  The probabilistic one-way distillable secret key is equal to the largest
expected rate at which perfect secret key bits can be probabilistically
distilled from a bipartite state by means of local operations and one-way
classical communication. Here we define the set of super two-extendible states
and prove that an arbitrary state in this set cannot be used for probabilistic
one-way secret-key distillation. This broad class of states includes both
erased states and all full-rank states. Comparing the probabilistic one-way
distillable secret key with the more commonly studied approximate one-way
distillable secret key, our results demonstrate an extreme gap between them for
many states of interest, with the approximate one-way distillable secret key
being much larger. Our findings naturally extend to probabilistic one-way
entanglement distillation, with similar conclusions.
