  We provide a necessary condition that a quantum measurement can be
implemented by the class of protocols known as Local Operations and Classical
Communication, or LOCC, including when an error is allowed but must vanish in
the limit of an infinite number of rounds, a case referred to as asymptotic
LOCC. Our condition unifies, extends, and provides an intuitive, geometric
justification for previous results on asymptotic LOCC. We use our condition to
answer a variety of long-standing, unsolved problems, including for
distinguishability of certain sets of states by LOCC. These include various
classes of unextendible product bases, for which we prove they cannot be
distinguished by LOCC even when infinite resources are available and
asymptotically vanishing error is allowed.
