Saturating the Data Processing Inequality for $\alpha -z$ Renyi Relative Entropy

It has been shown that the $\alpha -z$ R{'e}nyi relative entropy satisfies the Data Processing Inequality (DPI) for a certain range of $\alpha$’s and $z$’s. Moreover, the range is completely characterized by Zhang in `20. We prove necessary and algebraically sufficient conditions to saturate the DPI for the $\alpha -z$ R{'e}nyi relative entropy whenever $1<\alpha \le 2$ and $\frac{\alpha }{2}\le z\le \alpha$. Moreover, these conditions coincide whenever $\alpha =z$.