  Every measurement that can be implemented by local quantum operations and
classical communication (LOCC) using an infinite number of rounds is the limit
of a sequence of measurements each of which requires only a finite number of
rounds. This rather obvious and well-known fact is nonetheless of interest as
it shows that these infinite-round measurements can be approximated arbitrarily
closely simply by using more and more rounds of communication. Here we
demonstrate the perhaps less obvious result that (at least) for bipartite
systems, the reverse relationship also holds. Specifically, we show that every
finite-round bipartite LOCC measurement is the limit of a continuous sequence
of LOCC measurements, where each measurement in that sequence can be
implemented by LOCC, but only with the use of an infinite number of rounds.
Thus, the set of LOCC measurements that require an infinite number of rounds is
dense in the entirety of LOCC, as is the set of finite-round LOCC measurements.
This means there exist measurements that can only be implemented by LOCC by
using an infinite number of rounds, but can nonetheless be approximated closely
by using one round of communication, and actually in some cases, no
communication is needed at all. These results follow from a new necessary
condition for finite-round LOCC, which is extremely simple to check, is very
easy to prove, and which can be violated by utilizing an infinite number of
rounds.
