Classification of delocalization power of global unitary operations in terms of LOCC one-piece relocalization

We study how two pieces of localized quantum information can be delocalized across a composite Hilbert space when a global unitary operation is applied. We classify the delocalization power of global unitary operations on quantum information by investigating the possibility of relocalizing one piece of the quantum information without using any global quantum resource. We show that one-piece relocalization is possible if and only if the global unitary operation is a local unitary equivalent of a controlled-unitary operation. The delocalization power turns out to reveal a different aspect of the non-local properties of global unitary operations characterized by their entangling power.