Locally Discriminating Nonlocal Tripartite Orthogonal Product States with Entanglement Resource

In recent years, using entanglement resources to assist the local discrimination of orthogonal quantum states has attracted wide attention. However, many studies mainly focus on entanglement-assisted local discrimination in bipartite systems, and there are relatively few in multipartite states. In this paper, for the nonlocal set of 3d-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3d-3$$\end{document} orthogonal product states in d circle times d circle times d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\otimes d\otimes d$$\end{document}(d >= 3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d\ge 3)$$\end{document} constructed by Zhu et al. (Quantum Inf. Process. 21, 252, 2022), we propose a method of using an ancillary d circle times d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\otimes d$$\end{document} maximally entangled state to realize the local perfect discrimination. Firstly, with a 3 circle times 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\otimes 3$$\end{document} maximally entangled state as an auxiliary resource, we present a method to exactly identify the locally indistinguishable 6 orthogonal product states in 3 circle times 3 circle times 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\otimes 3\otimes 3$$\end{document} by local operations and classical communication (LOCC). Then the distinguishing method can be generalized to the 3d-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3d-3$$\end{document} states in d circle times d circle times d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\otimes d\otimes d$$\end{document}. These results not only reveal the phenomenon of less nonlocality with more entanglement, but also help us better realize the usefulness of entanglement in the local discrimination of quantum states.