Determination of locally perfect discrimination for two-qubit unitary operations

• Published in: Quantum information processing 2016-01, Vol.15 (1), p.529-549
• Main Authors: Cao, Tian-Qing, Gao, Fei, Yang, Ying-Hui, Zhang, Zhi-Chao, Wen, Qiao-Yan
• Format: Article
• Language: English
• Subjects: Data Structures and Information Theory; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Spintronics
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-015-1175-x

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 † 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 † 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 † U 2 = V are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.