Variational expressions and Uhlmann theorem for measured divergences

Abstract: We present a variational expression for measured $f$-divergences for convex functions $f$. Under a suitable operator convexity condition, this variational expression leads to a convex optimization problem recovering the known expression in the case of R\'enyi divergences. As an application, we establish an Uhlmann theorem for a wide class of measured $f$-divergences including measured $\alpha$-R\'enyi divergences for all $\alpha \geq 0$. The well-known Uhlmann theorem for the fidelity corresponds to the special case $\alpha = \frac{1}{2}$.