  The Rains relative entropy of a bipartite quantum state is the tightest known
upper bound on its distillable entanglement -- which has a crisp physical
interpretation of entanglement as a resource -- and it is efficiently
computable by convex programming. It has not been known to be a selective
entanglement monotone in its own right. In this work, we strengthen the
interpretation of the Rains relative entropy by showing that it is monotone
under the action of selective operations that completely preserve the
positivity of the partial transpose, reasonably quantifying entanglement. That
is, we prove that Rains relative entropy of an ensemble generated by such an
operation does not exceed the Rains relative entropy of the initial state in
expectation, giving rise to the smallest, most conservative known computable
selective entanglement monotone. Additionally, we show that this is true not
only for the original Rains relative entropy, but also for Rains relative
entropies derived from various R\'enyi relative entropies. As an application of
these findings, we prove, in both the non-asymptotic and asymptotic settings,
that the probabilistic approximate distillable entanglement of a state is
bounded from above by various Rains relative entropies.
