  Distillation, or purification, is central to the practical use of quantum
resources in noisy settings often encountered in quantum communication and
computation. Conventionally, distillation requires using some restricted 'free'
operations to convert a noisy state into one that approximates a desired pure
state. Here, we propose to relax this setting by only requiring the
approximation of the measurement statistics of a target pure state, which
allows for additional classical postprocessing of the measurement outcomes. We
show that this extended scenario, which we call virtual resource distillation,
provides considerable advantages over standard notions of distillation,
allowing for the purification of noisy states from which no resources can be
distilled conventionally. We show that general states can be virtually
distilled with a cost (measurement overhead) that is inversely proportional to
the amount of existing resource, and we develop methods to efficiently estimate
such cost via convex and semidefinite programming, giving several computable
bounds. We consider applications to coherence, entanglement, and magic
distillation, and an explicit example in quantum teleportation (distributed
quantum computing). This work opens a new avenue for investigating generalized
ways to manipulate quantum resources.
