Comparing the globalness of bipartite unitary operations: delocalisation power, entanglement cost and entangling power†

We compare three different characterizations of the globalness of bipartite unitary operations, namely, delocalization power, entanglement cost, and entangling power, to investigate global properties of unitary operations. We show that the globalness of the same unitary operation depends on whether input states are given by unknown states representing pieces of quantum information, or a set of known states for the characterization. We extend our analysis on the delocalization power in two ways. First, we show that the delocalization power differs whether the global operation is applied on one piece or two pieces of quantum information. Second, by introducing a new task called LOCC one-piece relocation, we prove that the controlled-unitary operations do not have the delocalization power strong enough to relocate one of two pieces of quantum information by adding LOCC.