Quantum Channels and Noise

A quantum channel is the most general deterministic transformation of a quantum state allowed by the standard finite-dimensional operational framework. Channels describe noise, unread measurements, state preparation, transmission, control errors, and reduced dynamics when inaccessible degrees of freedom are discarded.

The defining requirements are compact:

$$\begin{gathered} \Phi \text{ is linear and completely positive,} \\ \operatorname{Tr}[\Phi(\rho)] = \operatorname{Tr}\rho. \end{gathered}$$

Such a map is called completely positive and trace preserving, or CPTP. A selected measurement branch is instead completely positive and trace nonincreasing. The difference is probability bookkeeping, not a minor terminology choice.

This chapter develops three complementary tasks: deciding whether a proposed map is physical, changing between useful channel representations, and choosing a noise model whose assumptions match the experiment.

Required background. Use Density Operators to represent general quantum states by positive trace-one operators; use Partial Trace to derive reduced maps by discarding an environment.

Helpful background. Use Entangled States to understand why complete positivity tests ancillary extensions.

The Basic Vocabulary

Let

$$\Phi: \mathcal B(\mathcal H_{\mathrm{in}}) \longrightarrow \mathcal B(\mathcal H_{\mathrm{out}})$$

be a linear map on operators.

Term	Conditions	Operational role
positive map	$X\ge0\Rightarrow\Phi(X)\ge0$	preserves positivity on the system alone
completely positive map	$\Phi\otimes\operatorname{id}_R$ is positive for every reference $R$	remains physical when the input is entangled
trace-nonincreasing operation	$\operatorname{Tr}\Phi(\rho)\le\operatorname{Tr}\rho$	one selected branch or probabilistic filter
quantum channel	completely positive and trace preserving	deterministic state transformation
unital channel	$\Phi(I_{\mathrm{in}})=I_{\mathrm{out}}$ when dimensions match	preserves the identity and hence the maximally mixed state

Complete positivity and trace preservation answer different questions. Complete positivity controls output positivity in the presence of arbitrary reference systems. Trace preservation controls total probability. Unitality is a third, independent property and should not be inferred from either one.

Start with Quantum Operations for the common vocabulary of deterministic channels and selected branches.

Reading path

1. Read Quantum Operations for channels, selected operations, Kraus forms, and discarded-environment constructions.
2. Use complete positivity and the Kraus representation to verify physicality.
3. Compare the Choi representation with the Stinespring representation.
4. Apply the representations to the standard noise models, then use the validation workflow.
5. Continue to No-Broadcasting Theorem for a sharp limit on what one channel can reproduce.

Why Complete Positivity Is Required

A system can be entangled with a reference that the local process does not touch. If $\Phi$ acts only on the system, the extended transformation is

$$\Phi\otimes\operatorname{id}_R.$$

Physical consistency requires this map to preserve positivity for every reference dimension and every positive joint input. Positivity of $\Phi$ alone does not guarantee that property.

The transpose map is the standard warning. Transposition preserves the eigenvalues of a density operator and is therefore positive on an isolated system. Applied to one half of a maximally entangled state, however, partial transposition can produce a nonpositive operator. Transposition is not completely positive and cannot represent an arbitrary local physical channel.

In finite dimensions, complete positivity can be checked in several equivalent ways:

• construct a Kraus representation;
• show that the Choi matrix is positive;
• construct a Stinespring dilation;
• prove positivity of $\Phi\otimes\operatorname{id}_d$, where $d=\dim\mathcal H_{\mathrm{in}}$.

These tests are equivalent for finite-dimensional maps on the full input operator space. With initial system–environment correlations, however, a reduced map on arbitrary system states may not be well defined; its assignment domain and compatibility assumptions must be stated explicitly.

Kraus Representation

Every finite-dimensional completely positive map can be written

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

where

$$K_\alpha: \mathcal H_{\mathrm{in}} \longrightarrow \mathcal H_{\mathrm{out}}.$$

The map is trace preserving when

$$\sum_\alpha K_\alpha^\dagger K_\alpha =I_{\mathrm{in}},$$

and trace nonincreasing when the left-hand side is no larger than the identity.

