Mutually Unbiased Property of Special Entangled Bases

We study mutually unbiased bases formed by special entangled basis with fixed Schmidt number 2 (MUSEB2s) in ℂ3⊗ℂ4p(p∈ℤ+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{4p} (p\in \mathbb {Z}^{+})$\end{document}. Through analyzing the conditions MUSEB2s satisfy, a systematic way of the concrete construction of MUSEB2s in ℂ3⊗ℂ4p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{4p}$\end{document} is established. A general approach to constructing MUSMEB2s in ℂ3⊗ℂ4p(p∈ℤ+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{4p}(p\in \mathbb {Z}^{+})$\end{document} from MUSMEB2s in ℂ3⊗ℂ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{4}$\end{document} is also presented. Detailed examples in ℂ3⊗ℂ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{4}$\end{document}, ℂ3⊗ℂ8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{8}$\end{document} and ℂ3⊗ℂ12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{3}\otimes \mathbb {C}^{12}$\end{document} are given. Especially, by choosing special entangled basis with fixed Schmidt number 2 (SEB2) from [J. Phys. A. Math. Theor. 48245301(2015)], the limitation of 3∤p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$3\nmid p$\end{document} in [Quantum Inf. Process. 17:58(2018)] is successfully deleted.