  The difficulty in manipulating quantum resources deterministically often
necessitates the use of probabilistic protocols, but the characterization of
their capabilities and limitations has been lacking. We develop a general
approach to this problem by introducing a new resource monotone that obeys a
very strong type of monotonicity: it can rule out all transformations,
probabilistic or deterministic, between states in any quantum resource theory.
This allows us to place fundamental limitations on state transformations and
restrict the advantages that probabilistic protocols can provide over
deterministic ones, significantly strengthening previous findings and extending
recent no-go theorems. We apply our results to obtain a substantial improvement
in bounds for the errors and overheads of probabilistic distillation protocols,
directly applicable to tasks such as entanglement or magic state distillation,
and computable through convex optimization. In broad classes of resources, we
strengthen our results to show that the monotone completely governs
probabilistic transformations -- its monotonicity provides a necessary and
sufficient condition for state convertibility. This endows the monotone with a
direct operational interpretation, as it can exactly quantify the highest
fidelity achievable in resource distillation tasks by means of any
probabilistic manipulation protocol.