Kraus operators are a representation of the map, not generally unique physical events. Different Kraus families can describe exactly the same channel. A microscopic interpretation becomes justified only when an apparatus or environment model identifies a particular record basis.

The distinction from the preceding chapter is important. Measurement Kraus operators are grouped by reported outcome to define an instrument, whereas an unconditional channel forgets that record. At channel level, different Kraus families related by an isometry can represent the same map.

Choi Representation

Fix a basis $\{|i\rangle\}$ of the input space and define the unnormalized Choi matrix by

$$J(\Phi) = \sum_{i,j} \Phi(|i\rangle\langle j|) \otimes |i\rangle\langle j|.$$

With the output factor written first, the principal tests are

$$\begin{aligned} \Phi\text{ is CP} &\iff J(\Phi)\ge0, \\ \Phi\text{ is TP} &\iff \operatorname{Tr}_{\mathrm{out}}J(\Phi) =I_{\mathrm{in}}. \end{aligned}$$

The map is recovered through

$$\Phi(X) = \operatorname{Tr}_{\mathrm{in}} \left[ J(\Phi) \left(I_{\mathrm{out}}\otimes X^{\mathsf T}\right) \right].$$

The transpose and tensor-factor order depend on the stated convention. Mixing conventions is one of the most common sources of apparently contradictory Choi formulas.

The rank of $J(\Phi)$ equals the minimum number of Kraus operators. Dividing by the input dimension gives the normalized Choi state, which underlies channel–state duality, process tomography, and many convex optimization methods.

Stinespring Representation

Every finite-dimensional channel admits an isometric representation

$$V: \mathcal H_{\mathrm{in}} \longrightarrow \mathcal H_{\mathrm{out}}\otimes\mathcal H_E, \qquad V^\dagger V=I,$$

such that

$$\Phi(\rho) = \operatorname{Tr}_E \left[ V\rho V^\dagger \right].$$

Equivalently, after supplying an initialized environment and a sufficiently large output space, the isometry can be extended to a unitary interaction. The channel is what remains after the environment is ignored.

Choosing an environment basis $\{|\alpha\rangle_E\}$ gives Kraus operators

$$K_\alpha = {}_E\langle\alpha|V,$$

and changing that basis changes the Kraus representation without changing the channel. This makes the representation freedom physically transparent.

This is the working Stinespring construction. The theorem-level statement is that every completely positive map admits such an enlarged-space representation, with trace preservation corresponding to an isometry; unlike a Naimark dilation, it dilates a state transformation rather than only a measurement.

Equivalent Views, Different Jobs

Representation	Best suited to	Main caution
direct map $\Phi$	composition and abstract properties	physical constraints may be hidden
Kraus operators	state propagation and simple simulations	representation is not unique
Choi matrix	CP tests, optimization, and tomography	normalization and tensor order vary by convention
Stinespring isometry	environment interpretation and complementary channels	dilation is not unique
affine Bloch map	one-qubit geometry and parameter fitting	does not generalize unchanged to higher dimensions
master-equation generator	continuous-time dynamics	a valid finite-time map need not form a semigroup

Equivalent representations contain the same channel-level information, but they expose different structures. Good calculations change representation when the question changes.

Qubit Affine Form

A qubit state can be written

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

Every trace-preserving Hermiticity-preserving qubit map acts affinely on the Bloch vector:

$$\boldsymbol r' = T\boldsymbol r+\boldsymbol t.$$

Complete positivity imposes additional constraints on $T$ and $\boldsymbol t$. Unital channels have $\boldsymbol t=0$ and preserve the center of the Bloch ball. Amplitude damping is nonunital: it shifts the ball toward the ground-state pole. This geometric distinction is often more informative than memorizing a Kraus family.

