There seems to be numerous work about the distillation protocol of the $T$-magic state $$ \frac{1}{\sqrt{2}}(|0\rangle+e^{i\pi/4}|1\rangle). $$
Similarly, I am wondering if it is possible to distill a $\theta$-magic state $$ \frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle). $$
Can we achieve it for arbitrary angle $\theta$? If not, what are the conditions for distillable $\theta$?
