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 $$U_1$$ U 1 and $$U_2$$ 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 $$U_1^\dag U_2$$ U 1 † 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 $$U_1$$ U 1 and $$U_2$$ U 2 satisfying $$U_1^\dag U_2=V$$ U 1 † 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, $$U_1$$ U 1 and $$U_2$$ U 2 such that $$U_1^\dag U_2=V$$ U 1 † U 2 = V are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.
Quantum Information Processing – Springer Journals
Published: Nov 12, 2015
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