For a unital Pauli-diagonal channel,

$$(r_x,r_y,r_z) \longmapsto (\lambda_xr_x,\lambda_yr_y,\lambda_zr_z),$$

but the three contraction factors cannot be chosen independently. Complete positivity is equivalent to

$$1+\lambda_z\ge\lvert\lambda_x+\lambda_y\rvert, \qquad 1-\lambda_z\ge\lvert\lambda_x-\lambda_y\rvert,$$

which defines the allowed tetrahedral region.

Standard Noise Models

No standard channel is a universal synonym for noise. Each encodes a particular symmetry, record structure, and physical approximation.

Channel	Characteristic action	Typical interpretation	What it does not capture by itself
dephasing	suppresses selected coherences while preserving populations	phase noise or unread measurement in a preferred basis	energy relaxation
depolarizing	contracts all traceless qubit components equally	isotropic effective randomization	directional, coherent, or nonunital noise
amplitude damping	moves excitation toward a lower state	spontaneous emission or zero-temperature $T_1$ relaxation	finite-temperature upward transitions
Pauli channel	random Pauli conjugations	stochastic Pauli error model	coherent rotations or leakage
erasure	moves the state to an orthogonal flagged sector	known loss event	unflagged attenuation without an erasure record
bosonic loss	attenuates a mode through an environment port	optical loss or damping	arbitrary non-Gaussian noise
Gaussian channel	affine transformation of first and second moments	attenuation, amplification, or thermal Gaussian noise	non-Gaussian state changes in general

The table makes the assumptions comparable; the benchmark calculations below show how three common qubit models act on matrix elements and Bloch vectors.

Three Qubit Benchmarks

Dephasing

A dephasing channel with coherence factor $\lambda$ acts as

$$\Phi_\lambda(\rho) = \begin{pmatrix} \rho_{00} & \lambda\rho_{01} \\ \lambda^*\rho_{10} & \rho_{11} \end{pmatrix}, \qquad |\lambda|\le1.$$

It is basis dependent. Populations in the displayed basis remain fixed, while off-diagonal coherence contracts. The channel is unital and has every density matrix diagonal in that basis as a fixed state.

Depolarizing noise

One common qubit convention is

$$\Phi_p(\rho) = (1-p)\rho + p\frac{I}{2}, \qquad 0\le p\le1.$$

This convention makes $p$ the mixing weight with the maximally mixed state. Other sources use a Pauli-error probability instead, producing a different parameter range and conversion. Always state the convention before comparing fitted noise strengths.

Amplitude damping

With decay probability $\gamma$, zero-temperature amplitude damping acts as

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

The ground state is the attracting fixed point. Population relaxation also reduces coherence, so observed coherence decay need not be pure dephasing. The finite-temperature generalization must include upward as well as downward transitions.

Erasure, Leakage, and Loss

These words are often used interchangeably even though they define different effective models.

For a flagged erasure channel,

$$\Phi_\epsilon(\rho) = (1-\epsilon)\rho + \epsilon|e\rangle\langle e|,$$

where $|e\rangle$ lies in an orthogonal output sector. The receiver can tell that erasure occurred. Leakage moves population outside a chosen computational subspace, but the leaked sector may not be measured or flagged. Bosonic attenuation reduces field amplitude and mixes the system with an environment mode; it is not generally equivalent to a discrete flagged erasure.

Output Hilbert spaces and available records must therefore be stated explicitly, especially when loss is postselected, detector efficiency is folded into the model, or leaked population can later return.

Gaussian Channels

For bosonic modes, a Gaussian channel maps Gaussian states to Gaussian states. At the level of first moments $\boldsymbol d$ and covariance matrix $V$,

$$\begin{aligned} \boldsymbol d' &=X\boldsymbol d+\boldsymbol\delta, \\ V' &=XVX^{\mathsf T}+Y. \end{aligned}$$

The matrices $X$ and $Y$ must satisfy a convention-dependent complete-positivity condition involving the symplectic form. Attenuators, amplifiers, additive-noise channels, and thermal-loss channels are central examples.

