ defines the coherence-decay rate $\Gamma_\phi$. Complete dephasing occurs at $p=1/2$; $p=1$ is the reversible unitary $Z$, not maximal decoherence. See Dephasing Channel for complex coherence factors and dynamical assumptions.

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.

Dephasing is always defined relative to a basis. It preserves mean energy when its projectors commute with the system Hamiltonian; otherwise it can change energy. 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 displayed state formula assumes $\operatorname{Tr}\rho=1$; the linear map on arbitrary operators uses $q\operatorname{Tr}(\rho)I/d$. 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 derivation and the reference model card for a compact formula.

Amplitude Damping

Amplitude damping describes transfer from an excited state to a lower state, with $0\le\gamma\le1$ the transfer probability. 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

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

For $\gamma>0$, 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 derivation and the reference model card for a compact formula.

Thermal Damping

For a system weakly coupled to one stationary thermal bath, an appropriate detailed-balance master equation can describe relaxation toward the Gibbs state

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

For a qubit with level spacing $\hbar\omega>0$, both downward and upward transitions occur at finite temperature. For $\Gamma_\downarrow>0$, detailed balance gives:

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

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.

Zero-temperature amplitude damping omits these upward transitions. Strong coupling, driving, or multiple baths need not produce the Gibbs state of the bare system Hamiltonian.

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 $0\le p\le1$, with $\rho$ embedded in the non-erased output subspace,

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

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,

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

Here $0\le\eta\le1$, and $e_{\mathrm{in}}$ is an independent environment-mode annihilation operator, initially uncorrelated with the input. Its contribution is essential: it preserves $[a_{\mathrm{out}},a_{\mathrm{out}}^\dagger]=1$. Merely replacing $a$ by $\sqrt\eta a$ would not.

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

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

Here $p_\mu\ge0$, $\sigma_0=I$, 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. Nonunital relaxation and general coherent rotations are not Pauli channels; leakage also requires a larger state space. Spatially correlated Pauli errors are possible and must not be confused with independent single-qubit errors.

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.

Choosing a model from the experiment

Start with the states and records that the experiment can distinguish. If population leaves the modeled space, decide whether it becomes an observable erasure flag or an unobserved leakage state. If populations relax, determine whether upward transitions matter. If only phase coherence decays, identify the basis and test whether a single exponential describes the relevant time interval.

A one-step channel does not specify correlations between successive uses. Repeating it independently is an additional assumption, as explained in Channel Composition and Fixed Points. Noise, Channels, and Error Mitigation connects these distinctions to experimental diagnostics and correction strategies. For numerical examples, use Simulating Quantum Channels.

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 $p$.

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.