# 振幅と位相の減衰チャネルを組み合わせたKraus演算子を見つける

You can obtain the Kraus operators of the combined channel by taking products of the Kraus operators of the individual channels (using the notation from the paper you linked):

Amplitude damping:

$$E^{AD}_1 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-p_{AD}} \end{bmatrix}$$, $$E^{AD}_2 = \begin{bmatrix} 0 & \sqrt{p_{AD}} \\ 0 & 0 \end{bmatrix}$$

Phase damping:

$$E^{PD}_1 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-p_{PD}} \end{bmatrix}$$, $$E^{PD}_2 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{p_{PD}} \end{bmatrix}$$

Combined:

$$E^{D}_1 = E^{PD}_1 E^{AD}_1 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-p_{AD}}\sqrt{1-p_{PD}} \end{bmatrix}$$

$$E^{D}_2 = E^{PD}_1 E^{AD}_2 = \begin{bmatrix} 0 & \sqrt{p_{AD}} \\ 0 & 0 \end{bmatrix}$$

$$E^{D}_3 = E^{PD}_2 E^{AD}_1 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{1-p_{AD}}\sqrt{p_{PD}} \end{bmatrix}$$

This is the Kraus set given in the paper you linked. There is a fourth possible combination, which is

$$E^{D}_4 = E^{PD}_2 E^{AD}_2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

which is the null channel. Since we first destroy any $$|1\rangle$$ states, the phase damping channel only has $$|0\rangle$$ states to act on, which are sent to 0.

The order in which you apply amplitude and phase damping does not actually matter, that is

$$\mathcal{E}_{AD} \circ \mathcal{E}_{PD} (\rho) = \mathcal{E}_{PD} \circ \mathcal{E}_{AD} (\rho)$$.

Thus, you could swap the products in the Kraus terms defined above, which would result in a different Kraus set (now with four non-null elements), which would also describe the channel (the Kraus representation is not unique).

The other answer already uses this, but just to make the general fact more explicit: if $$\mathcal E=\mathcal E_A\circ\mathcal E_B$$, that is, $$\mathcal E(\rho)=\mathcal E_A(\mathcal E_B(\rho))$$, and the Kraus decompositions of the single channels read $$\mathcal E_A(\rho)=\sum_a A_a\rho A_a^\dagger, \qquad \mathcal E_B(\rho)=\sum_b B_b\rho B_b^\dagger,$$ then $$\mathcal E(\rho)=\sum_{a,b} C_{ab}\rho C_{ab}^\dagger$$, where $$C_{ab}\equiv A_a B_b$$ are the Kraus operators of the combined channel.