On the condition of conversion of classical probability distribution families into quantum families

The purpose of the paper is to study the condition for a probability distribution family to a quantum state family. This is an (relatively) easy example of quantum version of “comparison of statistical experiments”, which had turned out to supply deep insight into the foundation of classical and quantum statistics \cite{Strasser}\cite{Torgersen}. It turns out use of maximal quantum $f$-divergence is useful in characterizing the classical-quantum transformability.