Common Channels

A channel specifies the averaged output state for every input. This reference compares the effects and parameters of frequently used channels; the linked articles give their derivations. A channel alone does not specify which detector outcomes were recorded or which microscopic interaction produced it.

Required background. Completely Positive Maps explains why an untouched reference system matters; Kraus Representation supplies operator-sum formulas.

Qubit channels and their parameters

Use $|0\rangle$ for the ground state, $|1\rangle$ for the excited state, and $Z=|0\rangle\langle0|-|1\rangle\langle1|$. The parameters below describe a finite-time map; they are not decay rates.

Effect	Channel	Parameter meaning
Phase flips	$(1-p)\rho+pZ\rho Z$	$0\le p\le1$; coherence multiplier $1-2p$
Random Pauli errors	$\sum_{\mu=0}^3p_\mu\sigma_\mu\rho\sigma_\mu$	$p_\mu\ge0$, $\sum_\mu p_\mu=1$; $\sigma_0=I$
Isotropic mixing	$(1-q)\rho+qI/2$	Replacement probability $0\le q\le1$
Energy relaxation	$\sum_{r=0}^1K_r\rho K_r^\dagger$	Decay probability $0\le\gamma\le1$

Phase flips and dephasing. The populations are unchanged and $\rho_{01}\mapsto(1-2p)\rho_{01}$. Complete dephasing occurs at $p=1/2$. At $p=1$ the map is the unitary phase flip $Z\rho Z$: coherence magnitude has returned, with the opposite phase. Increasing a finite-time error probability is therefore not the same as increasing irreversible dephasing. For exponential coherence decay,

$$p(t)=\frac{1-e^{-\Gamma_\phi t}}2.$$

See Dephasing Channel and Pauli Channels. A dephasing basis need not be the energy basis.

Depolarizing. The table uses a literal replacement probability. For a qubit, the equivalent total probability of a nonidentity Pauli error is $p_X+p_Y+p_Z=3q/4$. The algebraic family remains completely positive for $0\le q\le4/3$, but $q>1$ no longer means a probability of replacement. More generally the $d$-dimensional map $(1-q)\rho+qI/d$ has CP range $0\le q\le d^2/(d^2-1)$ for $d\ge2$. See Depolarizing Channel.

Amplitude damping. A Kraus pair is

$$\begin{gathered} K_0=|0\rangle\langle0|+\sqrt{1-\gamma}\,|1\rangle\langle1|,\\ K_1=\sqrt\gamma\,|0\rangle\langle1|. \end{gathered}$$

It gives $\rho_{11}\mapsto(1-\gamma)\rho_{11}$ and $\rho_{01}\mapsto\sqrt{1-\gamma}\,\rho_{01}$. Unlike Pauli noise, it is nonunital for $\gamma>0$. A constant population-decay rate gives $\gamma(t)=1-e^{-t/T_1}$. See Amplitude-Damping Channel.

Thermal relaxation. In the interaction picture, constant upward and downward rates give

$$\begin{gathered} \Gamma_1=\Gamma_\uparrow+\Gamma_\downarrow>0, \qquad p_e^*=\frac{\Gamma_\uparrow}{\Gamma_1},\\ p_e(t)=p_e^*+[p_e(0)-p_e^*]e^{-\Gamma_1t},\\ \rho_{01}(t)=e^{-\Gamma_1t/2}\rho_{01}(0). \end{gathered}$$

For a single equilibrium bath, $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta\hbar\omega_0}$. These formulas omit additional pure dephasing. If both rates vanish, the map is the identity and no unique stationary population is selected. See Thermal Master Equations.

Dephasing in more than two levels

A basis-preserving dephasing map has the form

$$|n\rangle\langle m|\longmapsto \lambda_{nm}|n\rangle\langle m|.$$

It is CPTP exactly when the coefficient matrix $\Lambda=(\lambda_{nm})$ is positive semidefinite and $\lambda_{nn}=1$. Pairwise bounds $|\lambda_{nm}|\le1$ alone do not ensure complete positivity in dimensions above two. This is a useful check when fitting several coherence-decay factors independently. The general construction is in Common Noise Channels.

Erasure, loss, and Gaussian noise

An erasure channel includes an orthogonal output flag:

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

The direct sum separates the retained input space from the flag space. Ordinary unheralded loss is not automatically erasure: distinguishability of the loss event is part of the model. See Erasure and Loss Channels.

For one oscillator, use dimensionless quadratures with $[q,p]=i$, displacement vector $d$, and symmetrized covariance $V$; vacuum has $V=I/2$. The following channels assume an initially uncorrelated, zero-mean environment.

Channel	Displacement	Covariance
Attenuation, $0\le\eta\le1$	$\sqrt\eta\,d$	$\eta V+(1-\eta)V_E$
Phase-insensitive amplification, $G\ge1$	$\sqrt G\,d$	$GV+(G-1)(\bar n_E+1/2)I$

Pure loss has a vacuum environment, $V_E=I/2$; thermal attenuation has $V_E=(\bar n_E+1/2)I$. For vacuum loss, mean occupation decays as $\langle n\rangle\mapsto\eta\langle n\rangle$ and field amplitude as $\langle a\rangle\mapsto\sqrt\eta\langle a\rangle$. A thermal-loss semigroup gives $\eta=e^{-\kappa t}$ and $\langle n(t)\rangle=\bar n_E+[\langle n(0)\rangle-\bar n_E]e^{-\kappa t}$.

The amplifier table assumes a thermal environment; its quantum limit is $\bar n_E=0$. Squeezed environments require the appropriate quadrature reflection in the added-noise matrix. Removing added noise from a deterministic phase-insensitive amplifier with $G>1$ violates complete positivity. See Gaussian Channels for the matrix inequality and general conventions.

Choosing a finite-time model

Specify the retained Hilbert space, the dephasing basis, whether a loss flag is available, and whether parameters are probabilities or rates. A good fit to one prepared state does not determine a channel on all inputs. In particular, unchanged populations for a stationary input do not rule out relaxation.

Use Channel Composition and Fixed Points to study repeated use, Simulating Quantum Channels for numerical examples, and the Approximation Checklist when deriving a time-dependent model.

Exercises

Two phase flips. Compose two phase-flip channels with $p=1$. Compare this with two complete-dephasing channels.

Solution

Two unitary flips give $Z^2\rho Z^2=\rho$. Complete dephasing has $p=1/2$ and is idempotent: the first application removes the off-diagonal entries and the second changes nothing. A sign reversal and an irreversible loss of coherence are different operations.

An unchanged population. A thermal qubit has $\Gamma_\uparrow=\Gamma_\downarrow=\Gamma>0$. Prepare $|+\rangle$. Does unchanged population imply pure dephasing?

Solution

No. Its excited population remains $1/2$, already the stationary value, while coherence decays as $e^{-\Gamma t}/2$. A second preparation, such as $|1\rangle$, reveals relaxation toward $1/2$.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Thermal relaxation and finite-time maps.
• Nielsen, M. A., and I. L. Chuang. Quantum Computation and Quantum Information. 10th anniversary edition, Cambridge University Press, 2010. Standard finite-dimensional noise channels.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 2, channels and their representations. Author manuscript.
• Weedbrook, C., et al. “Gaussian quantum information.” Reviews of Modern Physics 84, 621–669 (2012). doi:10.1103/RevModPhys.84.621. Gaussian loss and amplification conventions.