  We study exact, non-deterministic conversion of multipartite pure quantum
states into one-another via local operations and classical communication (LOCC)
and asymptotic entanglement transformation under such channels. In particular,
we consider the maximal number of copies of any given target state that can be
extracted exactly from many copies of any given initial state as a function of
the exponential decay in success probability, known as the converese error
exponent. We give a formula for the optimal rate presented as an infimum over
the asymptotic spectrum of LOCC conversion. A full understanding of exact
asymptotic extraction rates between pure states in the converse regime thus
depends on a full understanding of this spectrum. We present a characterisation
of spectral points and use it to describe the spectrum in the bipartite case.
This leads to a full description of the spectrum and thus an explicit formula
for the asymptotic extraction rate between pure bipartite states, given a
converse error exponent. This extends the result on entanglement concentration
in [Hayashi et al, 2003], where the target state is fixed as the Bell state. In
the limit of vanishing converse error exponent the rate formula provides an
upper bound on the exact asymptotic extraction rate between two states, when
the probability of success goes to 1. In the bipartite case we prove that this
bound holds with equality.
