  The manipulation of quantum "resources" such as entanglement, coherence and
magic states lies at the heart of quantum science and technology, empowering
potential advantages over classical methods. In practice, a particularly
important kind of manipulation is to "purify" the quantum resources, since they
are inevitably contaminated by noise and thus often lose their power or become
unreliable for direct usage. Here we prove fundamental limitations on how
effectively generic noisy resources can be purified enforced by the laws of
quantum mechanics, which universally apply to any reasonable kind of quantum
resource. More explicitly, we derive nontrivial lower bounds on the error of
converting any full-rank noisy state to any target pure resource state by any
free protocol (including probabilistic ones)---it is impossible to achieve
perfect resource purification, even probabilistically. Our theorems indicate
strong limits on the efficiency of distillation, a widely used type of resource
purification routine that underpins many key applications of quantum
information science. In particular, this general result induces the first
explicit lower bounds on the resource cost of magic state distillation, a
leading scheme for realizing scalable fault-tolerant quantum computation.
Implications for the standard error-correction-based methods are specifically
discussed.