Before applying the complete-positivity constraint, fix the quadrature normalization, symplectic form, covariance convention, and tensor-factor order; otherwise factors of two and transposes are easily misidentified.

Composition and Fixed Points

If $\Phi$ acts first and $\Psi$ second, the composite channel is

$$\Psi\circ\Phi.$$

The order matters: distinct noise processes need not commute. Repeating a discrete-time channel gives $\Phi^n$, while continuous-time Markovian evolution may form a semigroup $\Phi_t=e^{t\mathcal L}$.

A fixed state satisfies

$$\Phi(\rho_*)=\rho_*.$$

Fixed states can represent preserved classical information, decoherence-free structure, thermal equilibrium, a reset target, or an engineered dissipative steady state. An invariant state need not be attracting, and a channel can have an entire fixed-point algebra rather than one fixed density operator.

For serial or repeated noise, keep the map order explicit, determine fixed states or fixed observables, and distinguish mere invariance from convergence to an attractor.

Channels, Instruments, and Master Equations

These descriptions differ by what information and time structure they retain.

Description	Keeps a classical record?	Time structure
instrument $\{\mathcal I_m\}$	yes, outcome $m$	one measurement step
channel $\Phi=\sum_m\mathcal I_m$	no	one finite transformation
family $\{\Phi_t\}$	no	finite-time evolution from an initial time
master equation $\dot\rho=\mathcal L_t(\rho)$	no	differential evolution
stochastic master equation	yes, a time-resolved record	conditional continuous evolution

A channel does not by itself specify how long the process took or whether it can be consistently divided into intermediate channels. Conversely, a generator must produce physical finite-time maps. A time-homogeneous Markovian semigroup has a Lindblad–GKSL generator, whereas microscopic reduced dynamics begins from an explicit system–environment state and need not define a semigroup.

Validating a Proposed Channel

For a finite-dimensional candidate map, use the following workflow.

1. State the input and output Hilbert spaces, including loss or leakage sectors.
2. Check linearity on operators, not only on normalized density matrices.
3. Test complete positivity, usually through a Choi matrix or Kraus form.
4. Check trace preservation for a deterministic channel or trace nonincrease for a selected branch.
5. Distinguish trace preservation from unitality.
6. State parameter ranges and conventions.
7. Identify fixed states, conserved observables, or attracting sectors.
8. Check whether compositions are applied in the correct order.
9. Compare the model’s symmetry and temperature assumptions with the physical mechanism.
10. If a microscopic story is claimed, state the environment preparation and dilation rather than reading it off a nonunique Kraus family.

For numerical work, positivity on a few sampled states is not a proof of complete positivity. Compute the Choi spectrum or use a structure-preserving parameterization.

Canonical boundaries

This chapter owns deterministic quantum operations, selected completely positive maps, their finite-dimensional representations, standard finite-time noise models, and channel composition. The Measurement and Open Systems gateway owns the wider physical context, while Partial Trace owns subsystem reduction itself.

Outcome-resolved instruments, microscopic bath derivations, continuous-time generators, error-correction protocols, and continuous-variable specializations remain separate planned topics. This chapter states the conceptual boundaries without advertising unfinished routes.

Common Mistakes

• Checking positivity only on the system and forgetting complete positivity.
• Calling a trace-decreasing selected operation a channel.
• Assuming trace preservation implies unitality.
• Treating Kraus operators as unique microscopic events.
• Mixing normalized and unnormalized Choi conventions.
• Reversing input and output factors in Choi formulas.
• Using depolarizing noise as a default description of every hardware error.
• Calling amplitude damping a Pauli channel.
• Confusing dephasing with energy relaxation.
• Treating flagged erasure, leakage, and attenuation as the same model.
• Comparing channel parameters without checking conventions.
• Inferring Markovianity from one physical finite-time channel.
• Ignoring initial system-environment correlations when assigning a reduced map.

