Entanglement as a resource to locally distinguish tripartite quantum states

Recently, Zuo et al. (Quantum Inf Process 20:382, 2021) construct the nonlocal sets of tripartite orthogonal product states with less members. Then, how much entanglement resource is sufficient to locally distinguish these states, and very little is known about entanglement-assisted state discrimination in multipartite case. In this paper, with only a 2⊗2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\otimes 2$$\end{document} maximally entangled state, we first prove the orthogonal product states in Cd⊗Cd⊗Cd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{d} \otimes {\mathbb {C}}^{d} \otimes {\mathbb {C}}^{d}$$\end{document} can be locally distinguished, where d⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqslant 3$$\end{document}. Then, considering Cd⊗Cd+1⊗Cd+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{d} \otimes {\mathbb {C}}^{d+1} \otimes {\mathbb {C}}^{d+2}$$\end{document} quantum system, we also present that the orthogonal product states are locally distinguishable by using a 2⊗2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\otimes 2$$\end{document} maximally entangled state, where d⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqslant 3$$\end{document}. Finally, we generalize the distinguishing method for general tripartite quantum systems. The above results can let us better understand the role of entanglement resource in quantum information processing, and also reveal the phenomenon of less nonlocality with more entanglement.