Local equivalence of quantum orthogonal arrays and orthogonal arrays

Two orthogonal arrays (OAs) are locally equivalent if they lead to locally equivalent quantum states. By studying permutations of the rows or levels of each factor, we present the local equivalence between two OAs. Using the tensor products of unitary matrices, we find that two infinite classes of OAs, OA(dn,n+1,d,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d^n,n+1,d,n)$$\end{document} and OA(d,n+1,d,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d,n+1,d,1)$$\end{document}, are locally equivalent. Therefore, we provide a positive answer to the open problem of which OAs are locally equivalent, i.e., OA(r,N,d,k)∼locOA(r′,N,d,k′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{OA}(r,N,d,k)\sim _{loc}\mathrm{OA}(r',N,d,k')$$\end{document}, in a sense that they lead to locally equivalent quantum states. In addition, an improved quantum orthogonal array (IQOA) is defined. The equivalence and local equivalence of IQOAs are investigated.