  In this paper, we consider the problem of discriminating quantum states by
local operations and classical communication (LOCC) when an arbitrarily small
amount of error is permitted. This paradigm is known as asymptotic state
discrimination, and we derive necessary conditions for when two multipartite
states of any size can be discriminated perfectly by asymptotic LOCC. We use
this new criterion to prove a gap in the LOCC and separable distinguishability
norms. We then turn to the operational advantage of using two-way classical
communication over one-way communication in LOCC processing. With a simple
two-qubit product state ensemble, we demonstrate a strict majorization of the
two-way LOCC norm over the one-way norm.
