$ \begin{aligned} M &= \sqrt p_0 \ketbra 00 + \sqrt p_1 \ketbra 11 \\ &= \sqrt[4]{p_0 p_1} \bigg( \sqrt[4]{p_0/p_1} \;\ketbra 00 + \sqrt[4]{p_1/p_0} \; \ketbra 11 \bigg)\\ &= \sqrt[4]{p_0 p_1}e^{\sigma_z \log \sqrt[4]{p_0/p_1}} \end{aligned} $ where $M \widetilde M = \sqrt{p_0 p_1}$.