  We study tree kinds of quantum fidelity. Usual Uhlmann's fidelity, minus of
f-divergence when $f(x)=-\sqrt{x}$, and the one introduced by the author via
reverse test. All of them are quantum extensions of classical fidelity, where
the first one is the largest and the third one is the smallest. We characterize
them in terms of convex optimization, and introduce their 'dual' quantity, or
the polar of the minus of the fidelity. They turned out to be monotone
increasing by unital completely positive maps, concave, and linked to its
classical version via optimization about classical-to-quantum maps and
quantum-to-classical maps.
