  Local pure states represent a fundamental resource in quantum information
theory. In this work we obtain one-shot achievable bounds on the rates for
local purity distillation, in the single-party and in the two-party cases. In
both situations, local noisy operations are freely available, while in the
two-party case also one-way classical communication can be used. In addition,
in both situations local pure ancillas can be borrowed, as long as they are
discounted from the final net rate of distillation. The one-shot rates that we
obtain, written in terms of mutual information-like quantities, are shown to
recover in the limit the asymptotic i.i.d. rates of Devetak [PRA, 2005], up to
first order analysis.