Exercises

Trace preservation from Kraus operators

Suppose

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

Show that $\sum_\alpha K_\alpha^\dagger K_\alpha=I$ makes $\Phi$ trace preserving.

Solution

By cyclicity of the trace,

$$\begin{aligned} \operatorname{Tr}[\Phi(\rho)] &= \sum_\alpha \operatorname{Tr} (K_\alpha\rho K_\alpha^\dagger) \\ &= \operatorname{Tr} \left[ \rho \sum_\alpha K_\alpha^\dagger K_\alpha \right] \\ &= \operatorname{Tr}\rho. \end{aligned}$$

The condition holds for every operator input, so total probability is preserved.

Complete positivity of a unitary channel

Let $\Phi_U(\rho)=U\rho U^\dagger$. Show directly that $\Phi_U\otimes\operatorname{id}_R$ is positive for every reference system $R$.

Solution

For every positive joint operator $X_{SR}$,

$$\begin{aligned} (\Phi_U\otimes\operatorname{id}_R)(X_{SR}) &= (U\otimes I_R) \\ &\quad{} X_{SR} (U^\dagger\otimes I_R). \end{aligned}$$

Unitary conjugation preserves positivity. Therefore every extension is positive and $\Phi_U$ is completely positive. It is trace preserving because $U^\dagger U=I$.

Composition of dephasing channels

Two dephasing channels have real coherence factors $\lambda_1$ and $\lambda_2$. Find the coherence factor of their composition and identify all fixed states when $|\lambda_1\lambda_2|<1$.

Solution

The first channel multiplies $\rho_{01}$ by $\lambda_1$ and the second multiplies the result by $\lambda_2$. The composite factor is

$$\lambda_{21} = \lambda_2\lambda_1.$$

When its magnitude is strictly smaller than one, a fixed density matrix must have zero off-diagonal elements. Every state diagonal in the dephasing basis is fixed because the populations are unchanged.

Trace preservation and unitality

Use the amplitude-damping formula above to compute $\Phi_\gamma(I)$. Is the channel unital for $\gamma>0$?

Solution

Insert $I=\operatorname{diag}(1,1)$ into the linear channel action:

$$\Phi_\gamma(I) = \begin{pmatrix} 1+\gamma&0\\ 0&1-\gamma \end{pmatrix}.$$

This differs from $I$ whenever $\gamma>0$, so amplitude damping is not unital. It is nevertheless trace preserving. The distinction reflects the shift of the Bloch ball toward the ground state.

Choi trace condition

With the output-first Choi convention used above, explain why

$$\operatorname{Tr}_{\mathrm{out}}J(\Phi) =I_{\mathrm{in}}$$

is the trace-preservation condition.

Solution

From the definition,

$$\operatorname{Tr}_{\mathrm{out}}J(\Phi) = \sum_{i,j} \operatorname{Tr} \left[ \Phi(|i\rangle\langle j|) \right] |i\rangle\langle j|.$$

If $\Phi$ preserves trace, then

$$\operatorname{Tr} \left[ \Phi(|i\rangle\langle j|) \right] = \operatorname{Tr}(|i\rangle\langle j|) = \delta_{ij}.$$

The sum therefore becomes $\sum_i|i\rangle\langle i|=I_{\mathrm{in}}$. Conversely, this matrix equality gives trace preservation on the operator basis and hence by linearity on every input.

Cross-links

• Measurement and Open Systems
• Density Operators
• Partial Trace
• Quantum Operations
• No-Broadcasting Theorem

References

• W. F. Stinespring, “Positive functions on $C^*$-algebras,” Proceedings of the American Mathematical Society 6, 211–216 (1955).
• M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra and its Applications 10, 285–290 (1975).
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, De Gruyter (2012).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• A. S. Holevo and V. Giovannetti, “Quantum channels and their entropic characteristics,” Reports on Progress in Physics 75, 046001 (2012).