Quantum Channels

A channel implementation should pass a test on the entire linear map, not only a few input states. Kraus completeness checks trace preservation; Choi positivity checks complete positivity. Together they expose errors that output-state normalization alone cannot detect.

Helpful background. Channel–state duality supplies the conceptual criterion; vectorization fixes the array convention used here.

Kraus and Choi representations

For operators $K_a:\mathbb C^{d_{\rm in}}\to\mathbb C^{d_{\rm out}}$,

$$\mathcal E(X)=\sum_a K_a X K_a^\dagger,\qquad \sum_aK_a^\dagger K_a=I_{\rm in}.$$

The second equation follows by moving $K_a^\dagger$ around the trace. Define the unnormalized, input-first Choi matrix by

$$J=\sum_{ij}|i\rangle\langle j|\otimes\mathcal E(|i\rangle\langle j|).$$

With column stacking, $J=\sum_a\operatorname{vec}(K_a)\operatorname{vec}(K_a)^\dagger$. Thus $J\geq0$, $\operatorname{Tr}_{\rm out}J=I_{\rm in}$, and $\operatorname{Tr}J=d_{\rm in}$. A normalized Choi state divides this matrix by $d_{\rm in}$; its partial trace is then $I_{\rm in}/d_{\rm in}$. Document which convention is used before comparing files or libraries.

The linked channel–state duality page places the output factor first. Its Choi matrix is related to the input-first matrix here by swapping the two tensor factors, not by complex conjugation. Both conventions express the same map, but their partial-trace axes and array entries differ.

The superoperator is $\mathsf E=\sum_a K_a^*\otimes K_a$. Conversion between $\mathsf E$ and $J$ is an index reshuffling, not ordinary matrix transposition.

An explicit conversion removes the ambiguity. With input indices $a,b$ and output indices $\alpha,\beta$,

$$J_{a d_{\rm out}+\alpha,\,b d_{\rm out}+\beta} =\mathsf E_{\alpha+d_{\rm out}\beta,\,a+d_{\rm in}b}.$$

Thus $\mathsf E$ has shape $d_{\rm out}^2\times d_{\rm in}^2$, while $J$ is square of size $d_{\rm in}d_{\rm out}$. Verify conversions on a rectangular channel, such as an isometric embedding from a qubit to a three-level system, as well as on square maps. Swapping dimensions can otherwise pass every shape check.

Amplitude damping as a complete test

For $0\leq p\leq1$, choose

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

Direct multiplication gives

$$\mathcal E(\rho)= \begin{pmatrix} \rho_{00}+p\rho_{11}&\sqrt{1-p}\rho_{01}\\ \sqrt{1-p}\rho_{10}&(1-p)\rho_{11} \end{pmatrix}.$$

In input-first ordering $(00,01,10,11)$, $J$ has diagonal entries $(1,0,p,1-p)$ and corner entries $\sqrt{1-p}$. Its nonzero eigenvalues are $2-p$ and $p$. This single example tests a nonunital channel, complex coherences, and both endpoint limits.

A code test should compare three independent actions: direct Kraus summation, the superoperator applied to column-stacked input, and reconstruction from the Choi matrix,

$$\mathcal E(X)=\operatorname{Tr}_{\rm in}[(X^{\mathsf T}\otimes I)J].$$

Use a complex input so that replacing the transpose by an adjoint fails visibly.

Composition and numerical tolerance

For $\mathcal F\circ\mathcal E$, Kraus operators are $F_bK_a$ and the superoperator is $\mathsf F\mathsf E$. The reversed product describes a different experiment when operations do not commute.

Measure Hermiticity and negative eigenvalue weight of $J$, as well as the norm of $\operatorname{Tr}_{\rm out}J-I$. Small negative values at roundoff scale can arise during eigendecomposition; a finite negative eigenvalue signals a nonphysical map or invalid reconstruction. Sampling many physical states cannot establish complete positivity.

To compress a redundant Kraus list, diagonalize the positive Choi matrix and reshape $\sqrt{\lambda_a}v_a$ into $d_{\rm out}\times d_{\rm in}$ operators using column order. Dropping small eigenvalues changes the channel and can violate trace preservation. In fact, keeping a subset of these Kraus operators gives $\sum_{a\in\mathrm{kept}}K_a^\dagger K_a\leq I$: the retained map is generally trace-decreasing. Renormalizing its output separately for each input would produce a nonlinear conditional update, not a repaired trace-preserving channel. Report discarded Choi weight and recheck completeness after any repair.

Exercise

Compose two amplitude-damping channels with probabilities $p$ and $q$. Is their composition another amplitude-damping channel?

Solution

The excited population is multiplied by $(1-p)(1-q)$, while each coherence is multiplied by its square root. Therefore $p_{\rm eff}=1-(1-p)(1-q)$. This gives a semigroup benchmark when $p(t)=1-e^{-\gamma t}$, and a test of channel multiplication order that should be supplemented with a noncommuting unitary.

References

• Choi, M.-D., “Completely positive linear maps on complex matrices,” Linear Algebra and its Applications 10, 285–290 (1975), doi:10.1016/0024-3795(75)90075-0.
• Watrous, J., The Theory of Quantum Information, Cambridge University Press (2018), author’s text — Choi operators and channel representations.