Geometric and topological considerations for the closure of LOCC

We point out 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 in the limit of an infinite number of rounds. A generalization of this condition is then proven to hold for any measurement that is in the closure of that set, ${\bar{\textrm{LOCC}}}$. This generalization unifies, extends, and provides a geometric justification for previously known results on ${\bar{\text{LOCC}}}$, reproducing their consequences with regard to practical applications. We have also used 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 vanishing error is allowed.