Quantum Channels and Noise

A quantum channel describes a deterministic transformation of a state, including noise, an unread measurement, state preparation, and transmission. In finite dimension it is a linear, completely positive, trace-preserving map. This chapter teaches how to recognize such a map, choose a useful representation, and connect its action to a physical noise mechanism.

Required background. Density Operators supplies positive trace-one states; Partial Trace explains reduced maps obtained by discarding an environment.

Helpful background. Entangled States explains why a local map must remain consistent when its input has an untouched reference system.

A channel preserves probabilities, including with a reference

A Kraus representation gives

$$\Phi(\rho)=\sum_\alpha K_\alpha\rho K_\alpha^\dagger, \qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I_{\rm in}.$$

The sum condition preserves trace. The representation guarantees complete positivity: applying the map to one part of any positive joint state still gives a positive operator. The output dimension may differ from the input dimension, as in a channel that includes an orthogonal loss flag.

A selected measurement operation instead satisfies $\sum_\alpha K_\alpha^\dagger K_\alpha\le I$. Its output trace is the success probability. Renormalizing after selection does not turn it into a deterministic linear channel.

Begin with Quantum Operations, then Completely Positive Maps and Trace-Preserving Maps. Keep the two requirements separate: positivity and normalization answer different questions.

Three representations for three kinds of calculation

Representation	When it is useful
Kraus Representation	Propagate states and construct simple channels while preserving complete positivity. A Kraus label is not automatically an observed physical event.
Choi Matrix and Channel–State Duality	Test complete positivity by a matrix spectrum and impose trace preservation by a partial-trace condition. Check tensor order and normalization.
Stinespring Representation	Realize the channel by an isometry into system plus environment, then discard the environment. The realization is not unique.

In the output-first convention of Conventions, a channel has $J\ge0$ and $\operatorname{Tr}_{\rm out}J=I_{\rm ref}$. The normalized Choi operator $J/d$ is therefore a density operator with the specific reference marginal $I/d$. An arbitrary density operator is not automatically the Choi state of a trace-preserving map.

Bloch-ball geometry is useful but not a complete positivity test

A qubit state has $\rho=(I+\boldsymbol r\cdot\boldsymbol\sigma)/2$. A trace-preserving, Hermiticity-preserving map acts affinely:

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

Unital channels have $\boldsymbol t=0$ and preserve the maximally mixed state. Amplitude damping shifts the center toward the ground state. Complete positivity imposes constraints stronger than merely keeping this ball inside itself.

For example, the proposed map $(r_x,r_y,r_z)\mapsto(r_x,r_y,0)$ maps the Bloch ball into its equatorial disk, so it is positive on isolated qubits. It is not completely positive. The Pauli-diagonal condition

$$1+\lambda_z\ge|\lambda_x+\lambda_y|$$

would require $1\ge2$. Its action on an entangled input fails even though every isolated qubit output looks valid. Pauli Channels derives the full allowed region.

Select a noise model by its action

Use Common Noise Channels for a first comparison, then inspect the model matching your mechanism:

• Dephasing Channel preserves populations in a specified basis and reduces coherence there. Energy preservation additionally depends on the Hamiltonian.
• Depolarizing Channel contracts all Bloch directions equally in its usual noise range. Its parameter convention differs from the total Pauli-error probability.
• Amplitude Damping Channel transfers excitation toward a lower state and is generally nonunital. A finite-temperature model also needs upward transitions.
• Erasure and Loss Channels distinguishes a flagged orthogonal erasure state from unflagged leakage or bosonic attenuation.
• Gaussian Channels describes bosonic attenuation, amplification, and noise with explicit quadrature and covariance conventions. Quantum fluctuations impose a minimum added-noise constraint.

Then read Channel Composition and Fixed Points. If $\Phi$ acts first and $\Psi$ second, the result is $\Psi\circ\Phi$. A fixed state need not attract every input, and repeated channels can cycle instead of converging. One valid finite-time channel also does not establish a continuous-time semigroup or an absence of memory.

For microscopic time dependence, continue to Open Quantum Systems. For applications to protecting transmitted information, continue to Quantum Channels for Quantum Information.

Exercises

Trace preserving, but not unital. A qubit amplitude-damping channel has decay probability $p$. What happens to $I/2$?

Solution

The excited population $1/2$ becomes $(1-p)/2$ and the ground population becomes $(1+p)/2$. Their sum is one, but the output differs from $I/2$ for $p>0$. Thus the channel is trace preserving and nonunital. Repeated damping with fixed $0<p\le1$ drives every input to the ground state.

Invariant does not mean attracting. Let $\Phi(\rho)=Z\rho Z$. Is $I/2$ fixed? Does repeatedly applying $\Phi$ drive $|+\rangle$ toward it?

Solution

$I/2$ is fixed under every unitary conjugation. But $|+\rangle$ and $|-\rangle$ alternate under $Z$, so the sequence does not converge to $I/2$. This is a valid channel with a fixed state but no attraction to that state from this input.

References

• Holevo, A. S. Quantum Systems, Channels, Information: A Mathematical Introduction. De Gruyter, 2012. Channel structure and bosonic examples.
• Nielsen, M. A., and I. L. Chuang. Quantum Computation and Quantum Information. 10th anniversary edition, Cambridge University Press, 2010. Qubit channels and representations.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Complete positivity, Choi conditions, and dilation theorems. Author manuscript.