Table 1. Kraus operators for the quantum channels: bit flip (BF), phase flip (PF), bit-phase flip (BPF), and generalized amplitude damping (GAD), where p and γ are decoherence probabilities, $0\lt p\lt 1$, 0 < γ < 1.
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| Channel | Kraus operators |
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| BF | ${E}_{0}=\sqrt{1-p/2}I,\,\,\,{E}_{1}=\sqrt{p/2}{\sigma }_{1}$ | | PF | ${E}_{0}=\sqrt{1-p/2}I,\,\,\,{E}_{1}=\sqrt{p/2}{\sigma }_{3}$ | | BPF | ${E}_{0}=\sqrt{1-p/2}I,\,\,\,{E}_{1}=\sqrt{p/2}{\sigma }_{2}$ | | GAD | ${E}_{0}=\sqrt{p}\left(\begin{array}{cc}1 & 0\\ 0 & \sqrt{1-\gamma }\end{array}\right),\,\,\,{E}_{2}=\sqrt{1-p}\left(\begin{array}{cc}\sqrt{1-\gamma } & 0\\ 0 & 1\end{array}\right)$ | | ${E}_{1}=\sqrt{p}\left(\begin{array}{cc}0 & \sqrt{\gamma }\\ 0 & 0\end{array}\right),\,\,\,{E}_{3}=\sqrt{1-p}\left(\begin{array}{cc}0 & 0\\ \sqrt{\gamma } & 0\end{array}\right)$ |
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