  Nonlocal properties (globalness) of a non-separable unitary determine how the
unitary affects the entanglement properties of a quantum state. We apply a
given two-qubit unitary on a quadpartite system including two reference systems
and analyze its "LOCC partial invertibility" under two-party LOCC. A
decomposition given by Kraus and Cirac for two-qubit unitaries shows that the
globalness is completely characterized by three parameters. Our analysis shows
that the number of non-zero parameters (the Kraus-Cirac number) has an
operational significance when converting entanglement properties of
multipartite states. All two-qubit unitaries have the Kraus-Cirac number at
most 3, while those with at most 1 or 2 are equivalent, up to local unitaries,
to a controlled-unitary or matchgate, respectively. The presented operational
framework distinguishes the untaries with the Kraus-Cirac number 2 and 3, which
was not possible by the known measure of the operator Schmidt decomposition. We
also analyze how the Kraus-Cirac number changes when two or more two-qubit
unitaries are applied sequentially.
