  We develop a resource theory of symmetric distinguishability, the fundamental
objects of which are elementary quantum information sources, i.e., sources that
emit one of two possible quantum states with given prior probabilities. Such a
source can be represented by a classical-quantum state of a composite system
$XA$, corresponding to an ensemble of two quantum states, with $X$ being
classical and $A$ being quantum. We study the resource theory for two different
classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum
channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS)
maps acting on $XA$. We introduce the notion of symmetric distinguishability of
an elementary source and prove that it is a monotone under both these classes
of free operations. We study the tasks of distillation and dilution of
symmetric distinguishability, both in the one-shot and asymptotic regimes. We
prove that in the asymptotic regime, the optimal rate of converting one
elementary source to another is equal to the ratio of their quantum Chernoff
divergences, under both these classes of free operations. This imparts a new
operational interpretation to the quantum Chernoff divergence. We also obtain
interesting operational interpretations of the Thompson metric, in the context
of the dilution of symmetric distinguishability.
