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We give a necessary condition that a separable measurement can be implementedby local quantum operations and classical communication (LOCC) in any finitenumber of rounds of communication, generalizing and strengthening a resultobtained previously. That earlier result involved a bound that is tight whenthe number of measurement operators defining the measurement is relativelysmall. The present results generalize that bound to one that is tight for anyfinite number of measurement operators, and we also provide an extension whichholds when that number is infinite. We apply these results to the famousexample on a $3\times3$ system known as "domino states", which were the firstdemonstration of nonlocality without entanglement. Our new necessary conditionprovides an additional way of showing that these states cannot be perfectlydistinguished by (finite-round) LOCC. It directly shows that this conclusionalso holds for their cousins, the rotated domino states. This illustrates theusefulness of the present results, since our earlier necessary condition, whichthese results generalize, is not strong enough to reach a conclusion about thedomino states.

Extended necessary condition for local operations and classical communication: Tight bound for all measurements