Fast protocols for local implementation of bipartite nonlocal unitaries

In certain cases the communication time required to deterministically implement a nonlocal bipartite unitary using prior entanglement and local operations and classical communication can be reduced by a factor of two. We introduce two such “fast” protocols and illustrate them with various examples. For some simple unitaries, the entanglement resource is used quite efficiently. The problem of exactly which unitaries can be implemented by these two protocols remains unsolved, though there is some evidence that the set of implementable unitaries may expand at the cost of using more entanglement.