  We develop two general approaches to characterising the manipulation of
quantum states by means of probabilistic protocols constrained by the
limitations of some quantum resource theory.
  First, we give a general necessary condition for the existence of a physical
transformation between quantum states, obtained using a recently introduced
resource monotone based on the Hilbert projective metric. In all affine quantum
resource theories (e.g. coherence, asymmetry, imaginarity) as well as in
entanglement distillation, we show that the monotone provides a necessary and
sufficient condition for one-shot resource convertibility under
resource-non-generating operations, and hence no better restrictions on all
probabilistic protocols are possible. We use the monotone to establish improved
bounds on the performance of both one-shot and many-copy probabilistic resource
distillation protocols.
  Complementing this approach, we introduce a general method for bounding
achievable probabilities in resource transformations under
resource-non-generating maps through a family of convex optimisation problems.
We show it to tightly characterise single-shot probabilistic distillation in
broad types of resource theories, allowing an exact analysis of the trade-offs
between the probabilities and errors in distilling maximally resourceful
states. We demonstrate the usefulness of both of our approaches in the study of
quantum entanglement distillation.
