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We consider bipartite LOCC, the class of operations implementable by localquantum operations and classical communication between two parties.Surprisingly, there are operations that cannot be implemented with finitelymany messages but can be approximated to arbitrary precision with more and moremessages. This significantly complicates the analysis of what can or cannot beapproximated with LOCC. Towards alleviating this problem, we exhibit twoscenarios in which allowing vanishing error does not help. The first scenarioinvolves implementation of measurements with projective product measurementoperators. The second scenario is the discrimination of unextendible productbases on two 3-dimensional systems.

When asymptotic LOCC offers no advantage over finite LOCC