Channel Composition

Noise rarely happens once. A qubit passes through gate after gate, idle period after idle period — each a channel. To reason about the cumulative effect we need to compose channels, and the good news is that the class of CPTP maps is closed under everything you would want to do: composing channels in series, in parallel, and convexly all yield channels again.

Series composition

Applying $\mathcal{E}_1$ then $\mathcal{E}_2$ gives the composite $\mathcal{E}_2\circ\mathcal{E}_1$ with $(\mathcal{E}_2\circ\mathcal{E}_1)(\rho) = \mathcal{E}_2\big(\mathcal{E}_1(\rho)\big)$. If $\mathcal{E}_1(\rho)=\sum_j A_j\rho A_j^\dagger$ and $\mathcal{E}_2(\rho)=\sum_k B_k\rho B_k^\dagger$, then

so the Kraus operators of the composite are all products $\{B_k A_j\}$. Completeness propagates:

using $\sum_k B_k^\dagger B_k=\mathbb{1}$ first, then $\sum_j A_j^\dagger A_j=\mathbb{1}$. The composite is CPTP. The number of Kraus terms can grow as the product of the two ranks, but the Kraus rank of the composite is often smaller and never exceeds $d^2$.

Composition need not commute

Order matters: in general $\mathcal{E}_2\circ\mathcal{E}_1 \ne \mathcal{E}_1\circ\mathcal{E}_2$. Dephasing then amplitude-damping is not the same process as amplitude-damping then dephasing. Two important families do commute among themselves — Pauli channels with each other, and dephasing channels with each other — because their Kraus operators share an eigenbasis.

Channels that compose cleanly

Some channel families are closed under composition with a simple parameter rule:

• Amplitude damping: $\mathcal{E}_{\gamma_2}\circ\mathcal{E}_{\gamma_1} = \mathcal{E}_{\gamma}$ with the survival probabilities multiplying, $1-\gamma = (1-\gamma_1)(1-\gamma_2)$. Two decays in a row are one decay with the combined survival factor.
• Phase damping: coherence-survival factors multiply, $\sqrt{1-\lambda} = \sqrt{1-\lambda_1}\sqrt{1-\lambda_2}$.
• Depolarizing: shrink factors multiply, $1-p = (1-p_1)(1-p_2)$.

In each case the family forms a one-parameter semigroup under composition — the continuous-time, Markovian limit of which is the Lindblad master equation.

Parallel composition

Acting on two systems independently gives the tensor product channel $\mathcal{E}_1\otimes \mathcal{E}_2$, whose Kraus operators are the products $\{A_j\otimes B_k\}$. This is how noise on a multi-qubit register is built: independent single-qubit channels tensor together. Convex mixtures $p\,\mathcal{E}_1 + (1-p)\,\mathcal{E}_2$ are channels too — "with probability $p$ apply $\mathcal{E}_1$."

Try it

Pass the excited state $\rho=|1\rangle\langle1|$ through two amplitude-damping channels in series, $\gamma_1 = 0.2$ then $\gamma_2 = 0.5$. Apply the first channel, then the second, using the Kraus operators $K_0=\operatorname{diag}(1,\sqrt{1-\gamma}),\,K_1=\sqrt{\gamma}\,|0\rangle\langle1|$, and return the surviving excited population $\langle1|(\mathcal{E}_{\gamma_2}\circ\mathcal{E}_{\gamma_1}) (\rho)|1\rangle$. It must equal $(1-\gamma_1)(1-\gamma_2) = 0.8\times0.5 = 0.4$.

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