Determination of locally perfect discrimination for two-qubit unitary operations

In the study of local discrimination for multipartite unitary operations, Duan et al. (Phys Rev Lett 100(2):020503, 2008) exhibited an ingenious expression: Any two different unitary operations \(U_1\) and \(U_2\) are perfectly distinguishable by local operations and classical communication in the single-run scenario if and only if 0 is in the local numerical range of \(U_1^\dag U_2\). However, how to determine when 0 is in the local numerical range remains unclear. So it is generally hard to decide the local discrimination of nonlocal unitary operations with a single run. In this paper, for two-qubit diagonal unitary matrices V and their local unitary equivalent matrices, we present a necessary and sufficient condition for determining whether the local numerical range is a convex set or not. The result can be used to easily judge the locally perfect distinguishability of any two unitary operations \(U_1\) and \(U_2\) satisfying \(U_1^\dag U_2=V\). Moreover, we design the corresponding protocol of local discrimination. Meanwhile, an interesting phenomenon is discovered: Under certain conditions with a single run, \(U_1\) and \(U_2\) such that \(U_1^\dag U_2=V\) are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.