  Two dual questions in quantum information theory are to determine the
communication cost of simulating a bipartite unitary gate, and to determine
their communication capacities. We present a bipartite unitary gate with two
surprising properties: 1) simulating it with the assistance of unlimited EPR
pairs requires far more communication than with a better choice of entangled
state, and 2) its communication capacity is far lower than its capacity to
create entanglement. This suggests that 1) unlimited EPR pairs are not the most
general model of entanglement assistance for two-party communication tasks, and
2) the entangling and communicating abilities of a unitary interaction can vary
nearly independently. The technical contribution behind these results is a
communication-efficient protocol for measuring whether an unknown shared state
lies in a specified rank-one subspace or its orthogonal complement.
