  This paper systematically develops the resource theory of asymmetric
distinguishability, as initiated roughly a decade ago [K. Matsumoto,
arXiv:1010.1030 (2010)]. The key constituents of this resource theory are
quantum boxes, consisting of a pair of quantum states, which can be manipulated
for free by means of an arbitrary quantum channel. We introduce bits of
asymmetric distinguishability as the basic currency in this resource theory,
and we prove that it is a reversible resource theory in the asymptotic limit,
with the quantum relative entropy being the fundamental rate of resource
interconversion. The distillable distinguishability is the optimal rate at
which a quantum box consisting of independent and identically distributed
(i.i.d.) states can be converted to bits of asymmetric distinguishability, and
the distinguishability cost is the optimal rate for the reverse transformation.
Both of these quantities are equal to the quantum relative entropy. The exact
one-shot distillable distinguishability is equal to the min-relative entropy,
and the exact one-shot distinguishability cost is equal to the max-relative
entropy. Generalizing these results, the approximate one-shot distillable
distinguishability is equal to the smooth min-relative entropy, and the
approximate one-shot distinguishability cost is equal to the smooth
max-relative entropy. As a notable application of the former results, we prove
that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum
states to another pair of i.i.d. quantum states is fully characterized by the
ratio of their quantum relative entropies.
