  The fidelity-based smooth min-relative entropy is a distinguishability
measure that has appeared in a variety of contexts in prior work on quantum
information, including resource theories like thermodynamics and coherence.
Here we provide a comprehensive study of this quantity. First we prove that it
satisfies several basic properties, including the data-processing inequality.
We also establish connections between the fidelity-based smooth min-relative
entropy and other widely used information-theoretic quantities, including
smooth min-relative entropy and smooth sandwiched R\'enyi relative entropy, of
which the sandwiched R\'enyi relative entropy and smooth max-relative entropy
are special cases. After that, we use these connections to establish the
second-order asymptotics of the fidelity-based smooth min-relative entropy and
all smooth sandwiched R\'enyi relative entropies, finding that the first-order
term is the quantum relative entropy and the second-order term involves the
quantum relative entropy variance. Utilizing the properties derived, we also
show how the fidelity-based smooth min-relative entropy provides one-shot
bounds for operational tasks in general resource theories in which the target
state is mixed, with a particular example being randomness distillation. The
above observations then lead to second-order expansions of the upper bounds on
distillable randomness, as well as the precise second-order asymptotics of the
distillable randomness of particular classical-quantum states. Finally, we
establish semi-definite programs for smooth max-relative entropy and smooth
conditional min-entropy, as well as a bilinear program for the fidelity-based
smooth min-relative entropy, which we subsequently use to explore the tightness
of a bound relating the last to the first.
