Examples of Quantum Channels

• Amplitude damping/spontaneous emission

\[K_0= \begin{pmatrix} 1 & 0\\ 0 & \sqrt{\eta} \end{pmatrix}, K_1= \begin{pmatrix} 0 & \sqrt{1-\eta}\\ 0 & 0 \end{pmatrix} \]

• Dephasing(Phase Shift, Phase Damping)

  \[K_0=\sqrt{\frac{1+\eta}{2}}I,K_1=\sqrt{\frac{1-\eta}{2}}\sigma_z \]

• Depolarizing

  \[K_0=\sqrt{1-p}I,\\ K_1=\sqrt{p/3}\sigma_x,\\ K_2=\sqrt{p/3}\sigma_y,\\ K_3=\sqrt{p/3}\sigma_z \]

• Bit flip

  \[K_0=\sqrt{p}I,K_1=\sqrt{1-p}\sigma_x \]

• Bit-phase flip

  \[K_0=\sqrt{p}I,K_1=\sqrt{1-p}\sigma_y \]

• Phase Damping

• \[K_0= \begin{pmatrix} 1 & 0\\ 0 & \sqrt{1-\eta} \end{pmatrix}, K_1= \begin{pmatrix} 0 & 0\\ 0 & \sqrt{\eta} \end{pmatrix} \]

  Same as Dephasing after unitary freedom.