$$ \begin{multline}\shoveleft \text{Kraus operators:}\space K_{0}=\begin{bmatrix} 1 & 0\\ 0 & \sqrt{1-p} \end{bmatrix} ,\space K_{1}=\begin{bmatrix} 0 & 0\\ 0 & \sqrt{p} \end{bmatrix} ,\space \text{Bloch vector:}\space \displaylines{ \begin{aligned} & x = x\sqrt{1-p}\\ & y = y\sqrt{1-p}\\ & z = z\\ \end{aligned} } \end{multline} $$

$$ \begin{multline}\shoveleft \text{Kraus operators:}\space K_{0}=\sqrt{1-p}\hat{\mathbb{1}} ,\space K_{1}=\sqrt{\frac{p}{3}}\hat{\sigma_{x}} ,\space K_{2}=\sqrt{\frac{p}{3}}\hat{\sigma_{y}} ,\space K_{3}=\sqrt{\frac{p}{3}}\hat{\sigma_{z}} ,\space \text{Bloch vector:}\space \displaylines{ \begin{aligned} & x = (1-\frac{4}{3}p)x\\ & y = (1-\frac{4}{3}p)y\\ & z = (1-\frac{4}{3}p)z\\ \end{aligned} } \end{multline} $$

$$ \begin{multline}\shoveleft \text{Kraus operators:}\space K_{0}=\begin{bmatrix} 1 & 0\\ 0 & \sqrt{1-p} \end{bmatrix} ,\space K_{1}=\begin{bmatrix} 0 & \sqrt{p}\\ 0 & 0 \end{bmatrix} ,\space \text{Bloch vector:}\space \displaylines{ \begin{aligned} & x = x\sqrt{1-p}\\ & y = y\sqrt{1-p}\\ & z = p+(1-p)z\\ \end{aligned} } \end{multline} $$