On maximization of measured $f$-divergence between a given pair of quantum states

This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probability density functions $p_{1}$ and $p_{2}$ over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right)$. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value $D_{f}^{\min}$ of measured measured $f$-divergence is also an interesting question. But so far the question is solved only for very restricted example of $f$. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.